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# ELECTRIC CURCUIT ANALYS III ECE 223

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This 120 page Class Notes was uploaded by Miss Chadrick Doyle on Tuesday September 1, 2015. The Class Notes belongs to ECE 223 at Portland State University taught by James McNames in Fall. Since its upload, it has received 70 views. For similar materials see /class/168227/ece-223-portland-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Portland State University.

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Date Created: 09/01/15

Overview of Communication Topics o Sinusoidal amplitude modulation 0 Amplitude demodulation synchronous and asynchronous 0 Double and singlesideband AM modulation 0 Pulseamplitude modulation 0 Pulse code modulation 0 Frequencydivision multiplexing o Timedivision multiplexing o Narrowband frequency modulation J McNames Portland State University ECE 223 Communications Ver 115 1 Handy Trigonometry Identities sin Slna cosb cos sin b cosa cosb COSCL b COSCL b s1na sinb cosm b cosm b Slna cosb 81HCL b l smm l b J McNames Portland State University ECE 223 Communications Ver 115 2 Introduction to Communication Systems 0 Communications is a very active and large area of electrical engineering 0 Experienced a lot of growth through the nineties with the advent of wireless cell phones and the internet 0 Still an active area of research 0 Fundamentals of signals and systems are essential to grasping communications concepts 0 The next two lectures will merely introduce some of the fundamental concepts 0 Will primarily focus on modulation and demodulation in continuoustime o Analogous concepts apply in discretetime J McNames Portland State University ECE 223 Communications Ver 115 3 Introduction to Amplitude Modulation 3160 W Ct yt ZEt 00 00 coswct l 60 0 Modulation the process of embedding an informationbearing signal into a second signal 0 Demodulation extracting the informationbearing signal from the second signal 0 Sinusoidal Amplitude modulation a sinusoidal carrier 00 has its amplitude modified by the informationbearing signal xt J McNames Portland State University ECE 223 Communications Ver 115 4 Fourier Analysis of Sinusoidal Amplitude Modulation For convenience we will assume 60 0 t Q coswct Cjw 7r6w wc6wwc 3115 05 39Ct Y max Cum Xjw gtllt 6w we Yjw J39w Xjw wc Xjw wcXjwwc 0 Thus sinusoidal AM shifts the baseband signal ZEt so that it is centered at iwc 0 Thus ZEt can be recovered only if we gt cam so that the replicated spectra don39t overlap J McNames Portland State University ECE 223 Communications Ver 115 5 Fourier Analysis of Sinusoidal Amplitude Modulation X000 1 I I I gt w w 0 wx Cjw T 71 T I I I gt w wc 0 we 1 WW I i I I I i I gt w 39wc39w 39wc 39wc i wx 0 wc39wx we wc i wx J McNames Portland State University ECE 223 Communications Ver 115 6 Example 1 Sinusoidal AM of a Random Signal Example of Sinusoidal Amplitude Modulation 02 a 0 02 Ct yt J McNames Portland State University Example 1 MATLAB Code function J AMTimeDomain N 2000 Z NO fc SOeS Z samples Carrier frequency fs 1e6 Z Sample rate k 1N t k1fs tMilliseconds t1000 Z Time index in units of milliseconds Xh randn1N Z Random highfrequency signal nwn ellipord0010020560 ba ellipnO560wn X filterbaXh Z Lowpass filter to create baseband signal c COS2pifCt y XC figure FigureSet1 LTX subplotS11 h plottMillisecondsx b seth LineWidth 12 xlimO maxtMilliseconds ylimOS 051 ylabel Xt title Example of Sinusoidal Amplitude Modulation box off J McNames Portland State University ECE 223 Communications Ver 115 AXisLines subplot812 h plottMillisecondsc g seth LineWidth O2 xlimO maxtMilliseconds ylim11 111 ylabel ct box off AXisLines subplot818 h plottMillisecondsy r tMillisecondsX g tMillisecondsX c seth1 LineWidth O2 seth28 LineWidth 12 xlimO maxtMilliseconds ylimOS 051 X1abel Time ms ylabel yt box off AXisLines AXisSet6 print depsc AMTimeDomain J McNames Portland State University ECE 223 Communications Ver 115 9 Synchronous Sinusoidal Amplitude Demodulation Transmitter Receiver yt yt wt yew f a Ct Ct o How do we recover the baseband signal 0 Why can39t we simply divide yt by coswct yt 9615 coswct wt yt coswct ZEt 3082wct ZEt cos2wct l cos2wct J McNames Portland State University ECE 223 Communications Ver 115 10 Synchronous Sinusoidal Amplitude Demodulation quotTr39 ffs39rh39i39tie39rquot Receiver yt yt wt we a C13 C13 yt ZEt coswct wt cos2wct o Synchronous demodulation requirements The carrier 00 is known exactly we gt W1 0 The xt can be extracted by multiplying yt by the same carrier and lowpass filtering the signal J McNames Portland State University ECE 223 Communications Ver 115 11 Fourier Analysis of Sinusoidal AM Demodulation I I r w wc 0 we T W Cjw T I I I gt w wc O we 1 WOW 2 i I i w 2wc 0 2000 2 A How gt w 0 1 f ROW I gt w J McNames Portland State University ECE 223 Communications Ver 115 12 Synchronous AM Demodulation Observations 060 Transmitter yt Ct f Ct Receiver yt wt a gt ft o The Iowpass filter H8 should have a passband gain of 2 o The transition band is very wide so the filter does not need to be close to ideal ie it can be low order 0 We learned how to design this type of filter in ECE 222 c We assumed the signal spectrum Xjw was real 0 The same ideas hold if Xjw is complex 0 Called synchronous demodulation because we assumed the transmitter and receiver carrier signals 00 were in phase J MCNarnes Portland State University ECE 223 Communications Ver 115 13 Synchronous AM Demodulation Carrier Phase Analysis Suppose the transmitter and receiver carrier signals differ by a phase shift A H V O 0 CD 8 Q as V CT R A A A H CTt 0315 308th 6 308th cosa cosb COSCL b COSCL b wt