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111W WRIGHT STATE UNIVERSITY Signals and Systems Lecture 07 ContinuousTime Fourier Transform AperiodicPeriodic Properties Erik Blasch 9379049077 erikblaschwpafbafmil httpwwwcswrightedueblasch TA Rehka Bangladore Kalegowda bangalorekalegow2wrightedu EE701 Erik Blasch HEW wmmsmlla Ch 4 Fourier Transform UNIVERSITY 41 Aperiodic Signal Representation By Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform 44 Signal Transmission Through LTIC Systems 45 Ideal and Practical Filters 46 Signal Energy 47 Communications Amplitude Modulation 48 Angle Modulation 49 Data Truncation Window Functions 410 Summary EE701 Erik Blasch 2 mm WRIGHT STATE EX 1 I Solvrng an ODE UNIVERSITY Consider the LTl system time impulse response he 2 ue b gt o to the input signal xt e aluU a gt O 1 TXM Transforming these signals into the frequency domain 1 ajw X060 How b 1 2 MULTIPLY and the frequency response is YjooHjooXjoo gt Yjw 1 bjaaja to convert this to the time domain express as partial fractions 1 1 1 Y 39a b J b a aja 09ij 72a 3 INVERSE FT Therefore tlie time doniain response is W 1 ca rum We EE701 Erik Blasch R WRIGHT STATE UNI VERSITY Transmission Distortionless just a shift in the signal Hm 15 E39jmta Output yf KfU td 111m Frijth Thus Ho k arm ka jf ti z Hoo w fd where toI is the delay slope m k Ilami Human Audition Perceive Amplitude distortion insensitive to phase distortion Human Vision Perceive Phase distortion insensitive to Amplitude distortion EE701 Erik Blasch 4 1 Ewan WRIGHTSTARTE Ch 4 Founer Transform VVRIGHTSTAE FE EX Filtering 1 UNIVERSITY UNIVERSITY 1 41 AperIodIc Slgnal Representation By FourIer Integral a k if Mt e atu a gt O 42 Transform of Some Useful Functions 7 I a t 43 Some Properties of the Fourier Transform XW foo xte jwtdt 0 e ate jwt dt 44 Signal Transmission Through LTIC Systems 00 0 elta2 wgt 45 Ideal and Practical Filters Z lt 1 ltajwgtt Z 1 a 3w 0 a 3w 46 Slgnal Energy 47 Communications Amplitude Modulation 1x003 612 212 4X0 gttag1ltwagt 1a 48 Angle Modulation 49 Data Truncation Window Functions 410 Summary EE701 Erik Blasch 5 EE701 Erik Blasch 6 Elm R I I I WRIGHT STATE EX Fllterlng WRIGHT STATE PerIOdIC Response USIng FOUHer UNIVERSITY UNIVERSITY Transforms ht e tut at 6 2tut 51375 M75 yt M75 51375 y t h t gtllt a t m HltjwgtXltjwgt 1 t YUw HjwXJw 1 1 The frequency response of a CT LTI system is simply the Fourier transform 1 l 70 2 l I of its impulse response a rational function of jw ratio of polynomials of jw 3305 ejwot HOW yt 1 Partial fraction expansion 1 1 Recall ejwot lt gt 27T6w tag Yjw HjwXjw Hjw27r6w too 27THjw06w tag ll inverse FT W Hjwo j 0t EE701 Erik Blasch 7 EE701 Erik Blasch 8 Y M 1jta 2jw U inverse FT W 6quott 6 2tlut WSlSlIERElTAtE Ideal Low Pass Filter Example WELSEEREEEE Filters H060 1 we Signal with Delay yt ft td W Bandlimited 1 ht eJWtw 27F wc Hlmll I l Sin wet LOW Pass dc DC a Wt ht Esme watt 1 3 7T 7T PaSS Ema M3 f1 rent m W E39jmt391 llllllll gt tell Wm EiI1IIrI rIIIdi SlIl7Tl9 393quot quot VAVAVA AVAVAV D ne SinC6 lHlmeil recttm i i iii Ht V V t e n g 7T6 we we Band Pas EE701 Erik Blasch 9 EE701 Erik Blasch 10 Mm l W l I WSlglgslltE Lowpass Fllter VVEL itIEREEEE Highpass Filter 39 Lowloass Filter continuous time Highpass Filter continuous time AHMJ AHa MW 1 Kw mw 1 Kw EJ 3 D 00 00 39500 00 O a lt a0 FX 0 a lt a0 YaltXa a0ltalta0 Yalt0 a0ltalta0 KO 600 lt a KXa 00 lt a EE701 Erik Blasch 11 EE701 Erik Blasch 12 HEW WRIGHT STATE UNIVERSITY Bandpass Filte Bandpass Filter continuous time Hw mo 1 r Yffo l q 3945 3950 01 02 V A DC 031 cog2 is called the center frequency of the bandpass filter rXa a1 lt wlt a2 Ya lt Xa k0 else a1ltalta2 EE701 Erik Blasch 13 Mimi WRIGHT STATE UNIVERSITY Cascading Nonideal Filters H1000 gt H2000 W H100 H200 eg39 H1050 2 H200 1 HGw H12ju hasa gt sharper frequency fx selectivity EE701 Erik Blasch 14 Emmi WRIGHT STATE UNIVERSITY sin 47ft sin 87ft gtllt 7 at 7ft mt ht x00 Hgoo ll 1 X l 4TC 47E 60 87c 87E 0 Cascading Gaussians 2 2 6 eTbt ll E 2 X E 4 ul2 L6 2 b M Gaussian gtlt Gaussian 2 Gaussian EE701 Erik Blasch 4 Cascading Ideal Filters 4TC i O IH Gaussian Gaussian 2 Gaussian 15 mi meHTsTATE Ch 4 Fourier Transform UNIVERSITY 41 Aperiodic Signal Representation By Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform 44 Signal Transmission Through LTIC Systems 45 Ideal and Practical Filters 46 Signal Energy 47 Communications Amplitude Modulation 48 Angle Modulation 49 Data Truncation Window Functions 410 Summary EE701 Erik Blasch 16 Wt WRIGHT STATE UNIVERSITY Signal Energy Power Periodic m Energy Finite exponential Rf i f Frmj2 Elm Energy Spectral Density EEf IranHFrijEm Fjl3Ejquot lm 39 II 39 t E 2 in f E 39 Iquot 7 7 V his i Power Spectral Density PSD Autocorrelation ESD Correlate perfectly no delay Low frequency some correlation w large delay High frequency lack of correlation with delay EE701 Erik Blasch 17 WEE ll WRIGHT STATE UNIVERSITY Signal Energy Ex Ex 416 Find Energy Determine W so that energy contributed of all frequencies below W is 95 o of E f 1 E39 amp an 1 i m mi jmr quot 2 I I Ef fl fjldI E III Mt L l 1 quot39I2 1 1 1Equot 1 3 1 Imam 13913 g 2mzdm natal 2 2 The band D O to D Wcontains 95 of the signal energy that is 0952a Therefore with 01 O and 92 W we obtain I195 1 w 1 1 1 m w 1 1W f E 2Elm tan tan at in U a m 1th if 3 It It I195 2 it 1 1E 2 W 123mg mars EE701 Erik Blasch 18 Ema WRIGHT STATE UNIVERSITY Signal EnergyPower Power Periodic Energy Finite exponential Energy Spectral Density EH33 E Um I Hm I Power Spectral Density PSD Elite 2 TLEE IEIIFTIm I2 Linear Io Egiw IHimIli grim Racer IHImIIi mm EE701 Erik Blasch 19 Him ll WIRES 3399 3 Energy Den Energy Finite exponential Given Find energy spectral density The magnitude of this curve is squared and divided by 21 to form the frequency spectrum Next we fold the energy frequency spectrum about the a 0 axis Amount of energy contained in the band of frequencies between 01 and 92 is the shaded area under the curve F iu if All ST S 2 2 Jmer 1 21 klw II 7 lFL F j I I i A T A11quot mi WI 31W I M Iquot T EE701 Erik Blasch 20 mm w s 39 Egg Power Spectral DenSIty IIEEI HT 11 Power Periodic w Need to truncate the signal Change Limits T2 to co m 6 TE quot fit in I n A mm A H V 71 g m E fermlidt i Zlmmjlidm V Il III L 1 1 1 m 1 F T lm leTimjl Elm P ffL lm Ftiwil dw II III As the duration of the rectangular pulse increases it can be seen that the energy of the signal will also increase 3920 p f PTrmjldm if rmde 1 1 2 rm TLEWEIFTrmJI III Average EE701 Erik Blasch 21 HEW meHTsTAE rE Power Spectral Densnty UNIVERSITY Power Periodic Need to truncate the signal Average p f PTrmjdm Fffmjdm 1 2 11111 2 L11 IFTEmJI D T mf For periodic signals the normalized average power can be determined from the Fourier series as P Z 1011i Imli 2 Ickli m k1 Normalized average power of a signal ft is written in terms of the Fourier transform as If ka 21Ck c 121113 in when l imlE k m f Inkmali 1 L E39IEk It is seen that for a periodic signal the power distribution over any band of frequencies can be determined from the Fourier transform of the signal EE701 Erik Blasch 22 VVRIGHWAE Power Spectral Densnty Ex UNIVERSITY Power Periodic Need to truncate the signal power spectral density can be displayed by squaring the magnitude of each discrete frequency component and dividing by 412 m I I l 1157331 F l 120 I I u 1 ml 1 I m g Ii 115331 m m 1 d El q M I n 11120 1 I 30 39 an m 21113 1U 39 I u 194 w quotl 1l11m am 4 l 139 l I 1 WM n tale1i 39 lee91 m3nmm1MJ 11 mm 20m m 39m39m m39mm 393 m m m m Iiiequch rails F39WMW my Normalized average power in the frequency band a s 1000 rads is found by summing the power of the discrete frequency components in that range 1167 2720130 2867W EE701 Erik Blasch 23 DEW meHTs TAi rE Ch 4 Fourier Transform UNIVERSITY 41 Aperiodic Signal Representation By Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform 44 Signal Transmission Through LTIC Systems l 45 Ideal and Practical Filters 46 Signal Energy 47 Communications Amplitude Modulation 48 Angle Modulation 49 Data Truncation Window Functions 410 Summary EE701 Erik Blasch 24 Wit Wit W l i iii ifff Modulation 1 WEEEIEREEAEE Modulatlon 2 yt ftcosw0t Example yt ft cosl0 t Fm Multiplication in the time domain is ft 393 an 39deal IOWpaSS S39gnal 1 convolution in the frequency domain Assume D1 ltlt DO 0Fa7r aw07r a a0 01 O 01 gt 0 Recall that 12 F coco0 Y 32 12 F 0 500 xt 6r J 5Txt rdr xt I T l I l XI I IO fm T IOXI TdZXI t0 w0w1 DO w0w1 O DO01 DO w0w1 So 1 1 Demodulation IS modulation followed by lowpass filtering Y60 FwwoEFwwo Similar derivation for modulation with sinl0 t EE701 Erik Blasch 25 EE701 Erik Blasch 26 DEW i m m ll militia Modulation and Demodulation Modulation Embedding an informative signal into another signal eg Amplitude Modulation AM Frequency Modulation FM Demodulation Extracting an informative signal from a signal Multiplexing Simultaneous transmission of multiple signals with overlapping spectra over the same channel eg FrequencyDivision Multiplexing FDM TimeDivision Multiplexing TDM Extensive usage of these concepts in communications Modulation Demodulation C 1 t 62 Mt xt X X W EE701 Erik Blasch 27 wwgkggla Amplitude Modulation AM 1 M Multiplication of a signal xt by a carrier signal Ct yt xt Ct xt is modulating signal yt is modulated signal Different types of AM depending on the carrier signal being used Complex Exponential AM Carrier signal is a complex exponential eg jact6lc Ct e Sinusoidal AM Carrier signal is a sinusoidal signal eg Ct coswct 96 EE701 Erik Blasch 28 Wt WRIGHT STATE Amplitude Modulation AM 2 Since xt is a factor for amplitude of ct this multiplication is referred to as amplitude modulation ct is selected such that frequency content of xt is shifted to a certain frequency band Two different signals are picked in order to make this frequency shift complex exponential or sinusoidal VVRIG39HTTSTATE Exponential AM 3 yt xtct The case when ct ejmc gc So yt amped a Ya effcxm ac In order to demodulate xt out of yt yt must be multiplied C 66 I by e quot9 xt yltrgte When xt is real these multiplications can be implemented as summations of two multiplications jact6lc coswct 66 In both forms of AM we is referred to as carrier Reyt frequency xt Imz yltl sinwct 66 EE701 Erik Blasch 29 EE701 Erik Blasch 30 from R m m ll W l it ifif Slnus0ldal AM WEEEIEREE EEDemodulatlon for Slnus0ldal AM yt xtct The case when Cf COSMIJ 96 When 96 0 yt xt cosact a Na 2 0 me Xa 06 coswct6 6 x W Important condition Suppose xt is bandlimited by W Le XoO for ogtoM then DC gt W must be true for safe recovery or demodulation of xt EE701 Erik Blasch 31 Demodulation of xt out of yt can be done by multiplying yt with ct again and passing it through a Lowpass Filter coswct 66 H w 2 W W 0 00 00 What s happening at the timedomain Is it possible to explain the above demodulation procedure in freq and timedomain 2m39 H H D r m l EE701 Erik Blasch 32 Emil W i t i emodulation for Sinusoidal AM It is possible to have phase differences between modulating and the demodulating signals Modulating and demodulating signals need to be synchronized or another method of demodulation is necessary We call the demodulation in the previous slide as Synchronous Demodulation Because it assumes that there is no phase difference What is the recovered signal when there is phase difference between the carrier signals Consider both the complex exponential and sinusoidal carrier cases 39 a 61 t cosact 66 61t em C 620 00 th C CZ t ejwct c Synchronous demodulation performs nicely in terms of signal quality however requires costly setup which is not desirable in systems meant to be cheap such as radios EE701 Erik Blasch 33 t l39i39nt r I 539 akaquot R f k Wl i iifi i emodulation for Sinusoidal AM 3 Asynchronous Demodulation It is possible to recover xt out of yt without requiring synchronization of modulator and demodulator Two conditions xt must always be positive Carrier frequency must be significantly larger than highest frequency of xt ie DC gtgt 9M xt can be recovered by an envelope detector as long as the two conditions are satisfied The second condition is satisfied for radios Xt is typically 15 to 20 kHz DC271 is SOOkHz to 2MHz The first condition can be satisfied by simply adding a constant A to xt while modulating such that xtA is always positive Then we can subtract A from the signal recovered by the envelope detector So let K be the maximum amplitude of xt ie xt lt K m KA is called as modulation index Apparently the lesser m means more possibility of including negative values of xt in the modulated signal EE701 Erik Blasch 34 Emit WRIGHT STATE UNI VERSITY TimeDIVISIon MultIpIeXIng TDM Multiplex in the time domain Divide the time into many slots Assign each slot to a particular signal At the receiver end when demuxing receive the signals such that they are being recovered from corresponding time slot Demultiplexer at the receiver must be synchronized with the multiplexer at the transmitter x 2 A Modulator Dcmodulatory t x I I n gt Modulator Dcmodulatm EE701 Erik Blasch 35 m m ll WEEEEEE EE EX TimeDIVISIon Multrplexrng AM with a PulseTrain Carrier The case when ct is a square wave or a pulse train as shown below new TT ITI Sketch out the frequency spectrum of modulated signal ytxtct assuming that xt is bandlimited TBiIsItechnique of modulating a pulsetrain can be used for EE701 Erik Blasch 36 mm R WRIGHT STATE UNIVERSITY 419 Tone Modulation modulating signal is pure sinusoid tone Sketch pm 2 for modulation indices of u 05 50 modulation and u 1 100 modulation when mt Bcos com 1 In this case m p B modulation index is u A B Hence B uAand mf Bcos emf uAcos oomf Therefore pm 1 A mt cos a C1 A1 ucoswm coswcf 24 3m In I I 3 39 F 39 I39 r t I 39 39 IMII39T I 12quot 05 EmmaIf 39lcosmmf I HEW R WRIGHT STATE UNIVERSITY mm I39 Spectrum u Ppm cm I mm 30st cnsmmtcnsmcf 43 I I 1113 m l Lppgggcm I 1f2cnsmc mm 105103 mmji l 5 bit DEB In m m at 31 1155 smm Edi High Pass Low Pass n If r II III IIIff 1f 9 a 11111 112115 111 w i EE701 Erik Blasch 37 EE701 Erik Blasch 38 WW 1 WW 1 was Sidebands was Ch 4 Fourier TranSIOIm 39 Spectrum Smell thmrmcmW 41 Aperiodic Signal Representation By Fourier Integral f WWW f m a 42 Transform of Some Useful Functions I g i I 43 Some Properties of the Fourier Transform wt 41 In I 1 W m 44 Signal Transmission Through LTIC Systems 45 Ideal and Practical Filters Emmi a T m 46 Signal Energy I I I 47 Communications Amplitude Modulation 39 mew tar m awl 48 Angle Modulation 49 Data Truncation Window Functions T T l M T M 410 Summary iff f39fl 39 r 1quotquot 339 3 ffr39fff 39 i a Angle Modulation can vary the bandwidth Amplitude Modulation bandwidth fixed EE701 Erik Blasch 39 EE701 Erik Blasch 40 HEW WRIGHT STATE UNIVERSITY Angle Modulation Exponential Modulation Angle Modulation Mimi WRIGHT STATE UNIVERSITY Angle Modulation Angle Modulation Exponential Modulation PEMU ACOSwct KW H pEMl Acoswct kwt Riff 1 new ht ut Frequency Modulation FM 1 ht 5t Phase Modulation t c i 1 l w ht ut Frequency Modulation FM 1 2 mm 5 WWW Iiimm f Hillquot am e Iquot Mmumm quoth pgMEI cos tact in f mm rill Fl WM I r mtg 39Ij ll l39 ht 8t Phase Modulation KPHIii Jquot391393393933939339rJ5 Err pfj g39Lf 39 FM Radio general angle modulation mm d N m rimm w noise suppression q r 3 Vary Frequency w time mm x x MMHMMQI39 Instantaneous Frequency 391 2 H EE701 Erik Blasch 41 EE701 Erik Blasch 42 mm ll mm l Wrists Angle Modulation Warns Angle Modulailon Angle Modulation Exponential Modulation PEM t Acosmct kwt ht ut Frequency Modulation FM ht 5t Phase Modulation FM and PM waveforms mm P mm I 211000 r l V i r J I mifi EE701 Erik Blasch w lillmrlll illllllllif l ill 43 Communication Topics Carrier Phase Modulation Phase Shift Keying PSK Carrier Frequency Modulation Frequency Shift Keying FSK mm Amsrw we tpm f Acusmcuskwrn Asmmcfsin wl f 9 limst Humming ital Bandwidth SE armquot3 Hz Angle modulated infinite However power amp energy is in finite band mm 5me and pm afar2 1 bandwidth of the modulated si nal is adjustable by choosing suitable value of 45F or the cons ant kk fIn FM or k p in PM Amplitude modulation lacks this feature Narrow Band A 5 ltlt Wide Band A 5 gtgt B wideninq a signal bandwidth makes the signal more immune to noise during transmission EE701 Erik Blasch 44 Dim R WRIGHT STATE FrequencyDivision Multiplexing Transmission of multiple signals over the same channel is desued This can be done by two wellknown techniques FDM TDM m Multiplex in the frequency domain Assign particular frequency bands to different signals Modulate the signals such that their frequency content is shifted to the assigned frequency band At the receiver end demodulate the signals after passing them through appropriate bandpass filters Bandpass filters must be adjusted based on the carrier frequencies of each signal EE701 Erik Blasch 45 Miami FrequencyDrvrsron MultIpIeXIng UNIVERSITY Transmission of multiple signals over the same channel is desired m 044 fulfil I m i k V A F I u 1 Ill emi g ah an if 7 7 E 39 I f l 39 i Demuuzi i quoth 3w 39 WWWH1 ialrur m j a I all I39 l39lr mam 5 m N hm quotIn T I I 6 m 7 7 a T Micahm fhilmnmlr EE701 IrIK Blascn 46 VVRIGPITsTAiE Ch 4 Fourier Transform UNIVERSITY 41 Aperiodic Signal Representation By Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform 44 Signal Transmission Through LTIC Systems 45 Ideal and Practical Filters 46 Signal Energy 47 Communications Amplitude Modulation 48 Angle Modulation 49 Data Truncation Window Functions 410 Summary EE701 Erik Blasch 47 R WRIGHT STATE UNI VERSITY Window Function Window wt Take Care of Sidelobes Wider the window smaller bandwidth Wider the window smaller spectral spreading at fttiwttli and Fwtm tletEliFtwLii ime For a practical design we may want to truncate ht beyond a sufficiently large value of t to make ht causal In signal sampling to eliminate aliasing we need to truncate the signal spectrum beyond the half sampling frequency cos2 using an antialiasing filter a 433 I lift rm fl 39 in in EE701 Erik Blasch 48 Emil WRIGHT STATE UNIVERSITY Window Function Window wt Take Care of Sidelobes EE701 Erik Blasch 49 Emil WRIGHT STATE UNIVERSITY Window Function Window wt Hanning and Hamming 1 wmtxl l w I Windmiir tuft i ML n uh ri mtg t 6 a 133 2 Bartlett mg i a 2 4155 3 Hanging lilti li mt39iil ji 1 e BE 7315 1i ilamming i1 ii d em 351 3 E 42 5 Hintimim i142 Mm ii tiiJEimtiF lie e 15 551 E i g a 5 iti war tr i 499 a El j 50 EE701 Erik Blasch mm WRIGHT smWindow Function Filter DeSIgn UNIVERSITY Lowpass Filter the impulse response ht WitsincWt is noncausal and therefore unrealizable Truncation of