Introduction To Analog And Digital C Ommunications

Introduction to Analog And Digital CommunicationsSecond EditionSimon Haykin, Michael Moher

Chapter 5 Pulse Modulation :Transition from Analog to Digital Communications5.1 Sampling Process5.2 Pulse-Amplitude Modulation5.3 Pulse-Position Modulation5.4 Completing the Transition from Analog and Digital5.5 Quantization Process5.6 Pulse-Code Modulation5.7 Delta Modulation5.8 Differential Pulse-Code Modulation5.9 Line Codes5.10 Theme Examples5.11 Summary and Discussion

Ø Some parameter of a pulse train is varied in accordance with the message signalØ Analog pulse modulation§ A periodic pulse train is used as the carrier wave§ Some characteristic feature of each pulse is varied in a continuous manner in accordance with the corresponding sample value of the message signalØ Digital pulse modulation§ The message signal is represented in a form that is discrete in both time and amplitude§ Its transmission in digital form as a sequence of coded pulseØ Lesson1 : Given a strictly band-limited message signal, the sampling theorem embodies the conditions for a uniformly sampled version of the signal to preserve its information contentØ Lesson2 : Analog pulse-modulation systems rely on the sampling process to maintain continuous amplitude representation of the message signal. In contrast, digital pulse-modulation system use not only the sampling process but also the quantization process. Digital modulation makes it possible to exploit the full power of digital signal-processing techniques.3

5.1 Sampling Processv Instantaneous Sampling and Frequency-Domain ConsequencesØ Sample the signal g(t) instantaneously and at a uniform rate,Ø Instantaneously (ideal) sampled signal§ The signal obtained by individually weighting the elements of a periodic sequenceof Dirac delta functions : gδ (t ) g (nTs )δ (t nTs ) (5.1)Fig. 5.1n Ø Reproduce the relationships listed at the bottom of the right-hand side of the table 5.1§ The process of uniformly sampling a continuous time signal of finite energy resultsin a periodic spectrum with a repetition frequency equal to the sampling rate. g (nT )δ (t nT ) f G( f mf ) g (nT ) exp( j 2πnT f ) G ( f )sn sssn ssδ(5.2)n Table. 5.14

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v Sampling TheoremØ A discrete-time Fourier transform of the sequence n jπnf Gδ ( f ) g exp (5.3)2WW n Gδ ( f ) f s G ( f ) f s G ( f mf s )m m 0Ø For a strictly band-limited signal, under the two conditions1.G( f ) 0 for f W2. f s 2W1Gδ ( f ), W f W (5.4)2W1 n jπnf G( f ) gexp , W f W 2W n 2W W G( f ) Fig. 5.2(5.5)7

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Ø The sequence {g(n/2W)} has all the information contained in g(t).Ø Reconstructing the signal g(t) from the sequence of sample values.g (t ) G ( f ) exp( j 2πft )df1 W 2WW n jπnf g exp exp( j 2πft )df W n 2W n 1g (t ) g 2W 2Wn n exp j 2πf t df W 2W W (5.6)Ø The interpolation formula for reconstructing the original signal g(t) from the sequence of sample values {g(n/2W)} . n g (t ) g sin c(2Wt n), t (5.7)n 2W 9

Ø The sampling theorem for strictly band-limited signals of finite energy in two equivalent parts§ § Analysis : A band-limited signal of finite energy that has no frequency components higher than W hertz is completely described by specifying the valuesof the signal at instants of time separated by 1/2W seconds.Synthesis : A band-limited signal of finite energy that has no frequency components higher than W hertz is completely recovered form knowledge of itssamples taken at the rate of 2W samples per second.Ø Nyquist rate§ The sampling rate of 2W samples per second for a signal bandwidth of W hertzØ Nyquist interval§ 1/2W (measured in seconds)10

v Aliasing PhenomenonØ The phenomenon of a high-frequency component in the spectrum of thesignal seemingly taking on the identify of a lower frequency in the spectrum of its sampled version.Ø To combat the effects of aliasing in practices§ § Prior to sampling : a low-pass anti-alias filter is used to attenuate those high-frequency components of a message signal that are not essential to the information being conveyed by the signalThe filtered signal is sampled at a rate slightly higher than the Nyquist rate.Fig. 5.3Ø Physically realizable reconstruction filter§ The reconstruction filter is of a low-pass kind with a passband extending from –W to W§ The filter has a non-zero transition band extending form W to fs-WFig. 5.411

