Chapter 3 Pulse Modulation 3.1 Introduction

rate sampling:1

period sampling : where

(3.1) )( )()(

signal sampled ideal thedenote )(Let

ss

s

sn

s

Tf

T

nTtnTgtg

tg

Chapter 3 Pulse Modulation3.1 Introduction

n

s

mm

sss

sn

s

mss

mss

m ss

ns

W

n fj

W

ngfG

WTWffG

mffGffGffG

nf TjnTgfG

mffGftg

mffGf

T

mf

TfG

nTtt

(3.4) )exp()2

()(

21 and for 0)( If

(3.5) )()()(or

(3.3) )2exp()()(

obtain to(3.1) on Transformier apply Fourmay or we

(3.2) )()(

)(

)(1

)(

)()g(

have weA6.3 Table From

0

2

)( of ninformatio all contains )2

(or

for )2

(by determineduniquely is )(

(3.7) , )exp()2

(2

1)(

as )( rewritemay we(3.6) into (3.4) ngSubstituti

(3.6) , )(2

1)(

that (3.5) Equationfrom find we

2.2

for 0)(.1

With

tgW

ng

nW

ngtg

WfWW

nfj

W

ng

WfG

fG

WfWfGW

fG

Wf

WffG

n

s

3

)( offormula ioninterpolat an is (3.9)

(3.9) - , )2(sin)2

(

2

)2sin()

2(

(3.8) )2

(2exp 2

1)

2(

)2exp()exp()2

(2

1

)2exp()()(

havemay we, )2

( from )(t reconstruc To

tg

tnWtcW

ng

n Wt

n Wt

W

ng

dfW

nt fj

WW

ng

df f tjW

n fj

W

ng

W

dfftjfGtg

W

ngtg

n

n

n

W

W

W

Wn

4

rate. samplinghigher haveor bandwidth signal

limit themay wealiasing, avoid .To occurs aliasing

sampling)(under limited-bandnot is signal theWhen2

1 intervalNyquist

2 rateNyquist

)2

( from recovered completely be can signal The.2

.)2

(by described

completely be can , tolimited is whichsignal1.a

signals limited-bandstrictly for Theorem Sampling

W

W

W

ng

W

ng

WfW

5

Figure 3.3 (a) Spectrum of a signal. (b) Spectrum of an undersampled version of the signal exhibiting the aliasing phenomenon.

6

Figure 3.4 (a) Anti-alias filtered spectrum of an information-bearing signal. (b) Spectrum of instantaneously sampled version of the signal, assuming the use of a sampling rate greater than the Nyquist rate. (c) Magnitude response of reconstruction filter. 7

(3.14) )()()()(

have we,property sifting theUsing

(3.13) )()()(

)()()(

)()()()(

(3.12) )()()(

is )( of versionsampledously instantane The

(3.11)

otherwise

Tt0,t

Tt 0

,

,02

1,1

)(

(3.10) )( )()(

as pulses top-flat of sequence thedenote )(Let

s

n

s

s

n

s

s

n

s

nss

sn

s

nTthnTmthtm

dthnTnTm

dthnTnTm

dthmthtm

nTtnTmtm

tm

th

nTthnTmts

ts

3.3 Pulse-Amplitude Modulation

8

(3.18) )()()(

(3.17) )()(M

(3.2) )()( (3.2) Recall

(3.16) )()()(

(3.15) )()()(

is )( signal PAM The

kss

kss

mss

δ

fHk ffMffS

k ffMff

mffGftg

fHfMfS

thtmts

ts

9

與 idea sampling 比較多 H(f)

Pulse Amplitude Modulation – Natural and Flat-Top Sampling

The circuit of Figure 11-3 is used to illustrate pulse amplitude modulation (PAM). The FET is the switch used as a sampling gate.

When the FET is on, the analog voltage is shorted to ground; when off, the FET is essentially open, so that the analog signal sample appears at the output.

Op-amp 1 is a noninverting amplifier that isolates the analog input channel from the switching function.

Figure 11-3. Pulse amplitude modulator, natural sampling.


Op-amp 2 is a high input-impedance voltage follower capable of driving low-impedance loads (high “fanout”).

The resistor R is used to limit the output current of op-amp 1 when the FET is “on” and provides a voltage division with rd of the FET. (rd, the drain-to-source resistance, is lo

w but not zero)


The most common technique for sampling voice in PCM systems is to a sample-and-hold circuit.

