Communication Systems, 5e

Transcription

1 Communication Sytem, 5e Chapter 6: Sampling and pule modulation A. Bruce Carlon Paul B. Crilly 00 The McGraw-Hill Companie

2 Chapter 6: Sampling and pule modulation Sampling theory and practice Pule-amplitude modulation Pule-time modulation 00 The McGraw-Hill Companie

3 Sampling Theory and Practice The pectrum of a ampled ignal The time domain and pectrum of a ampling waveform Minimum ampling frequency Baed on the maximum allowable aliaing error, meage BW, LPF characteritic, etc. The Nyquit Rate Practical ampling veru ideal ampling Signal recontruction PAM, PDM and PPM

4 Sampling Multiplicative, periodic ampling waveform x t xt pt Fx t Fxt Fp t t p pule T T T T p t t nt p t T n T pule p t f n p t n pule

5 Convolution with Sampling Spectrum x t Fxt Fp t F x t x t p t Spectral replication of F[x(t)] pule If x(t) not band limited, there will be pectral aliaing The hape of the ampling pule may change the magnitude and phae of the pectral replica! T n f n T

6 Spectral Replication Reduce the ampling rate to the minimum x t t xt t Sf Xt X X t T n f X f n T X t f f W W f f freq f X Replication Interval f f W W f f freq

7 Copyright The McGraw-Hill Companie, Inc. Permiion required for reproduction or diplay. Nyquit Rate f W f, min W W f W Figure 6.-3, Spectra for witching ampling (a) meage (b) ampled meage (c) ampled meage

8 Aliaing by an ADC The deired baeband ignal pectrum prior to ampling, typical definition Filter Deign for baeband ignal or complex ignal Center frequency i zero F F F F

9 Practical Filter Contruction f paband f paband W f tranition f topband f f paband ftranition f paband W f f f W paband topband For practical application, we uually ample at greater than the Nyquit rate. Thi allow for a guard band around the ignal of interet (SOI) If ADC ytem are involved, many time we ue 4 x W ampling, time the Nyquit rate or even higher.

10 Bandpa Aliaing by an ADC The deired baeband ignal pectrum prior to ampling. Filter deign for bandpa ignal ampling. Signal-of-interet center frequency i f /4

11 Sampling in Matlab Uing the interpolated meage from before A ampling rate of f/4 can be ued 0 Sequenctial FFT of the meage -0 Sequenctial FFT of the Sampled Waveform Power (db) Power (db) Frequency (Hz) x Frequency (Hz) x 0 4 Quetion, what if I wanted to hift a baeband ignal to a higher frequency ample(?) and BPF!

12 Aliaing () The frequency domain repone of the perfect ampling function i: P j exp j n T n T k k j j T 3 F F F 0 F F 3 F Convolve with the input ignal pectrum

13 Aliaing () The frequency band that could be aliaed when ampled are G p T j G j j Gj jk k k T T k T 3 F F F 0 F F 3 F

14 Aliaing Example () Predict the aliaing reult for F F F F 3 F F

15 Aliaing Example () F F F F 3 F Original pectrum F 3 F 5 F f F f 3 F F f F 3 F F f Aliaed Baeband Spectrum F F F 0 F

16 Interpolation-Filter Example -tone tet ignal (30&60 Hz, f = 000 Hz) fft caled to maintain power interpolation x4 (upample function) caled by 4 o interpolated ignal ha the ame power interp-filter (interp function) filter include interpolation gain of x4

17 Continuou Time Recontruction Recontruction of a dicrete time waveform into a continuou time waveform. If it digitized, it probably need to be retored to an analog waveform at ometime. The ampled repreentation Shown a a pectrum from f / to + f / or for DSP people from 0 to f (it hown aliaed!?) It theoretically contain all frequency replica! G p T j G j j Gj jkt k k T T k 3 F F F 0 F F 3 F

18 Continuou Time Recontruction Impule output at the ample time Perfect Recontruction Filter to eliminate replica Derived from rect in the frequency domain The convolution of the ample with a inc function H f f K rect exp j f h B t K B inc B t t delay t delay Let K B t x t ht xˆ xˆ t t delay t xnt t nt inc n T

19 Recontruction () Perfect Recontruction Filter Derived from rect in the frequency domain The convolution of the ample with a inc function xˆ t t delay t xntt nt inc n T xˆ delay t xntinc n T t nt t Let t delay 0 xˆ t T t xnt inc n n

20 Recontruction (3) Perfect Recontruction Filter Each ample caue a time offet inc All the inc are ummed to form the continuou ignal xˆ t T t xnt inc n n Individual inc function with T delay

21 Recontruction (4) Perfect Recontruction Filter xˆ t T t xnt inc n n Individual inc Reulting ummation

22 Alternate Function Ued for Recontruction Zero Order Hold (like a Digital to analog converter) Should be followed by LPF xˆ rect t xnt rect n Firt Order Hold (Triangle) xˆ rect t nt T t nt T t xnt tri n

23 Copyright The McGraw-Hill Companie, Inc. Permiion required for reproduction or diplay. ZOH Signal recontruction from ample (a) ZOH Figure