ht by a suitable window Fig 448a makes it realizable although the resulting filter is now an approximation to the desired ideal filter We shall use a rectangular window wRt and a triangular Bartlett window wTt to truncate ht and then examine the resulting filters The truncated impulse responses hRt and hTt for the two cases hn1 h1 WRt and Mt h1 WTU EE701 Erik Blasch 51 WWI I I I I WRIGHT smWindow Function Filter DeSIgn UNIVERSITY LPF hRt ht wRt and hTt ht wTt m i ii 3 writ i E ll 1 i a is Him m I 93 w iii iv in Im 5 El EE701 2 must ah a g 52 wm G m TE Ch 4 Cont Time Fourier Transform 1 UNIVERSITY 41 Aperiodic Signals ContinuousTime Fourier Transform Development of the Fourier Transform Representation Convergence of Fourier Transforms Examples of ContinuousTime Fourier Transforms 42 The Fourier Transform for Periodic Signals 43 Properties of the ContinuousTime Fourier Transform Linearity Time Shifting Conjugation and Conjugate Symmetry Differentiation and Integration Time and Frequency Scaling Duality Parseval s Relation 44 The Convolution Property 45 The Multiplication Property FrequencySelective Filtering with Variable Center Frequency 46 Tables of Fourier Properties and Basic Fourier Transform Pairs 47 Systems Characterized by Linear ConstantCoefficient Differential Wit W i iifg iftommon Fourier Transform Pairs TABLE 42 COMMON FOURIER TRANSFORM PAIRS 1 w if 1 lt2 ECHZH Uu 1 U5 141 lt 8 Jim MU lt gt mm 1w 31 31 60 c H iii c any real number m 4 I ti iv 33gt c u Jim b b 0 aim H 2mm mg no any real number J i H 1 sins El If 1 sun H 2 39 2f 391 90 as 5 sins2 E 2 zit 2 sun 43 923139 1 T prim cos 0J0 H dw w l 3a wQJ cos wot 9 H Ede malice mm amok 90 sin w rerrD n mg lm 010 Equations sin mgr 6 1 j effarm on ef cif39w wig EE701 Erik Blasch 53 EE701 Erik Blasr 54 Brawl Emmi reefs Fourier Integral WEEEEREE EE Fourler Transform Gf fmgtej2 ft dt Xa J xtejm dt go Lanewdf we Emmaww Signal Processing Communication Systems Conditions for the Fourier transform of gf to exist Dirichlet conditions Xt is singlevalued with finite maxima and minima in any finite time interval Xt is piecewise continuous ie it has a finite number of discontinuities in any finite time interval Xt is absolutely integrable f lgtdt lt 00 EE701 Erik Blasch 55 What system properties does it possess x Memoryless x Causal Linear x Timeinvariant doesn t apply What does it tell you about a signal Answer Measures frequency content What doesn t it tell you about a signal Answer When those frequencies occurred in time EE701 Erik Blasch 56 331W 1 VVRIGHT STATE UNIVERSITY Useful Functions 1 rectgx i 0 x gt E 1 1 l l gt x rectx lt 5 ix E 12 0 12 1 x lt l 2 What does rectx a look likeL Unit triangle function W i itfg ffff Useful Functions 2 Sinc function sin x sincx x How to compute sinc0 sincx As x a O numerator and 1 A x denom1nator are both gomg 7Z39 yz 27 o 7 27 W to 0 How to handle it f 0 Mgtl Evenfunc on I A 12x Mi Zero crossings at xi ai2 ai3 w 42 390 12 gt x 2 Amplitude decreases proportionally to 1x EE701ErikBasch 57 EE701ErikBasch 58 mm 1 Him ll nggglggygla Fourier Transform Pairs 1 ngeglgggla Fourier Transform Pairs 2 F I rect je wdt fie Wait UT UT 1 2811117 anquot KW2 emf2 zquot sinc7j 2 21 1quot w 2 f0 Fun I 1 f ltgt t rV AN 452 0 152 6T7r 4Tgt 2T 0 r 6T7 EE701 Erik Blasch 59 From the sampling property of the impulse F5t J 5tejmdt e jot 1 f0 1 F1 2 1 80 1 f ltgt 2 O gt t I O gt a 2717 means that the area under the spike is 2 EE701 Erik Blasch 6O Wt WRIGHT STATE UNI VERSITY Fourier Transform Pairs 3 1 i jwr 2i jcoor F 6a w0 2 160 00e do 2 e Z ejwot ltgt 5w w0 or 61W ltgt 5w w0 7 Since COSaOt ejwot e jwot cosa0t ltgt 7r5a 00 5w 600 t Fl 7t W a l w l I V o oo 390 no 0 WStStIERETfEE Fourier Transform Pairs 4 mi 1 t O sgnr1 t i33e atut earu Fsgnt lingeatut Featu t sgnt are 1 1 1 gt I 11n1 6H0a0 a Ja 1 an a a JG EE701 Erik Blasch 62 Ema ng lggw Laplace Transform Generalized frequency variable 3 o in Fs foofte dt 1 00 ft FSe ds 27 00 Laplace transform consists of an algebraic expression and a region of convergence ROG For the substitution 3 jet or s 2 E f to be valid the ROC must contain the imaginary axis EE701 Erik Blasch 63 nggtlggf ouner vs Laplace Transform ft F S Region of Convergence Foo e39atut 1 ReS gt Rea 1 39t39 251 Rea lt Res lt Rea 2a 8t 1 complex plane 1 27quot9931Eg iummmF 22 ut i ReS gt O atom 1000 S cosn0t 1t8o 00 8a 00 sint0t j1t8oo no 80 00 quotquot S 51 Assuming that Rea gt O EE701ErikBlasch 64 W it WRIGHT STATE UNIVERSITY g HILL a f y 7 Ch 2 CT Time Domain UNIVERSITY 21 Introduction LeCtUre O4 meDomam AnalySIS Of 22 Sys Response to Internal Conditions CT Systems Z l R Stablllty ero I nput esponse 23 The Unit Impulse response hit 24 Sys Response to External Input ZeroState Res E n k 25 Classical Solution of Differential equations 26 System Stability k bl ngowow fb f 27 Intuitive Insights into System Behavior er39 39 aSC Wpa 3961 39m39 28 Appendix 21 Determining The Impulse Response httpwwwcswnghtedueblasch 29 Summary TA Rehka Bangladore Kalegowda bangalorekalegow2wrlghtedu EE701 Erik Blasch 1 EE701 Erik Blasch 2 m it WW ii Wings Ch 2 LTI Systems MEWS LTI Systems 21 DiscreteTime LTI Systems The Convolution Sum c Reca for System with input fand output y Representation of DiscreteTime Signals in Terms of lmpulses I Biscrete lt39irtte Unit Impulse Response and the ConvolutionSum Homogeneity O f 1 gt C y t epresen a Ion 22 ContinuousTime LTI Systems The Convolution Integral Tlme39anarlanCe C f 39 T gt C Yt 39 T Representation of ContinuousTime Signals in Terms of lmpulses ContinuousTime Unit Impulse Response and the Convolution Integral Addmg addltIVIty39 23 Properties of Linear TimeInvariant Systems m m Commutative Property Distributive Property F t 2 Cl ft gt 2 Cl yt Y I Associative Property With and without Memory i1 i1 avittlifi39ttv ilelTS39 Systems Tl usif39i tvSt R If a Signal Ft can be expressed as a sum of a I I y or ys ems e m ep esponse 24 Causal LTI Systems Differential and Difference Equations Shl ed t39 and We39ghted Ci Coples Of a Slmpler Linear ConstantCoefficient Differential Equations Signal Kt Linear ConstantCoefficient Difference Equations Block Diagram Representations of FirstOrder Systems 25 Singularity Functions Unit Impulse as an ldealized Short Pulse Defining the Unit Impulse through Convolution Unit Doublets and Other Singularity Functions A Common ChOICe for f 393 the ImPUISe EE701 Erik Blasch 3 EE701 Erik Blasch 4 we easily find a system s output Yt to that signal if we only know system s output yt to that simpler signal Wit VVRHSHH STATE UNIVERSITY Because of additivity of LTI systems yn ZHxk5nk kz m Because of homogeneity of LTI systems yin ixtkiHi in kil kz oo Because of timeinvariance of LTI systems yin ixikihin k kz oo EE701 Erik Blasch ml VVRHSHH STATE UNIVERSITY Then output signal yt will be ya How H 2 xkA5A t mm kz oo Because of additivity of LTI systems W Z HxkA6At kAA kz oo Because of homogeneity of LTI systems W inkAinAa kmm kz oo Because of limeinvariance of LTI systems yin inkihin k kz oo EE701 Erik Blasch 333w ll wggggggla Properties of Convolution Commutative Property Switch the order Xin yln yin Xin X 1 yl 1 y 1 Xl 1 Distributive Property Separate out additionsubtraction XnY1n Y2n Xn1n Xn2n Xty1t y2t Xty1t Xty2t Associative Property Reorder the operations ie like matrix algebra Xn y1n y2n Xn y1n y2n Xl y1l y2i X 1 W1 1 WM EE701 Erik Blasch mm Emil Commutative Property UNIVERSITY Convolution is a commutative operator in both discrete and continuous time ie xn hn hn xn i hkxn k kz oo xt ht ht xt foo hz39xt 0dr For example in discretetime xn hn i xkhn k i xn rhr hn xn and similar for continuous time Therefore when calculating the response of a system to an input signal xn we can imagine the signal being convolved with the unit impulse response hn or vice versa whichever appears the most straightforward EE701 Erik Blasch i Wee sell Commutative Property WRIGHT STATE AssoCIative Property UNIVERSITY UNIVERSITY Serial Systems Another property of LTI convolution is that it is associative ilt hn 2 Mn ilt xnh1n h2n xn h1n Mn Elft yn hln yn Xthith2t xth1th2t Again this can be easily verified by manipulating the Proof summationintegral indices Therefore the following four systems are all equivalent and yn yn min t W g elmlhln m1 Xnh1nh2n is unambiguously defined Let rt m I then Xlt Mt Mi 72 W W may72w W yn at Mn 2 lhl Z hln l 2 Mn at l l h2th1t 39 h2i hiU 39 This is not true for nonlinear systems y1n 2Xn y2n X2n EE701 Erik Blasch 9 EE701 Erik Blasch 1o W1 wages Associative Property ll nggglgggi Commutative Assomative Commutative Property yn wlnl htlnl h2ln rclnl htlnl h2lnl Switch the order yn yn Xn yn yn Xn quot hlln t hzlnl quot quot hlln h2ln quot Xtyt ytxt Associative Property Proof Reorder the operations ie like matrix algebra n y n yzlnll Xi7 W1 3971 yzln yn Z Zwlmlhtll ml h2lnl m htll mlh2l l X 1 l m m I X 01 Y2t Xt Yiu Y2 Let k l m to obtain yn Essie Zhiikiiein m k1 Zeimihin mi m k m II y l39i iu i 39h in gt nl taint h hlnl niinireini e yln xiii u where Mn 2 h1n h n EE701 Erik Blasch 1 1 EE701 Erik Blasch 12 VVRIEITT STARTE Distributive Property WRIE FSTARTE DIStrlbUtlve Property UNIVERSITY UNIVERSITY Distributive Property Another property of convolution is the distributive property Separate out additionsubtraction Xn gtllt mm h2n Xn Min Xn hzm y1n yzm Xn y1n yzlnll Xn y1n Xn y2n xt h1 r 1120 m mg m me y1t y2t Xt1t Y2t Xt1t Xt2t This can be easil verified Convolution distributes over addition so that in discrete y time Therefore the two systems yt Xfh1f Xfh2t W X t y t X t y t T 439 h1th2t 39quotquot hifti Y2 I quot 1 Vi are equivalent The convolved sum of two impulse Th f responses is equivalent to considering the two equivalent quot f xt1 gti himr nziii t parallel system equivalent for discretetime systems EE701 Erik Blasch 13 EE701 Erik Blasch 14 mm W5 V R IT E Ex Distributive Property ti Wrists DIStflbUt39Ve Property Let yn denote the convolution of the following two sequences yn mlnl hllnl h2lril 3Eli39s hllnl mlnl I12 n Xn O5quotun 2quotu n hn un n yn 33M r him yn Xn is nonzero for all n We will use the distributive property quot h1 h2 39 2 to express yn as the sum of two simpler convolution problems h2n Let X1n O5 un X2n 2 un it follows that yn x1n x2n hn Proof and yn yiiniy2ini where yiini IX1iI7hini yiini Xiinihini yn Zmmh1n m I h2n 1 05n1 quot quot m y1n 3 Ewimihlinm Zmimih2Wm 1 05 E 2quot1 n S O hug i 2 n21 quoti EE701 Erik Blasch 15 EE701 Erik Blasch n 16 WRIGHT STATE WI Convolution Properties 105 05 Mt Mt 05 1105 I130 h1t h2t ya th1t MG 905 11305 i110 MW 905 11305 h1i5 3305 h2i5 th T1 T2 3if T1 W T2 ylnmt mslog I hmt The last property implies that differentiating the input n times and the impulse response m times results in an output that is differentiated n m times Commutative Associative Distributive Delay accumulation Derivative accumulation EE701 Erik Blasch 17 WI WRIGHT STATE UNIVERSITY LTl System Memory An LTl system is memoryless if its output depends only on the input value at the same time ie yn kxn W kW For an impulse response this can only be true if hn k5n hr k5r Such systems are extremely simple and the output of dynamic engineering physical systems depend on Preceding values of xn1 xn2 Past values of yn1 yn2 for discretetime systems or derivative terms for continuous time systems EE701 Erik Blasch 18 WRIGHT STATE WWI v1 nialJr 5 quott1 quot5 UNIVERSITY Remember a causal system only depends on present and past values of the input signal We do not use knowledge about future information For a discrete LTl system convolution tells us that hn 0 fornltO as yn must not depend on xk for kgtn as the impulse response must be zero before the pulse 71 xnhn Zxkhn k 2 00 xt ht 00 mm rm Both the integrator and its inverse in the previous example are causal This is strongly related to inverse systems as we generally require our inverse system to be causal If it is not causal it is difficult to manufacture EE701 Erik Blasch 19 m mi WRIGHT STATE UNIVERSITY Causality 1 A system is causal if its output depends only on the past and the present values of the input signal not the future values Consider the following for a causal LTl system yn ixikihin ki kz m Because of causality hn k must be zero for kgtn So n klt O for the LTl system to be causal Then hn O for n lt O EE701 Erik Blasch 20 Wt WRIGHT STATE UNIVERSITY So the convolution sum for a causal LTl system becomes yn Zxkhn k ihkxn k kz m Similarly the convolution integral for a causal LTl system t 00 yn Jx7ht Td7 Jh7xt 7dr oo 0 So if a given system is causal one can infer that its impulse response is zero for negative time values and use the above simpler convolution formulas Also can solve which operation is easier h or X EE701 Erik Blasch 21 WRIGHT STATE Stability and lnvertibility Stability A system is stable if it results in a bounded output for any bounded input ie boundedinputboundedoutput BIBO If Xt lt k1 then yl lt k2 Example xtdt 0 lnvertibility A system is invertible if distinct inputs result in distinct outputs If a system is invertible then there exists an inverse system which converts output of the original system to the original input yn 100xn xt Wt Inverse WW XW System V 7 System W Examples t yt Ixtdt fl 1 W 4xt yn Zch 1 k 4021370 wnyn yn 1 WU 51371 dt EE701 Erik Blasch 22 DEW ii VVRIGHT STATE System lnvertibility Does there exist a system with impulse response h1t such that yltXt xt Mr W quot W Mi Widely used concept for control of physical systems where the aim is to calculate a control signal such that the system behaves as specified filtering out noise from communication systems where the aim is to recover the original signal xt The aim is to calculate inverse systems such that hnfan 5M ham 60 The resulting serial system is therefore memoryless EE701 Erik Blasch 23 m m ll WRIGHTSTATE Ex lnvertible Accumulator Sys UNIVERSITY Consider a DT LTl system with an impulse response MMmm Using convolution the response to an arbitrary input Xn yin ixtkihtn k kz oo As unk O for nklt0 and 1 for n kzo this becomes yin ixtk kz oo ie it acts as a running sum or accumulator Therefore an inverse system can be expressed as W2 Xn xin 1 A first difference differential operator which has an impulse response Mn 5M 5M 1 EE701 Erik Blasch 24 Elm WRIGHT STATE UNIVERSITY LTl System Stability A system is stable if every bounded input produces a bounded output Therefore consider a bounded input signal Xn lt B for all n Applying convolution and taking the absolute value 2 hkxn k kz oo lynl Using the triangle inequality magnitude of a sum of a set of numbers is no larger than the sum of the magnitude of the numbers lyinl s Zhtk1xtn k 2 00 3 B ilhkl k 00 Therefore a DT LTI system is stable if and only Continuoustime if its impulse response is absolutely summable ie System II WRIGHT STATE Ex System Stability Are the DT and CT pure time shift systems stable hn 5n n0 ht 6t t0 n0 1lt oo k koo Therefore both the CT and DT systems are stable all finite input signals produce a f hz39dz39 f l67 t0d71ltoo finite output signal Are the discrete and continuoustime integrator systems stable hn un n0 ht ut t0 MH ill4k non uk 2 oo k ltgt k 00 kzno Therefore both the CT and DT systems are unstable at least one finite input causes kgmlmk lt I hTldT lt 0 DMTMT 0 Tt0 dT um dT 00 an infinite output signal EE701 Erik Blasch 25 EE701 Erik Blasch 26 Em A gfgcm u w hi R C h I I W T ST T 5 W s 2 T D t i mff SEES Ime Omaln BoundedInput BoundedOutput Stability yn can be expressed in terms of the input 1n as yn Z hlmlwln m m oo For a 8180 stable system if n lt 00 then yn lt 00 Therefore 00 lt Z hmn mlltmam Z lhlmll m cgtc mZ oe Z m mZ oo where 33mm is the maximum value of am Therefore if amem lt co and if 00 Z lhimil lt 00 7112 00 then y39n lt 00 This condition is both necessary and sufficient EE701 Erik Blasch 27 21 Introduction 22 Sys Response to Internal Conditions ZeroInput Response 23 The Unit Impulse response ht 24 Sys Response to External Input ZeroState Res 25 Classical Solution of Differential equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21 Determining The Impulse Response 29 Summary EE701 Erik Blasch 28 HEW WRIGHT STATE UNIVERSITY Forced Response Forced Response Analysis sinusoid Standard Solutions assume the form nput 1111115121111 Input t3 Forced Response me wit a Exponential 11111111 a jm fullf HEW 39539 t I I39m S tusuitlal Input f cos to at 33433 RE IHEF39EEI I E39 j m iH my Jim IH mjl cn3mt H t j tjcusmie flatti lHtfmIl E slmt 5 thin EE701 Erik Blasch 29 Mimi WRIGHT STATE UNIVERSITY Ex Differential Equations 7 Forced Response Analysis sinusoid 02SD2yf Dff 1 Initial conditions yO 2 and d yO d t 3 and the input is 10 cos3 f 30 1 H02 ma lefmIl custom 6 were i 90 63 2 y f file t 3 e Et Jim l lH 3H cus3f3 39 H 3 Wham H0 3 m 2363 2 2 Tfijg 221 31321 ma a mg tjnerefnre lH 39Ejl n253 H 3 419 Mag amp Phase and Mn 1001253 cos 313un 4199 263pm 31 199 The complete solution is 111 quott 1 e quotfquot 253 ms 3 we EE701ErikBIasch Natural Forced Response 30 Emmi WRIGHTSTATE 2 UNIVERSITY 21 Introduction 22 Sys Response to Internal Conditions ZeroInput Response 23 The Unit Impulse response ht 24 Sys Response to External Input ZeroState Res 25 Classical Solution of Differential equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21 Determining The Impulse Response 29 Summary EE701 Erik Blasch 31 Emmi WRIGHTSTATE 2 UNIVERSITY 21 Introduction 22 Sys Response to Internal Conditions ZeroInput Response 23 The Unit Impulse response ht 24 Sys Response to External Input ZeroState Res 25 Classical Solution of Differential equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21 Determining The Impulse Response 29 Summary EE701 Erik Blasch 32 mm t I I mm t W lGHT STATE W IGHT STATE I r39 n ENIVERSITY SNIVERSITY g Cuttoff Frequency f 1 Th Bandwidth Duration of m ulse T T c p r h BandWIdth f if r h I Passes sInusoIds below fc ya f Jenmfg 139 m a Attenuates sinusoids above fc if H Area under a t 39n T t integral I Th M U 2 lil T r 1 h In f g 1 at High Freq b r3sz gating 39 391 1 m M i n a 0 In Th I Th HIS539 if l w t I It 1 Low Freq impulse response s effective width m V a 33 smaller time constant system responds quickly to an input 395 39 relatively large time constant System is sluggish Communication information transmission oc Bandwidth EE701Erik Blasch 33 EE701Erik Blasch at a rate of fc pulses second not to exceed 1 T h 34 W l wgtgtggglaLlnear TimeInvariant System Any linear timeinvariant system LTI system continuoustime or discretetime can be uniquely characterized by its Impulse response response of system to an impulse Frequency response response of system to a complex exponential efZWfor