As seen in Figure 11-4, the instantaneous amplitude of the analog (voice) signal is held as a constant charge on a capacitor for the duration of the sampling period Ts.

This technique is useful for holding the sample constant while other processing is taking place, but it alters the frequency spectrum and introduces an error, called aperture error, resulting in an inability to recover exactly the original analog signal.


The amount of error depends on how mach the analog changes during the holding time, called aperture time.

To estimate the maximum voltage error possible, determine the maximum slope of the analog signal and multiply it by the aperture time T in Figure 11-4.


Figure 11-4. Sample-and-hold circuit and flat-top sampling.



Figure 11-5. Flat-top PAM signals.

Recovering the original message signal m(t) from PAM signal

s

amplitude distort

Where the filter bandwidth is

The filter output is .

Note that the Fourier transform of ( ) is given by

(3.19)

( ) ( )( (3.18))

( ) sinc( )exp( )

W

f M f H f k o in

h t

H f T f T j f T

ion delay

aparture effect

Let the equalizer response is

(3.20)

Ideally the original signal ( ) can be recovered completely.

2

1 1

( ) sinc( ) sin( )

T

f

H f T f T f T

m t

10

3.4 Other Forms of Pulse Modulation

a. Pulse-duration modulation (PDM) (PWM)

b. Pulse-position modulation (PPM)

PDM and PPM have a similar noise performance as FM.

11

In pulse width modulation (PWM), the width of each pulse is made directly proportional to the amplitude of the information signal.

In pulse position modulation, constant-width pulses are used, and the position or time of occurrence of each pulse from some reference time is made directly proportional to the amplitude of the information signal.

PWM and PPM are compared and contrasted to PAM in Figure 11-11.

Pulse Width and Pulse Position Modulation

Figure 11-11. Analog/pulse modulation signals.


Figure 11-12 shows a PWM modulator. This circuit is simply a high-gain comparator that is switched on and off by the sawtooth waveform derived from a very stable-frequency oscillator.

Notice that the output will go to +Vcc the instant the analog signal exceeds the sawtooth voltage.

The output will go to -Vcc the instant the analog signal is less than the sawtooth voltage. With this circuit the average value of both inputs should be nearly the same.

This is easily achieved with equal value resistors to ground. Also the +V and –V values should not exceed Vcc.


Figure 11-12. Pulse width modulator.


analog 比較大時

A 710-type IC comparator can be used for positive-only output pulses that are also TTL compatible. PWM can also be produced by modulation of various voltage-controllable multivibrators.

One example is the popular 555 timer IC. Other (pulse output) VCOs, like the 566 and that of the 565 phase-locked loop IC, will produce PWM.

This points out the similarity of PWM to continuous analog FM. Indeed, PWM has the advantages of FM---constant amplitude and good noise immunity---and also its disadvantage---large bandwidth.


Since the width of each pulse in the PWM signal shown in Figure 11-13 is directly proportional to the amplitude of the modulating voltage.

The signal can be differentiated as shown in Figure 11-13 (to PPM in part a), then the positive pulses are used to start a ramp, and the negative clock pulses stop and reset the ramp.

This produces frequency-to-amplitude conversion (or equivalently, pulse width-to-amplitude conversion).

The variable-amplitude ramp pulses are then time-averaged (integrated) to recover the analog signal.

Demodulation

Figure 11-13. Pulse position modulator.


Demodulation

As illustrated in Figure 11-14, a narrow clock pulse sets an RS flip-flop output high, and the next PPM pulses resets the output to zero.

The resulting signal, PWM, has an average voltage proportional to the time difference between the PPM pulses and the reference clock pulses.

Time-averaging (integration) of the output produces the analog variations.

PPM has the same disadvantage as continuous analog phase modulation: a coherent clock reference signal is necessary for demodulation.

The reference pulses can be transmitted along with the PPM signal.

This is achieved by full-wave rectifying the PPM pulses of Figure 11-13a, which has the effect of reversing the polarity of the negative (clock-rate) pulses.

Then an edge-triggered flipflop (J-K or D-type) can be used to accomplish the same function as the RS flip-flop of Figure 11-14, using the clock input.

The penalty is: more pulses/second will require greater bandwidth, and the pulse width limit the pulse deviations for a given pulse period.

Demodulation

Figure 11-14. PPM demodulator.