24 Copyright The McGraw-Hill Companie, Inc. Permiion required for reproduction or diplay. FOH Signal recontruction from ample (b) FOH Figure

25 Aliaing Error in Recontruction Etimated a the amount of filter power that come from the firt adjacent pectral replica. t order Butterworth LPF and pectral replica Integrate the aliaed portion of the filter in the paband t order etimate: the aliaing filter power at the paband bandedge Figure 6.-9 (a) output of RC filter, (b) after ampling Butterworth LPF jw n H w w 0

26 Aliaing Computation Figure 6.-9: (a) output of RC filter, (b) after ampling Dr. Bazuin aumption: Paband ha negligible attenuation (book ue W with B at the 3 db point) Aliaing at paband W not B (book ue B) A digital filter will be ued to clean-up the tranition band. Atten Voltage Copyright The McGraw-Hill Companie, Inc. Permiion required for reproduction or diplay. error voltage f W B errorpower n f W B n

27 Telephony Example Maximum voice W=3.4 khz, find the ample rate Paband 0.5 db ripple, N=4 th order Butterworth filter Find the filter cutoff (3 db) frequency Stopband Attenuation -40 db, find the topband frequency Compute the ample rate H B n n , Let khz 3.4 khz B 4. 43kHz H B n n 3.6, Let 3.6 f W f W 4.43kHz khz B 985 f W B f 3.985kHz 3.4kHz kHz

28 Telephony Example () Maximum voice 3.4 khz, find the ample rate Paband 0.5 db ripple, N= t order Butterworth filter Find the filter cutoff (3 db) frequency Stopband Attenuation -40 db, find the topband frequency Compute the ample rate H B n n , Let khz 3.4 khz B kHz H B n n , Let f W B 9.75kHz 9.733kHz kHz f W B kHz 3.4kHz kHz

29 Pule-Baed Modulation A long a ampling i performed at appropriate ample rate, any communication ignal that convey the ampled value during the ample time interval can communicate a continuou waveform. The carrier doen t have to be continuou, it can be different a long a the ample value can be recovered. Pule Communication Pule with amplitude (PAM) Pule with duration (PDM or PWM) Pule with a poition in the ample time frame (PPM)

30 Pule-Amplitude Modulation The pule output from an intantaneou ampler x PAM x p PAM t xnt pt nt n t n T ptt n T t xntp t t n T p t xntt n T n x PAM X PAM t pt x t f Pf X f n Aperture Effect reult from the time aperture p(t)

31 PAM Pule For unipolar ignal the pule may appear a an AM modulation amplitude AM with pule intead of a carrier x PAM x p t mt t mnt pt nt n

32 Analog ignal and correponding PAM ignal Figure 6.- Copyright The McGraw-Hill Companie, Inc. Permiion required for reproduction or diplay.

33 Copyright The McGraw-Hill Companie, Inc. Permiion required for reproduction or diplay. Flat-top ampling (a) ample & hold circuit (b) waveform Figure 6.- PAM time delayed by ½ the PAM width If tranmitted, doe the time delay matter? probably not

34 PAM Waveform The modulation i dependent upon the pule amplitude. Signal near zero may be hard to detect Can the receiver detect poitive and negative pule Therefore, ue an AM like offet for the amplitude Called uni-polar flat top in the text p t A xk T pt k T x 0 k where k T 0 x

35 PAM Spectral Content Convolution of the pule and the AM-like waveform Fx t Fxt Fp t x t A0 mt Fxt X f A f M f F 0 p t f n in c T n T T Spectrum baed on widet: frequency element, typically the ymbol period or ample pule. Ue inc null-to-null a Bandpa Bandwidth, B T T W B T W

36 PAM Application Rarely ued for ingle channel communication ytem, but ued in conjunction with intrumentation, data telemetry, and intrumentation ytem One element of a Time-diviion multiplexing (TDM) ytem A bai for other digital modulation ytem 00 The McGraw-Hill Companie

37 Pule-Time Modulation PAM receiver require amplitude to be determined for brief pule, can we tranlate the ampled ignal into a form that might be eaier to receive? One not dependent upon amplitude? Pule-Duration Modulation, PDM (alo called Pule-Width Modulation, PWM) The length of the pule width Nominally centered on the periodicity Pule-Poition Modulation, PPM The poition of the pule relative to the pule period Pule width are fixed

38 Copyright The McGraw-Hill Companie, Inc. Permiion required for reproduction or diplay. Type of pule-time modulation Figure 6.3-

39 Generation of PWM and PPM (a) Generation of PDM and PM ignal, (b) waveform 00 The McGrawHill Companie

40 PWM and PPM Bandwidth Baed on the minimum pule width, but The value i dependent upon accurately meauring time. Therefore, the fater the receiver rie-time in tracking the ignal, the more accurate the analog meaurement. t r T W B T t r W Note : For PPM, if the pule location are decribe uing probability, the power pectral denity can be computed a the product of the PSD and inc. Note : For PWM, the pule center and width hould both be decribed probabilitically.

41 Converion of PDM or PPM into PAM for Demodulation Figure Deired PAM Amplitude Copyright The McGraw-Hill Companie, Inc. Permiion required for reproduction or diplay.