ZEt cos6 o cos2wct l 6 l on t cos6 o 3082th 6 22 J McNames Portland State University ECE 223 Communications Ver 115 14 Synchronous AM Demodulation Carrier Phase Comments wt cos6 d 0032th 6 l o If 6 p then we have the same case as before and we recover 9615 exactly after a lowpass filter with a passband gain of 2 o If l6 pl we lose the signal completely 0 Otherwise the received signal is attenuated o The phase relationship of the oscillators must be maintained over time o This type of careful synchronization is difficult to maintain o Phaselocked loops PLL can be used to solve this problem In ECE 323 you will design and build PLL39s o The carrier frequency we of the transmitter and receiver must also be the same and remain so over time J McNames Portland State University ECE 223 Communications Ver 115 15 Introduction to Asynchronous AM Demodulation o Asynchronous modulation does not require the carrier signal Ct be available in the receiver 0 Thus there is no need for synchronization o Asynchronous Modulation Assumptions we ZEt gt 0 for allt 0 However it does require that the baseband signal ZEt be positive o In this case the envelope of the modulated signal yt is approximately the same as the baseband signal ZEt 0 Thus we can recover a good approximation of xt with an envelope detector misnomer J McNames Portland State University ECE 223 Communications Ver 115 16 Example 2 Asynchronous Amplitude Modulation Smusoidal AM Modulation Xt ll l jg lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ct d a CD 5 J McNames Portland State Un1vers1ty function 1 N 2000 Z No samples fc 50e5 Z Carrier frequency fs 1e6 Z Sample rate k 1N t k1fs Xh rand1N05 Z Random highfrequency signal limited to 05 05 nwn ellipord0020080560 ba ellipn0560wn X filterbaXh Z Lowpass filter to create baseband signal X X 02 Z Convert to positive signal c COS2pifCt y XC t t1000 Z Conver time to milliseconds figure Example 2 MATLAB Code AAMTimeDomain FigureSet1 LTX subplotS11 h plottX b seth LineWidth 12 Xlim0 maXt1 ylim01 041 ylabel Xt J McNames Portland State University ECE 223 Communications Ver 115 18 title Example of Asynchronous Sinusoidal AM Modulation box off AxisLines subplot812 h plottc g seth LineWidth O2 x1imO maxt ylim11 111 ylabel ct box off AxisLines subplot818 h plotty r tx b tx b seth LineWidth O2 seth28 LineWidth 12 x1imO maxt ylimO89 0591 x1abel Time ms ylabel yt box off AxisLines AxisSet8 print depsc AAMTimeDomain J McNames Portland State University ECE 223 Communications Ver 115 19 Diodes lug Ideal Model Real Model l I l 7 gt V gt V 0 07 0 07 0 To do asynchronous amplitude demodulation we need an envelope detector 0 Diodes can be used as a key component in envelope detectors 0 Like resistors Diodes do not have memory The currentvoltage relationship is independent of time 0 Key idea diodes only allow current to flow in one direction 0 If on V 2 07 V they act like an ideal voltage source 0 If off V lt 07 V they act like an open circuit J McNames Portland State University ECE 223 Communications Ver 115 20 Example 3 Full Wave Rectifier Draw the equivalent circuits for Us gt 0 and Us lt 0 assuming an ideal model of the diode with a threshold voltage of 0 V J McNames Portland State University ECE 223 Communications Ver 115 21 Example 3 Workspace J McNames Portland State University ECE 223 Communications Ver 115 22 Envelope Detectors N 1 VI 740 W 2 C R mt o A diode can be used to extract the upper half of the modulated signal 0 This is called a halfwave rectifier o Roughly speaking when yt gt 0 et yt o By connecting a capacitor in parallel with the resistor the RC circuit acts like a firstorder lowpass filter 0 This smoothes the received waveform o A fullwave rectifier that recovers both the negative and positive peaks has better performance J McNames Portland State University ECE 223 Communications Ver 115 23 Example 4 Envelope Tracking Example of Envelop Tracking of an Asynchronous AM Signal 8 21111111111111111111111111111111mm11vavA1111111111111mm I I I n u nII l l l lI III II I I I IIII D II I I IIII l h ll i ni l l l l l j 11 1 1391 1 1 11 11 1 1 1 11 1I I x lllilllilllI l lly lll y1 11 1 1 1 1 1 11 1XIIII I I IllIllx Ill lll llIlllixllll l1WI I III I1111 1391 1 1 1 11 11 1 1 1 I I H H I V 08 09 1 11 12 Timtims14 15 16 17 18 Example 4 MATLAB Code function J EnvelopeTracking N 2000 Z No samples fc 50e5 Z Carrier frequency fs 1e6 Z Sample rate al 095 Z Firstorder filter parameter k 1N t k1fs tMilliseconds t1000 rand state 10 Xh rand1N05 Z Random highfrequency signal limited to 05 05 nwn ellipord0010020560 ba ellipn0560wn X filterbaXh Z Lowpass filter to create baseband signal X X 02 Z Convert to positive signal c COS2pifCt y XC eh yygt0 rh filter1al1 alehmeanehmeaneh ef absy rf filter1al1 alefmeanefmeanef figure FigureSet1 LTX subplotS11 J McNames Portland State University ECE 223 Communications Ver 115 25 h plottMi11isecondsy r tMi11isecondsx b tMi11isecondsx b seth1 LineWidth O2 seth28 LineWidth O5 xlimO8 181 ylimO89 0591 ylabe1 yt tit1e Example of Envelop Tracking of an Asynchronous AM Signal box off AxisLines subplot812 h plottMi11isecondseh r tMi11isecondsrh b seth1 LineWidth O2 seth2 LineWidth O5 xlimO8 181 ylim002 0591 ylabe1 Halfwave Rectifier box off AxisLines subplot818 h plottMi11isecondsef r tMi11isecondsrf b seth1 LineWidth O2 seth2 LineWidth O5 x1imO8 18 ylim002 0591 ylabel Fu11wave Rectifier xlabe1 Time ms box off AxisLines AxisSet6 print depsc EnvelopeTracking J McNames Portland State University ECE 223 Communications Ver 115 26 Asynchronous Amplitude Modulation Terminology 05 905 A Ct 0 Most baseband signals will not be positive 0 We can make amplitudelimited signals g xmax positive by adding a constant A such