all possible transform of impulse Given one of the three we can find other two provided that they exist EE701 Erik Blasch 35 response lgw m 111mg WRIGHT STATE UNI VERSITY Signals and Systems Lecture 05 Fourier Series Continuous Time Erik Blasch 937 9049077 erikblaschwpafbafmil mpwwwcswrightedueblasch TA Rehka Bangladore Kalegowda bangalorekalegow2wrightedu Wk WRIGHT STATE UNI VERS 1 TY Ch 3 Fourier Series 31 Signals and Vectors 32 Signal Comparison Correlation 33 Signal Representation by Orthogonal Signal Set 34 Trigonometric Fourier Series 35 Exponential Fourier Series 36 Numerical Computation of D n 37 LTIC System response to Periodic Inputs 38 Appendix 39 Summary 3 EE701 Erik Blasch 1 EE701 Erik Blasch DEW R I I HEW R h F ngggglg w Organ ization CT Wg ggggli C 3 ou rier Series TntalRespnnse ejelit mitten High Band Low Pass DT Filters FIR IIR ejwteHeHQwMjm V 1 539t zeroinput component 2 Q a J gt ejwnaHanjw em 11 Bernstate Compnnent it Mi l 5 Complex Frequency Tlme Domaln Domain Frequency 1 r Domain A ContiIIUOUS DiSCthe I Continuous Discrete Continuous Discrete Time Time g Time Time i Time Time 0 k i S Z i 0 52 Z Laplace 00 27t T 00 271 N Transform Transform 6 St Z n CT Fourier DiscreteTime S D Z e j 0 Transform Fourier Transform Convolution jw Fourler Serles a k a H 051 k a k 9 H6 a k k 39k 2 T xt Zake W Zake E koo 2 00 Z akejkwon Synthesis equation kltNgt k 1 ak Lxgk J quotOtdt Lxte ingDidi ak N Z nk 3ka Analysis equaiton nltNgt EE701 Erik Blasch 3 31 Signals and Vectors 32 Signal Comparison Correlation 33 Signal Representation by Orthogonal Signal Set f 34 Trigonometric Fourier Series 35 Exponential Fourier Series 36 Numerical Computation of D n 37 LTIC System response to Periodic Inputs 38 Appendix 39 Summary EE701 Erik Blasch if 39 7 F39 I ialili N 7 DEW ii W i nggglg gfgli Signals and Vectors WIgggglgRngAgE Signals Energy Error Si nal Inner Product f o x f x cos 9 9 iii im ii m Engage 12 L n h x 2 x x 9 qt I I Energy ofaSignal j 3 j 2 Projection Ei 2 it it n cm iii 2 1 if1 Error vector 9 f C x Minimize derivative 1 if 3 E ii exrii if I fC1IEl 63x33 f1 Second Derivative Min i 2 f t 33 if I f I tf E1 J fit xiii iii 1 1 I Complex Conjugate J Kim ii 1 33 ii Innerproduct e E friixtriiai fand x are perpendicular t f1 52 EX Ey fl 1 rth 9 quota39 a f39 x O 395 I I f39 I The energy of the sum of two orthogonal signals is Mm39m39zes the mean39square error equal to the sum of the energies of the two signals EE701 Erik Blasch X 5 EE701 Erik Blasch 6 DEW l ll vvggcgglgkgmla Example Orthogonal vaggglgRgng Correlation Sin Wave Correlation similarity measure between i amp X ft 2 csin t O s t 3 2n Correlation Coefficient 1 3 cs 1 21quot M39n39m39ze Energy xiii 3i11t and at sin Eli iii ii En CUBE 2 x i lfl III Find c m n m i 1 f i39jisiniftji iii 1 f swiftch f sir1tdt 39w i 2 ti t i i ii Usrng the Schwarz Inequality j iixiiiaii 5 gig Minimize Energy area of analysis E 1 If imde ft e 4 7t sin 1 Ef e a u 1 a i an f rte tilich Complex j 39 EfE x I l If the correlation O the signal is orthogonal EE701 Erik Blasch 7 EE701 Erik Blasch 8 DEW DEW I I k I I 1119ngng Correlation Ex wgggggggla Signal Detection 1 quot a Given 2 srgnals En arm tit 1 Correlation degree of srmllarlty 1 Determine the Energy Can be used to measure alignment of 2 signals 2 Find the inner product Ex sonar radar ie target present coherent 3 Solve Trick determine a signal buried in noise mi 5 5 flirt 5 Transmit orthogonal signals E fxtmd fatr 5 E Iquot 39 5qu f d 1 Use threshold detectorto determine signal from noise F1 r b 5 39339 D I 1 5 quot I L 1 11 d 393 Mm f gi r 393 x 5 u l 1 1 t1 H 5 D5H5 E d WE r Signals same corrO I arm But not full pulse over time xix 39 i 1 l39 l n l 1 5 V E if F 31112th m T m H Wati if rage I z Signals same with corrected time shift corr O EE701 Erik Blasch 9 EE701 Erik Blasch 10 mm 1 mm 1 195333 Signal Detection vaggglgggli Ch 3 Fourier Series Correlation 31 Signals and Vectors Convolution 32 Signal Comparison Correlation 33 Signal Representation by Orthogonal Signal t 4 34 Trigonometric Fourier Series 35 Exponential Fourier Series igtt tiwtti f frown rid f reign rid Autocorrelation self correlation in time 35 Numerica39 COmPUtatiOquot 0 D n 37 LTIC System response to Periodic Inputs we 5 f m Jim nan 38 Appendix 39 Summary EE701 Erik Blasch 1 1 EE701 Erik Blasch 12 WRIGHT STATE UNIVERSITY Orthogonal Series Orthonormal Normalized orthogon 39 f fxmrrixnrridr L 2 2 1 H slit Z Cnxnttfi 111 Want to minimize the error 2 fft xnt cft 1 r E fft xnt cft E fx mdr 3 1 t Z cnxntt Have a complete set Fourier Series EE701 Erik Blasch 13 Mimi WRIGHT STATE UNIVERSITY Energy View Signals are equal Energywise W W Z39f Energy W 310 0 Energyrwrt T Iwlttgt2dt Fourier Series Preserves Energy Parsevals 1 00 lwitl2dt 2 air T J 00 V v Energy in the Signal Energy km harmonic EE701 Erik Blasch 14 WRIGHT STATE UNIVERSITY Orthogonal Series Signal Energy Fourier Series r r in cuntt Ea ff if di EH 3112511 111 111 31 Sum of the energies of all the orthogonal components 3 ff t dt I 3125731 62232 2 En En 31 111 Parseval s theorem Recall that the signal energy area under the squared value of a signal is analogous to the square of the length of a vector in the vectorsignal analogy ln vector space we know that the square of the length of a vector is equal to the sum of the squares of the lengths of its orthogonal components The above equation 342 is the statement of this fact as it applies to signals EE701 Erik Blasch 15 Mimi WRIGHT STATE UNI VERSITY Ch 3 Fourier Series 31 Signals and Vectors 32 Signal Comparison Correlation 33 Signal Representation by Orthogonal Signal Set m 34 Trigonometric Fourier Series 35 Exponential Fourier Series 36 Numerical Computation of D n 37 LTIC System response to Periodic Inputs 38 Appendix 39 Summary EE701 Erik Blasch 16 mm R R W W STATE W IGHT STATE f Fourier Series 1ENIVJERSITY 1ENIVERSITY Use 5675 Mt yt Eigenfunction property to 00 jka t 00 jkwot sin1plify LTI W Z are a W 2 H JkW0alt system k2 OO 00 koo analysis H jkwo hte3k 0tdt Compress N Data using x jlbuhjt truncated 3917 t N E ake Fourier Series N Joseph Fourier 17681830 EE701 Erik Blasch 17 EE701 Erik Blasch 18 ii WRSTATE Analogy Taylor s Series UNIVERSITY UNIVERSITY S If LTI 1 The idea of using trigonometric sums p X1m 1 Systgm ana 3815 was used to predict astronomical a 1 events Combine Nonlinear constant rarnp parabola 39 Euler StUdiefj Vibrating Strings N1 750 which are Signals where linear h f2 displacement was preserved with time 0 2 3E 3 me f in Fourier described how such a series could be applied and showed that a 2 periodic signals can be represented as 11 E1 f 39 Elf 51 PD f 391 7521 H3 the integrals of snnusonds that are not a 3 39 a harmonically related Problam must 1 b6 Sim 1i abla Now widely used to understand the p structure and frequency content of 4 2 be linear arbitrary signals a 3 contributions of the simpler solutions to the total Fourier Series COined after his Work solution must become neolliolible after considering by anOther aUthor EE701 Erik Blasch Oan a few terms 19 EE701 Erik Blasch 20 lm it VVRIGHT STATE UNIVERSITY ft is periodic if for some positive constant TO For all values of t ft ft To Smallest value of T0 is the period of ft Example Sin27tf0t sin21tf0t 2717 sin21tf0t 4717 Answer period 2 A periodic signal ft Unchanged when timeshifted by one period Twosided extent is te oo oo May be generated by periodically extending one period Area under ft over any interval of duration equal to the period is the same eg integrating from O to T0 would give the same value as integrating from TO2 to T0 2 EE701 Erik Blasch 21 HEW ll VVRIGHT STATE UNIVERSITY f0t CO cos2 713 f0 t 90 fnt On cos2 7t n fOt en The frequency n f is the nth harmonic of f0 Fundamental frequency in Hertz is fO Fundamental frequency in rads is o 2 713 f0 0 cosn no t 9 0 cosOn cosn no t Cnsin9 sinn no t an cosn no t bnsinn no t EE701 Erik Blasch 22 Emit VVStv ii ERETfYTEWhy Fourier Theory Important For a particular system what signals kt have the property that X kt W rm1 System gt Then mm is an eigenfunction with eigenvalue xik If an input signal can be decomposed as X 2k ak kt Then the response of an LTl system is Y 2k air1k kt For an LTl system mm est where SE C are eigenfunctions EE701 Erik Blasch 23 ng rlggfrtEWhy Fourier Theory Important Fourier transforms map a timedomain signal into a frequency domain signal Simple interpretation of the frequency content of signals in the frequency domain as opposed to time 2 4 1 1110 o39 t 10 EJ4 2 o 2 4 Design systems to filter out high or low frequency components Analyse systems in frequency domain N1 Invariant to F0 l gh 0 gt gt equency signals EE701 Erik Blasch 24 J J 4 ya th it ifftVhy Fourier Theory Important iii If FXt Xja Then Fx t jaXja a is the frequency So solving a differential equation is transformed from a calculus operation in the time domain into an algebraic operation in the frequency domain see Laplace transform becomes a2Yjaj2aYja3Yja0 and is solved for the roots aNB complementary equations 02 j2w 3 O and we take the inverse Fourier transform for those a EE701 Erik Blasch 25 WW II WRIGHTSWE Intro to System Eigenfunctions UNIVERSITY Lets imagine what basis signals kt have the property that X kt W 1amp0 System gt 7 ie the output signal is the same as the input signal multiplied by the constant gain 1k which may be complex For CT LTI systems we also have that W 2k aklk kt gt W Zkak ki Ln 7 System Therefore to make use of this theory we need 1 system identification is determined by finding KZk 2 response we also have to decompose xt in terms of kt by calculating the coefficients ak This is analogous to eigenvectorseigenvalues matrix decomposition EE701 Erik Blasch 26 Em ll VVRIGHT STATE UNIVERSITY Eigenfunctions of any CT LTI System Consider a CT LTI system with impulse response ht and input signal xt t est for any value of se C yt hz39xt z39d7 immeW dr image m e I h7e d7 Hs J h7e dz39 Assuming that the integral on the right hand side converges to Hs this becomes for any value of SE C W H 56 Therefore teSI is an eigenfunction with eigenvalue x1Hs EE701 Erik Blasch 27 WmflT39STAIEEx 1 Time Delay amp Imaginary Input UNIVERSITY Consider a CT LTI system where the input and output are related by a pure time shift W 2603 Consider a purely imaginary input signal xt 6th Then the response is ya gnu 3 e j6ej2t em is an eigenfunction as we d expect and the associated eigenvalue is HjZ e 16 The eigenvalue could be derived more generally The system impulse response is ht 5t 3 therefore I Hs foo6r 3e dre 3S So HjZ equot6 EE701 Erik Blasch 28 W i WRIGHTSTATE UNIVERSITY Note that the corresponding input e 12 t has eigenvalue e16 so lets consider an input cosine signal of frequency 2 so that cos2t gem 6quot By the system LTl eigenfunction property the system output is written as cos2t 3 So because the eigenvalue is purely imaginary this corresponds to a phase shift time delay in the system s response If the eigenvalue had a real component this would correspond to an amplitude variation EE701 Erik Blasch 29 WRIGHT STAEX 2 Time Delay amp Superposrtion UNI VERSITY Consider the same system 3 time delays and now consider the input signal Xt cos4tcos7t a superposition of two sinusoidal signals that are not harmonically related The response is obviously yt cos4t 3 cos7t 3 Consider Xt represented using Euler s formula Xt ej4t ej4t ej7t ej7t Then due to the superposition property and Hs e 3 S j21 j7t 3912 394 3912 394 3921 397 yt e e e e e e e e i 2 4 4 6Jl3eJl3ej7l3e 1703 cos4t 3 cos7t 3 While the answer for this simple system can be directly spotted the superposition property allows us to apply the eigenfunction concept to more complex LTl systems EE701 Erik Blasch 30 45 1 WWquot quotI Wt 39 wggggggls Fourier Representations Exponential Lathi Section 35 Combined trigonometric Compact 34 Trigonometric Lathi Section 34 Coefficients E Egaattn Ettpnmcfttal Ekef39hw a iillgei g a g rigmm t tit E l tlme it i l l f i li a trigonometric wt Elk Wf g metrfc att ts Z j g g gg a HE 553 km fuse EEE E 31 E jig Eli Art Coef cients Qt 5 if rflr le m f kfl 1iFri r 39 EE701 Erik Blasch 31 Wt wggggggygla Ex Periodic Harmonics o l lm I 3 ft 2A7t Funda i t teut ml cos it t 90 f ui Second Harmonic Third Ht irmmtlc 12cos27t t 90 13cos 371 t 90 111999315751 99321 ft 2A1t cos it t 90 12cos27t t 90 13cos 371 t 90 14cos 471 t 90 EE7 1 Erik Blasch 32 DIM k I R I meHTsTATE Useful FS Signals 1 meHTsTATE Useful FS Signals2 UNIVERSITY UNIVERSITY Name I Waeeform Ck k if I Comment5 h t I a I r 1 Eli Sillfete In k g Illa 1 Ci quot i f I x k dd x pt were 51quot x T i j r ark itevem Halfe wmre D quot i1 m Ln L teeti ed 1quot 3931 1 395 I l W lled Li V F El Sawtooth j ink I Remangular sine Ema FEW n W 1 wave Till Tim 2 2 TD 3 Triangular Ell HE C E 393 I were 393 wk Eleven Ianpuls 11 Fullsweve l quot j l 39 rac fied l W A 7 iquot 239T T l Tn zrnar alr I EE701 Erik Blasch 33 EE701 Erik Blasch 34 W if were FS Demo 113 if were F3 Mat39ab quot 5 nev wauemrm Of the Slgnal Choose the signal type l H m J39 grief 139 r r fl Fi I syf i d SW Elm Q Triang U Iar wave 1 fury I ve were I 3 J LI for k 1 3 fiends Em 3quot W0 2I0i ker exp ikW0t 39 megzrrezmaugmeze2freea Ck int 2tker 0 05 int 2 1 t ker 05 1 simplifyCk E M comm mm 4 to n 5 0 end Number of Fourier Coefficients Phase spectrum r ooeffi39freq T I l Hemp or Sawtooth J r J J FullWave Heetifiecl Sine Full39II39II39eue HeotIerzl Cosme l HalfWave Heetifiecl Sine n I I I Halfmule Fteotifiezl Cosine 15 1l 5 I 5 ll 15 Number of Fourier Coefficients EE701 Erik Blasch 35 EE701 Erik Blasch 36 WW 1 VVRIGHT STATE Fourier Series ml Wit FS Example 1 f t n r l r r n i n 00 2 39 2 Ge e a e10 eee tat o fta0 261 00802than 8111016000 1 f0 a0 gancos ntbsm m of a periodic Signal 2 e 1 I 2 7 1 0 l l 610 re 56h 62 1z 0504 610sz ftdt n l0 7 7r 7 0 I L Four39lelr ser39es 2 Fundamental period an 3 16 20082ntdt05041 126 2 coeffICIents an 2 coshwo dt To n 7139 n 2 8n I 20 0 Fundamental frequency bu 6 Sln2ntdt 0504116n2j compaCt Fourler 19 J ftSinnwotdt f0 To 13 HZ an andbn decreasein amplitude asn aoo series 00 To 00 27tT 2 rads ft CO Zen cosn00t 9 n1 00 2 Whare CO a0 Cn 1a bnz and 6 tan1L bn ZmCOS2Jlt 4n Sln2nt n an n1 EE701 Erik Blasch 37 EE701 Erik Blasch 38 DEW 1 DEW l W G T ST T W G T ST T Enigma FS Exam ple 2 gnaww FS Exam ple 2 t 0 t Fundamental period fta0Zan cosnntbn sin7rnt To 2 n1 A a0 0 by inspection of the plot A Fundamental frequency 0 VI an 0 because it is Odd symmetric 0 VI f0 17 0 12 HZ A 12 A 9713 12Atsinnntdt w02 T0 radS 12 Fundamental eriod 32 p 3 8111075711 dt EA 1 1 1 T 2 723912 t Esmart Esmi tttEsm51tt Esm739ttt Fundamental fre uenc q y 1 Wl th Phase Shlf t isin kt 003kt 90 2 1 7 1 21 13912 E531 Illi 3 311 mo 275 TO 75 radS bn lt nz z n 1 5 9 13 t cus1tt 9H 503131 t 9H 21 5 nus in t 90quot mis t t Q j 8A 1 nz z n 371115 EE701 Erik Blasch 39 EE701 Erik Blasch 4o W l mm a WGTSTT WGTSTT Emma FS Example 3 Emma FS Example 4 M Fundamental period at 1 TO 2 quotr 39 Fundamental frequency g 1 m Ta 3m W in in 7t2 7t2 it in f0 1275 HZ 4quot 00 2717 T 1 rads Fundamental eriod co Tri n m ri F rm 393quot w p fltrgtcozcncosltnmengt 9quot 1 at C O I T0 23975 quot1 mitt E Ems El taut 90quot Fundamental frequency C0 1 7 MW I I w a g f0 2 0 IE i Esi k cant ism awn u a 3 5 L 7 even 11 Ila11 a 00 2717T 1 rads C 2 2 n nodd I 75 quot Exponential Form I J a a 45mg m a u l 39 t 6 0 f0ra11n 371115 I z i EEMEM I I 1 l 7r n 371115 M t at 40 EE701 Erik Blasch 41 EE701 Erik Blasch 42 Him WRIGHT STATE UNI VERS I TY Signals and Systems Lecture 07 ContinuousTime Fourier Transform AperiodicPeriodic Properties Erik Blasch 937 9049077 erikblaschwpafbafmil ipwwwcswrightedueblasch TA Rehka Bangladore Kalegowda bangalorekalegow2wrightedu WI WRIGHTSTATECh 2 UNIVERSITY 21 Introduction 22 Sys Response to Internal Conditions ZeroInput Response 23 The Unit Impulse response ht 24 Sys Response to External Input ZeroState Res 25 Classical Solution of Differential equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21 Determining The Impulse Response 29 Summary CC yin e adikilffiim ki e riffII W W xremewiae e Z 1 i VT k W 39 gt L4 21 L1 Ale kit db k2 EE701 Erik Blasch EE701 Erik Blasch 2 HEW 3 I i I HEW 3 I I WRIGHT STATE C h 3 F S WRIGHT STATE 0 g 1 UNIVERSITY UNIVERSITY r I II I CT Filters TotalRespnnse I cje Jt f 2fji jwt I 390 I h B d L P III e aHaHOweJ ig an ow ass 31 SIgnaIs and Vectors I um n DT Filters FIR llR 32 Signal Comparison Correlation p C 9 if 30 39 6 aHe He 9quot zerostate nmpunent f t 33 Signal Representation by Orthogonal Signal Set T D In 5 Complex Frequency 5 Frequency 1 lme omam v 34 Trigonometric Fourier Series I Doma Doma 35 Exponential Fourier Series conimuous DISCrete i continuous Discme i Continuous Discrete 3 5 N 39 I C t t39 f D Tlme Tlme Time Time Time Time umerica ompu a ion 0 I I t k i S Z 0 Q 37 LTIC System response to Periodic Inputs 5 5 Laplace Z i kh k h l i 02712T 0 27iiN 3 8 Appendix yin kgoxi I in I xn n I Transform Transform I 0 0 3399 summary Mt EOXWWU TMT 340 6 St Z n CT Fourier Discrete Time 39 S D z e j w E Transform Fourier Transform Convolutlon o 2 t I E Z I 25 IT E gt QT 632 6T6 lt I k d Fourler Serles a k a H 051 k a k a He 051 k x a L v i 7 not 7 I jkwon S th 39 t39 f L I r j I J XE3917ng x Zakejk Zakek2 T kvgt ake yn 818 equa 10H k oo koo ak Lxtejkwotdt Lxtejk2 Ttdt ak Z mnejkw0quot Analysis equaiton EE701 Erik Blasch EE701 Erik Blasch ltNgt 4 wmmsmiia Ch 4 Fourier Transform UNIVERSITY 41 Aperiodic Signal Representation By Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform 44 Signal Transmission Through LTIC Systems 45 Ideal and Practical Filters 46 Signal Energy 47 Communications Amplitude Modulation 48 Angle Modulation 49 Data Truncation Window Functions IIIWk EFT I Ii39l l39f lJllli mi WRIGHT STATE I UNIVERSITY Fourier Series of CT Periodic Signas A periodic CT signal can be represented by its Fourier Series Fourier series coefficients form the spectrum or the frequencydomain representation of the periodic signal Fourier Series for Example Signals JHU 88 and C Demo 410 Summar i y Convergence and Gibbs phenomenon EE701 Erik Blasch 5 EE701 Erik Blasch DEW ii Him ii VVIgi iIEREWEFourier Series of CT Periodic Signals Properties of the Fourier Series Time Shift xt t0ekw0t0ak Multiplication CtytFS i albkl 2 00 FS of an Impulse Train Va i 5t nT is ak 712 00 1 2 2 Parseval s RelationP 2 ixt dt Z lakl t k SteadyState Response of LTl Systems to Periodic Signals and Filtering System Function HS j MTV d Frequency Response Hja Fourier Series Coefficients of output of LTI System ya ZZZ m akHjkw06jW bk 2 akHjk00 EE701 Erik Blasch 7 WRIGHT STATE I I I