Demodulation

Pulse Code Modulation (PCM)

Pulse code modulation (PCM) is produced by analog-to-digital conversion process.

As in the case of other pulse modulation techniques, the rate at which samples are taken and encoded must conform to the Nyquist sampling rate.

The sampling rate must be greater than, twice the highest frequency in the analog signal,

fs > 2fA(max)

3.6 Quantization Process

function. staircasea is whichstic,characteriquantizer thecalled is

(3.22) )g( mapping The

size. step theis , levels tionreconstrucor tionrepresenta theare

L,1,2, , where isoutput quantizer the then )( If

3.9 Figin shown as )(

amplitude discretea into )( amplitude sample

theing transformof process The:onquantizati Amplitude

. thresholddecision or the level decision theis Where

(3.21) ,,2,1 , :

cell partition Define

1

1

m

mm

tm

nT

nTm

m

Lkmmm

kk

s

s

k

kk

kννJ

J

kkk

k

12

Figure 3.10 Two types of quantization: (a) midtread and (b) midrise.

13

Quantization Noise

Figure 3.11 Illustration of the quantization process. (Adapted from Bennett, 1948, with permission of AT&T.)

14

(3.28) 12

1

)(][

(3.26) otherwise

2

2 ,0

,1

)(

levels ofnumber total: ,

(3.25) 2

is size-step the

typemidrise theofquantizer uniforma Assuming

(3.24) )0][( ,

(3.23)

valuesample of variable

random by the denoted beerror onquantizati Let the

2

2

2

22

2

222

max max

max

dqqdqqfqQE

qqf

LmmmL

m

MEVMQ

mq

qQ

QQ

Q

When the quatized sample is expressed in binary form,

(3.29)

where is the number of bits per sample

(3.302

2

log

RL

R

R L

o

)

(3.31)

(3.32)

Let denote the average power of ( )

( )

max

2 2 2max

2

2

21

23

R

RQ

Q

m

m

P m t

PSNR

o

2

o

FM

3 ( )2 (3.33)

(SNR) increases exponentially with increasing (bandwidth).

Page 147 FM (SNR)

2max

22 2

30

3( ) (2.149)

2

R

cf

P

m

R

A Pk f

N W

Conditions for Optimality of scalar Quantizers

Let m(t) be a message signal drawn from a stationary process M(t)

-A m A

m1= -A mL+1=A

mk mk+1 for k=1,2,…., L

The kth partition cell is defined as

Jk: mk< m mk+1 for k=1,2,…., L

d(m,vk): distortion measure for using vk to represent values inside Jk .

kk

kk

M

M

L

km k

L

kkL

kk

mmd

mf

dmmfmdDk

by zedcharacteridecoder a andby zedcharacteri

encoder an: components twoof consistsquantizer

owever thesolution.H form closed havenot may

whichproblemnonlinear a is onoptimizati The

(3.38) )( ),(

commonly used is distortion square-mean The

pdf theis )( where

(3.37) )(),(

distortion average the

minimize that , and sets two theFind

,

2

1

11

J

J

J

condition.neighbor nearest called is This

allfor holds ),(),(

if , boundlower theattain To

)(),(min)())(,(

have that wesuch

,,1,2 ,)g(

mappingnonlinear by the definedencoder thefind toisThat

. Dminimizes that set thefind , set theGiven

decoder givena for encoder theof Optimality . 1 Condition

(3.42)

)41.3(

(3.40)

1m

11

kjmdmd

dmmfmddmmfmgmdD

Lkm

jk

M

L

k

kA

A M

k

L

kkL

kk

k

J

J

k 1

(3.43)

Condition 2 .Optimality of the decoder for a given encoder

Given the set , find the set that minimized .

For mean-square distortion

1 1

2( ) ( ) ,

k

L L

k kk k

L

k Mm

D

D m f m dm

J

J

k 1

opt

(3.44)

(3.45)

(3.47)

Us

2

,

1

2 ( ) ( ) 0

( )

( )

k

k

k

L

k Mmk

Mmk

Mm

k k

Dm f m dm

m f m dm

f m dm

E M m m m

J

J

J

ing iteration condition I, condition II ,

until D reaches a minimum

先用再用重複

Probability Pk (given)

Figure 3.13 The basic elements of a PCM system.