that A gt xmax o The envelope detector then approximates ZEt l A o xt can then be extracted with a highpass filter to remove A the DC component 0 The ratio m xmaXA is called the modulation index 0 If expressed in percentage 100xmaXA it is called the percent modulation 0 The spectrum of yt contains impulses to account for A J McNames Portland State University ECE 223 Communications Ver 115 27 Spectrum of Asynchronous Amplitude Modulation X000 1 I I I gt w w 0 wx Toma T 71 T I I I gt w wc O we A71 YUW l 2 I I I I I I I w 39wc39wx 39wc wcw 0 wc39wx we wc i wx J McNames Portland State University ECE 223 Communications Ver 115 28 Asynchronous Amplitude Modulation Tradeoffs 0 yt A Ct o In most applications the FCC limits the transmission power 0 For asynchronous AM transmitting the carrier component requires a portion of this power 0 Thus asynchronous AM is less efficient than synchronous AM 0 However the receiver is easier and cheaper to build 0 As m gt 1 more of the transmitter power is used for the baseband signal ZEt 0 As m gt 0 the signal is easier to demodulate with an envelope detector J McNames Portland State University ECE 223 Communications Ver 115 29 Single Sideband AM 1 3060 I i I I I i I gt w 39wc39w 39wc 39wc l wx 0 wc39wx we wc l wx 0 Let us define the bandwidth of the signal as cum the highest frequency component of the signal 0 The signal transmitted requires twice the bandwidth 2 0 Near we the signal content for both negative and positive frequencies is transmitted c We don39t need this much information to reconstruct Xjw o If we know Xjcu for either positive or negative frequencies we can use symmetry to construct the other part 0 Thus we only need to transmit one of the sidebands J McNames Portland State University ECE 223 Communications Ver 115 30 Single Sideband AM Continued I I I I I gt w 39wc wcwx 0 wc39wx we 0 What we have discussed so far uses doubleSideband modulation 0 We can use singleSideband modulation by removing the upper or lower sidebands o Requires only half the bandwidth 0 An obvious approach lowpass to retain lower sidebands or highpass filter to retain upper sidebands o Requires a nearly ideal highfrequency filter o 558 modulation increases the cost of the transmitter o If asynchronous modulation is used it also increases the cost and complexity of the receiver J McNames Portland State University ECE 223 Communications Ver 115 31 FrequencyDivision Multiplexing 1TX1jw 1TX2UW XSUW 0 We can transmit multiple signals using a single transmitting antenna with frequencydivision multiplexing FDM 0 Each baseband signal is shifted to a different frequency band 0 Thus multiple baseband signals can be transmitted simultaneously over a single wideband channel 0 The different modulated signals yl y2t and y3t are simply summed before sending to the antenna 0 To recover a specific signal the corresponding frequency band usually is extracted with a bandpass filter J McNames Portland State University ECE 223 Communications Ver 115 32 TimeDivision Multiplexing o The sampling theorem tells us we can represent any bandlimited signals by its samples 96nT as long as ws gt 2wx 0 Thus we can convert multiple bandlimited signals into discretetime signals 1E1t gt 961 1E2t gt 962 113305 1133lnl o TimeDivision multiplexing interleaves these signals to form a composite signal 1l0203l01121311l2l22l32l o A different time interval is assigned to each signal 0 We could then form a continuoustime signal using bandlimited interpolation o If M signals are multiplexed and each signal has a bandwidth of cum the multiplexed signal yt will require a bandwidth of M X cum J McNames Portland State University ECE 223 Communications Ver 115 33 Example 5 TimeDivision Multiplexing 1 Example of TimeiDiVision Multiplexing 39 0 it 05 I 0 O T T C T T I 0 1 2 3 4 5 6 7 8 9 10 11 1 o I I l 0 i l l 139 r T 1 0 1 2 3 4 5 6 7 8 9 10 11 yn J McNames Portland State University ECE 223 Communications Ver 115 34 Example 5 MATLAB Code function J TDMultipleXing close all N 10 Z No samples k 1N X1 randN1 X2 randN1 X8 randN1 y zerosSN1 k1 15SN k2 25SN k8 SSSN yk1 x1 yk2 x2 ykS XS wc pi T 1 Sample rate t 0001SN1 n 1SN yr zerossizet Reconstructed signal for cnt 1lengthn yr yr wcTpiycntsincwctncntTpi J McNames Portland State University ECE 223 Communications Ver 115 35 end figure FigureSet1 LTX subplot411 h stemkx1 g seth1 MarkerSize 2 seth1 MarkerFaceColor g hold off xlimO N11 y11m0 1051 ylabel x1n title Example of TimeDivision Multiplexing box off subplot412 h stemkx2 b seth1 MarkerSize 2 seth1 MarkerFaceColor b hold off xlimO N1 y11m0 1051 ylabel x2n box off subplot418 h stemkx5 r seth1 MarkerSize 2 seth1 MarkerFaceColor r hold off xlimO N11 y11m0 1051 ylabel x5n J McNames Portland State University ECE 223 Communications Ver 115 36 box off subplot414 h plottyr k hold on h stemk1yk1 g seth1 MarkerSize 2 seth1 MarkerFaceColor g seth5 Visible 0ff h stemk2yk2 b seth1 MarkerSize 2 seth1 MarkerFaceColor b seth5 Visible 0ff h stemk8yk5 r seth1 MarkerSize 2 seth1 MarkerFaceColor r seth5 Visible 0ff hold off Xlim0 SN1 ylimOS 151 ylabel yn box off AXisLines AXisSet6 print depsc TDMultipleXing J McNames Portland State University ECE 223 Communications Ver 115 37 Pulse Amplitude Modulation W Z xnTpt WT TL OO T rem 9V ya 00 190 o In modern communication systems the baseband signal 9615 is first sampled to form 96nT in accord with the sampling theorem 0 In a pulseamplitude modulation PAM system each sample is multiplied by a pulse pt o Timedivision multiplexing can be easily combined with PAM 0 Thus we could use pt 81nd to ensure yt is bandlimited to w 0 We require 111 gt 2wx 2T to satisfy the sampling theorem J McNames Portland State University ECE 