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This 0 page Class Notes was uploaded by Cassidy Effertz on Monday November 2, 2015. The Class Notes belongs to ECE 4270 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/233923/ece-4270-georgia-institute-of-technology-main-campus in ELECTRICAL AND COMPUTER ENGINEERING at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
a c 113 ECE Fundamentals of DSP Lecture 23 LinearPhase Systems School of Electrical and Computer Engineering Centerfor Signal and Image Processing Georgia Institute of Technology Linearphases tems ys Generalized linear phase systems Chapter 6 lssues in implementing LTl systems Signal ow graphs and block diagrams Frequency respons Jam wlltwc Hem 0 wcltco Magnitude and phase lHWL 1 39 39lt 0 AHlteJ gt7wa 0 ac ltco 39 Impulse response hnw 7wltnltw 7rnia AmplitudL hm s Sumpl numbm m Amphmdu sin4 quot745 h quot 45 04 1 I 7 1 l 1 V39Y I J39 L Q L 7 5 I II I Ampmudc 7 sin 4750 7 43 W m 7 43 s Sump numlm n De a 2 1 v m2 m2 Hz 718827 204z7 88z7 4 3 2 z 88z 204z 88z71 111 4 Z Hz 1 7z4 88z3 2042 88z71 Hz 1z4Hz Hz00 ltgt Hz510 um o M to an a 9 I a g Modumr modmp U zmg a uxmmu mud mm waud r Wm M 0 ya ca 20 u an 99 M to an In uz U WWW m2H mm awnw lt Warwww w v M433Vm3aV wm3aVm3H Zr42mzsooz401300st v039z Mam A 42 3 20 Ila I Wm2H a Eu r42 m388 170 Z W424 M42884 a El 1 5 5 a n 1 szJ mi WmiaH J 1702 M288 Zm34 WM 4 W288 wmvoz 388414 m3H HM Aeltef gte J MZ Aelte1 gtAelte J gt hn 0 S n S M HM 1Aaltefwgte f MZ Age 0 714 TypeslllampV hM7 n Hz 1 72M 12 Type m u even quot 1 quot mtegerde ay 4 f f l Type W 1 Mo ha fsamp e de ay nce equauon M yn Zak n ek1 Zbk n 7k 111 k SStem mncuon k M 71 Haw k1 N 7k 1 110de k1 T erewsadwrect correspond nc between me cmerence equauon and the system m e CUOH when the numerator and denommaor are Wrmen as po ynorma s m 21 ECE4270 3511 Fundamentals of DSP Lecture 9 DiscreteTime Random Sign School of Electrical and Computer Engineering Center for Signal and Image Processing Georgia Institute of Technology Stationary random processes Time averages and ergodic random processes The Bernoulli random process MATLAB simulation of random processes The Uniform ran dom process Introduction for the next Lecture Linear systems with random inputs The proba ity distributions do not change with time mek an px an xnxmmgt x mm Thus mean and variance are constant mx E x 2 ax EIXK 7 mm 7 m I And the autocorrelation is a onedimensional function ofthe time difference mm mm mm E nmx Timeaverages ofa ra variables themse Kn 381an Ives 1 L 11m Z Lgtw2Lln 1 L 11m Z Lam2L1n ndom process are random Kn L 381an L Time averages ofa single sample function 471 xrz mxrz 1 L 11m Z Lam2L1n 1 L 11m 2 Lgt ltn2L1n XVl L XD I mxn L Timeaverages are equal to probability averages 1 Xquot ngnw2L1n ltXnmxil1iimn A 1 L m x Ln0 PX xnx11 ux 1 V 1 pxnxn 1 Z 961quot BPXK x n L 2 KM EXn mx 7L 1 Z Xnmx L EXnmx Mm Estimates from a single sample function 1 71 3x it m 1 L l Z xnm1x n Ln0 6x11 6x 1 1 3 A histogram shows counts of samples that fall in certain bins If the boundaries of the bins are close together and we use a sample function with many samples the histogram provides a good estimate of the probability density function of an assumed stationary random process Suppose that the signal takes on only two different values 1 or1 with equal probability PXquot xmn 05uxn 1 05uxn 71 pxn xn n 055xn 1 055xn 1 Furthermore assume that the outcome at time n is independent of all other outcomes Pxnxmltxmnxmmgtngxmn gxmm i Miss