UNIVERSITY ContinuousTime Fourier Transform Fourier Transform representation of an aperiodic signal Derived by defining an aperiodic signal as the limit of a periodic signal as the period T becomes arbitrarily large Properties Differentiation Extijjw 2 oo 1 I Integration iXTdTj w X050 73X 050 Convolution ICU yt FT XjaYja Multiplication xtyt2ix jwYjw 7T EE701 Erik Blasch W i WW 1 WRIGHT STATE Fourier Series to Fourier Transform WRIGHT STATE Ch 4 Fourier Transform UNIVERSITY UNIVERSITY For periodic signals we can represent them as linear 41 Aperiodic Signal Representation By Fourier Integral combinations of harmonically related complex exponentials 42 Transform of Some Usefu Functions To extend this to nonperiodic signals we need to consider 43 Some Properties of the Fourier Transform 1 aperiodic signals as periodic signals with infinite period 44 Signal Transmissmn Through LTIC Systems As the period becomes infinite the corresponding frequency components form a continuum and the Fourier series sum 45 Ideal and Practlca39 Fllters becomes an integral like the derivation of CT convolution 45 Signa Energy However we ll qeneralise this to nonperiodic signals that 47 Communications Amplitude ModuIation which have a Fourier transform representation a complex valued function Instead of looking at the coefficients a harmonically related 48 Angle Modulation 49 Data Truncation Window Functions Fourier series we ll now look at the Fourier transform 410 Summary which is a complex valued function in the frequency domam EE701 Erik Blasch 9 EE701 Erik Blasch 10 ii ii WRIGHTSTATE Ch 4 Cont Time Fourier Transform WRIGHTSTATE ApenodIC Signal UNIVERSITY UNIVERSITY 41 Aperiodic Signals ContinuousTime Fourier Transform o Aperiodic signa can be M Development of the Fourier Transform Representation expressed as a continuous A in Convergence of Fourier Transforms 10 Examples of continuousTime Fourier Transforms Fig 41 Construction ofaponodio signal by poiiioiiio extension of t 42 The Fourier Transform for Periodic Signals everiaSimg exponeniiais I if If we let TO e co the pulses o 43 Properties Of the ContinuousTime Fourier Transform in the periodic siqnal repeat III 1 i L T m Inearltv ImeS Itlng after an infinite interval Conjugation and Conjugate Symmetry E I 39 Differentiation and integiaiion The exponential Fourier series notation ii39iiii Z Dtiiwii Time and Frequency Scaling En quotm 1 Tm I Duality Parseval s Relation iii E ii 5 41 ifiniilfiimii ii 44 The Convolution Property Integrating fTO t over TO 2 TO 2 is the same as integrating ft 45 The Multiplication Property FrequencySelective Filtering with Variable Center Frequency over 00 0039 IIIIII 46 Tables of Fourier Properties and Basic Fourier Transform Pairs on 2 iii nits fifties Emma gt Fist 2 If ne quotimam 47 Systems Characterized by Linear ConstantCoefficient Differential Til its Ti 39 39 m Equa ons EE701 Erik Blasch 1 1 EE701 Erik Blasch 12 WRIGHT STATE Aperiodic Signal 2 UNIVERSITY Fourier coefficients D are 1 To times the samples of FD uniformly spaced at intervals of 90 Therefore 1TOFt is the envelope for the coefficients D n We now let To 9 0 by doubling TO repeatedly Doubling TO halves the fundamental frequency 00 Envelope I J 39 T Fm t WRIGHT STATE UNI VERSITY Fourier series analysis switch to Ace Aperiodic Signal Gill Fquot 2 my Page fT HJE e E Mwnt wz FT fitjg Efjn mit ma em TIE no Amount of the component of frequency nAoo is FnAwAoo21t 1 ctr h i I I E i i i i f it T1112 fig g t gmi E v an we e a if I i Envelope WEI33 l l ll il m i e t e t 39 iwt 1quotlquot39lquotalil l l 7 I I HrI fhr r y JaI39l39lrli39rrulIII 1quotrii m g H D am Flint i r gt Etta Ftri t l l39 wfg im Spectrum progressively becomes denser while its magnitude becomes smaller Vii Spectrum is so dense that the spectral components are spaced at zero i Vt infinitesimal intervals f a7i EE701 Erik Blasch 13 EE701 Erik Blasch mite at e 14 Matlab WRIGHT STATE WRIGHT STATE Fou rier Sen es to F Transform 1 Matlab Periodic Signals 69 Fourier Series To use the CT Fourier transform you need to have the symbolic toolbox for Matlab installed If this is so try typing gtgt syms t gtgt fouriercost gtgt fourierltcos2t gtgt fouriersint gtgt fourierexp tA2 Note also that the ifourier function exists so gtgt ifourierfouriercost EE701 Erik Blasch 15 Aperiodic Signals 69 Fourier Transform Consider an aperiodic signal xt Now consider periodic version of xt Xpt where xt is repeated over a period T We can say that xpt is the limiting case of xt as T approaches to infinity ie xt ting xp t Since xpt is periodic we can write Fourier series representation for it 00 o 1 T2 kat kat XP I Zake 0 ak Ix te 0 dt kw T2 where w027tT EE701 Erik Blasch 16 ng glg gfgli Fourier Series to F Transform 2 Since xt 0 outside of the interval T2T2 1 00 jkwot ak xte dt 00 Define an envelop function MM Mae Wait 1 SO ak Xw0 Now substitute that in the Fourier series representation of xpt 00 1 v w xplttgt Z Xw0ek t k EE701 Erik Blasch 17 Marsala Fourier Series to F Transform 3 UNI VE R 51 TY I Since 21tTo0 or1T DO27t 1 39wt xpt E Xw0 k 0600 kz oo As T900 this sum will approach to an integral because there will a continuum of Fourier terms in the Fourier Finally the Fourier Transform of xt 00 Definition X 1 60 acme Wt envelop 0 function And the Inverse Fourier Transform of xt xt JXjaeJ dd Over all o 27 Oo EE701 Erik Blasch 18 meHTs TAlE Definition of Fourier Transform UNIVERSITY We will be referring to functions of time and their Fourier transforms A signal xt and its Fourier transform Xja are related by the Fourier transform synthesis and analysis equa ons Xja j Mme Welt Fxt and xt l Xltjwgtejwtdw F 1Xltjwgt We will refer to xt and Xja as a Fourier transform pair with the notation F xr9Xja As previously mentioned the transform function X can roughly be thought of as a continuum of the previous coefficients A similar set of Dirichlet convergence conditions exist for the Fourier transform as for the Fourier series T oooo EE701 Erik Blasch 19 ii WRIGHT sm Fourier Transform vs Series UNIVERSITY We will be referring to functions of time and their Fourier transforms A signal xt and its Fourier transform Xja are related by the Fourier transform synthesis and analysis equations and Xo39w Lace Walkman xt l Xltjwgtejrdw F 1Xltjwgt F xraX jw Founersenes Zakejk27Z39TZ kz oo kz oo ak Lxte jkwotdt Lxtejk27rTtdt Difference Series versus transform 12712 EE701 Erik Blasch 20 Whigs Convergence Fourier Transform3W Similar to Fourier Series convergence conditions apply for Fourier Transform The signal must be absolutely integrable Jxtdt lt oo Over a finite interval of time the signal must have finite number of maxima and minima or vana ons Over a finite interval of time the signal must have finite number of discontinuities Also those discontinuities must be finite EE701 Erik Blasch 21 WW ll WHERE Mechanical Analogy Loaded Beam an aperiodic signal is not easy to visualize because it has a continuous spectrum The continuous spectrum concept can be appreciated by considering an analogous more tangible phenomenon Example of a continuous distribution is the loading of a beam Load at every point the load at any one point is zero A meaningful measure of load in this situation is not the load at a point but rather the loading density per unit length at that point Let Fx be the Force 11 in WE39f39iEI quot i 1 ED alg EFEn L1 i E1 EE701 Erik Blasch 22 33m ll Wtfiffsif Signals Analogy For an aperiodic signal the spectrum becomes continuous that is the spectrum exists for every value of a but the amplitude of each component in the spectrum is zero The meaningful measure here is not the amplitude of a component of some frequency but the spectral density per unit bandwidth W is SyntheSiZGd by adding exponentials of the form ejnAODt But the contribution by exponentials in an infinitesimal band Au located at a n A0 is 121tFn AwAw and the addition of all these components yields ft in the integral form W m a quot 39 39 fi wflf jet m t gw Z stimuli are 2 Male at a nape m Contribution by components within a band do is 12717FD do Fo dF where dF is the bandwidth in hertz Foo the Fourier spectrum EE701 Erik Blasch 23 l I WSW Time Frequency Analysrs We can transform between Time and Frequency flirt1 FM Flwlgf Mftj39emdt lt Ffm gjm fr imt oo Note Sign change 39 1 jwt fifl imFlwle rim And represented in Magnitude and Phase For Fiwleiitiri IFE wll E lFlfwll Flw a Fi w EE701 Erik Blasch 24 DEW ii MW ll wrsrverrlir arge EX 1 Decaying Exponential yvrsjrverrgrglrgeEx 2 Smgle Rectangular Pulse Consider the nonperiodic rectangular pulse at zero 1 ltlltT1 xt O t2T1 The Fourier transform is Consider the nonperiodic signal xt e aluU a gt O Then the Fourier transform is u Xja garage War feajwtdt if K Xja Lane War le Walt 1 ajat 2 6 1 a tartar i E aja Fife rr ef j if 1 We 2 e Z 1 jw T CH 10 a 1 2 1 1 I 2 Note the values are real 7 3 o 5 lFlwll e w U 395 393 m 5 E 1 E 2 I E T1 1 E U Ffw tamal 3 CI 2 I y I I m 395 3 f 5 3912 1o 0 a 10 2o EE701 Erik Blasch 25 EE701 Erik Blasch 26 WW it ll W s 39 W S 39 39 Elfitfmffff EX 3 ImPU39Se Signal SEES Time vs Frequency Domain 39 The Fourier tranSfOFm Of the impUSe Signal Can be Fourier Analysis Series or Transform is in fact a way of calculated as follows determining a given signal s frequency content ie move from xt 5m timedomain to frequency domain It is always possible to move back from the frequencydomain to Xja f 5tequot dt 1 timedomain by either summing the terms of the Fourier Series or by Inverse Fourier Transform 39 TherefOres the FOUHQ trlansmrm 0f the 39mPUSe funCt39On Given a signal xt in timedomain its Fourier coefficients ak or has a constant contribution for all frequenCIes its Fourier Transform Xo are called as its frequency or line X I A spectrum 60 lf ak or Xo is complex then frequency spectrum is observed by their magnitude lakl or Xo and phase 4ak or zXo plots gt eg w Xa A616 Xa A 4mm t9 EE701 Erik Blasch 27 EE701 Erik Blasch 28 DEW r VVRIGHT STATE LTIC System Response Inputoutput pair ip it MEN e m s H Ifn wje m w W penest g FWRWHlR MMMJEstneujs 21f 39 2st Using Linearity Elli m 1a A m sweetest Emmy Em F tees1W lemmas ssure rlnl39 f I 2i rw 3W TEECw rm input it nutpit will 1 RS 39 a 39 z a l l Wthh leads to W sir net 2 FWWJHW E W R s engage W t WRIGHT STATE UNIVERSITY Timedomain input ft is a sum of its impulse components Frequencydomain input is a sum of everlasting exponentials 1 For the timedomain case en 2 Isis the impulse response ef the system is Islij t I x 5Q 9 ti espresses t es a sum ef impulse eempnnents III 1 flit J st Isis I d r expresses flit as a sum efrespenses tn impulse enmp enents 2 For the frequencydomain case 139 T 3th 2 HIij pm the system respense tn ejmtis Hire ejm EIII 1 1 ea FE lwgd t a f Fth e a t t1 re shews at as a sum ef everlasting exponential eempeuents T E w an e3 w f39i Tri es 4 FmHim 2ft Les s 1 I m YE w aw flit a f Fth H tejl en 1 re flit 1s a sum efrespenses te espenennel eempenents v 39 er equot 39 393 EE701 Erik Blasch 21 Em n 29 EE701 Erik Blasch 30 em r 111 it meTsmE Ch 4 Cont Time Fourier Transform UNIVERSITY 41 Aperiodic Signals ContinuousTime Fourier Transform Development of the Fourier Transform Representation Convergence of Fourier Transforms Examples of ContinuousTime Fourier Transforms 42 The Fourier Transform for Periodic Signals 43 Properties of the ContinuousTime Fourier Transform Linearity Time Shifting Conjugation and Conjugate Symmetry Differentiation and Integration Time and Frequency Scaling Duality Parseval s Relation 44 The Convolution Property 45 The Multiplication Property FrequencySelective Filtering with Variable Center Frequency 46 Tables of Fourier Properties and Basic Fourier Transform Pairs 47 Systems Characterized by Linear ConstantCoefficient Differential Equa ons EE701 Erik Blasch 31 WRIGHT STATE UNIVERSITY For all t Xt T Xt Xt is a period signal Periodic signals have a Fourier series WHI representation Cf jxte dt Cquot computes the projection components of Xt having a frequency that is a multiple of the fundamental frequency 1 139 EE701 Erik Blasch 32 VVRIEPTT STA fE EX 4 39 Periodic naIS WRIE PTTMSTARTE CT Fourier Transforms Of UNIVERSITY 9 UNIVERSITY Periodic Signals A periodic Signal Violates condition 1 of the Dirichlet conditions for the Fourier transform to exist integral may not work 5111010086 However lets consider a Fourier transform which is a single impulse of X0 5W 0 area 27zat a particular harmonic frequency a 00 T Xjw27r5w w0 oo t i jwtd i jwot The corresponding signal can be obtained by 5 7 2 00 w 0 6 w 2W6 xt i f 2725co 00ej da 6W That is which is a complex sinusoidal signal of frequency 00 new lt gt 2mm W0 More generally when X060 szk5w kwo kz oo Then the corresponding periodic signal is More generally xt Z akekaot 2 00 jkwot ltgt 39 The Fourier transform of a periodic signal is a train of impulses at the Mt k ake wa k 2mk6w IMO harmonic frequencies with amplitude 27zak 00 00 EE701 Erik Blasch 33 EE701 Erik Blasch 34 mm 3133 Ex Fourier Transform of Cosme WRIGHT STATE UNIVERSITY xt COS wot lejwot le jwot Consider the periodic signal xt below 2 2 A I X001 7T60 00 7T6w LL20 39 A t I V X000 T 39T1 0 T1 T It It We know that the Fourier coefficients for xt will be I 2T1 10 0 Do i ak lt 9 k O Slnka0T1 k 72 O k7 EE701 Erik Blasch 35 EE701 Erik Blasch 36 lm it waster Ex Rectangular Pulse 2 Sketch ak on the k axis The sketch forms a discrete version of the sinc function For each value of k the signal xt has a periodic component with weight ak So we way that the above sketch shows the frequency content of the signal xt EE701 Erik Blasch 37 mm Wmlggw Ex Rectangular Pulse 3 Now sketch ak on the oaxis ak 2T1T T I I I a l 11 12030000000020001 11 l On the oaxis distance between two consecutive as is now w027tT the fundamental frequency EE701 Erik Blasch 38 Elm it th it ifff Ex Rectangular Pulse 4 As the period T oc the fundamental frequency 1090 So the distance between the two consecutive aks becomes zero and the sketch of ak becomes continuous what is called as Fourier Transform At the other side as T oc the signal xt becomes aperiodic and takes the form xt A V T o T V This means the Fourier Transform can represent an aperiodic signal on the frequencydomain EE701 Erik Blasch 39 DEW meHTsTAlE Ch 4 Fourier Transform UNIVERSITY 41 Aperiodic Signal Representation By Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform a 44 Signal Transmission Through LTIC Systems 45 Ideal and Practical Filters 46 Signal Energy 47 Communications Amplitude Modulation 48 Angle Modulation 49 Data Truncation Window Functions 410 Summary EE701 Erik Blasch 4o Tm T mm i WTTGTTT STATE Transform Fu notions WTTGTTT STATE Transform Fu notions UNIVERSITY UNIVERSITY Gate Sinc ci clg sin it I rest a g Sincx sin X X is the sine over argument i ill 2 1 i This function plays an important role in signal processing It is also m J E known as the filtering or interpolating function T J I I I T 1 SincX is an even function of X f q 2 SincX 0 when sin X 0 except at X 0 where it appears indeterminate 1 3 i Hquot 1 g 14 This means that sinCX O for X in at i31t 2 all 2 W 3 Using L Hopital s rule we find SincO 1 4 SincX is the product of an oscillating signal sin X of period 271 and a Triangle amp g El Imi 3 monotonically decreasing function 1X Therefore Sincx exhibits E T quot 1 2M M g 1 sinusoidal oscillations of period 271 with amplitude decreasing continuously I I 2 as I X I If i The quotTE 39 TIT quotinn 1 I a T 2 quot EE701 Erik Blasch 41 EE70 mast 42 33m 3 lm it WTTGTTT STATE Transform Fu notions WTTGTTT sTATT Transform Fu notions UNIVERSITY UNIVERSITY Other iii mm L egmtuff 1 a T 0 a jw 2 out at l a 29 l I jw in 3a 3 5 it a 3 ill 4 to tuff Q ij t1 2 D 1m 11 n 5 f e tuft Em jtmjn1 at 2 i To gift 1 T l 2m w E gjw t Ettg LLFIB 39939 ma a th ft 5 m um Mm l mu W 57111 writ jfrf lfm Than aim ETTng EE701 Erik Blasch 43 EE701 Erik Blasch 44 W I WW I wgggggIgITIAgE Transform Functions wgggggIgITIAgE Ch 4 Fourier Transform 1 41 Aperiodic Signal Representation By Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform I v 44 Signal Transmission Through LTIC Systems l 13 cos whim 14 Sin mat irt 15 at 5 i M it If f if D 45 Ideal and Practical Filters 16 Fatwa viaMfr a 2 46 Signal Energy 1 reel g wine 4 47 Communications Amplitude Modulation ifs ggimwn ma 2 48 Angle Modulation 1399 a It gaining mi 49 Data Truncation Window Functions 23 gaining 410 summary no DD 21 2 Mt fiT mu Z Slim main HI 7 22 eiirw amrrfwfi EE701 Erik Blasch 45 EE701 Erik Blasch 46 gm I I DEW I I WRIGHTSTATE Ch 4 Cont Time Fourier Transform wmamsm Fourier Transform UNIVERSITY UNIVERSITY 41 Aperiodic Signals ContinuousTime Fourier Transform A Signal X and its Fourier transform qu are related by Development of the Fourier Transform Representation I Convergence of Fourier Transforms X 2 f Xjaefwtda Examples of ContinuousTime Fourier Transforms 42 The Fourier Transform for Periodic Signals X010 0 De WC 43 Properties of the ContinuousTime Fourier Transform This iS denoted by Linearity Time Shifting F Conjugation and Conjugate Symmetry xt a X 13960 Differentiation and Integration Time and Frequency Scaling Duality Parseval s Relation 39 For example 1 1 44 The Convolution Property eatum 45 The Multiplication Property a ja FrequencySelective Filtering with Variable Center Frequency I I I I I I Remember that the Fourier transform is a denSIty function 46 Tables of Fourier Properties and Ba3ic Fourier Transform Pairs 47 Systems Characterized by Linear ConstantCoefficient Differential you must 39ntegrate 39ts rather than summ39ng Up the d39Screte Equations Fourier series components EE701 Erik Blasch 47 EE701 Erik Blasch 48 DEW if L It f F T f HEW ll u WiggiiETREITgE Inearl y 0 OUrler ranS orm VVIEZEIPIIEREEE f F 39f xltrgteXltjwgt xt e Xa F Then jwtoX d F xt t0He 10 a y emw P f 00 xt 1M 1 50de Then F Now replacing tbyt f IOU to axtbyteaX 1wbY 10 HMa EXWW 6 I f e j 0Xjaerw This follows directly from the definition of the Fourier transform as the integral operator is linear It is easily 0 Recognizing this as Fxtt0 e J39wl oXUw extended to a linear combination of an arbitrary number A I h h Mt d t of Signals Slgl la W lC IS S l e In ime Does not change magnitude of the Fourier transform Does shift the phase of the Fourier transform by D tO so tO is the slope of the linear phase EE701 Erik Blasch 49 EE701 Erik Blasch 5o Wit WRIGHT STATE EX Linearity amp UNIVERSITY ng ll WRIGHT STATE Frequencyshifting Property Consider the signal linear sum of two time shifted steps xt 1t 25x2t 1 39 ejwotft927zFw w0 e jwotfte 27rFaa0 where X1t i of width 1 91 0 x1 1 1 1 39 cosa0t f e 7rFa w07rFa 00 Using