223 Communications Ver 115 38 Pulse Amplitude Modulation Continued T 00 9 Q y ya Z xltnTgtpltt m 190 n00 o This looks like bandlimited interpolation 0 However the pulse need not be a sinc function 0 Could be a modulated sinc function or similar that shifts the signal to a high frequency band J McNames Portland State University ECE 223 Communications Ver 115 39 PulseCode Modulation T T Modulation M i mm w 100 o In practice digital systems encode discretetime signals with discrete amplitudes 0 Most digital signal processing DSP uses discretevalued signals 0 Continuousvalued signals are converted to discretevalued signals using analogtodigital ADC converters o Discretevalued signals can be encoded using binary 139s and 039s 0 These discrete signals can be transmitted over a communications channel by transmitting 0 Transmit pt 1 Transmit pt J McNames Portland State University ECE 223 Communications Ver 115 40 Example 6 PCM Create a random digital discretetime and discretevalued signal consisting of fifty 039s and 139s Encode the baseband signal ZEt such that the bandwidth is limited to 100 Hz What is the minimum time required to transmit the signal Plot the discretetime signal the baseband encoded signal and an eye diagram of the overlapping received pulses Assume that the channel does not cause any distortion and that the receiver and transmitter sampling times are synchronized Hint recall that W tW sinc ltgt PWjw 7t 7t J McNames Portland State University ECE 223 Communications Ver 115 41 Example 6 Workspace J McNames Portland State University ECE 223 Communications Ver 115 42 Example 6 Plot of and 9625 Example of PulseCode Modulation 08 E 06 x 04 02 0 L L L L L L L 0 1 2 3 4 5 6 7 8 9 10 11 2 0 001 002 003 004 005 006 007 008 009 01 J McNames Portland State University ECE 223 Communications Ver 115 43 Example 6 Eye Diagram 1 In v In 0 In T 0002 0004 0006 0008 001 7 01 70008 70006 70004 70002 m NO Tlme sec Ver 115 44 ity ECE 223 Communications Portland State Univers J McNames function J PCMEX close all N 50 n 1N Xd randN1gtO5 we 2pi50 Z T piwc Z TS 00005 t OTsN1T figure FigureSet1 LTX subplot211 h stemnxd b hold off xlimO 11 ylimO 1051 ylabel x1n Z Z Z Limit Example 6 MATLAB Code No samples Discretetime index Digital signal pulse bandwidth to 100 Hz 50 to 50 Sample period seth1 MarkerSize 2 seth1 MarkerFaceColor b title Example of PulseCode Modulation box off subplot212 xc zerossizet Z Modulated signal Xt for cnt lzlengthn J McNames Portland State University ECE 223 Communications Ver 115 45 s 1xdcnt0 1Xdcnt1 p ssincwctncntTpi plottp b hold on XCXCp end plottxc g plotnT1xd01xd1 ro MarkerSize 2 MarkerFaceColor r hold off XlimO 11Tl ylabel xt box off AXisLines AXisSet6 print depsc PCMSignals figure FigureSet2 LTX for out 1lengthn k roundTTsroundTTs plotkTsxcroundncntTTsk1 hold on end hold off XlimminkTs maxkTs ylabel xt Xlabel Time sec title Eye Diagram box off AXisSet6 AXisLines J McNames Portland State University ECE 223 Communications Ver 115 46 print depsc PCMEyeDiagram J McNames Portland State University ECE 223 Communications Ver 115 47 function J PCMEX close all N 50 n 1N Xd randN1gtO5 we 2pi50 Z T piwc Z TS 00005 t OTsN1T figure FigureSet1 LTX subplot211 h stemnxd b hold off xlimO 11 ylimO 1051 ylabel x1n Z Z Z Limit Example 6 MATLAB Code No samples Discretetime index Digital signal pulse bandwidth to 100 Hz 50 to 50 Sample period seth1 MarkerSize 2 seth1 MarkerFaceColor b title Example of PulseCode Modulation box off subplot212 xc zerossizet Z Modulated signal Xt for cnt lzlengthn J McNames Portland State University ECE 223 Communications Ver 115 48 s 1xdcnt0 1Xdcnt1 p ssincwctncntTpi plottp b hold on XCXCp end plottxc g plotnT1xd01xd1 ro MarkerSize 2 MarkerFaceColor r hold off XlimO 11Tl ylabel xt box off AXisLines AXisSet6 print depsc PCMSignals figure FigureSet2 LTX for out 1lengthn k roundTTsroundTTs plotkTsxcroundncntTTsk1 hold on end hold off XlimminkTs maxkTs ylabel xt Xlabel Time sec title Eye Diagram box off AXisSet6 AXisLines J McNames Portland State University ECE 223 Communications Ver 115 49 print depsc PCMEyeDiagram J McNames Portland State University ECE 223 Communications Ver 115 50 Example 7 Noise Tolerance of PCM Repeat the previous example but this time assume that the channel adds noise that is uniformly distributed between 05 and 05 Can you still accurately receive the signal J McNames Portland State University ECE 223 Communications Ver 115 51 Example 7 Plot of and 9625 Example of PulseCode Modulation 1 E 05 xvi i 0 001 002 003 004 005 006 007 008 009 01 4 2 lt2 0 2 I I I I I I I I I I 0 001 002 003 004 005 006 007 008 009 01 J McNames Portland State University ECE 223 Communications Ver 115 52 Example 7 Eye Diagram Eye Diagram E 0 m 391 v x o L y w Mu v 0 t i Vquot wtint quot i 39 vilil iv39ii viquot p 1 m w w W quot quot 1 00 Aquot 7001 70008 70006 70004 70002 0 0002 0004 0006 0008 001 Time sec J McNames Portland State University ECE 223 Communications Ver 115 53 Example 7 MATLAB Code function J PCMNoisEX close all N 50 Z No samples n 1N Z Discretetime index Xd randN1gtO5 Z Digital signal we 2pi50 Z Limit pulse bandwidth to 100 Hz 50 to 50 T piwc Z Sample period Ts 00002 t OTsN1T nt nTTS figure FigureSet1 LTX subplotS11 h stemnxd b seth1 MarkerSize 2 seth1 MarkerFaceColor hold off xlimO 11 ylim0 1051 ylabel x1n title Examp1e of PulseCode Modulation box off Ibo subplotS12 xc zerossizet Z Modulated signal Xt J McNames Portland State University ECE 223 Communications Ver 115 54 for out 1lengthn s 1xdcnt0 1Xdcnt1 p ssincwctncntTpi plottp b hold on XCXCp end plottxc g plotnT1xd01xd1 ro MarkerSize 2 MarkerFaceColor r hold off XlimO 11Tl ylabel xt box off AXisLines subplot818 r xc