frigid iE eii 7 DM MATLAB s rand function is useful for such simulations gtgt d rand1N uniform dist Between 0 amp1 gtgt k ndxgt5 nd 1s gtgt x ones1N make vector of all 1s 39 Mean mx x056x 1o55x mx I 05x6x1dx I 05x6x 1dx x 7 0quot5 l 05r0 gtgt Xk ones1lengthk insert 1s Variance gtgt subplot211 hanstem0Nplt1X1Nplt 2 2 gtgt sethan markersize 3 O39x I x m 05505 1 05505 DW gtgt subplot212 histxNbins hold on gtgt stem11N5539r39 add theoretical values iuimiiiigmmi iDiieiiiri lziuiiiiiwi LEii i iwiiliiiiin i r i f 39 Bernoulli Distributed White NOiSe Sequence Witn Unit Variance PX xmn Probxn S xn o iiiiiiiiiiiHiiiiHimiiHiiiiiiiiiiiiiiii iiii JilliiiiiiimmiHHiJiiJiiiiJiiJi 50 80 1 Histogram or 15000 Samples of a White Noise Sequence ann xn 7 pxn xn n Vi 3xquot 4 Suppose that the signal values are uniformly 1 x2 1 distributed between 1 and 1 6 IX05dx 1 e1 05 1 lt xquot lt1 x n px quot 0 otherw1se I Furthermore assume that the outcome at time n 39 var39ance3 2 1 2 x3 1 05dx Is Independent of all other outcomes 03 I x mx 6 Pxnxm xngtquotgtxmgtmPxn 3quotme xmgtm 71 71 MATLAB s rand function is useful for such simulations WOW D str med Wme NO Se Sequence gtgtsethltandN 1 gtgt x 2randmvmU uniform dist 1 to 1 ii in h l i ll ll lull ll l l i n i lliiil r x i ill ll gtgt subplot211 hanstem0Nplt1x1Nplt 1ill ll l l m H UH l N l l l l gtgt sethan markersize 3 7 gtgt subplot212 histxNbins hold on V 20 40 60 80 1 gtgt add theoreticaI VEIUES m Histogram of 15000 Samples ofa White Noise Sequence gtgt plot1111N0550Nbins39r39 CSIP ECE 39 39 Fundamentals ariation of SNR with Signal Level Lecture 20 9 Ov rsampling can be used to reduce quantization noise Properties of Systems Use ofztransform in analysis of LTl systems School of Electrical and Computer Engineering Centerfor Signal and Image Processing Georgia Institute of Technology Q0414 idn Assume 2M levels i m guinti Then using a probabi Xl XmJr Eh listic analysis we obtain 39 l J iliil ilvil Ilulwlul 2X 3 2 2725 2 1 2 A e 2 Xm 2 0399 1 e I Error is uncorrelated with the input Error is uniformly distributed overthe interval 7A2ltenSA2 5 Error is stationary white noise ie at spectrum SNR 1010gioU e 1010g10 X X AZ 1 A2 P l0392 I 52d5 v Wis 1087201 e 9 MA 12 0310 UK V tv step Size noise powa Therefore the quantizer SNR is Hm l in mfv Assumptions QgtX7XgtLn0 mpg Z 2mdeowgff Z lwlgz Quantization of signal values and results of computation is unavoidable in a digital system We can analyze quantization error using a random noise model The more bits in the number representation the lowerthe noise A soft stated theorem is that the signaltonoise ratio increases 6 dB with e ch added bitquot however remember that ifthe signal level decreases while kee ing the quantizer step 39 s sIze the same it is like throwing awayblt SNR6OZB1081010310M72010310 J 397 241 Imam wn y w i d Since we can consider the effects of magnitude and phase separately it follows that linear phase of the form 4H 70nd implies delay of nd samples Thus an ideal Iowpass filter with delay has YlteJ Hlte gtXltergt bl Hplawful My leltwc 4Ye1 4HlteJ 4XlteJ gt p H 0 WW Consider a general difference equation of the for me HejweZH6M N M ED akyn k E0 kan k Logmagnitude in dB Rational s stem function of a causal and stable LTl 2010g10 Hew system y Phase in radians 1mg argHeJ Group delay in samples 1a grd He139 d 4Hef artlal fractIon expansion of the We can make a p rational system function MrN 1v Hz z Brz zi1 Hgtmf idil 0 17de nw System function 172 17 2 VH27JE r5192quot17rs 192 