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5.2 Pulse-Amplitude Modulationv Pulse-Amplitude Modulation (PAM)Ø The amplitude of regularly spaced pulses are varied in proportion to thecorresponding sample values of a continuous message signal.Ø Two operations involved in the generation of the PAM signal§ § Instantaneous sampling of the message signal m(t) every Ts seconds,Lengthening the duration of each sample, so that it occupies some finite value T.14

v Sample-and-Hold Filter : AnalysisØ The PAM signal is s (t ) m(nTs )h(t nTs ) (5.8)n Ø The h(t) is a standard rectangular pulse of unit amplitude and duration T 1, 0 t T t 1h(t ) rect 2 , t 0, t T T 2 0, otherwise(5.9)Ø The instantaneously sampled version of m(t) is mδ (t ) m(nTs )δ (t nTs ) (5.10)n Fig. 5.515

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Ø To modify mδ(t) so as to assume the same form as the PAM signal mδ (t ) h(t ) mδ (τ )h(t τ )dτ m(nT )δ (τ nT )h(t τ )dτssn n m(nTs ) δ (τ nTs )h(t τ )dτ (5.11)δ (τ nTs )h(t τ )dτ h(t nTs )Ø The PAM signal s(t) is mathematically equivalent to the convolution ofmδ(t) , the instantaneously sampled version of m(t), and the pulse h(t) mδ (t ) h(t ) m(nTs )h(t nTs ) (5.12)n s(t ) mδ (t ) h(t ) (5.13)S ( f ) M δ ( f ) H ( f ) (5.14) M δ ( f ) f s M ( f kf s ) (5.15) k S ( f ) f s M ( f kf s ) H ( f ) (5.16)k 17

v Aperture Effect and its EqualizationFig. 5.6Ø Aperture effect§ The distortion caused by the use of pulse-amplitude modulation to transmit an analog information-bearing signalØ Equalizer§ Decreasing the in-band loss of the reconstruction filter as the frequency increases§ The amplitude response of the equalizer is11πf H ( f ) T sin c( fT ) sin(πfT )Fig. 5.7Ø The noise performance of a PAM system can never be better than direct transmission of the message signalØ For transmission over long distances, PAM would be used only as a means ofmessage processing for time-division multiplexing.18

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5.3 Pulse-Position Modulationv PDM (Pulse-duration modulation)Ø Pulse-width or Pulse-length modulation.Ø The samples of the message signal are used to vary the duration of the individual pulses.Ø PDM is wasteful of powerv PPM (Pulse-position modulation)Ø The position of a pulse relative to its unmodulated time of occurrence isvaried in accordance with the message signal. s (t ) g (t nTs k p m(nTs )) (5.18)n g (t ) 0,t (Ts / 2) k p m(t ) maxk p m(t ) max (Ts / 2) (5.20)(5.19)Fig. 5.821

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5.4 Completing the Transition from Analog to Digitalv The advantages offered by digital pulse modulationØ Performance§ Digital pulse modulation permits the use of regenerative repeaters, when placed along the transmission path at short enough distances, can practically eliminate the degrading effects of channel noise and signal distortion.Ø Ruggedness§ A digital communication system can be designed to withstand the effects of channel noise and signal distortionØ Reliability§ Can be made highly reliable by exploiting powerful error-control coding techniques.Ø Security§ Can be made highly secure by exploiting powerful encryption algorithmsØ Efficiency§ Inherently more efficient than analog communication system in the tradeoff between transmission bandwidth and signal-to-noise ratioØ System integration§ To integrate digitized analog signals with digital computer data23