the ectangular pulse exa 2 X1jwinw2 j 2 I x2tl 2 cosw0tfteFww0Fw w0 Then using e linearity and time hift i 39 i Fourier transfo m properties 0 Slnw0tft9 277Fww0 JEFW 00 0 2 xt 4 sina0t fine Fw e0 Fw 00 X M e ijz sina 2 25in3a 2 J J EE701 Erik Blasch 51 EE701 Erik Blasch 52 DEW i VVRIGHT STATE Fourier Transform of RealValued Signals Conjugation If Xt is realvalued it is X 60 X gtllt Hermitian s nimetr Moreover y y X co X 60 XP j argX 60 whence E i ii whitish Differentiation Property yt 2 d2 an LTI system FT Differentiatlon property gt E jw U HUW jw 1 Amplifies high frequencies enhances sharp edges 2 7t2 phase shift j 81772 I X 60 39239 X I and sin wot wo cos wot wg sinw0t argX argX cos wot w0 sin wot wo cosw0t EE701 Erik Blasch 53 EE701 Erik Blasch 54 mm i ii I I I I VVI i iiIERETTAJE Differentiation amp Integration WEEEEREE EE Tlme leferentlatmn PFOPertY By differentiating both sides of the Fourier transform Conditions synthesis equation 00 I dlt 2 iijjaequot quotda t oo Therefore In ijUw This is important because it replaces differentiation in the time domain with multiplication in the frequency domain Integration is similar xz39dz39 X jco 7rX 05w 00 M The impulse term represents the DC or average value that can result from integration EE701 Erik Blasch 55 From the chain rule recall that Judi2m Iiidbl Let u 2 EM and dv dft so du j0 EM 3a e jmftm grewW jwfmfte j dt ja ft 0 when t a co ft is differentiable Derivation of property Given ft ltgt FD Let Ba Fi ft dt Ba JEtejmdt ltgt mm Egg WW digit e 1w Fw Note Frequency Differentiation mg a jdXwdw EE701 Erik Blasch 56 Em R 1351ng Time Integration Property Find J f xdx ltgt From the property of time convolution j fxdx i fxut xgtdx f t MU Fa7r 5a Ja 7 F 5w Therefore Fw 0 fxdx ltgt 7239 FO 5a WK wmoorsm Ex Fourier Trans of Step Signal UNIVERSITY quotiiio Lets calculate the Fourier transform Xja of Xt ut making use of the knowledge that F gt 5t Gjw 1 and noting that xlttgt gwwf Taking Fourier transform of both sides GUCU ja using the integration property Since Gja 1 Xltjcogt 7r5lta2gt Ja Xja 7zGO5a We can also apply the differentiation property in reverse 50 dug ja7r5a1 dt 16 EE701 Erik Blasch 57 EE701 Erik Blasch 58 wmoHTsmi rE Timereq Scaling WISIAggREEEE Ex Scaling Desugn Tradeoffs UNIVERSITY Narrow In Time Wide In Frequency Same as Laplace ftHFw l lire c nC I e lie fatiF2j q y iai 6 n transform scaling property h x a gt 1 compress time axis expand frequency axis a lt 1 expand time axis compress frequency axis Effective extent in the time domain is inversely proportional to extent in the frequency domain aka bandwidth ft is wider a spectrum is narrower ft is narrower a spectrum is wider EE701 Erik Blasch 5g EE701 Erik Blasch 60 DEW i 39 meHTsmE Fourier Transform Duality UNIVERSITY MW ll WRIGHT STATE EX I D U a39 Ity UNIVERSITY I I Forwardinverse transforms are similar Fourier Transform Synthesns and Analysns formulas jt 1 jCUt Fw foofte dt ft JFae da f0 9 Fw 33t H XUw 5109275 0 Example recttt a C sincw C 2 oo Apply duality C sinct 12 a 2 7t rectwt Xjw t 9mdt FT rect is even I sinct 12 a 2 717 rectwt 00 f FUD T 1 00 1 23t Xjwej tdw Inverse FT 27T OO 0 42 0 12 t E JEN 0 291 6quot EE701 Erik Blasch 61 EE701 Erik Blasch 62 WRIGHT STATE r0 er eS ummar WRIGHT STATE p y N VERSPrOper39tleS of the Fourier Transform The Fourier transform is widely used for designing filters You can design systems with reiect hiqh frequency noise and just retain the low frequency components This is natural to describe in the frequency domain Linearity we Xw wetco Important properties of the Fourier transform are axt yt e 05X 0 6Ya 1 Linearity and time shifts axtbytaXj60bYj60 Left or in o 2 Differentiation dxm ij 1w an dt xt t0 9 X 06 3 Convolution Mt htxfYjw HjwXjw scaling 1 0 Some operations are simplified in the frequency domain but there are a xltat a X number of signals for which the Fourier transform do not exist this leads a a naturally onto Laplace transforms EE701 Erik Blasch 63 EE701 Erik Blasch 64 EEEEUI R VWHGHTSDME UNIVERSITY a m g Properties of the Fourier Transform Time Reversal x t a X 60 Multiplication by a Power of t EEEEUE a VWUGHTSDUE UN39VERSBroperties of the Fourier Transform Multiplication by a Sinusoio Modulation xt sina0t H Xa coo Xa 00 xt cosa0t H Xa 00 Xa 00 n n dn t 14091 M X 60 Differentiation in the Time Domain Multiplication by a Complex Exponential dn a 39 EEEEEQR at MEEEUE R WRIGHT STATE UNIVERSFYroperties of the Fourier Transform Integration in the Time Domain t j mm a X 0 75X 05m 1a Convolution in the Time Domain xt W H X 60Y60 Multiplication in the Time Domain xtyt H X 60 Y 60 EE701 Erik Blasch 67 VHUGHTSDHE UNIVERSFYroperties of the Fourier Transform Parseval s Theorem Jxt WW 9 i JXaYada 27 E if yt w I w I2 dt 9 i I Xo Izda Duality X t H 27rx a EE701 Erik Blasch 68 DEW i Wii ittgrbperties of the Fourier Transform Summary WRIGHT STA vmmm ectangular Form of Fourier Txm Consider TABLE 41 PROPERTIES OF THE FOURlER TRANSFORM 00 Property Transform PairProperty at Linearity axr bvt 3 uXcu iicu X J J a E D Right or left shift in time xt c gt Xme i 9 Time scaling xat lt gt X g a gt 0 00 Time reversal x r lt gt X cu I I I Multiplication by a power oft t xt Hfquot er n 1 2 O S I n In 9 e n e IS a m p I Multiplication by a complex exponential xte quotquot lt gt Xw too a real I I Multiplication by sin a xt sin agree Xa 0J0 X039 200 n O n 3 U S I n g E U I e r S U I a Multiplication by cos on xt cos wot H X u mg Xai 110 00 00 Differentiation in the time domain 341 gt Urn Xw n 1 2 Integration f xfi dAHXw 3rX0r5w X J Convolution in the time domain xt gtr ut X to Wm OO 00 Multiplication in the time domain xrvt lt gt 51 X to gt1 Vw Y J y Y J Parseval s theorem xrvt dr Mlm do R I Special case of Parseval s theorem x20 dt J lXml2 do X R Duality Xr lt gt brad m EE701 Erik Blasch 69 EE701 Erik Blasch 7o DEW R WRIGHT STATE UNIVERSITY wgivggg gp la Ex Polar Exponential Signal Consider the signal xT mug be D Its Fourier transform is given by X 0 Ra 110 can be XW Jemutejwdt expressed in a polar form as so X 0 X 0 exp j argX 60 Ole War eltbfwgtt bjw 2 200 If b lt O Xoo does not exist If b O xt ut and Xoo does not exist either in the ordinary sense 2 2 X0 R w1 0 o fbgtOitis argXw arctanj where I amplitude spelctrum phase spectrum X 0 IXquot Rt 0 19 ja EE701 Erik Blasch 71 EE701 Erik Blasch 72 argX 0 arctan new WRIG HTSTATE Ch 4 Cont Time Fourier Transform UNIVERSITY 41 Aperiodic Signals ContinuousTime Fourier Transform Development of the Fourier Transform Representation Convergence of Fourier Transforms Examples of ContinuousTime Fourier Transforms 42 The Fourier Transform for Periodic Signals 43 Properties of the ContinuousTime Fourier Transform Linearity Time Shifting Conjugation and Conjugate Symmetry Differentiation and Integration Time and Frequency Scaling Duality Parseval s Relation 44 The Convolution Property 45T he Multiplication Property FrequencySelective Filtering with Variable Center Frequency 46 Tables of Fourier Properties and Basic Fourier Transform Pairs 47 Systems Characterized by Linear ConstantCoefficient Differential Equa ons EE701 Erik Blasch 73 wmamsmla Impulse Train Sampling Function UNIVERSITY xt 51375 Z 6t nT TLZ OO Note period in t T period in D 27tT 471 2n 0 27 47 n EE701 Erik Blasch 74 m i W i ffg iw Convolution Property W Mt 9375 H 3000 HOWXUW Where Mt lt gt Hjw A consequence of the eigenfunction property 1106 2 foo ixgw ejwtdw oo W coef cient a ii superposition ya foo Homixw ejwtdw HltLgtoa 1 00 HjwXjw eJWtdw 27139 00 W YUW EE701 Erik Blasch 75 UNIVERSITY WRIGHTSTATQOHVOIU EIOH in Frequency Domain With a bit of work next slide it can show that ya hltrgtfxltrgt rltjwgt HltjwgtXltjwgt Therefore to apply convolution in the frequency domain we just have to multiply the two functions 39 NOte ht a Ha Gimme de To solve for the differentialconvolution equation using Fourier transforms 1 Calculate Fourier transforms of xt and ht 2 Multiply Hja by Xja to obtain Yja 3 Calculate the inverse Fourier transform of Yja Multiplication in the frequency domain corresponds to convolution in the time domain and vice versa EE701 Erik Blasch 76 7 By the time shift property oo the bracketed term Y 160 MTV WHUOMT is e 10 Hja so 00 Hja jx7ejmd7 WW VVRICFIT STARTE EX 1 SOIVlng an UNIVERSITY Consider the LTl system time impulse response hr e b ug b gt o to the input signal xt 6atltl a gt O 1 TXM Transforming these signals into the frequency domain 1 H060 X060 b aJw 2 MULTIPLY and the frequency response is 1 39 39 39 gt Y 39a Yj on How Xj n J bjwajw to convert this to the time domain express as partial fractions 1 1 1 Y 39a J b a aja bja b 3 INVERSE FT Therefore tlie time doniain response is W 1 ca um We H 39a X 39a EE701 Erik Blasch 77 EE701 Erik Blasch 78 w 1517mm w IGHT STATE Ch 4 Cont Time Fourier Transform ENIVERSITY 2 a ENIVERSITY H Lets design a low pass lter HU 41 Aperiodic Signals ContinuousTime Fourier Transform 1 I 0K 0 Development of the Fourier Transform Representation Hja C Convergence of Fourier Transforms 0 lwlgt 06 600 00 w Examples of ContinuousTime Fourier Transforms The impulse response of this filter is the inverse Fourier transform IN g emdw 8111th which is an ideal low pass filter Noncausal how to build The timedomain oscillations may be undesirable How to approximate the frequency selection characteristics F Consider the system with impulse response tum H aJw Causal and nonoscillatory time domain response and performs a degree of low pass filtering EE701 Erik Blasch 79 42 The Fourier Transform for Periodic Signals 43 Properties of the ContinuousTime Fourier Transform Linearity Time Shifting Conjugation and Conjugate Symmetry Differentiation and Integration Time and Frequency Scaling Duality Parseval s Relation 44 The Convolution Property 45 The Multiplication Property FrequencySelective Filtering with Variable Center Frequency 46 Tables of Fourier Properties and Basic Fourier Transform Pairs 47 Systems Characterized by Linear ConstantCoefficient Differential Equa ons EE701 Erik Blasch 8O W i WRIGHT STATE UNIVERSITY Parseval s Relation gt0 lwtl2dt i 00 Xjw2dw OO 27l OO 1 2 W k V J lX 3wl Total energy Total energy 27139 in the time domain in the frequency domain Spectral denslty Multiplication Property Since FT is highly symmetric 00 atF 1 Xjwejwtdw Xjw i xte jquotquottdt 27139 00 thus if m gtlt W H W You then the other way 1 75 t X39wgtlltY39w around is also true 2W 9 9 Z Xj6Yjw 6d6 Definition of convolution in D A consequence of Duality W gla Convolution and Multiplication Properties UNIVERSITY Convolution in timedomain corresponds to multiplication in frequencydomain xt W 9 XwYw Similarly multiplication in timedomain corresponds to convolution in frequencydomain xtyt e iXa Ya i OX6 Ya 6 d6 27 27 00 EE701 Erik Blasch 81 EE701 Erik Blasch 82 i ll VVI iv ii EREitiE Ex Multiplication Property 1 Wt i i a Ex Multiplication Property 2 3jwA rot so ow H Rm gum Pawn For pt coswot lt gt Pjw 776w wo 776w l 000 Row 59w w0 sltjltww0gtgt EE701 Erik Blasch 83 75 8t cosw0t quot 31 031 u Amplitude modulation AM P POGO Fl 1 Drawn assuming ROW iSJw wo Sjw w0l Riw21 TESiwPicol w w gt0 0 1 A A2l do gt L01 l A 030 T T 030 w 39030 39 031 3900 01 030 39 01 00 01 EE701 Erik Blasch 84 mm R WRIGHT STATE EX Filtering 1 UNIVERSITY 1 ei 330 e atue a gt 0 la t 00 00 t JWtdt e ate ywt OO 0 W e aijw lt 1 e ltajwgtt Z 1 39Xi0 1a2 12 2 ZXGCD tan391coa 1a WK WRIGHT STATE UNI VERSITY mm HltjwgtXltjwgt 1 1310 39 lt2 1310 a rational function of jw ratio of polynomials of jaw ii Partial fraction expansion 1 1 1 jw 2 jw ii inverse FT W W 6 2tluit Yle EE701 Erik Blasch 85 EE701 Erik Blasch 85 Periodic Pies onse Usin Fourier Watts P 9 Watts Ideal Low Pass Filter Example Transforms 9305 W wit W wit H J D 1 we wt You HjwXjw 1 ht W 69 w U I I sin tact The frequency response of a CT LTI system is simply the Fourier transform ooC DC 00 Wt of its impulse response ha w w t Csino C gt 5W ejwot r HUM Wit no 7r 7r 5 Recall Sin 7T6 6360075 C00 VAVAVA AVAVAV 6 Yew HjwXjw Howgt2w6ltw we 27rHowogt6ltw we Vi W t T we 0c ii inverse FT 905 Hijwokjwot EE701 Erik Blasch 87 EE701 Erik Blasch 88 mm R DEW R VVIELC IETREITgE C ascad n g N O n deal e rs WIggglgRnggE Cascadlng Ideal FlltCr S sin 47Tt sin 8777f gtIlt at at xt ht X003 Haw YQOJ 1L 1 X 1 1 gt H1000 gt H2000 gt 4n 47139 D 87 8n 0 4TC 47 03 W H100 quw Cascadmg Gauss1ans 2 2 6 e m e39g H100 2 H200 ll 1 H000 High has a f u2 w2 7T w2 sharper frequency E gtlt 64b eT 0 f gt selectivity 2 b V ab Gaussian gtlt Gaussian 2 Gaussian Gaussian Gaussian 2 Gaussian EE701 Erik Blasch 89 EE701 Erik Blasch 90 K R mum rr quotl 391 l mmmI Vkltilrll lnl39l Lecture 1 Outline Course Outline C I f t Lecture 01 Introduction to enpaging Signals and Systems Class Philosophy Found 10 textbooks with Matlab Typically and undergrad course k But material is used in every discipline Mathematics 9379049077 Trigonometry Vector erlkblaschwpafbafmll 39 Probability Distributions Signals Overview Systems Overview httpwwwcswrightedueblasch TA Rehka Bangladore Kalegowda bangalorekalegow2wrightedu EE701ErikBlasch 1 E901 Erik Blasch 2 IE c Max K 39lLEll l SI39 k 39 VRlGlll39 911 l Acknowledgments COURSE OUTLINE 07 Slides From Week 1 Ch01 Intro Signals and Systems RPI httpnetworksecserpieduyuksemss Week 2 Ch02 Time Domain Continuous MIT Willsky Week 3 Ch03 Fourier Series Convolution Manchester Dr Martin Brown Week 4 Ch04 Fourier Transform Univ Texas Dr Brian L Evans Week 5 Ch05 Sampling WWWWW PF 9 as rdm 39 Week 5 Ch06 Laplace transform Continuous Onine Demos Week 6 Ch08 DiscreteTime Signals Projects Links to Excellent Demos Pages week 7 Chog Time Domain 39 Discrete WWWquot quot391 humus J quot39 h39m39 Week 8 Ch10 Discrete Fourier Transform tii39jm39e39zm Dem a m Week 9 Ch11 ZTransform Discrete Georgia Tech 2 Domain CLTI DLTI FourierSeries Week 10 Ch 7 Frequency Response Analog Filters ere anterh Week 10 Ch 12 Frequency Response Digital Filters EE701 Erik Blasch 3 EE701 Erik Blasch 4 K k cou RSE GOALS Signals and Systems Appreciation of the Exponential Fourier Analysis Continuous and Discrete TimeFrequency Analysis Implementation Sampling Transforms made easy Laplace Z transform EE701 Erik Blasch 37 mlWClass Grading Selection 15 Homework Work in groups of three Establish Regular Schedule Solve and some in MATLAB work Wquot others Stay up with material Verballze Ideas 20 Projects 4 4 Get an early start 5 Project of your own 4 Investment forFuture 30 Midterm with Matlabtakehome 30 Final with Matlabtakehome Instead of Midterm Final Can select own if experience in the field Expect conference quality report and code Grades 85 A 75 B 65 c D 50 Fgt50 In the eld of experimentation chance favors only the prepared mmd Lours Pasteur 1854 EEIO1 Erik Blasch Focus on the basics Find your own approach r l l 39fil39lnfl39 l Course Policy 1 From the Professors of Past Learning Why you are here The only thing that will never fail us is learning Leonardo Di Vinci Education What you are here to get Education is what is left after the facts are forgotten Origin Unknown How to proceed What you will do Tell me and I will forget Lectures Show me and lwill remember HandoutsCode Let me do it and lwill understand Project Attributed to Confucius IIU I rm blasch 13 l wmum HAM llJ IRlquotl Ch B Mathematics B Background B1 Complex Numbers B2 Sinusoids B3 Sketching Signals B4 Cramers Rule B5 Partial fraction expansion B6 Vectors and Matrices 3 Miscellaneous EE701 Erik Blasch J g What IS Signal llz VRIUlll sir Ch 1 Intro to Systems Engineering 39w Telephoneieceivei 11 Size of a Signal Z i gilgjn ff AUdlo V fortransmssmn Images i Biological i Q quot pllmal l messenger 0 e o 12 Classification of Signals 13 Some Useful Signal Operations 14 Some Useful Signal models 15 Even and Odd Functions 16 Systems 17 Classification of Systems 18 System Model Inputoutput Description 19 Summary i at Electricalsignal is converted back i sound onthe otherend Ligand Recap or Q Signal intimation via securiil gene expression EE701 Erik Blasch 9 EE701 Erik Blasch Wireless Communications Engineering Examples Contm39lers Time Frequency Space TempSpatialSpectral Aircraft Antilock Brakes Engine control Chemical Plant Pumps mm mm CW Communication Systems Timefrequency Audio CD Samples at 441 kHz 5 Fourier analysis Human hearing is from about 20 Hz to 20 kHz Digital communication Sampling theorem We will cover this later sample analog W as Mquot increases 39 39 39 39 NEW DIMENSIONSPACE Signal at a rate of more than tWIce the highest frequency in 2 SNRcapaCIty the analog signal Apply a filter to pass frequencies up to 20 kHz called a lowpass filter and reject high frequencies eg a coffee lter passes water through but not coffee grounds E Antenna array adds further increase in SNR 5 amp capacity by using Lowpass lter needs 10 of cutoff frequency to roll off to zero filter g be can reject frequencies above 22 kHz WW Spatlal dlverslty Sampling at 441 kHz captures analog frequencies of up to but not SPACEDIVISIONMULTIPLEACCESS iSDMAJ 25G and 3G systems including 2205 kHz Picture byProf MuratTorlak UTDallas amp EE701 Erik Blasch EE701 Erik Blasch E 39 Slgnals A signal is a pattern of variation of a physical quantity as a function of time space distance position temperature pressure etc These quantities are usually the independent variables ofthe function defining the signal time frequency A signal encodes information which is the variation itself Continuoustime signals are functions of a real argument xt where t can take any real value xt may be 0 for a given range of values oft Discretetime signals are functions of an argument that takes values from a discrete set xnwhere n e 3210123 We sometimes use index instead of time when discussing discretetime signals Values for X may be real or complex EE701 Erik Blasch I V39RIGHI Il Hlhlll Analog vs Digital The amplitude of an analog signal can take any real or complex value at each timesample AR V Amplitude of a digital signal takes values from a discrete set 1 EEIO1 Erik Blasch V wth Alp mum l Ex CT versus DT Signals W u t in is u 5 It I nglml Mll hymnHmmm m n unis munquot mintutumnylcnmmm plottx EE701 Erik Blasch stemnx I Wilhlll Vlll liJlIRlquotl Systems A system is a transformation from one signal called the input to another signal called the output or the response Continuoustime systems with input signal x and output signal y aka the response W Xt Xt1 w yt ytx2t quot i Discretetime system examples yn xn xn 1 xn I yn Yin len r Hybrid system examples ADC EE701 Erik Blasch Electrical Systems 0 Examples of 1D RealValued CT Signals Temporal Evolution of Currents and Voltages in Electrical Circuits M i meii Ill IIUIIKHIT Mechanical Systems 