Pulse Code Modulation

(3.51) 1

1

10

)1(

log1

(3.50) 1

1

10

log1

)log(1log1

)(

law-A

(3.49) )1()1log(

(3.48) )1log(

)1log(

law-

mA

Am

mAA

A

d

md

mA

Am

A

mAA

mA

md

md

m

Quantization (nonuniform quantizer)

Figure 3.14 Compression laws. (a) -law. (b) A-law.

Encoding

1. Unipolar nonreturn-to-zero (NRZ) Signaling

2. Polar nonreturn-to-zero(NRZ) Signaling

3. Unipor nonreturn-to-zero (RZ) Signaling

4. Bipolar nonreturn-to-zero (BRZ) Signaling

5. Split-phase (Manchester code)

Line codes:

Figure 3.15 Line codes for the electrical representations of binary data. (a) Unipolar NRZ signaling. (b) Polar NRZ signaling.(c) Unipolar RZ signaling. (d) Bipolar RZ signaling. (e) Split-phase or Manchester code.

Page 39Fig 1.6

Page 49Fig 1.11

Differential Encoding (encode information in terms of signal transition; a transition is used to designate Symbol 0)

Regeneration (reamplification, retiming, reshaping )

Two measure factors: bit error rate (BER) and jitter. Decoding and Filtering

3.8 Noise consideration in PCM systems

(Channel noise, quantization noise)

(will be discussed in Chapter 4)

Time-Division Multiplexing

Synchronization

Figure 3.19 Block diagram of TDM system.

Example 2.2 The T1 System

3.10 Digital Multiplexers

Page 216 Example 3.3 AT&T M12

3.11 Virtues, Limitations and Modifications of PCM

Advantages of PCM

1. Robustness to noise and interference

2. Efficient regeneration

3. Efficient SNR and bandwidth trade-off

4. Uniform format

5. Ease add and drop

6. Secure

3.12 Delta Modulation (DM) (Simplicity)

size step theis and , of version quantized the

is ,output quantizer theis where

(3.54) 1

(3.53) ) sgn(

(3.52) 1

is signalerror The

).( of sample a is )( and period sampling theis where

,2,1,0 , )(Let

ne

nenm

nenmnm

nene

nmnmne

tmnTmT

nnTmnm

qq

qqq

q

q

ss

s

Figure 3.23 DM system. (a) Transmitter. (b) Receiver.

The modulator consists of a comparator, a quantizer, and an accumulator

The output of the accumulator is

(3.55)

)sgn(

1

1

n

iq

n

iq

ie

ienm

Two types of quantization errors :Slope overload distortion and granular noise

.)( of slope local the torelative large toois

size step whenoccurs noisegranular hand,other theOn

(3.58) )(

max (slope)

require we, distortion overload-slope avoid To

signalinput theof difference backward

first a isinput quantizer the,1for Except

(3.57) 11

have we, (3.52) Recall

(3.56)

, by error onquantizati theDenote

tm

dt

tdm

T

nq

nqnmnmne

nqnmnm

nq

s

q

Slope Overload Distortion and Granular Noise

( differentiator )

Delta-Sigma modulation (sigma-delta modulation) The modulation which has an integrator can

relieve the draw back of delta modulation (differentiator)

Beneficial effects of using integrator:

1. Pre-emphasize the low-frequency content

2. Increase correlation between adjacent samples

(reduce the variance of the error signal at the quantizer input )

3. Simplify receiver design

Because the transmitter has an integrator , the receiver

consists simply of a low-pass filter.

(The accumulator in the conventional DM receiver is cancelled by the differentiator )

Figure 3.25 Two equivalent versions of delta-sigma modulation system.

3.13 Linear Prediction (to reduce the sampling rate)

Consider a finite-duration impulse response (FIR)

discrete-time filter which consists of three blocks :

1. Set of p ( p: prediction order) unit-delay elements (z-

1)

2. Set of multipliers with coefficients w1,w2,…wp

3. Set of adders ( )

(3.62)

2

have we(3.61) and (3.60) (3.59) From

minimize to,,, Find

(3.61) error) square (mean

be eperformanc ofindex Let the

(3.60) ˆ

iserror prediction The

(3.59) )(ˆ

is )input theof preditionlinear (Theoutput filter The

1 1

1

2

21

2

1

p

j

p

kkj

p

kk

p

p

kk

knxjnxEww

knxnxEwnx EJ

Jwww

neEJ

nxnxne

knxwnx

equations Hopf-Wiener called are (3.64)