randsizexcO5 Add noise to the received signal plottr b hold on plotnTr1nroundTTs ro MarkerSize 2 MarkerFaceColor r plottxc g hold off XlimO 11Tl ylabel rt box off AXisLines AXisSet6 print depsc PCMNoiseSignals figure FigureSet2 LTX for out 1lengthn J McNames Portland State University ECE 223 Communications Ver 115 55 k roundTTsroundTTs plotkTsrncntroundTTsk1 hold on end hold off XlimminkTs maxkTs ylabel xt X1abel Time sec title Eye Diagram box off AXisSet6 AXisLines print depsc PCMNoiseEyeDiagram J McNames Portland State University ECE 223 Communications Ver 115 56 Example 8 Communication System Design a system transmitter and receiver that transmits a stereo audio signal through a channel in the frequency band of 12 MHz i40 kHz Discuss the design tradeoffs of different approaches to this problem and sketch the spectrum of signals at each stage of the process J McNames Portland State University ECE 223 Communications Ver 115 57 Example 8 Workspace 1 J McNames Portland State University ECE 223 Communications Ver 115 58 Example 8 Workspace 2 J McNames Portland State University ECE 223 Communications Ver 115 59 Sinusoidal Angle Modulation Ct Acos wet 90 005 Acos 605 So far we have discussed different types of amplitude modulation Angle Modulation alters the angle of the carrier signal rather than the amplitude Define the instantaneous angle of the carrier signal 005 as 6t There are two forms of angle modulation Phase modulation PM 6t wet 60 kipxh Frequency modulation FM tt we krfxt Note that for FM 6t 7E we l kifxtt J McNames Portland State University ECE 223 Communications Ver 115 60 Angle Modulation Versus Amplitude Modulation Angle Modulation say FM versus Amplitude Modulation AM l One advantage of FM is that the amplitude of the signal transmitted can always be at maximum power l FM is also less sensitive to many common types of noise than AM However FM generally requires greater bandwidth than AM J McNames Portland State University ECE 223 Communications Ver 115 61 Example 9 Angle Modulation Create a random signal bandlimited to 1 Hz and amplitude limited to one eg g 1 Use use PM and FM to modulate the signal with a carrier frequency of 3 Hz Use kip 3 and krf 47139 Plot the baseband signal the carrier signal and the modulated signals J McNames Portland State University ECE 223 Communications Ver 115 62 Example 9 Signal Plot WWWWWW J McNames Portland State University ECE 223 Communications Ver 115 63 Example 9 MATLAB Code qunction AngleModulation close all N 500 Z No samples fc 5 Z Carrier frequency Hz fs 50 Z Sample rate Hz fx 1 Z Bandlimit of baseband signal kp 5 Z PM scaling coefficient kf 2pi2 Z FM scaling coefficient wc 2pifc k 1N t k1fs Xh randn1N Z Random highfrequency signal nwn ellipordO95fxfs2fXfs2O560 ba ellipnO560wn X filterbaxh Z Lowpass filter to create baseband signal X XmaxabsX Z Scale so maximum amplitude is 1 c coswct Z Carrier signal yp coswct kpx theta cumsumwc kfxfs Z Approximate integral of angle yf costheta figure J McNames Portland State University ECE 223 Communications Ver 115 64 FigureSet1 LTX subplot411 h plottx b seth LineWidth O2 xlimO maxt ylabel xt box off AxisLines subplot412 h plottc b seth LineWidth O2 xlimO maxt ylabel ct box off AxisLines subplot418 h plottyp b seth LineWidth O2 xlimO maxt x1abel Time s ylabel PM box off AxisLines subplot414 h plottyf b seth LineWidth O2 xlimO maxt x1abel Time s ylabel FM box off AxisLines J McNames Portland State University ECE 223 Communications Ver 115 65 AXisSet6 print depsc AngleModulation J McNames Portland State University ECE 223 Communications Ver 115 66 Relationship of Angle and Frequency Modulation d9 t ea wet 60 kpxt we krfxt 0 These two forms are easily related dac t PM With Et IS equivalent to FM With 0 FM with ZEt is equivalent to PM with rth d7 0 For Ct instantaneous frequency is defined as d6t 239 75 w dt 0 Frequency modulation w t we l krfxt 0 Phase modulation w t we l kip digit J McNames Portland State University ECE 223 Communications Ver 115 67 Frequency Modulation 0 Consider a sinusoidal baseband signal ZEt Acosth o This models a bandlimited signal limited to iwx 0 Then w t we l kacoswmt o The instantaneous frequency varies between we l MA and we ka o The modulated signal is then of the form t A yt cos wet l krf ZE739 d7 cos wet w sinth l 60 0 Define the following variables Aw kifA m m is called the modulation index J McNames Portland State University ECE 223 Communications Ver 115 68 Narrowband Frequency Modulation yt coswctmsinwmt coswct cosm sinwmt sinwct sinm sinwmt When m is small m lt this is called narrowband FM modulation and we may use the following approximations cosm sinwmt 1 sinm sinwmt msinwmt Thus yt coswct msinwct sinwmt coswct COSth wmt l COSth l wmt J McNames Portland State University ECE 223 Communications Ver 115 69 Narrowband Frequency Modulation Continued A Yew T T 2 T I i I I I i I gt w wcw wc wcw 0 wcwx We wc l wx YUw m7r 2 w w T T wclwx cl x T T gt w 39wc39wx 39wc l l w c wc l wx Like AM spectrum contains sidebands Unlike AM sidebands are out of phase by 1800 Bandwidth is twice that of ZEt like doublesideband AM Carrier frequency is present and strong J McNames Portland State University ECE 223 Communications Ver 115 70 Example 10 Angle Modulation Create a sinusoidal baseband signal with a fundamental frequency of 1 Hz and a carrier sinusoidal signal at 12 Hz Plot these signals and the modulated signals after applying amplitude modulation and frequency modulation Use a scaling factor krf 1 and a modulation index of m 05 Solve for the baseband signal amplitude A J McNames Portland State University ECE 223 Communications Ver 115 71 Example 10 Signal Plot WWWWWW mfgWWWij J McNames Portland State