1ampan Maw kg w imam 0 Examples of 1D RealValued CT Signals quot 39 Temporal Evolution of Some Physical Quantities in Mechanical Systems Figure lad l ciruuil ya mi i i 3 i I l i EE701 Erik Blasch EEIO1 Erik Blasch Vik WW 394 l Image Processmg 5 Audio Processmg lllll lll Mmmf HJIRlquotI 0 Example of 3D RealValued Digital Signal with a 2D Domain A Digital Color Image rn1n2 xn1n2 gn1n2 bn1nz bn1n2 gnl n2 171 2ED2 rnlnz EE701 Erik Blasch 0 Example Communications 0 Example of 1D RealValued Digital Signal Digital Audio Signal xin CIT EE701 Erik Blasch mum 39l39ll iil HUM Ch 1 Intro to Systems 11 Size of a Signal Energy Power 12 Classification of Signals 13 Some Useful Signal Operations 14 Some Useful Signal models 15 Even and Odd Functions 16 Systems 17 Classification of Systems 18 System Model Inputoutput Description 19 Summary M I men Ill llUIIKHIl Signal Size Area Under Signal Size Signal Energy Duration 1 Amplitude Finite Energy EE701 Erik Blasch 21 EEIO1 Erik Blasch 22 51 3 39ltlilll SLY I l WKltilll Vii u Power and Energy of Signals1 we WWEIectrical Signal Energy amp Power Energy accumulation of absolute of the signal T no Em iii 1 xt2dt j xt2dt 7T foo A N no Em letnilz letnllz nrN Poo Power average of absolute of the signal A 1 T 2 E Poo 11m Ixt dt11m Hm 2T if H 2T A 1 N EDo Poo lim xn2 lim Naw2N1PN Naw2N1 EE701 Erik Blasch It is often useful to characterise signals by measures such as energy and power For example the instantaneous power of a resistor is 1 W Vtlt EVZU and the total energy expanded overthe interval tm t2 is t2 t2 1 2 t dt I v t dt 1 pltgt 1 R 0 o and the average energy is 1 f 1 t 1 2ptdt I2 v2tdt lz l1 1 lz tl 1 R o How are these concepts defined for any continuous or discrete time signal EE701 Erik Blasch K mum l A mm wwwGeneric Signal Energy and Power Total energy ofa continuous signal xt over 1 12 is E xt2dt where H denote the magnitude ofthe complex number Similarly for a discrete time signal xn over n n2 E 2 By dividing the quantities by 121 and nZn1 respectively gives the average power xn2 Note that these are similar to the electrical analogies voltage but they are different both value and dimension EE701 Erik Blasch 37 l lflfil n nergy and Power over Infinite Time For many signals were interested in examining the power and energy over an infinite time interval as n These quantities are therefore defined by Ew 11mm Jxt2dt jxt2dt Ew hmw ZiLNIxinf 2ixini2 If the sums integrals do not converge the signal energy is infinite Pw 11mm zl Tjxz2dz R hm ziiwlxinif Two important subclasses of signals 1 Finite total energy and therefore zero average power 2 Finite average power and therefore infinite total energy Signal analysis over infinite time all depends on the tails Easimiiugmehaviow ion x 39llhl ll l39 Power and Energy of Signals 2 Energy signal iff 0 lt E lt 00 and so P 0 eg t 0 tlt 0 x 6quot t2 0 Power signal iff 0 lt P lt ac and so Eoo eg xn 11 11 11 Neither energy nor power when both E and P are infinite eg xtez It Finite Energy gt Area under signal Finite Power Time average of Energy EE701 Erik Blasch izI w 39 v lilil l xh l w Power and Energy of Signals 3 Find the measures of Energy and Power a Since signal amplitude gt 0 as t gt oo Choose Energy 5f 2 j from U m f 2305 f 4e d D 1 44z b Since amplitude does not gt 0 as t gt 00 However it is periodic and therefore its power exists periodic signal repeats regularly each period 2 seconds in this case 1 m ffmm gt1 1 1 2 1 239rd73 EESignal power is the sguare of its RMS value RMS k I KNEE it llltl39g Power and Energy of Signals4 Example Average Power of a Sinusoidxt C cosa0t 6 A T Pm 1imi jxt2dt hmg pm 2 7T 7 2T 1 72 Pm lim I C2 cos2a0t 6dt Tam T 772 wruml I39l Ch 1 Intro to Systems 11 Size of a Si nal I 12 Classification of Signals I 13 Some Useful Signal Operations 14 Some Useful Signal models 15 Even and Odd Functions 2 T 2 coszu1cos2u aims T f 1cos2w0t26dt 16 Systems I C2 7 I C2 m 17 Classification of Systems HEEJzd l iZCOSQWHQM 18 System Model Inputoutput Description C2 19 Summary 2 EE701 Erik Blasch EEIO1 Erik Blasch 30 M DL w 13 39llllil ll 39quotl l WRlGlll l Signal Classmcatlons x Contmuous Discrete 1 Continuoustime and discretetime signals 2 Analog and digital signals 3 Periodic and aperiodic signals 4 Energy and power signals 5 Deterministic and probabilistic signals 55701 Erik Blasch 31 Discretetime signals Continuoustime Continuous time at analog Discrete time at digital 55701 Erik Blasch 32 I muuiii 39lll lll llUl Analog Digital 139 I in m I Continuoustime 39 39 fit I 7 d Lilli H 39 Discretetime Analog Digital u i H 39iitfii39iti Periodic Signal Definition a signal Xt is said to be periodic with period T if Xt 77 W Tis called the fundamental period EXAMPLE A cos o t e a rad sec 6 rad fz 1secHz 27 EE701Erik Blasch 33 E901 Erik Blasch W 39 WW MW quot W39quot 39 j i c a 1 l iiiiiriii Periodic Signal 2 iiiiit ii Periodic Signal 3 Definition a signal Xn is said to be periodic with period N if c Q Is the Sum of Periodic Signals Periodic Xn Xn N Let X1t and X2t be two periodic signals with N is called the fundamental period EXAMPLE N 3 r H l l w iii iii l I l quot ll EE701 Erik Blasch periods T1 and T respectively Then the sum X1t X2t is periodic only if the ratio T1T2 can be written as the ratio qr of two integers q and r In addition if rand q are coprime then TrT1 is the fundamental period of X1t X2t Q What is the power for Periodic e 13903 Answer 1 see text E T0 EE701 Erik Blasch mum 39I 39ll Periodic and Aperiodic Signals Periodic signals are such that 1tT at for all t The small est value of 1 that satisfies the definition is called the period Below on the left below is an aperiodic signal with a periodic signal shown on the right i It t 0 T t Aperiodic Periodic EE701 Erik Blasch rm Periodic Signals 1 m I Vklkilll Ill lelllihlll An important class of signals is the class of periodic signals A periodic signal is a continuous time signal xt that has the property xt xt T where T gt 0 for all t 5m Examples cost27t cost sint2n sint Are both periodic with period 27 xii NB for a signal to be periodic the relationship must hold for all t EEIO1 Erik Blasch 39lirlquotl l Ch 1 Intro to Systems xviw 11 Size of a Signal 12 Classification of Si nals l 13 Some Useful Signal Operationsl 14 Some Useful Signal models 15 Even and Odd Functions 16 Systems 17 Classification of Systems 18 System Model Inputoutput Description 19 Summary 55701 Erik Blasch 39 WKltiill lv l ww39iiiransformations of Time Variable Three possible time transformations Time Flip or reverse x t xn Flips the signal overthe vertical axis Time Shift xtaxna On horizontal axis shifts to the rightwhen alt0 shifts to the left when agt0 Time Scale xat xan for agt0 On horizontal axis scales the signal length down when agt1 scales it up when alt1 EE701 Erik Blasch 3 a f 2 11 quot5339393939i lquotime Shift Signal Transformations l i ili39in39mquot A central concept in signal analysis is the transformation of one signal into TranSformatlonS Of Tlme Variable another signal Of particular interest are simple transformations that involve a transformation of the time axis only Time flip example A linear time shift signal transformation is given by X Xt yt xat b 1 Where b represents a signal offset from O and the a parameter represents a signal stretching if agt1 compression if 0ltalt1 and a reflection if alt0 xll M X 15 1 1 1 05 v 05 9 o 2 92 o 2 t t 15 15 r 5 1 a 1 Y o 3 0 5 ya 05 2 o 2 2 o 2 EE701 Erik Blasch t EE7D1 Erik Blasch a VKlel ll S39l 39l39l39 uin IIHII a Time Inversion 395339539 I39I39n3139 gt39 39 Time Shift fun ft 2 no 0 I 0 Delay I N In H I I Inversion 7 5 3 1 I ft T 3 Note 0 osite of intuition f7 t f 2quot pp I l l 1 3 5 7 1 Advance ftHT t D N EE7D1 Erik Blasch EE7D1 Erik Blasch I k I its39 t39 quotrl39ransformations of Time Variable 2 l i l39 ltquot Time Scale Timescale example m f t 2t tZ X X Note does not have to 1 1 be symmetric n o T t g 112 4 3 2 1 Compression f 21 Mi f2l Combinations are possible T r r gt X t3 7L 7 Expansion 39 2 39ltIgtI7 2r 0 21a 1 I EE701 Erik Blasch EEIO1 Erik Blasch f Combined Analysis Ex quotA i 39ltlilll 3939 l E WRlGlll l39ll tinyM Ch 1 Intro to Systems Most general operation is fat b which is realized in two possible sequences of operation 1 1 Size of a Si nal 39 quot 1 Timeshift ft by b to obtain ft b 39 g Now timescale the shifted signal ft b by a 12 Classification of Signals that is replace twith at to obtain fat 7 b o 2 Timescale m by ato obtain fat 13 Some Useful Signal Operations Now timeshift fat by ba 14 Some Useful Signal models that is replace twith ti b a to obtain fa ti b a fati b In either case if a is negative time scaling involves time inversion 15 Even and Odd Functions 16 Systems EXAMPLE f2t 6 b bt39 d39 t 4 can e 0 am In W0 ways I 17 Classmcation of Systems timecompress ft by factor 2 to obtain f2t then delay 18 System Model Inputoutput Description this signal by 3 replace twith t 3 to obtain f2t 19 Summary 2 delay ft by 6 to obtain ft 6 and then timecompress this signal by factor 2 replace twith 2t to obtain f2t 6 EE701 Erik Blasch EE701 Erik Blasch 48 E R N39thilll quot39ll l I Phil Real and Complex Signals An important Class of signals are a CT signals of the form 11 a o DT signals of the form zn zquot where z and 5 are complex numbers For both exponential CT and DT signals x is a complex quantity and has 0 a real and imaginary part or o a magnitude and an angle What is most convienient depends on the analysis EE701 Erik Blasch TJE 39lLiiii 39i lll Klll WW Real and Complex Signals Ex l l i3939 tlquot Building Block Signals Eternal Complex Exponentials m Xe for all t o zn X2quot for all 17 where X s and 2 are complex numbers We illustrate the rich ness of this class of functions for CT signals DT signals are similarly rich In general 9 is complex and can be written as where 7 and w are the real and imaginary parts of s EEIO1 Erik Blasch 3 so M For example suppose s jwS and z 61773 then the real parts are 3mm miaquotL8cosms Rzn reeiquotquot8cosm8 EE701 Erik Blasch l mucm Sixiii ILJ I RI39i Exponential Signals Key Function is the complex exponential Integrates easily Represents periodic signals Acusmgz Ai2gj gj Aizje39jte39jwo Euler s Formula em Boston jsin mgr Real Acnsmot 4 A 1R3ej maginary Asintoo 4 A1mej EE701 Erik Blasch K A 3 Exponential Signals Exponent39a39 S39gnals vllllh Continuous Representation Key Function is the complex exponential Ge lex Exponential Signals ContinuousTime at Want bounded for decaying signal 951 Ce Replace C and awith C i Ci e16 and a rja0 e l mot e l 030 n x C deem4i C Erieaw Distinct signals for distinct values Identical signals for values of no Apply EUIBFYS VBIatloni 0 mo separated by mUltipleS 0f 27 xt l C l equot cosa0t i9 j l C l equot sina0t 6 Periodic for any choice of no Periodic only if m0 2 urnN Im II Rext l C l equot coswot 8 for some integers N gt 0 and m 5 Fundamental frequency 030 Fundamental frequency w 0rn E at rquot K 3 xx Fundamental period Fundamental period r 5 h CW F quot f 030 0 undefined m0 0 undefined 1 39 m0 O21rm0 m0 OI n21rm0 h l 1 2 Assumes that rn and N do not have any factors in common T E EE701 Erik Blasch EEIOl Erik Blasch 39ET m Wm LK Exponential and SinUSOIdaI Signals Perlodlc Complex Exponential amp milirmiri llllIRli39i Exponential and sinusoidal signals are characteristic of Smusmdal S39gnals realworld signals and also from a basis a building block Considerwhen a is purely imaginary xa Cejwot for other signals By Euler s relationship expressed as A generic complex exponential signal is ofthe form 2 quotnt cos m0tj sin not xt Ceat This is a periodic signals because i where C and a are in general complex numbers Lets 81m COS WotT J39Sin 00 investigate some special cases of this signal cos w0t jsin wot em t Real exponential signals when T27dw0 a Exponential grovvth 5 Exponential decay A closely related signal sinusoidal signal gt0 at i i s lt0 xtcosa0t mo 27if0 C gt 0 l i P C gt 0 E in Sit We can always use g i i g 1wut 2 r 239 I ACOSWOH39 ASR 8 T0 isthe fundamental time period 39 39 quot i 39 quot a is the fundamental fre uenc quot1T s d 5 ii f io 5 5 A51na ot A 3 W 0 q y EE701 Erik BI K EE701 Erik Blasch IE 4 VIULH l39l39 H39l A l39l JI39I39RSrl Exponential amp Sinusoidal Signal Properties Periodic signals in particular complex periodic and sinusoidal signals have infinite total energy but finite average power A 1 l I Consider energy over one period E To to 2 0391 39 Epmod 0 e 0 dt 10 5 t0 5 10 To A 1 v I 2 0 WWW Therefore Ego 00 1 1 10 5 0 5 10 Avera e ower t g p Pperiod T Eperiod 1 1 I 0 a Useful to consider harmonic signals 0 o 0 Terminology is consistent with its use in music 2110 5 5 10 t where each frequency is an integer multiple of a fundamental frequency EE701 Erik Blasch wmm miluGeneral Complex Exponential Signals HH39f F i l l So far considered the real and periodic complex exponen al C ICIej Now conSIder when C is complex Let us express C is polar form and a in rectangular form 80 Ceaf C6139 erjwot lC Using Euler s relation Ceat ZICIej erfa0t ert COSaO These are damped sinusoids a r ja0 erteja0 t ert sina0 t 23 o 3 EE701 Erik Blasch quot390 0 0 quot0 l 0 K a I 1315gg39vpg Exponential Function Exponential Function s t o Exponential Signal e where S iS the complex freguencx lISlIEIllllll lll hllillwullllll Therefore st Gj0t Gt jmt 6 il ll l ili iii 1 l llillllll ill II 3quot39quot I i l II nquot3quot33Iliumiil 3i a imilquot31Lil ifi iL l i li eje 0056 sin 6 If s G j o the conjugate of s then 0036 142 9 59 2 st eG j0t eote jmt st 6t 1 a e e COScot jSlncot 5mgexeee19 0 and 1 thCOSOJt 12 eSt e8 t o Compare this to Euler s formula shows that e St is a generalization of the function e J 0 where the frequency variable jog is generalized to a complex variable 3 6 jco EE701 Erik Blasch 31 wmui l39l39 Nix I39T UNH39l RIli Exponen alFunc on t SpeCIal cases of e S v 1 Constant k k e 0 s 0 3 Sinusoid cos on t 2 Monotonic exponential e t 03 O s 0 0 0 S i 13903 6W gt0 o 0 a E r c 00 0 a 6t 6 cosltnt sij03 EE701 Erik Blasch 39 Stable Unstable LE 39liilll I ll lll llUl l V Complex F Determine response Assess the response S o m Radian frequency Exponential Function requency Domain t ofeS Stable Unstable Left Half Plane Is Right Half Plane Q imaginary part of s is a indicates the frequency of m oscillation of e S Neper frequency the real part 0 Gives rate of increase or decrease of the amplitude Constant DC signal S 0 a a 0 gt Exponentially g Exponentially Regimens 5399 S39Qna eaIA islty ll Sinusoid imaginary axi m axiswhere o 0 equott 1 EE701 Erik Blasch I I mum Ill Hillhill Complex Frequency Domain Exponen alFunc on Cos For ut Xequot Rwi Xe coswl is plotted for different values of s superimposed on the complex s plane EEIO1 Elm niusui u Complex s plane l 39ltLilll 39 l llllll1lli Exponen a Complex F l Function Sin requency Domain For zL Xequot xml Xe sinwt is plotted for different values of s superimposed on the complex s pane ll l 100 Complex splane tar Milli l l Wh l 0 ill EE701 Erik Blasch 74 t lltlltilll mm llJl IRll39i Complex Frequency Domain Examples e cos5t e ea ls em twith complex frequencies 2 15 and 2 7 Left Half Plane Stable e39 cos5te Exponen alFunc on Right Half Plane Unstable my 6 el 2l5 eH l5 with 39 complex frequencies 72 15 and e 2 39 cos 5t e 0 0t e cos 5t e V 6715 Wit frequencies 15 5t eJ jm EE701 Erik Blasch Real Axis 6 7m 6 1 mount 39lll lll llUll Exponential Signals Ex1 Oppenheim 115 o It is sometimes desirable to express the sum of two complex exponentials as the product of a single complex exponential and a single sinusoid For example suppose we wish to plot the magnitude of the signal xt 6m em X 612st 67105 eiost which because of Euler s relat X0 26125 cos 051 Magnitude of xt xt 2 cos05t Xf Fullwave rectified sinusoid Here we have used the fact that the magnitude of the complex exponential e1 2 5 is always unity Ixm 2 EE701 Erik Blasch I VRILHH Ill millile Exponential Signals Discrete Representation General Complex Exponential Signals DiscreteTime xn Caquot Replace Cand a with C l C l 616 and a Idlem xnCej6a 6Mquot Ca Apply Euler s relation xn lCHa quot quot ejquot nquot9 quot cosa0n 6 j Clla quot sina0n 6 EEIO1 Erik Blasch m 39ltlilll 39 l lllll lll Pulse Types Impulse narrowing Step Function 50 E a 40 f 50 d 1 7 gt g Exponential Triangular Gaussian 1 e u m39 a 632 m 0H 0quot 60 0 o 1 I 39m d 1 M 1 Area under 1 Area under 1 D EE701 Erik Blasch 3 l WRKilll ltl39ll llJ IRlquotl Unit Impulse Aka the delta function or Dirac distribution It is de ned by 6t0 0 8 jamdz 1 Vs gt 0 The value 60 isingot defined in particular 50 7 00 Which is a step small width It mun u rm utmxmn in Am EE701 Erik Blasch K Unit Step Discrete Discrete Unit Step um 1 n 2 0 11II 0 nlt0 O Q m o s a F a I 239 i a o C 2 F I F a 39o unk 1 nZk 0 nltk EE701 Erik Blasch win Iv39I Unit Step Continuous Continuous Unit Step ut 1 tgt0 1 t U 0 tlt0 t Continuous Shifted Unit Step ut E 1 tgt r 1 utT 0 tltr I t E EEIO1 Erik Blasch 7 A 39lihlii i mii Hmii Unit Step cont d Continuous Unit Step is discontinuous att 0 so is not differentiable Define delayed unit step M 1 tgt82 u t 0 tlt 82 t 1 otherwise 8 2 0 M Jr is continuous and differentiable W 133140 dugt dt g2lttltg2 otherwise EE701 Erik Blasch 37 I Witiuili iw IiiIR quotSignals as Step Functions m y EE701 Erik Blasch I K i f 39 iglngiykygn Sifting Property Impulse Sampling Itfl II quot Unit Impulse Sampling orsiftingpropertyofthe unit impulse Discrete Unit impulse 5n Given function t and impulse 51 I W 50 d rm f m d gt f w 60 d in 5M 2 1 n w 0 means that the area under the product of a function with an impulse 51 is equal to the value of that function at the instant where the unit impulse is located Discrete Shifted Unit Impulse 39 Also at a point in time no 5n k f rose r d ma 1 nzk And the unit case w n k y FI 0 I l i k i M fame U1 ut gt in S E gt Z 2 EE701 Erik Blasch EEIO1 Erik Blasch eer sown LE l a 39ltLilII I f VRHJHI VIH f H H i in i I i am Unit Impulse cont d 5J0 Unit Impulse cont d Properties of discrete Unit Impulse functions 39 continuous unit Impu39sa 6M 2 M 14 1 6m 0 i un iaik 5 3 H i 6tdt 1 t xn6n x06n is 60 xn n kxk n k dug l y lttlt 890 ch g 2 2 t Signal Impulse 60 ling Xn k2 xk6n k at each time gt0theVWise HO step k EE701 Erik Blasch EE701 Erik Blasch E VRlLill39l Sl3939l39l mu Hum Unit Impulse cont d Continuous Shifted Unit Impulse 517 Properties of continuous unit impulse du t 50 2 dt ut j rdr t 5a xt5t x05t xt t r xr t r EE701 Erik Blasch oo xt jxr5t rdr foo xrz i xk n7 k k421 jig l fm tg lj39fjlll39nff ContinuousTime Signals 1 t 2 0 Unitstep function 111 0 K 0 l l 2 0 Unitramp function rt 0 t lt 0 uul Figure L3 3 