Introduction to Analog And Digital Communications Second Edition

Simon Haykin, Michael Moher

Chapter 5 Pulse Modulation : Transition from Analog to Digital Communications

5.1 Sampling Process 5.2 Pulse-Amplitude Modulation 5.3 Pulse-Position Modulation 5.4 Completing the Transition from Analog and Digital 5.5 Quantization Process 5.6 Pulse-Code Modulation 5.7 Delta Modulation 5.8 Differential Pulse-Code Modulation 5.9 Line Codes 5.10 Theme Examples 5.11 Summary and Discussion

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Ø  Some parameter of a pulse train is varied in accordance with the message signal Ø  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 accord

ance with the corresponding sample value of the message signal Ø  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 pulse Ø  Lesson1 : Given a strictly band-limited message signal, the sampling theorem e

mbodies the conditions for a uniformly sampled version of the signal to preserve its information content

Ø  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.

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5.1 Sampling Process

v  Instantaneous Sampling and Frequency-Domain Consequences Ø  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 sequence of Dirac delta functions :

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 results

in a periodic spectrum with a repetition frequency equal to the sampling rate.

∑∞

−∞=

−=n

ss nTtnTgtg )1.5()()()( δδ

∑ ∑∑∞

−∞=

∞

−∞=

∞

−∞=

=−=−⇔−n n

ssn

ssss fGfnTjnTgmffGfnTtnTg )2.5()()2exp()()()()( δπδ

Table. 5.1

Fig. 5.1

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Fig.5.1 Back Next

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table.5.1 Back Next

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v  Sampling Theorem Ø  A discrete-time Fourier transform of the sequence

Ø  For a strictly band-limited signal, under the two conditions

∑∞

−∞=

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

n Wnfj

WngfG )3.5(exp2

)( πδ

∑∞

≠−∞=

−+=

0

)()()(mm

sss mffGffGffGδ

WffG ≥= for 0)(.1Wfs 2.2 =

)4.5(),(21)( WfWfGW

fG <<−= δ

)5.5(,exp22

1)( WfWWnfj

Wng

WfG

n

<<−⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛= ∑

∞

−∞=

π

Fig. 5.2

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Fig.5.2 Back Next

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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.

Ø  The interpolation formula for reconstructing the original signal g(t) from the sequence of sample values {g(n/2W)} .

∫ ∑

∫

−

∞

−∞=

∞

∞−

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

=

W

Wn

dfftjWnfj

Wng

W

dfftjfGtg

)2exp(exp22

1

)2exp()()(

ππ

π

)6.5(2

2exp21

2)( df

Wntfj

WWngtg

W

Wn

∫∑−

∞

−∞=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛= π

)7.5(),2(csin 2

)( ∞<<∞−−⎟⎠⎞

⎜⎝⎛=∑

∞

−∞=

tnWtWngtg

n

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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 comp

onents higher than W hertz is completely described by specifying the values of 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 its samples 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 h

ertz Ø  Nyquist interval

§  1/2W (measured in seconds)

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v  Aliasing Phenomenon Ø  The phenomenon of a high-frequency component in the spectrum of the

signal 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 hig

h-frequency components of a message signal that are not essential to the information being conveyed by the signal

§  The filtered signal is sampled at a rate slightly higher than the Nyquist rate.

Ø  Physically realizable reconstruction filter §  The reconstruction filter is of a low-pass kind with a passband extending fro

m –W to W §  The filter has a non-zero transition band extending form W to fs-W

Fig. 5.3

Fig. 5.4

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Fig.5.3 Back Next

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Fig.5.4 Back Next

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5.2 Pulse-Amplitude Modulation

v  Pulse-Amplitude Modulation (PAM) Ø  The amplitude of regularly spaced pulses are varied in proportion to the

corresponding 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 val

ue T.