University ECE 223 Communicati o n s V e r 1 1 5 72 Example 10 Relevant MATLAB Code qunction AMFM close all fx 1 Z Signal frequency Hz fc 15 Z Carrier frequency Hz fs 100 Z Sampling frequency kf 1 Z FM scaling coefficient m 05 Z Modulation index wx 2pifx Z Signal frequency rads wc 2pifc Z Carrier frequency rads A mwxkf Z Modulating amplitude t O750001O75 X Acoswxt Z Baseband signal c coswct Z Carrier signal ya XC Z Amplitude modulated signal yf coswct msinWXt figure FigureSet1 LTX subplot411 h plottx b seth LineWidth O2 Xlimmint maxt ylabel xt J McNames Portland State University ECE 223 Communications Ver 115 73 title Example of Sinusoidal AM and FM Modulation box off AxisLines subplot412 h plottc b seth LineWidth O2 x1immint maxt ylabel ct box off AxisLines subplot418 h plottya b tx r tx g seth LineWidth O2 x1immint maxt x1abel Time s ylabel AM box off AxisLines subplot414 h plottyf b seth LineWidth O2 x1immint maxt x1abel Time s ylabel FM box off AxisLines AxisSet6 print depsc AMFM J McNames Portland State University ECE 223 Communications Ver 115 74 Summary 0 Modulation is the process of embedding one signal in another with desirable properties for communication 0 Most forms of modulation are nonlinear o Sinusoidal AM is relatively simple and inexpensive o Synchronous AM is more efficient than asynchronous AM but is also more expensive 0 FM is more tolerant of noise than AM but requires more bandwidth and cost 0 Filters and frequency analysis using the Fourier transform have a crucial role in communication systems 0 Frequency FDM and timedivision TDM multiplexing can be used to merge multiple bandlimited signals into a single composite signal with a larger bandwidth J McNames Portland State University ECE 223 Communications Ver 115 75 Overview of Sampling Topics Shannon sampling theorem 0 Impulsetrain sampling 0 Interpolation continuoustime signal reconstruction o Aliasing 0 Relationship of CTFT to DTFT o DT processing of CT signals 0 DT sampling 0 Decimation 84 interpolation J McNames Portland State University ECE 223 Sampling Ver 120 1 Amplitude versus Time o In this class we are working only with continuousvalued signals 0 Signals with discrete values that have been quantized are called digital signals 0 AnalogtoDigital converters ADC convert continuousvalued signals to discretevalued signals These often also convert from CT to DT o Digitaltoanalog converters DAC convert discretevalued signals to continuousvalued signals These often also convert from DT to CT 0 Signals that are both continuousvalued and continuoustime are usually called analog signals 0 Signals that are both discretevalued and discretetime are usually called digital J McNames Portland State University ECE 223 Sampling Ver 120 2 Overview of DT Processing of CT Signals Ts Ts 0 Many systems 1 Sample a signal 2 Process it in discretetime 3 Convert it back to a continuoustime signal 0 Called discretetime processing of continuoustime signals Most modern digital signal processing DSP uses this architecture The first step is called sampling 0 The last step is called interpolation is the ztransform of the system39s impulse response Mn J McNames Portland State University ECE 223 Sampling Ver 120 3 Signals xt gt gt x t X M a Xe a me For now just consider the two conversions Sampling CT gt DT Interpolation DT gt CT 0 Suppose we want to be as close to ZEt as possible Need to consider relationships in both time and frequencydomains 0 Three signals three transforms J McNames Portland State University ECE 223 Sampling Ver 120 4 Guiding Questions and Objectives xlnl 05 ltnTs o The operation of sampling CT gt DT conversion is trivial o The challenge is to understand the limits and tradeoffs c We will focus on three questions 1 How is Xjw the CTFT of related to Xej9 the DTFT of xnTS 2 Under what conditions can we fully recover ZEt from gt 9615 3 How do we perform this DT gt CT conversion J McNames Portland State University ECE 223 Sampling Ver 120 5 Consider three signals Need for a Bridge Signal Type Time Domain Frequency Domain Original Signal CT 3615 Xjw Sampled Signal DT Xej9 Bridge Signal CT 36505 X5jw 0 Goals to determine the relationship of 9615 to 96nTs in the 1 time and 2 frequency domains 0 Goal 1 In the time domain the relationship 96nTS is clear 0 Goal 2 But what is the relationship of Xjw to Xej9 o The transforms d Xej9 is peri ifFer in character odic Q has units of radians per sample J McNames Portland State University ECE 223 Sampling Ver 120 6 Use of the Bridge Signal 9615 gt 96in XUW gt XWQ o A bridge signal 96515 is the only way that I know of to determine how Xjw is related to Xej9 1 Define the bridge signal by relating X5jw to Xej9 2 Determine how ZEt is related to 1E5t in the time domain 3 Determine how Xjw is related to X5jw 4 Use the relationships of Xej9 and Xjw to X5jw to determine their relationship to one another we 1E5t mm Xjw X5010 Xej9 J McNames Portland State University ECE 223 Sampling Ver 120 7 The Bridge Signal o The bridge signal 96515 is a CT representation of a DT signal o It is defined as having the same transform within a scale factor as the DT signal 0 Suppose Aejmn s o What is the CT equivalent 0 Let us pick ZEt Ae wt s o How do we relate the DT frequency 2 to the CT frequency w ad has units of radianssecond Q has units of radianssample 0 Let us use the conversion factor of TS f8 1 secondssample 2 radianssample w radianssecond gtlt TS secondssample w radianssecond 2 radianssample gtlt fS samplessecond J McNames Portland State University ECE 223 Sampling Ver 120 8 Bridge Signal Definition 2cm Xequot 9wTs mews o The bridge signal is a CT representation of a DT signal 0 We define it by equating the CTFT and DTFT