Unltrstcp and b unitrramp Indians EE701 Erik Blasch wwrlmy PleceWIse Continuous Signal HIlli 39 The Rectangular Pulse Function prl ulT2 ul T2 12 0 EE701Erik Blasch ll Figure 19 Rectangular pulse l39unction Fm will Piecewise Continuous Signal quot quotquot The Pulse Train Function repeats I repeats n o u o a i I l l l I l l l i v l 5 4 3 2 1 l0 1 2 3 4 5 Figure 110 Signal that is discontinuous at I U 1 2 4 This is a series of step functions EE701 Erik Blasch tz t a39 5 i Wltltill l Nillit Vltltill i39 ll l l WW W Discrete Unit Impulse Step Signals Successive Integrations of the Unit Impulse Function 0 The discrete unit impulse signal is de ned 0 n 0 Successive integration of the unit impulse yields a family of func x 5W I n20 1 tions l 0 Useful basis for analyzing othersignals tm Integration on t g Unit impulse Unit step Unit ramp Unit parabola The discrete unit step signal is defined 39 l E m tn l 0 nlt0 Ill t on ut tut aup mun Mqu 1 MO 9 3 0 Note that the unitimpulse is the first it difference derivative of the step signal lq0 m l l l l l 6in1uini uin li 39 quot l 39 0 Similarly the unit step is the running sum integral of the unit impulse EE701 Erik Blasch EE701 Erik Blasch igigglgpgyeontinuous Unit lmpulse Step Signals t 53llipv Signal composition Ex 0 The continuous unit impulse signal isdefined t l l l l l l 0 0 amzefatcosww ate 7tcoswtu t t r r x 00 0 l 0 Note thatitis discontinuousat tO t o The arrow is used to denote area l i l l i t l l rather than actual value 0 Again useful for an infinite basis xt ut 5mm l 1 1 oo o Continuousunitstepsignal is l l 1 quot l 12 quot 0 flt0 WW 1 rgt0 EE701 Erik Blasch EE701 Erik Blasch 35 u quot quoti39 U n It m p u lse Ex Derivative for getting the Impulse Consider the function de ned as 2t1 O tlt1 1 1 tlt2 39 xt it3 zgtg3 0 all other mquot g an i x o The ordinary derivative of xt at all texcept t 012 3 is 2utiutil 7ut727ut73 Its generalized derivative is dxt 7x0 7x0 5tx1 7x1 H5071 1 72 EE7D1 Erik Blasch 11 R Vmun r m l39i lll Ch 1 Intro to Systems 1 1 Size of a Sinal 1 2 Classification of Sinals 1 3 Some Useful Sinal Operations 14 Some Useful Sinal models 15 Even and Odd Functions 1 6 Systems 1 7 Classification of Systems 1 8 System Model Inputoutput Description 1 9 Summary EE7D1 Erik Blasch as Odd and Even Signals An even signal is identical to its time reversed signal ie it can be reflected in the origin and is equal to the original V39thjll l 39l39l l39 IJ I Rkll xit xt l Examples c5 If 393 xt cost a xt c quot f 1 An odd signal is identical to its negated time reversed signal ie it is egpiab ij8negative reflected signal f f R Examples 539 7quot1 7 xt sint If quota f Xt t quotquot m o This is important because any signal can be expressed as the sum of an odd signal and an even signal EE7D1 Erik Blasch 15 l VRIUH l39 S39l39l39l l ll J RKI ll Even Odd Signals Even Odd Functions Even Function oft fe t fe 7 t Odd Function oft fo 1 7 f0 7 t 10 a 10 even 7 odd Nu 390 o r 0 z r Properties J even function x odd function odd function odd function x odd function even function even function x even function even function Every function can be decomposed into even and odd ft 12ff 7170 12ft ft even odd EE7D1 Erik Blasch iaz Willkilll l ll Even and Odd Functions quoti f i39i39ii39i vquot Even and Odd Signals even function x odd function odd function odd function x odd function even function xn is even if XnX39n even function x even function even function Xn is Odd if XnXn Any signal xn can be divided into two parts Evxn xn X n 2 1mm Q 2 UR O OdX Xln1Xlnl2 even add ZE Every function is the sum of these The arguments above are also valid for continuous signals Exercise Divide the following signals into even and odd parts fem GENE at 140 tut0 Xn y foo 112e39 urn 2 deg EE701 Erik Blasch 89 EE701 Erik Blasch 1 Even and Odd Signais2 Signals Analysis Continuous Time domain analysis Signals and systems in continuous and discrete time i Convolution finding system response in time domain 0 I 3 l 39IUH ll ii39tiili uiii a M Time shifting invertibility and scaling Even Odd Generalized frequency domain analysis Discrete Laplace and z transforms of signals Transfer functions of linear timeinvariant systems Tests for system stability eqxinipl 3122 Frequency domain analysis I gt0 Fourierseries I I E I 3 z 1 Fouriertransform of continuoustime signals 4 2 D 1 2 3 4 2 1 0 1 2 3 39 I I 2 3 quot Frequency responses of systems I Causality xlnl1 quot2 lelnll nlto no 039 lt quotgt0 A m Nl Even Odd EE701 Erik Blasch EE701 Erik Blasch 21E V39RIGHI Ill lellhll x k mum mnSummary1 Signal Classuflcatlon Exponential Signals Signal It 1 Continuoustime signal continuum of values of Independent variable time t Use MatLab t0 Rext l C l e coswot 8 A discretetime signal is speci ed only at a nite or a countable set oftime instants 2 An analog signal is a signal whose amplitude can take on any value over a continuum A digital signal whose amplitudes can take on only a nite number of values The terms discretetime and continuoustime gualify the nature of a signal along the time axis horizontal axis t 0100123 The terms analog and digital qualify the nature of the signal amplitude vertical axis C 20 r 1 0 W0 2 i 3 A periodic signal ft is de ned by the fact that ft ft f TD for someTD X Cexprt cosw0t x 4 plottx39k39tCexprt39ki39tCeXPrt39kquot I luv wmu A eriodic si nal remains unchanged when shilted b an inte ral multi le ofits eriod Ageriodic signal by de ni ion must exist over the eryitire timeginterval rpoo lt tlt Xlabel39t39 ylabellx1t v A signal is aperiodic ifit is not periodic An everlasting signal starts at t r so and continues forever to t no Hence periodic signals are everlasting signals A causal signal is a signal that is zero for t lt 0 4 energysignal A signal with nite energy is an Investigate the behavior Of running MatLab power signal signal with a nite and nonzero power mean square value script and the values of C r and W0 A signal can either be an energy signal or a power signal but not both 5 deterministic signal physical description is known completely in a mathematical form ndom signal is known only by probabilistic description such as mean value EE701 Erik Blasch EElOl Erik Blasch Elm R WRIGHT STATE UNIVERSITY Lecture 03 TimeDomain Analysis of CT S stems Impulse Diff q Convolution Erik Blasch 9379049077 erikblaschwpafbafmil httpwwwcswrightedueblasch TA Rehka Bangladore Kalegowda bangalorekalegow2wrightedu EE701 Erik Blasch 1 ii a V VVRIGHTSTATE 2 UNI VER s 1 TY 21 Introduction 22 Sys Response to Internal Conditions ZeroInput Response 23 The Unit Impulse response ht 24 Sys Response to External Input ZeroState Res 25 Classical Solution of Differential equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21 Determining The Impulse Response 29 Summary EE701 Erik Blasch 2 ll w ms m ContinuousTime Differential Eq UNIVERSITY A general Nthorder LTl differential eouation is ak dkyt 19k dkxt k k k0 dt k0 dt If the equation involves derivative operators on yt Ngt O or Xt it has memory Expanded tiny I21111 d y dmf linn1f d f dr11 n1dr 1i 391dri 39 Ji39l3 Elm dimbm1dim1b1drEz39 i39 And for this book 00 yt PD fl 311 a 1 n1 mg a jyri 535 am1om391 215 sums Note use differentiation Magnifies low frequency if use integration Magnifies High Frequency Noise 3 EE701 Erik Blasch Him I I I I f quotquotquot39 I WRIGHTSTAIE ContinuousTime Differential Eq UNIVERSITY A general Nthorder LTl differential equation is N dkyt M dkxt Zak k Zbk k k0 dt k0 dt If the equation involves derivative operators on yt Ngt O or Xt it has memory The system stability depends on the coefficients ak For example a fsi order LTl differential equation with a01 dyt dt m0 0 W Aeal lf a1gt0 the system is unstable as its impulse response represents a growing exponential function of time lf a1lt0 the system is stable as its impulse response corresponds to a decaying exponential function of time EE701 Erik Blasch 4 wmammlEDiscreteTime Difference Eq UNIVERSITY A general Nthorder LTl difference equation is N M ZakyDI k ZbkaI k k0 k0 If the equation involves difference operators on yn NgtO or xn it has memory The system stability depends on the coefficients ak For example a 1St order LTl difference equation with a01 yn alyn 1 O yn A611quot lf a1gt1 the system is unstable as its impulse response represents a growing power function of time lf a1 lt1 the system is stable as its impulse response corresponds to a decaying power function of time EE701 Erik Blasch 5 VVRIGHTSTATE 2 Wit 1M M9 355 quot39 Ia UNIVERSITY quot i11 any 7 l 21 Introduction 22 Sys Response to Internal Conditions ZeroInput Response 23 The Unit Impulse response ht 24 Sys Response to External Input ZeroState Res 25 Classical Solution of Differential equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21 Determining The Impulse Response 29 Summary EE701 Erik Blasch 6 mm WRIGHT STATE UNI VERSITY DHquua ons Causal Systems The output of a causal system depends only on the present and past values of the input to the system Impulse response Continuoustime systems Input and output are related through a linear constant coefficient differential equation Used to describe behavior overtime Discretetime systems input and output are related through a linear constant coefficient difference equation Used to describe the sequential behavior EE701 Erik Blasch 7 meHTsTATE Differential and Difference E UNIVERSITY q 39 Two extremely important classes of causal LTl systems 1 CT systems whose inputoutput response is described by linear constantcoefficient ordinary differential equations with a forced response dyt RC circuit with yt vCt Xt vst a b 1RC 2 DT systems whose inputoutput response is described by linear constantcoefficient difference equations ayt bxt yn 6M 1 2 mm Simple bank account with a101 b 1 Note that to solve these equations for yt or yn we need to know the initial conditions Examine such systems and relate them to the system properties just described EE701 Erik Blasch 8 W1 WRIGHT STATE UNI VERSITY Differential Eq A general Nthorder LTI differential equation is N 61km M dkxt Zak k 217k k k0 dt k0 dt If the equation involves derivative operators on yt Ngt O or Xt it has memory Expanded 311 a1 391 3153 mayo Erm m tzB1 im391 215 bnj f System Response W ll 39 VVRIGHTSTATE EX leferentlal EQUatIOnS UNIVERSITY For an electrical circuit I 3E m an H rill u m l 2 m m LF It 1 in l H fifth 339 a j 3F Zerostate component of yt resulting from the input ft assuming that all initial conditions are zero that is yO vCO O When the input terminals are shorted at 2 O the inductor current is still zero and the capacitor voltage is still 5 volts Thus yOO O Loop equation flirt Total response zeroinput response zerostate response fratrjl Hyatt mm I Set t O KDO KDgtO wm3mmmmn Internal External Since yOO O and VC 0 5 then d yOO d t 5 EE701 Erik Blasch g EE701 Erik Blasch 10 mm ll 31 ll rpm Block Diagrams 1 Simplify the understanding Ex Firstorder differential equation 1211 cf t 3 cry f may I Ti 1 1t 1 Operations 1 f if addition a w multiplication by a coefficient and ill i 313W differentiator d D I Need to represent as 1 if E ym 1 Extti quotll39 It139 wa J 39 BUT differentiators are both difficult in to implement and extremely 39 5 D sensitive to errors and noise l n if cit EE701 Erik Blasch 1 1 ll Wlsnftsffn Block Diagrams 2 Simplify the understanding Ex Firstorder differential equation gym 2 Mm m m artf Use an integrator Need to represent as T ya J bxei ayoaidn 39I EE701 Erik Blasch 12 W i crews Block Diagrams 3 Simplify the understanding Ex Firstorder difference equation yn ayn 1 bXn i a y VVRIGHTSTATE 2 UNIVERSITY m a 21 Introduction 22 Sys Response to Internal Conditions ZeroInput Response gfg ggns39 g 23 The Unit Impulse response ht multiplication by a coef cient and KM Mn 24 Sys Responseto External Inpiut ZeroState Res delay relation between yn and yn1 I V gt 25 Classmal Solution of Differential equations izi39i xn1xp Need to represent as B I l 2396 System Stab39l39ty 27 Intuitive Insights into System BehaVIor 39r W11 28 Appendix 21 Determining The Impulse Response n a n 1 b n y y X 29 Summary 39 rim 1 EE701 Erik Blasch 13 EE701 Erik Blasch 14 mm 1 lm i WRIGHT STATE Impulse Response Impulse response of a system is response of the system to an input that is a unit impulse Le a Dirac delta functional in continuous time Linear constant coefficient differential equation QD yt PD f0 When initial conditions are zero this differential equation is LTI and system has impulse response ht b0 50 PD y0ltrgtlultrgt b0 is coefficient could be 0 of b0 DN ft on righthand side N is highest order of derivative in differential equation EE701 Erik Blasch 15 Unit Impulse ht WRIGHT STATE UNIVERSITY At time O we are looking for the response but note that the denominator PD gives the characteristic equation 00 PD fz yt ht Hg i t characteristic mctlc tcrziris IE I Which can be expressed as FEEi Emmii Pt lrnt Hill where b n is the coefficient of the nthorder term in PD and ynz is a linear combination of the characteristic modes of the system subject to the following initial conditions rnm39lji l 1 and yniin 15hrch Enrniyn39i 39iirni c EE701 Erik Blasch 16 WW II VVRIGHT STATE Impulse Response In following plug where did bn come from ht 190 60 PD yo 0 ut Lathi 223 In solving these differential equations for t2 0 f t g0 mt yt mt mt Funny things happen to y t and y t W m39t mt mt 5m yquot r mquot r mt 2 m39 r 5r mt 539 r In differential equations class solved for mt Likely ignored at and 51 terms WRIGHT mills Ex Differential Equations 2 UNI VERSITY Need to solve for particular and homogenous natural response solutions Find the zeroinput response of 02 30 2 yt D ft Initial conditions yOO O d yOO d t 5 1 characteristic polynomial of system is A2 3 2 7L 1 7L 2 O 2 characteristic roots of the system are A1 1 and A2 2 and the characteristic modes of the system are e t and e Zt So y0z c1e t 02e 2t 3 Determine the constants c1 and 02 we differentiate to obtain dyOOdt c1e t 201e 2t 4 Substituting the initial conditions O c1c2 and 5 c1 2c2 Solving c1 5 025 Solution for mt is really valid for t2 0 5 Thus you 5 et 5 e gt NOTE in Matlab use dsolve EE701 Erik Blasch 17 EE701 Erik Blasch 18 meant Ex Differential Equations 3 wm sm na EX leferentlal Equa ElOnS 4 UNIVERSITY UNIVERSITY Get the impulse response ht MATLAB 02 SD2 yt D ft Initial conditions yOO O d yOO d t 1 1 characteristic polynomial of system is A2 3 2 7L 1 7L 2 O 2 characteristic roots of the system are A1 1 and A2 2 and the characteristic modes of the system are e t and e Zt So y0z c1e t 02e 2t 3 Determine the constants c1 and 02 we differentiate to obtain dyntdt c1e t 2c1e 2t 4 Substituting the initial conditions O c1c2 and 1 c1 2c2 Solving 011 02 1 5Thus ynt e t e 2t 6 Solve for Impulse ht I PltDgt ynlttgt i W Bruit Tin ti t 2 7 Then ht e t 2e 2tut NotebnO EE701 Erik Blasch 19 V7 Determine the impulse response ht for an LTIC system specified by the differential equations 02 3 02 yt Dft This is a secondorder system with b n b 2 0 First we find the zeroinput component for initial conditions yO O and 1 Yzi dsolve D2y 3Dy 2y 0 y0 0 Dy0 1 t Yzi exp2texp t Since PD D we differentiate the zeroinput response PYzi symdiffYzi PYzi 2exp2t expt Therefore ht b25t Dy0tuf 2 e 2t e t uz EE701 Erik Blasch 20 Wit WRIGHTSTATE Ex Differential Equations 8 UNIVERSITY 1 A ung 131 MATALB Forced Response Analysis 02 30 2yt Dft for the input ft 51 3 f 5l 3 mpa f f yt dsolve D2y3Dy2y f yO2 DyO 3 t yt 9452t 194exp 2t9exp t EE701 Erik Blasch 21 WSIV IEREE EELTI Systems Impulse Response Any continuousdiscretetime LTI system is completely described by its impulse response through the corvolutionio z k j IMT r This only holds for LTI systems as follows Example The discretetime impulse response h n ls completely described by the following LTI 0 otherwise yin xn WI 1 However the following systems also have the same impulse response yn xn xn 12 M maxxn xn 1 Therefore if the system is nonlinear it is not completely characterised by the impulse response EE701 Erik Blasch 22 FEW it I I IVEI it ERETfEE Unit Impulse Functional Mathematical idealism for an instantaneous event Dirac delta as generalized 1quot tquot function aka functional 8 3 5t1imPgt Unit area 150 5121 H0 Sifting J gt t dt gO provided gt is defined aft O Scaling f 5atdt ifa 0 a Note that 50 is undefined g g 6r1igg11r 23 WW7 II I WEEEEREE EE Unit Impulse FUHCIIOHa39 Generalized sifting 1 ifaltTlta Lat T dt 2 Assuming that agt O O ifTlta0rTgta By convention plot Dirac delta as arrow at origin Undefined amplitude at origin Denote area at origin as area Height of arrow is irrelevant Direction of arrow indicates sign of area 1 O l With 5t O for t7 0 it is tempting to think Wltlf695ltk Simplify unit impulse WW under integration only EE701 Erik Blasch 24 WK iviiitistitEUnit Impulse FUNCTiOnal WWII 39 ww wum Impulse Functional we can simplify 5t Other exampes Relationship between unit impulse and unit step under integration w 5tejmdt1 o rlto d Ho W6 7 d7 i0 ltgt 150 J t5tdr 0 5005C08d 0 1 l o it What about 0 2 2 2 2 ll 1 e 3 62 tdte x m t5tdt quot What happens at the origin for ut Answer 0 What about at origin uO O and uO 1 but uO can take any value What about 0 5mm 7 Common values for uO are 0 12 1 uO 12 is used in impulse invariance filter design fw t5t T dt By substitution of variables J rT5rdt T EE701 Erik Blasch 25 EE701 Erik Blasch 26 L B Jackson A correction to impulse invariance IEEE Signal Processing Letters vol 7 no 10 Oct 2000 pp 273275 Wit Him ll WSlv ii ERETttE Step Response WRIGHT STAT UN VERS TyESteP ReSPOHSe Impulse Responseaf Unit Step Response Output of a system when it is given a Relationship Between Step and Impulse Response unit step as an input Represented by sn or st sn Zn Mk Step is the integration of impulse 0 Fully characterizes an LTI system just like the Unit Impulse koo Response does Mn 2 sn sn 1 Impulse is derivative of step t all Sy em hlt Sm JimMT Step is the integration of impulse dst W System Sm ht S t Impulse 13 derivative of step H You can prove the above relationships EE701 Erik Blasch 27 EE701 Erik Blasch 28 when Useful Functions VV5i I fEI Inear TimeInvarlani System Unit gate function aka unit pulse function Any linear timeinvariant System LTI System mm IO M continuoustime or discretetime can be uniquely 1 a x 1 characterized by Its l j 437 N X W25 Impulse response response of system to an 40 0 12 x T 11 ixlltl impulse o does rectltx a look Frequency I GSPOFISGI ISSPOHSG Of system t0 a Unit triangle function complex exponential eIZWIfor all possrble freq U 9 n C39 93 f u filfgi f ii i39fii39TZ will 1m 0 W1 Transfer function Laplatransf of impulse 