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v  Sample-and-Hold Filter : Analysis Ø  The PAM signal is

Ø  The h(t) is a standard rectangular pulse of unit amplitude and duration

Ø  The instantaneously sampled version of m(t) is

∑∞

−∞=

−=n

ss nTthnTmts )8.5()()()(

)9.5(

otherwise ,0

,0,21

0 ,1

2)(

⎪⎪⎩

⎪⎪⎨

⎧

==

<<

=⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ −= Ttt

Tt

T

Ttrectth

)10.5()()()( ∑∞

−∞=

−=n

ss nTtnTmtm δδ

Fig. 5.5

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Fig.5.5 Back Next

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Ø  To modify mδ(t) so as to assume the same form as the PAM signal

Ø  The PAM signal s(t) is mathematically equivalent to the convolution of mδ(t) , the instantaneously sampled version of m(t), and the pulse h(t)

)11.5()()()(

)()()(

)()()()(

τττδ

τττδ

τττδδ

dthnTnTm

dthnTnTm

dthmthtm

nss

nss

−−=

−−=

−=∗

∑ ∫

∫ ∑

∫

∞

−∞=

∞

∞−

∞

∞−

∞

−∞=

∞

∞−

)()()( ss nTthdthnT −=−−∫∞

∞−τττδ

)12.5()()()()( ∑∞

−∞=

−=∗n

ss nTthnTmthtmδ

)13.5()()()( thtmts ∗= δ

)14.5()()()( fHfMfS δ=

)15.5()()( ∑∞

−∞=

−=k

ss kffMffM δ

)16.5()()()( ∑∞

−∞=

−=k

ss fHkffMffS

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v  Aperture Effect and its Equalization Ø  Aperture effect

§  The distortion caused by the use of pulse-amplitude modulation to transmit an analog information-bearing signal

Ø  Equalizer §  Decreasing the in-band loss of the reconstruction filter as the frequency increases §  The amplitude response of the equalizer is

Ø  The noise performance of a PAM system can never be better than direct transmission of the message signal

Ø  For transmission over long distances, PAM would be used only as a means of message processing for time-division multiplexing.

)sin()(csin1

)(1

fTf

fTTfH ππ

==

Fig. 5.6

Fig. 5.7

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Fig.5.6 Back Next

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Fig.5.7 Back Next

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5.3 Pulse-Position Modulation

v  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 i

ndividual pulses. Ø  PDM is wasteful of power

v  PPM (Pulse-position modulation) Ø  The position of a pulse relative to its unmodulated time of occurrence is

varied in accordance with the message signal.

)18.5())(()( ∑∞

−∞=

−−=n

sps nTmknTtgts

)19.5()()2/(,0)(max

tmkTttg ps −>=

)20.5()2/()(max sp Ttmk < Fig. 5.8

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Fig.5.8 Back Next

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5.4 Completing the Transition from Analog to Digital

v  The advantages offered by digital pulse modulation Ø  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 n

oise and signal distortion Ø  Reliability

§  Can be made highly reliable by exploiting powerful error-control coding techniques. Ø  Security

§  Can be made highly secure by exploiting powerful encryption algorithms Ø  Efficiency

§  Inherently more efficient than analog communication system in the tradeoff between transmission bandwidth and signal-to-noise ratio

Ø  System integration §  To integrate digitized analog signals with digital computer data

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5.5 Quantization Process

v  Amplitude quantization Ø  The process of transforming the sample amplitude m(nTs) of a baseband

signal m(t) at time t=nTs into a discrete amplitude v(nTs) taken from a finite set of possible levels.

Ø  Representation level (or Reconstruction level) §  The amplitudes vk , k=1,2,3,……,L

Ø  Quantum (or step-size) §  The spacing between two adjacent representation levels

)21.5(,...,2,1},{: 1 LkmmmI kkk =≤< +

)22.5()(mgv =

Fig. 5.9

Fig. 5.10

25

Fig.5.9 Back Next

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Fig.5.10 Back Next

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5.6 Pulse-Code Modulation

v  PCM (Pulse-Code Modulation) Ø  A message signal is represented by a sequence of coded pulses, which is accom

plished by representing the signal in discrete form in both time and amplitude Ø  The basic operation

§  Transmitter : sampling, quantization, encoding §  Receiver : regeneration, decoding, reconstruction

v  Operation in the Transmitter 1.  Sampling

1.  The incoming message signal is sampled with a train of rectangular pulses 2.  The reduction of the continuously varying message signal to a limited number of discr

ete values per second 2.  Nonuniform Quantization

1.  The step size increases as the separation from the origin of the input-output amplitude characteristic 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.