with an appropriate scaling factor for frequency 0 Note that this is a highly unusual CT signal The CTFT is periodic What does this tell us about 96505 c We know what the bridge signal39s transform is but what then is the bridge signal 965t J McNames Portland State University ECE 223 Sampling Ver 120 9 Solving for the Bridge Signal Xej9 Z e jf2n X5010 Xej 9wTs Z e ijsn 515605 g X5jw 6a to z 6t Tsn g e ijsn OO 00 Z 95in 505 Tm Lg EM e jWTsn x5 Z 96in 6t Tsn J McNames Portland State University ECE 223 Sampling Ver 120 10 What just happened 96nTS c We generated from 9615 96nTS W from m Xauw Her Mama o This seems reasonable 0 Why wasn39t 96515 more similar to 9615 Ver 120 ECE 223 Sampling J McNames Portland State University Why Isn t the Inverse Transform Similar to 9615 1 DTFT m Xe99 e99 do 27i 271 1 00 CTFT ZEt 2 Xjw 69 dw 7 OO 0 Consider the range of the CTFT and DTFT synthesis equations 0 The DTFT synthesizes out of a finite range of frequencies 0 The CTFT synthesizes ZEt out of all frequencies 0 This is why 605 7 1X6jw f1X j TS 7E 05 J McNames Portland State University ECE 223 Sampling Ver 120 12 What have we accomplished 96nTS X501 Xe l9wTs 96515 Z 6t Tsn 965t Z 96nTS 6t Tsn 0 We have established four relationships between the three signals and 96515 c We now know how X5jw is related to Xej9 0 We can also now relate ZEt to 1E5t J McNames Portland State University ECE 223 Sampling Ver 120 13 Impulse Sampling HTW MHt 4Ts 3Ts 2Ts Ts 0 Ts 2T5 3T5 4Ts 5T5 pa 2 6t nTs 35525 Z xnTs6t Tsn 9615 pt o The creation of 1E5t directly from ZEt is called impulse sampling 0 We can model sampling by multiplication with a periodic impulse train J McNames Portland State University ECE 223 Sampling Ver 120 14 Rectangular Window and Impulse Train Notation HHlpmmmt 4Ts 3Ts 2Ts Ts 0 Ts 2T5 3T5 4Ts 5T5 Note that a similar symbol was used for rectangular windows PTt 1 ltl lt T 0 Otherwise but pt is an impulse train pTt is a rectangular window They are not the same J McNames Portland State University ECE 223 Sampling Ver 120 15 Impulse Sampling Conceptual Example 31005 V 4Ts 3Ts 2Ts Ts 6 TS 2T5 3Ts 4Ts 5Ts 1905 TTTTITTTTT 4Ts 3Ts 2Ts Ts 5 Ts 2T5 3T5 4T5 5T5 00290 T TTT 4Ts 3Ts 2Ts Ts 0 TS 2T5 3Ts 4Ts 5Ts V V J McNames Portland State University ECE 223 Sampling Ver 120 Impulse Sampling Terminology T T S S OO x5t 9615 pt Z xnTs6t nTs TL OO o The impulse train pt is called the sampling function pt is periodic with fundamental period TS Ts the fundamental period of pt is called the sampling period fs and cos 27th are both called the sampling frequency J McNames Portland State University ECE 223 Sampling Ver 120 17 Fourier Transforms of Periodic Signals Overview 00 96515 Z 96nTS 6t Tsn xtpt me 52cm Pm 0 To determine how X5jw is related to Xjw we need to calculate Pjw pt is a periodic and a power signal 0 The CTFT clearly doesn39t converge ls easier to Calculate the Fourier series coefficients PM for pt Solve for Pjw from PR using the general relationship between the CTFS and the CTFT Recall that periodic signals have a CTFT that consists of impulses J McNames Portland State University ECE 223 Sampling Ver 120 18 Fourier Transforms of Periodic Signals Recall the Fourier series representations of periodic signals 9615 Z Xk ej m k oo 39 7 eJWOt ltgt 27T 6w Lao xt f Xkejkw0t g 27T f Xk 6w kwo k oo k oo J McNames Portland State University ECE 223 Sampling Ver 120 19 Example 1 CT Fourier Transform of an Impulse Train Solve for the Fourier transform of the impulse train Hint the impulse train is periodic 00 Xk Txtejkw0tdt Xjw xte jwtdt OO J McNames Portland State University ECE 223 Sampling Ver 120 20 Example 1 Workspace J McNames Portland State University ECE 223 Sampling Ver 120 21 The Relationship of Xjw to X5jw 37 1 xtpt ltgt gXUw PW 7 27f 00 27139 Pw ltgt EZ6w k k oo w 2 2 8 TS 7 1 27v 0 xtpt ltgt gXQw gtllt E Z 6w kws k oo Xjw6wwo Xjww0 7 1 00 xlttgtplttgt gt T Z X ma m S ki oo J McNames Portland State University ECE 223 Sampling Ver 120 22 Summary Sampling 96nTS Definition X5jw Xej9i9wT Inverse CTFT 35525 Z xin6t nTs impulse Sampling 35525 Z xnTs6t nTs xtpt 1 Multiplication Property X5jw XUw gtllt Pjw CTFT Pjw 31 f 6t kws S k oo X6001 Z Xw kws S k oo J McNames Portland State University ECE 223 Sampling Ver 120 23 Conceptual Diagram of the Relationship A 1 X000 i i M l i i i L i i i i i i i i V was 3900 001 wS 27r T s T i i i i i i i i i gt 39ws 39wx 0 Wm ws A 1 X600 i i i i i i 1 7 39ws39wx 39ws wswx 39wx Wm ws39wx ws ws l wx A L Xem Ts i i i i 7 7 l i 39Qs39Qx Qs 39Qs l Qx Qs39Qx 95 Qs 2x J McNames Portland State University ECE 223 Sampling Ver 120 24 Observations 7 1 00 xlttgtplttgt gt f 2 me kws S k oo 1 0 wTs Xe gt me kws o What is 28 2 as COSTS 2 1 27v 0 Thus ads in CT corresponds to 27139 radianssample in DT Recall that the fastest DT signal is ejm 1 and oscillates at 7T radianssample This is the fastest signal we can observe due to em ejmiw o What is this frequency in the CT domain J McNames Portland State University ECE 223 Sampling Ver 120 25 Expressing DT Transform in Terms of CT Transform OO 1 X TS Z XJ39w has S k oo 2 wTS w 2 TS 1 00 399 Q Xe gt i Z X xi 1w k oo 28 27T Us TS TS 1 00 2 k327r X 99 X 39 ltegt Ts Z 9 TS ki oo J McNames Portland State University ECE 223 Sampling Ver 120 26 What Does this Mean 00 xlttgtplttgt gtTis Z me kws k oo 0 Sampling with an impulse train results in a signal with a CTFT of X5jw that is a periodic function of w c There are replicas of Xjw at each multiple of ws amp 0 Note that the replicas will not overlap if cam lt 2 o In this case Xjw could be recovered from X5jw by applying a lowpass filter with gain TS and a cutoff frequency we such that W1 lt we lt ws Cum 0 This is a surprising result 0 