1721105027 r 2 quot t 2 re Xzire Difference equation n Mossynay in 21 an Impulse response 1 smmmn Um sm 9 law 4 ECE4270 C311 Fundamentals of DSP Lecture 1 Quantization in AtoD Conversion School of Electrical and Computer Center for Signal and Image Processing Georgia Institute of Technology Oversampling to ease ltering from last lecture AtoD conversion Probabilistic analysis of quantization Mod e Signaltonoise ratio HM X09gtHlte TgtXauo EX 399 lQlgtQ What is the overall effectiveaffelquency response Xa 1399 fIaaUQ XCOQ gt X1109 0 forIQI EQN x09 HlteJmgtHnoagtXcom H Xa 1399 HaaUQXcUQ Choose HaaUQ 0 for pl 2 MEN 3 Xao39Q 0 forLQ2MQN wur m t39 um I W I39mch u mmmtmt e e e e V Wmmm mm Practtcat constderattons m tmptementattons put stgnat cannot be perfec y bandttmted e ArtceD and at A converters have ttnteprectston output and mput respecttvety e omy ttmteprectston anthmeuc ts ma ab e for computauons Ouarmzauon Stepr tze forES1rbtt quanuzer 9 Dunn le Hml k s L muamum savrwlea a Qumul a runal 51mm me hold V7 7 7 7 r 39 1me 039 M amp cmvvzaler x Ungmal i 1 I Slgllal V i a 3 i I E lt A y 11 V 1 4 i r i u l39 u E I I v x ml l ull I 10 4 hill SI IIVII J 20 I 40 sample mum xllll lunnhgcd 39mehrmlur l lulu mwg oxnm 0 EU All Evil SD IUD 12le Ill sample mlnlu Implu i m mswa d autumnn 1 mquot u muquot Elma mwm vam QLUJ EL l fDZ lCQU 11er 3 39Asl l k g 39 39 quot quot wgumiEEL39EMW WW llmpg Li39 qweamf zar amplitude 0 20 40 60 80 100 120 140 sample index 5000 amplitude o 5000 0 20 40 60 80 100 120 140 sample index 5 quot39739 v a lt 1quot39u i p we v mmiiji l r39yiil i QL HJ W1 llll 33 f uupllnudc amplitude H 20 m m MI 00 my 0 0 20 40 60 80 100 120 140 sqii39Illllc allIlvl sample index IN V Quzinlnwd 39mel nnu uur l lln 2000 d n amplitude e um pl mule 2000 0 20 40 60 80 100 120 140 sample index I 10 40 i0 SLI lJlZI lllil Milli nlil pic mitlu quotquotlll39l illlilfll will 3 ml r Each sample is quantized and each sample has a quantization error defined as Laplaefenm I 39 t ettrtlmttiom v are Since each sample falls in an interval of length A for and the quantized sample falls in the middle of P CQi that interval V Amplitude Distribulion oi 3bit Ouantized Signal g 15 a We call this quantization noise because it 1 qmamttzed seems to vary randomly Clearly the strength 05 J V eleme l mee w power of this noise is proportional to A ie quot mWm a L a 5 i 3 1 1131 39l 2 3 d E lmm r CSIP 5654270 39 39 quot Fundamentals of DSP Lec re 16 DT Filtering of CT Signals Changing the Sampling Rate Using Digital Filtering School of Electrical and Computer E ineering Centerfor Signal and image Processing Georgia institute of Technology DT ltering of CT signals Derivation of general formula Illustration Examples The need to change sampling rates Decimation E xlnk39m anemt ml my my me Hymnal 1 YQH21Q Hl E 24097ka T kg lfthe input is bandlimited such that X009 0 for IQi 2 Q 27 72 29 then the overall input and output are related by x09 mammal H39JQH2 mX2 m 7 V we Helmmxjngmm QT l HeffQH2J lQllt T 7 in 39 Note The effective cutoff frequency deandsonthesammtel W 1 v u we H9H2 Q 3T AKA9499 lfthe input is bandlimited such that X609 0 forIQIEQ and 2172 29 then the overall input and output are related by QT 1119 HM XCUQ HmUQVQ 9 Difference equation yn ayn 1 bx Frequency response Hea 1 b jw Overall frequency response b H09gtHltefmgtm 10k runu xln lt2 X91 kiamp1nigj n lt2 22197H2197X21m 2m lt2 2M W L M W 7 MTIEJWKQ MT 1 Mquot 0772 w 7 X J W M E e
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