(3.64) 21 ,

022

(3.63) 2

as simplify may We

)(

ationautocorrel The

)(

0)]][[( mean zero withprocess stationary is )( Assume

1

1

1 11

2

2

222

,p,,kkRkRjkRw

jkRwkRw

J

jkRwwkRwJ

J

knxnxEkRkTR

nxE

nxEnxE

nxEtX

p

jXXXj

p

jXjX

k

p

j

p

kXkj

p

kXkX

XsX

X

2min

1

12

02

1

2

11

2min

210

1

01

thanless always is 0,

(3.67)

2

yields(3.63) into (3.64) ngSubstituti

,, 1, 0

021

201

110

]][],...,2[ ],1[[

,,, where

(3.66) exists if , as

XXXTX

XXTXX

TXX

p

kXkX

p

kXk

p

kXkX

XXX

XXX

XXX

XXX

X

TXXXX

Tp

XXX

J

kRw

kRwkRwJ

pRRR

RpRpR

pRRR

pRRR

pRRR

www

rRr

rRrwr

R

r

w

rRwR

For convenience, we may rewrite the Wiener-Hopf equations

The filter coefficients are uniquely determined by

on.presentati of

econveniencfor is 2

1 andparameter size-stepa is where

(3.69) 21 ,2

11

1 update Then . n iterationat value thedenotes

(3.68) 21 ,

ectorgradient v theDefine

descentsteepest of method theusing iteration Do2.

valuesinitialany starting ,,,2,1 , Compute 1.

sensefollow thein adaptive ispredictor The

,p,,kgnwnw

nwnw

,p,,kw

Jg

pkw

kkk

kk

kk

k

kRXLinear adaptive prediction (If for varying k is not available)

algorithm square-mean-lease called are equations above The

(3.73) (3.60)(3.59)by ˆ where

)72.3( ,,2,1 , ˆ

ˆˆ1ˆ

)71.3( ,,2,1 , 22ˆ

n)expectatio the(ignore

k]]-x[nfor E[x[n] use wecomputing hesimplify t To

(3.70) ,,2,1 , 22

22

1

1

1

1

1

jnxnwnxne

pkneknxnw

jnxnwnxknxnwnw

pkknxjnxnwknxnxng

knxnx

pkknxjnxEwknxnxE

jkRwkRw

Jg

p

jj

k

p

jjkk

p

jj

p

jj

P

jXjX

k

k

k

Substituting (3.71) into (3.69)

Differentiating (3.63), we have

Figure 3.27Block diagram illustrating the linear adaptive prediction process.

3.14 Differential Pulse-Code Modulation (DPCM)

Usually PCM has the sampling rate higher than the Nyquist rate .The encode signal contains redundant information. DPCM can efficiently remove this redundancy.

Figure 3.28 DPCM system. (a) Transmitter. (b) Receiver.

Input signal to the quantizer is defined by:

(3.78)

(3.77) ˆ

isinput filter prediction The

error. onquantizati is where

(3.75)

isoutput quantizer The

value.predictiona is ˆ

(3.74) ˆ

nqnmnm

nm

nqnenmnm

nq

nqnene

nm

nmnmne

q

q

q

From (3.74)

) (minimize G maximize filter to predictiona Design

(3.82) G Gain, Processing

(3.81) )SNR(

is ratio noise onquantizati-to-signal theand

error sprediction theof variance theis where

(3.80))SNR(

))(((SNR)

and 0)]][[( of variancesare and where

(3.79) (SNR)

is system DPCM theof (SNR) The

2

2

2

2

2

2

2

2

2

2

o

22

2

2

o

o

Ep

E

Mp

Q

EQ

E

Qp

Q

E

E

M

QM

Q

M

G

nqnmEnm

Processing Gain

3.15 Adaptive Differential Pulse-Code Modulation (ADPCM)

Need for coding speech at low bit rates , we have two aims in mind:

1. Remove redundancies from the speech signal as far as possible.

2. Assign the available bits in a perceptually efficient manner.

Figure 3.29 Adaptive quantization with backward estimation (AQB).

Figure 3.30 Adaptive prediction with backward estimation (APB).

Date post:	06-Jan-2016
Category:	Documents
Upload:	nellis
View:	47 times
Download:	1 times

Chapter 3 Pulse Modulation 3.1 Introduction

Documents