This is the sampling theorem J McNames Portland State University ECE 223 Sampling Ver 120 27 The Sampling Theorem Let ZEt be a bandlimited signal with Xjw 0 for lwl gt cam Then ZEt is uniquely determined by its samples xnTS n 0 i1 i2 if ws gt 2wx where ws 0 Thus we can reconstruct any bandlimited signal ZEt exacty by creating a scaled impulse train and Iowpass filtering 0 This theorem is sometimes called the Shannon sampling theorem 0 min ws 2wx is called the Nyquist rate 0 max aux is called the Nyquist frequency 0 In other words we must obtain at least two samples per a cycle of the fastest sinusoidal component 0 This should sound familiar 0 Recall that the fastest perceivable frequency in discretetime signals is 7T radians per sample ie 05 cycles per sample J McNames Portland State University ECE 223 Sampling Ver 120 28 Signal Relationships Summary WM Xltej9gtTi i 44 S k oo x5t Z mes an Tsn X5jw Ti Z X jw ms n oo S k OO 965t Z 6t Tsn X5jw X eijs 0 We now know the relationships between all of the three signals 0 Note that 96515 was just a means to determining the relationship of Xe99 to Xjw J McNames Portland State University ECE 223 Sampling Ver 120 29 Guiding Questions and Objectives Revisited xln mes mews Z X jw ms k oo 0 Recall our guiding questions How is the CTFT of ZEt related to the DTFT of 96nTs Under what conditions can we synthesize 96t from How do we do this DT gt CT conversion 0 Only the last remains McNames Portland State University ECE 223 Sampling Ver 120 30 Example 2 Equivalent Sinusoids Suppose is a DT sinusoidal signal 308971 l 6 Solve for all of the CT sinusoids ZEt coswt l o that satisfy the relationship thnTs IBM Which of these CT sinusoids satisfy the sampling theorem criterion J McNames Portland State University ECE 223 Sampling Ver 120 31 Example 2 Workspace J McNames Portland State University ECE 223 Sampling Ver 120 32 Example 2 Equivalent Sinusoids L39 m A L39 A A 1 J V K A2 NA h A NA V 1 V V 1V V 3 cosco t27 t 6 o Time s J McNames Portland State University ECE 223 Sampling Ver 120 33 Example 2 MATLAB Code function J EquivalentSinusoids t S61000S n floormintceilmaxt ph 2pirand w O6pi figure FigureSet1 Slides for ci1z5 switch cl case 1 Xt coswt ph yAXisLabel cosomega ttheta case 2 Xt cos2piwt ph yAXisLabel cos2pi omegat thetaJ case S xt cosw2pit ph yAXisLabel cosomega t2pit thetaJ end subplot81cl h plottxt hold on h stemncoswn ph k hold off J McNames Portland State University ECE 223 Sampling Ver 120 34 seth MarkerSize 12 box off ylabelyAxisLabel ylim105 1051 AXisLines if cl X1abel Time s end end AXisSet6 print depsc EquivalentSinusoids J McNames Portland State University ECE 223 Sampling Ver 120 35 Interpolation Introduction 0 CT gt DT conversion is trivial xnTS c We now know that we can reconstruct ZEt from exactly 0 This should be surprising c There are many signals ZEt that satisfy the constraint 96nTS 0 Which one is the original 3605 J McNames Portland State University ECE 223 Sampling Ver 120 36 Interpolation Possibilities Signal scaled 02 i i i i i i i 0 Time s J McNames Portland State University ECE 223 Sampling Ver 120 37 Bandlimited Interpolation o Intuitiver there are 00 signals that are equal to ZEt at evenly spaced TS samples 0 However if The samples were taken from a bandlimited signal Xjw 0 for w gt cam The conditions of the sampling theorem are satisfied ws gt 2wx then there is only one bandlimited signal that passes exactly through the samples J McNames Portland State University ECE 223 Sampling Ver 120 38 DT Processing of CT Signals Ts Ts 0 Since completely represents we can process in discretetime 0 However once we process the discretetime signal and generate a discretetime output we need to create a continuoustime output yt c We assume that represent samples from a continuoustime signal yt that is bandlimited the same as ZEt J McNames Portland State University ECE 223 Sampling Ver 120 39 Interpolation 0 There are many forms of interpolation Piecewise constant Linear point to point Splines piecewise cubic o For bandlimited signals conceptually we can Construct the bridge signal 96515 Apply a Iowpass filter to eliminate duplicates of the spectrum at higher frequencies 0 Called bandlimited interpolation J McNames Portland State University ECE 223 Sampling Ver 120 40 Recall the Spectral Relationships A 1 X000 l l l I l l l l L l l l l l l l l r was 3900 001 wS 27r T s T l l l l l l l l l gt was wax 0 cum ws A 1 X600 1 l l l l l l 1 7 39ws39wx 39ws wswx 39wx Wm ws39wx ws ws l wx A Xem Ts l l l l l l 7 l l 39Qs39Qx Qs 39Qs l Qx Qs39Qx 95 Qs l Qx J McNames Portland State University ECE 223 Sampling Ver 120 41 Interpolation Simplified and Made More Practical x5t 2 mm 6t nTs x7415 1E5t gtllt ht x505 739 h7 d7 00 00 Z gamma FMS h7d7 Z xn 6t T nTs hm d7 Z xnht nTS J McNames Portland State University ECE 223 Sampling Ver 120 42 Interpolation with Ideal Filters OO xnxnTs X5ltjwgtTis Z me kws k oo If the terms of the sum do not overlap an ideal lowpass filter can extract Xjw from X5jw o The passband gain must be TS Recall that the ideal lowpass filter with cutoff frequency we and gain TS has an impulse response given by Tswc sinwct Tswc wet smc 7T wet 7T 7T W 0 Thus for ideal interpolation W Ewan J McNames Portland State University ECE 223 Sampling Ver 120 43 Notes on Bandlimited Interpolation xt i an sine 7T n oo o If the sampling theorem criterion is satisfied 9670t ZEt 0 However this assumes that the frequency components of are between 7r and 7r o If the sampling criterion is not satisfied highfrequency components will appear as lowfrequency components J McNames Portland State University ECE 223 Sampling Ver 120 44

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