5.5 Quantization Processv Amplitude quantizationØ The process of transforming the sample amplitude m(nTs) of a basebandsignal m(t) at time t nTs into a discrete amplitude v(nTs) taken from a finite set of possible levels.I k : {mk m mk 1}, k 1,2,., L (5.21)Ø Representation level (or Reconstruction level)§ The amplitudes vk , k 1,2,3, ,LFig. 5.9Ø Quantum (or step-size)§ The spacing between two adjacent representation levelsv g (m) (5.22)Fig. 5.1024

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5.6 Pulse-Code Modulationv PCM (Pulse-Code Modulation)A message signal is represented by a sequence of coded pulses, which is accomplished by representing the signal in discrete form in both time and amplitudeØ The basic operationØ § § v Transmitter : sampling, quantization, encodingReceiver : regeneration, decoding, reconstructionOperation in the Transmitter1.Sampling1. The incoming message signal is sampled with a train of rectangular pulses2. The reduction of the continuously varying message signal to a limited number of discrete values per second2.Nonuniform Quantization1. The step size increases as the separation from the origin of the input-output amplitudecharacteristic is increased, the large end-step of the quantizer can take care of possible excursions of the voice signal into the large amplitude ranges that occur relatively infrequently.27

Ø Compressor§ A particular form of compression law : µ-lawlog(1 µ m )(5.23)log(1 µ )d m log(1 µ ) (1 µ m ) (5.24)dvµv § µ-law is neither strictly linear nor strictly logarithmic§ A-law : A m1 1 log A , 0 m A v (5.25) 1 log( A m ) , 1 m 1 1 log AA1 1 log A,0 m d m AA (5.26)1d v (1 log A) m , A m 1Fig. 5.1128

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Fig. 5.123. Encoding1. To translate the discrete set of sample vales to a more appropriate form of signal2. A binary code§ § The maximum advantage over the effects of noise in a transmission medium is obtained by using a binary code, because a binary symbol withstands a relatively high level of noise.The binary code is easy to generate and regenerateTable. 5.230

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v Regeneration Along the Transmission PathThe ability to control the effects of distortion and noise produced by transmitting a PCM signal over a channelØ EqualizerØ § Ø Timing circuitry§ § Ø Provides a periodic pulse train, derived from the received pulsesRenewed sampling of the equalized pulsesDecision-making device§ Ø Shapes the received pulses so as to compensate for the effects of amplitude and phasedistortions produced by the transmissionFig. 5.13The sample so extracted is compared o a predetermined thresholdideally, except for delay, the regenerated signal is exactly the same as the information-bearing signal1. The unavoidable presence of channel noise and interference causes the repeater to make wrong decisions occasionally, thereby introducing bit errors into the regeneratedsignal2. If the spacing between received pulses deviates from its assigned value, a jitter is introduced into the regenerated pulse position, thereby causing distortion.33

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v Operations in the Receivers1. Decoding and expanding1. Decoding : regenerating a pulse whose amplitude is the linear sum of all thepulses in the code word2. Expander : a subsystem in the receiver with a characteristic complementaryto the compressor1. The combination of a compressor and an expander is a compander2. Reconstruction1. Recover the message signal : passing the expander output through a low-pass reconstruction filter35

5.7 Delta Modulationv Basic ConsiderationØ DM (Delta Modulation)§ An incoming message signal is oversampled to purposely increase the correlation between adjacent samples of the signal§ The difference between the input signal and its approximation is quantized into only two levels - corresponding to positive and negative differencese(nTs ) m(nTs ) mq (nTs Ts ) (5.27)eq (nTs ) Δ sgn[e(nTs )] (5.28)mq (nTs ) mq (nTs Ts ) eq (nTs ) (5.29)Fig. 5.1436

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