1 szz 2 response lrtmfmanIm hf 42 390 12 we HM W Given one of the three we can find other two provided that they exist EE701 Erik Blasch 29 EE701 Erik Blasch 30 W i mm l nggglgggla Ex Frequency Response ngglgggli Ch 2 CT Time Domain System response to complex exponential eiw for all possible frequencies wwhere a 2 zf me39 2mm passband stOvalld i I D 39T39i If I M battle foulr 39 eIsrw39 i i I I l in a If iii 4 7l J i w w H r il DJ ill m 41 3912 1 u If Ls il39iiiil IfI Tiff ii 1i i i j 39f739ll Passes low frequencies aka lowpass filter passband g A wag V S t019196lmil stopband EE701 Erik BlasehuS Dp D D 31 21 Introduction 22 Sys Response to Internal Conditions ZeroInput Response 23 The Unit Impulse response ht 24 Sys Response to External Input ZeroState Res 25 Classical Solution of Diff 39 equations 26 System Sta 39 27 Intuitive Ins 28 Appendix 2 29 Summary m Behavior ermining The Impulse Response EE701 Erik Blasch 32 ll WRIGHTSTATE Ex Differential Equations 2 UNIVERSITY Need to solve for particular and homogenous natural response solutions Find the zeroinput response of 02 30 2 yt D ft Initial conditions yOO O d yOO d t 5 1 characteristic polynomial of system is A2 3 2 7L 1 7L 2 O 2 characteristic roots of the system are A1 1 and A2 2 and the characteristic modes of the system are e t and e Zt So y0t c1e t 02e 2t 3 Determine the constants of and 02 we differentiate to obtain dyOOdt c1e t 201e 2t 4 Substituting the initial conditions email Ex Differential Equations 5 UNIVERSITY Now we want to convolve Find the loop current yt of the RLC circuit in for the input ft 10 e 3t ut when all the initial conditions are zero ht 2e 2t e t ut The input is ft 10 e 3t ut and the response yt is yt ft ht 10 e 3tut2e 2t e tut Using the distributive property of the convolution we obtain yt 10 e 3tug 2e 2tut 1O e 3tute tut yt 20 e 3tute 2tut 10e 3tute tut Using the convolution table O 5 2 3t 2t 3t t Solving c1 5 025 ywzEfEIIH quot3 Mm 3t1E W Mm Zeroin ut result rt Elli239 E39Ethrtt 5 e 3t nit 11243 5Thus y0t 5e t 5e2u up NOTE in Matlab use dsolve yo r 52 1 Hoe ft 15 2quot fern 5 31sz sf fquot EE701 Erik Blasch 33 EE701 Erik Blasch 34 m it ll WRIGHT STATE EX Differential Equations WRIGHT STATE UNIVERSITY UNIVERSITY o n it flit hit ful l it felt hit ftff Total Response Z cj eljt r Unit j1 1 t 51 T fit T 11 551 zero input component Z c e J j1 J 2 emulft off 1 EM oft zerostate Component r Emit g 3 ctr orquotth rum Total current Eire t 5egtj ESE t Elle Et l e t M H Hungry I I zeroinput current zerostate current 4 E 1 m E i it In u Sum A 9 M Total current 1o quotf 25 2 393 r 15 e 3931 re 0 5 fut f mflil few natural response ynl f forced response 33 p mum 5 Mn 1 gamma 39 39f innit eyuff w It39ll u 39 at a tmuft mgr tm um g teylgu nial eh e quotzit 311 lake uh EE701 Erik Blascl 35 it l2 EE701 Erik Blasch 36 Egggg i WRIGHT sma Convolution Table UNIVERSITY Common 139 m it 39 it 1 m l M t a Lift t a tilt nm t a ut m in w i l i k39quot t liim n Himm tetit 11 i a nit i e 3 art 1me 315le MPH nit j EvElfiknlizmTlalll39inai ej i A1 a A 2 Him 5 mg g MM utti k 39 F i L H H urinalE M E GEMB Muir engulf ma a mg E nit w n A 32 at tant tlz rtc All All 4 v hat d 13 rgttiutti chin E 1 axMg HE A gt Re ll a 1 A g A 7 till Ell3t H a mi t a 31u t fet A Jr EE701 Erik Blasch 37 W ll WRIGHT STATE Convolution Demos Johns Hopkins University Demonstrations httpwwwihuedusignals Convolution applet to animate convolution of simple signals and handsketched signals Convolve two rectangular pulses of same width gives a triangle Some conclusions from the animations Convolution of two causal signals gives a causal resuH Nonzero duration called extent of convolution is the sum of extents of the two signals being convolved EE701 Erik Blasch 38 CHEW mesTAiE Discrete Unit Impulse System UNIVERSITY A very important way to analyse a system is to study the output signal when a unit impulse signal is used as an input n System 6 gt hn Loosely speaking this corresponds to giving the system a kick at n 0 and then seeing what happens This is so common a specific notation hn is used to denote the output signal rather than the more general yn The output signal can be used to infer properties about the system s structure and its parameters 9 EE701 Erik Blasch 39 ii 39 ngv glggfgmlntrOdUCtlon to Convolution Definition Convolution is an operator that takes an input signal and returns an output signal based on knowledge about the system s unit impulse response hn Xin an yn hn Xln J ln System System hn The basic idea behind convolution is to use the system s response to a simple input signal to calculate the response to more complex signals This is possible for LTI systems because they possess the superposition property xn 2k akxk n axn a2x2 n a3x3 n yn Zkakkaz a1y1na2y2na3y3n EE701 Erik Blasch 4o Wt WRIGHT STATE System Response Signals as sum of impulses Fm Frz i FnATrectt lm AT nz oo FtAliTr 10 i FnAT rectt AnTATj F7 50 7 A tnAc But we know how to calculate the impulse response ht of a system expressed as a differential equation Ft fmFr6t Tdrgt fmFz39ht z39dz39 Yt Therefore we know how to calculate the system output for any input Ft EE701 Erik Blasch 41 Emmi WRIGHT STATE UNIVERSITY Continuoustime convolution yt 2 0 hr xt T d7 For every value of t we compute a new Discretetime convolution 00 yn Z hmxnm mz oo For every value of n we compute a new summation integral xn hn am W ht AW LTI system LTI system represented represented by its impulse by its impulse response response EE701 Erik Blasch 42 Emil WRIGHT STATE Continuous Conv Summary UNIVERSITY A continuous signal xt can be represented via the sifting property xt J xt395t t39dt39 Any continuous LTl system can be completely determined by measuring its unit impulse response ht Given the input signal and the LTl system unit impulse response the system s output can be determined via convolution via better explained yt f Mme rm from discrete Note that this is an alternative way of calculating the solution yt compared to an ODE ht contains the derivative information about the LHS of the ODE and the input signal represents the RHS Sure we can solve the CT but it is EE701 Erik Blasch 43 DEW Discrete Conv Summary Any discrete LTl system can be completely determined by measuring its unit impulse response hn This can be used to predict the response to an arbitrary input signal using the convolution operator yn ixtkihtn k kz oo The output signal yn can be calculated by Sum of scaled signals Discrete Example 1 Nonzero elements of h Discrete Example 2 The two ways of calculating the convolution are equivalent Calculated in Matlab using the conv function but note that there are some zero padding at start and end EE701 Erik Blasch 44 Wt WRIGHT STATE UNI VERSITY Convolving Two Signals Array Write Xn and hn as sequences Create a two dimensional array A where AijXlhj Then formula for ynis Y Aij where ijn Graphica Consider hk Timeflip it and obtain h k Then timeshift it by n and obtain hn k Since n can be an arbitrary number in oocgto that means slide hnk across Xn while calculating the convolution sum for that 00 particular value of n W kzoxmwm k EE701 Erik Blasch 45 WRIGHTSTATE Kronecker Impulse Function UNIVERSITY Let 5n be a discretetime impulse function aka the Kronecker delta function 5n 1n0 5n 1i 0 11720 n Impulse response hn response of a discretetime LTI system to a discrete impulse function EE701 Erik Blasch 46 Ema wm stiscrete lmpulsesTime Shifts UNIVERSITY Basic idea use a infinite set of of discrete time impulses to represent any signal Consider any discrete input signal Xn This can be written as the linear sum of a set of unit impulse signals x 1 n 1 x 15n 1 o n 72 1 XO n 0 WWW o n 75 o x16n 1 XE 1 1 actual value Impulse time fl shifted Signal Therefore the signal can be expressed as Xn x 25n 2 x 15n 1 x05n x15n 1 In general any discrete signal can be represented as Xn ixk5n k kz m The sifting property EE701 Erik Blasch 47 i WRIGHT STATE UNI VERSITY Discrete lmpulses Linear superposition Scaled and shifted impulses 0 W0 XU ELI2 lt DD I15 1 I 1 PE 1 Decomposition Xn X 35n3 X 26n2 X 15n1XO5nx16n 1 X26n 2 X35n 3 intin 7 EDIE139 1 4 391 g I 3i iiiii I i 39 39 1i 1 V7 7 gt V I a I 1 1 i sifting property the summation sifts through the sequence of values Xk and preserves onym Km i ma a valuecorrespondinoi to k n this EE701 Erik Blasch 48 Wit Wests Exam Pie The discrete signal Xn WiggetlggittETypes of Unit Impulse Response Causal stable finite impulse response yn Xn O5Xn1 O25Xn2 Looking at unit impulse l 1 responses allows you to 1395 39 Ie decomposed into the I I I determine certain system 39E39 following additive components propert39es f I l I I 05 Causal stable infinite impulse response 0 e I g 10 X3Jn3 X2Jn2 X1Jn1 lFi yn Xn O7yn 1 n i i i i i i i i i i i i v x i 1 r w I r w 15 15 quot w e e 1 10 I 0539 5 I 394 2 i 6 1 5 4 s 5 2 l a l g 4 e 5 2 i a i s 4 0 5 10 0 5 10 n n Causal unstable infinite impulse response EE701 Erik Blasch 49 EE701ErikBlasch n Xn 13Yn391 5o i T t t wggggg glegla Convolution Integral veveimiaraphioal Convolution Methodsa Commonly used in engineering science math tf2tE foo rlfzv rm Convolution properties Commutative ft f2t f2t ft Distributive ft f2t f3t ft f2t ft f3t Associative ft f2t f3t ft f2t f3t SNWH wQwcw then ft f2t 7 ft 7 f2t Ct 7 Convolution with impulse ft 6t ft Convolution with shifted impulse im ortam later in modulation EE701 Erik lasch 51 Width Property l39 t 3 39539 quot7 r 3951 J 1 ii 31 L51 From the convolution integral oonvolution is equivalent to f1t f2 t i 32 712 iii elm Rotating one of the functions about yaxis Shifting it by t Multiplying this flipped shifted function with the other function Calculating the area under this product Assigning this value to f1t f2t at t EE701 Erik Blasch 52 jtw TELLih I 39iquot 4 F L 9 quotl wg gy raphical Convolution Methods From the convolution integral convolution is equivalent to f1t is r if 3a 7 is Rotating one of the functions about yaxis Shifting it by f Multiplying this flipped shifted function with the k l lt in i r is 739 s WEl E fER Graphical Convolution Ex Convolve the following two functions ii ft A gt 3 2 gtt gt i 2 I I I I Replace twith z in ft and gt Choose to flip and slide gz39 since it is simpler other function and Symmetric 3 gm Calculating the area under this product Functions overlap iike this 2 m Assigning this value to f1f f2f at f T Wigging Graphical Convolution Ex2 wggggg raphical Convolution Ex 3 Convolve the following two functions A A gt 3 gtllt t lt t 2 2 3 gt39 2quot l T t 39 L T 2t 2t EE701 Erik Blasch 55 Convolution can be divided into 5 parts flt 2 E3gto Two functions do not overlap 2 flts Area under the product of the 4 12 s 7 functions is zero 2t 2t 2 S tlt O 3 gt Part of gt overlaps part of ft 2 Area under the product of the ltM functions is 2H ZH 39 T 2t T2 j3 r 2m 3 27 0 2 EE701 Erik Blasch 56 W i Em i WRIGHT STATE WRIGHT STATE UNIVERSITY UNIVERSITY Ill 0 S tlt 2 Result of convolution 5 Intervals of interest Here gt completely overlaps ft A3 gU o Area under the product IS Just 2 0 f0ftlt 2 2 12 2 Kfw t26 for ZSIltO J3 72d73 27 6 lt 12 gt T 2 o 2 0 2t 2t ytftgtlt6 forOStlt2 N 2 g k 4 A 3t2 12t24 for2 tlt4 Part of gt and ft overlap 3 g 3 K0 fort24 Calculated similarly to 2 s tlt O 2ltf I l I A I 2 t39z 2 t7 T 4 gt 370 E gt and ft do not overlap k I Area under their product is zero I 39I i f I t EE701 Erik Blasch 57 EE701 Erik Blasch 2 O 2 4 58 I 39 ll 39 mam Convolution Matlab WRIGHTSTATE Convolution Matlab UNIVERSITY UNIVERSITY C24Lathi Example 27 so I mars o dz 024lLaihil 39Examp39e 27 err I mars t 11 N no 2 Bert segment m wt 2 l I gm 232t segmentE l I I m I 11 2 E EliT SEEI ent hf at E WEE I I I I g 2 lam I segment E h I Shh t1 momquot 0 t1 t1 m m 1 2ex 2t1 39339 f I git t t 33931 t Eli t3 at 2 OO011Ot2t2 j g2 2expt2 7 iii z it git ti it t tIt2 g 91921 a e k cm 2 E f zerossizegtonessize92 get a Em 53 f alarm 11 c 001 convf g 393 Et t 20 0015 t t I i If 3393 t39 r it 393 plottc1lengtht q EE701 Erik Blasch 59 EE701 Erik Blasch 60 WK WSIEEEQEIIIE Continuous Convolution Considering the relationship I l I between discrete and My a continuous systems I The convolution sum for discrete 91 I 39 systems was based on the I sifting principle the input signal 05 can be represented as a I superposition linear quot 0 two2 I combination of scaled and shifted impulse functions This can be generalised to 9 continuous signals by thinking fourth of it as the limiting case of arbitrarily thin pulses l I I 39 I 39 I 39 I 39 I 39 I I 0 t a 02 til2M 5 t2 l t WWII WRIGHTSTATSIQnaI Staircase Approximation UNIVERSITY As previously shown any continuous signal can be approximated by a linear combination of thin delayed pulses 1 1AT 6A0 O S t lt A 6w A l O otherw1se I A r Note that this pulse rectangle has a unit integral Then we have 3cm ixkA5At kAA kz oo Only one pulse is nonzero for any value of t Then as A a O xt 152 kzwxkA5A t kAA When AeO the summation approaches an integral xt f xT5t rm This is known as the sifting property of the continuoustime impulse and there are an infinite number of such impulses 5t z wm s mlla Alternat39Ve Der39Vat39O 0f wm FsmthontInuous TIme Convolution UNIVERSITY Sifting Property The unit impulse function 5t could have been used to directly derive the sifting function 6t r 0 t7 7 if so z39dz39 i Therefore xT5t TO I727 fWJL t dz39 j at adv xt j 50 z39dz39 xt The previous derivation strongly emphasises the close relationship between the structure for both discrete and continuoustime signals EE701 Erik Blasch 63 UNIVERSITY Given that the input signal can be approximated by a sum of scaled shifted version of the pulse signal 5At kA 3cm ixkA5At kAA kz oo The linear system s output signal yis the superposition of the responses but which is the system response to 5At kA From the discretetime convolution A 0 A yt ZxkAhkAtA kz oo What remains is to consider as A e O In this case y lAiinggooxkAhkAtA J xrhrdr EE701 Erik Blasch 64 Wit WRIGHTSWE Ex Discrete to Continuous Time Linear Convolution The CT input signal red xt is approximated blue bA quot39lllllllii39 y xlttgt ZxltkAgt6Altt kAgtA 15 39 a 39 39 a koo Each pulse signal 0 IT 02 5AtkA x 215 o t 1 14 K 006 o t 1 14 generatesaresponse if 1 39 39 EDS 39 hm g 05 0 Therefore X 2115 E t 1 14 936 if t 1 14 convolution response is ltrgt immn mmx kz oo Which approximates the CT convolution response ya fmxcmxodr Kit Dc 4 4 arm 34ft 11ft 15 11 t 1 14 1113 D t 1 14 W ll WRIGHT STATE Linear Time Invariant Convolutio For a linear time invariant system all the impulse responses are simply time shifted versions mm hr r Therefore convolution for an LTl system is defined by 9 Iquot f yt j xTht TdT This is known as the convolution integral 1 or the superposition integral Algebraically it can be written as gnawW To evaluate the integral for a specific value of t obtain the signal ht r and multiply it with xr and the value yt is obtained by integrating over rfrom oo to co Demonstrated in the following examples W5 V R IT FE EX 1 CT Convolution nggglggggli EX 2 COHVOIUtIOn Let Xt be the input to a LTl system 1 1 with unit impulse response ht DB 44 4 DE 1113 xt mlKt a gt O E 44 ff I14 112 I12 ha MU l32 u a I 4 El2 For tgt O 6 6 0 lt T ltt De us x7ht 7 DE DE O othchISc 4 14 3 t 1 E 14 J 04 We can compute yt for fgt O 112 42 T 1 T t U2 III 2 I 41 U2 El 2 I 4 Cl a yt e dT a6 0 1 I18 2 1 6 6quot Em In this example gt Tm a 1 So for all t 0 2 I 39 yltrgt11 e ultrgt a 2 r 4 EE701 Erik Blasch 67 Calculate the convolution of the 1 l followrng Signals 08 8 2t ue ne E114 E114 1312 02 94 2 1 t 2 4 2 393 t 2 4 The convolutW becomes 1 39l 3 118 as Mt 27d7620 3 e06 TUE 00 375134 E04 02 32 For t 3 2 O the product XZ39ht z 11 Q 3 2 4 t4 2 two L 4 is nonzero for oo lt Tlt 0 so the convolution integral becomes yt fwezfdr ii I I I I f EE701 Erik Blasch DO W I WRIGHT STATE Discrete Conv Ex DSP Examples Sin wave with LF rise and HF noise It Low pass Filter 135 4 I I I I I I I I I I I I gg5 IlzI5IIllI I 3quotquotIquot39Iquot39I39quot39Iquot39Iquot39I39quot39Iquot39I39quotI39quot39Iquot39 I I I I I I I I I I I I Ii VIII I 51 I I I I I I I I r 7 H 7I rr r L 3FI I I I I 2 I Er II II It I39I Iq P I a Aquot a IIII39 433 i i I III I so 5 ample utter 439 125 I I 1 4 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 3 IIII1II1IIIIIIII rII llv 39lrIIE39 3 IIIIIIIII139IIIIIII1IIIIIIrIIIII I I 21 2 IIII1IIIIIIll IIIIlIu 175 llllrlllll 2 II IIIIIIIIIIIIIIIIIIIII4IIIIIIFIIIII I I I I I II I I I I I I I I I I I I I I I I E L r I1 l lrlll IHIIII I E gul I PIIII I E I I II II I I I I I I I F r Illlullllllll lj l L I I I I I 05 I I I I I I I I I r I r I I u r I I III Bil 3 lII EEI I til 1 ZZI 3 le number Sample ber EE701 Input Signal Impulse Response 013mm Signal 69 mm sky aw 7 ll WRIGHT STATE Discrete Conv Ex DSP Examples a Inserting Attenuator II 3393 I I 39 I I I I I I 39 39 i i i I i i I i i I 3 IIIIILII I I 3 IIIIIIIIrIIIIIIIIIIIIIIIIIIIIIIIIIIII I I I I I I 393939 lllllllll I I I I I I I I I I 2III4I4I lFIIFII I J IIIIIIIIIFIIIIIIIIIIIIIIIIIIIIIIIIIIIII I I I 39 iIII quot I I 41 III II llllIIIIIIIIIILIIIIIIIIIIILIIIIIII F IiiIiHII Illh Ilrlllrll 39 I39 39 II PI 39 39 39 39 39 39 39 39 I II I I h 39 I l IIJI M4IIIIIIIII PF I I I I I I 39 EEIIrIII I IUII I I I I I I I I I I I I I I I I I I I I I I I l l I I I rI 23 539 il 7r sI m I III thi 33 393 1339 393 quot339 5393 5393 539 E 3393 1 quot Sample member Sills lll39 f sample 39 1 are It 39239 I i i 39 39IZ rIr I I I I I I I I quot3 I I I I I I I 39139 1 1i 139 II39 39I lt3 5 7 EC C 13 E 139quot I 13 5 339 El 53 53 Til 3239 539 3131153 Sample number Sample ll fhEi39 Art I i 39I I I It Lu a I L I I I I I I I I I I I i l 1 I I I I I l I I I II III I I l I I l I I in ta Iquot Input Signal I m ulse Respo se Dutpttt EE701 Erik Blasch 7O wmmmimLInear Time Varying Systems UNIVERSITY If the system is time varying let hkn denote the response to the impulse signal ink because it is time varying the impulse responses at different times will change Then from the superposition property of linear systems the system s response to a more general input signal Xn can be written as Input signal Xnixk5nk kz oo System output signal is given by the convolution sum yin Zxkhkn k Ie It IS the scaled sum of Impulse responses EE701 Erik Blasch 71 WRIGHT STATE N VERSUnit SampleImpulse Responses of Different Classes of Systems Memoryless Mn 5n Mt 5t El Causal hn 0 for III lt1 0 Mt 0 for t lt 0 l i I I 3130 stable 2 IIIII lt oo 00 htcft lt oo t J n The condition for the unit impulsesample response for a 8180 stable system requires some justification EE701 Erik Blasch 72
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