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Ø  Compressor §  A particular form of compression law : µ-law

§  µ-law is neither strictly linear nor strictly logarithmic

§  A-law :

)25.5(11,

log1)log(1

10,log1

⎪⎪⎩

⎪⎪⎨

⎧

≤≤+

+

≤≤+

=

mAA

mAA

mA

mA

v

)23.5()1log()1log(

µµ+

+=

mv

)24.5()1()1log( mvdmd

µµ

µ+

+=

)26.5(11,)log1(

10,log1

⎪⎪⎩

⎪⎪⎨

⎧

≤≤+

≤≤+

=m

AmA

Am

AA

vdmd

Fig. 5.11

29

Fig.5.11 Back Next

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3.  Encoding 1.  To translate the discrete set of sample vales to a more appropriate form of si

gnal 2.  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 regenerate Table. 5.2

Fig. 5.12

31

Fig.5.12 Back Next

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table.5.2 Back Next

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v  Regeneration Along the Transmission Path Ø  The ability to control the effects of distortion and noise produced by transmitti

ng a PCM signal over a channel Ø  Equalizer

§  Shapes the received pulses so as to compensate for the effects of amplitude and phase distortions produced by the transmission

Ø  Timing circuitry §  Provides a periodic pulse train, derived from the received pulses §  Renewed sampling of the equalized pulses

Ø  Decision-making device §  The sample so extracted is compared o a predetermined threshold

Ø  ideally, except for delay, the regenerated signal is exactly the same as the information-bearing signal

1.  The unavoidable presence of channel noise and interference causes the repeater to make wrong decisions occasionally, thereby introducing bit errors into the regenerated signal

2.  If the spacing between received pulses deviates from its assigned value, a jitter is introduced into the regenerated pulse position, thereby causing distortion.

Fig. 5.13

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Fig.5.13 Back Next

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v  Operations in the Receivers 1.  Decoding and expanding

1.  Decoding : regenerating a pulse whose amplitude is the linear sum of all the pulses in the code word

2.  Expander : a subsystem in the receiver with a characteristic complementary to the compressor 1.  The combination of a compressor and an expander is a compander

2.  Reconstruction 1.  Recover the message signal : passing the expander output through a low-pas

s reconstruction filter

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5.7 Delta Modulation

v  Basic Consideration Ø  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 differences

)27.5()()()( ssqss TnTmnTmnTe −−= )27.5()()()( ssqss TnTmnTmnTe −−=

)28.5()](sgn[)( ssq nTenTe Δ=

)29.5()()()( sqssqsq nTeTnTmnTm +−=

Fig. 5.14

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Fig.5.14 Back Next

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v  System Details Ø  Comparator

§  Computes the difference between its two inpus Ø  Quantizer

§  Consists of a hard limiter with an input-output characteristic that is a scaled version of the signum function

Ø  Accumulator §  Operates on the quantizer output so as to produce an approximation to the message

signal.

(5.30) )(

)()()2(

)()()(

1∑=

=

+−+−=

+−=

n

isq

sqssqssq

sqssqsq

iTe

nTeTnTeTnTmnTeTnTmnTm

Fig. 5.15

39

Fig.5.15 Back Next

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v  Quantization Errors Ø  Slope-overload distortion

§  The step size is too small for the staircase approximation to follow a steep segment of the original message signal

§  The result that the approximation signal falls behind the message signal

Ø  Granular noise §  When the step size is too large relative to the local slope characteristic of th

e original message signal §  The staircase approximation to hunt around a relatively flat segment of the

message signal.

Fig. 5.16

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Fig.5.16 Back Next

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v  Delta-Sigma Modulation (Sigma-delta modulation) Ø  A delta modulation system that incorporates integration at its input Ø  Benefit of the integration

§  The low-frequency content of the input signal is pre-emphasized §  Correlation between adjacent samples of the delta modulator input is incre

ased §  Design of the receiver is simplified

Fig. 5.17

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Fig.5.17 Back Next

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5.8 Differential Pulse-Code Modulation

v  Prediction Ø  If we know the past behavior of a signal up to a certain point in time, it

is possible to make some inference about its future values

Ø  Tapped-delay-line filter (discrete-time filter) §  A simple and yet effective approach to implement the prediction filter §  With the basic delay set equal to the sampling period

Ø  The quantizer output may be expressed as

)31.5()()()( sssq nTqnTmnTm +=

)32.5()()()()( ssssss TnTqTnTmnTmnTe −−−−=

)34.5()()()( sss nTmnTmnTe∧

−=

)35.5()()()( sssq nTqnTenTe +=

)36.5()()()( sqssq nTenTmnTm +=∧

)37.5()()()()( ssssq nTqnTenTmnTm ++=∧

)38.5()()()( sssq nTqnTmnTm +=

Fig. 5.18

Fig. 5.19

45

Fig.5.18 Back Next

46

Fig.5.19 Back Next

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Ø  Comparing the DPCM with DM system, §  The use of a one-bit (two-level) quantizer in the DM system §  Replacement of the prediction filter in the DPCM by a single delay element

Ø  Noise is concerned §  DPCM, like DM, is subject to slope-overload distortion whenever the input

signal changes too rapidly for the prediction filter to track it §  Like PCM, DPCM suffers from quantization noise

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5.9 Line Codes

v  Several line codes 1.  On-off signaling 2.  Nonreturn-to-zero (NRZ) 3.  Return-to-zero 4.  Bipolar return-to-zero (BRZ) 5.  Split-phase (Manchester code) 6.  Differential encoding

Fig. 5.20

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Fig.5.20 Back Next

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5.10 Theme Examples

v  Time-division Multiplexing Ø  Enables the joint utilization of a common communication channel by a

plurality of independent message sources without mutual interference among them

Ø  Highly sensitive to dispersion in the common channel – a non-constant magnitude response of the channel and a nonlinear phase response.

v  Synchronization Ø  Keep the same time as a distant standard clock at the transmitter Ø  One possible procedure to synchronize the transmitter and receiver cloc

ks is to set aside a code element or pulse at the end of a frame and to transmit this pulse every other frame only

Fig. 5.21

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Fig.5.21 Back Next

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v  Impulse Radio Ø  Information is sent by means of very narrow pulses that are widely separated i

n time Ø  A form of a ultra-wideband (UWB) radio transmission

Ø  Gaussian monocycle §  One type of pulse used for impulse radio

Ø  PPM is one method for digitally modulating such an impulse wave

)39.5(exp)(2

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

τπ

τttAtv

Fig. 5.22

Fig. 5.23

Fig. 5.24

54

Fig.5.22 Back Next

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Fig.5.23 Back Next

56

Fig.5.24 Back Next

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Ø  Good aspect §  The signal power is spread over a large bandwidth, the amount of power th

at falls in any particular narrowband channel is small Ø  Bad aspect

§  The power falls in all such narrowband channel

Ø  Due to the limitation on transmit power, §  Ultra-wideband radio is restricted to short-range applications ( less than a

few hundred meters )

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5.11 Summary and Discussion Ø  Sampling : which operates in the time domain ;

§  The sampling process is the link between an analog waveform and its discrete-time representation

Ø  quantization : which operates in the amplitude domain; §  The quantization process is the link between an analog waveform and its dis

crete-amplitude representation v  Sampling theorem

Ø  A strictly band-limited signal with no frequency components higher than W Hz is represented uniquely by a 2W samples per second.

Ø  The sampling process is basic to the operation of all pulse modulation systems

v  Analog pulse modulation results from varying some parameter of the transmitted pulses

v  Digital pulse modulation systems transmit analog message signals as a sequence of coded pulses

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v  The advantage of DM (delta modulation) is simplified circuitry v  Differential pulse-code modulation employs increased circuit compl

exity to improve system performance v  Adaptivity

Ø  Is used in delta modulation to improve noise performance Ø  Is used in differential pulse-code modulation to reduce bandwidth requi

rement v  Pulse modulation

Ø  lossy in the sense that some information is lost as a result of the signal representation that they perform

v  Source-encoding strategies (PCM, DM, and DPCM) Ø  Whose purpose is to convert analog signals into digital form

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Fig.5.25 Back Next

Fig. 5.25

61

Fig.5.26 Back Next

Fig. 5.26

62

Fig.5.27 Back Next

Fig. 5.27

63

Fig.5.28 Back Next

Fig. 5.28

64

Fig.5.29 Back Next

Fig. 5.29