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Chapter 3 Pulse Modulation
3.1 Introduction
Let g (t ) denotetheideal sampledsignal
g (t ) 

 g (nT )  (t  nT )
n  
s
s
where Ts : samplingperiod
f s  1 Ts : samplingrate
(3.1)
From T able A6.3 we have

g( t )   (t  nTs ) 
n  
1
G( f ) 
Ts


 ( f
m  

 f G( f
m  

s
m
)
Ts
 m fs )
g ( t )  f s

 G( f
m  
 m fs )
(3.2)
or we may apply Fourier T ransformon (3.1) t o obt ain
G ( f ) 

 g (nT ) exp( j 2 nf T )
n  
s
or G ( f )  f sG ( f )  f s
s

 G( f
m  
m 0
 m fs )
(3.3)
(3.5)
If G ( f )  0 for f  W and Ts  1
2W

n
j n f
G ( f )   g (
) exp(
)
2W
W
n  
(3.4)
2
With
1.G ( f )  0 for f  W
2. f s  2W
we find from Equation(3.5) that
1
G( f ) 
G ( f ) ,  W  f  W
(3.6)
2W
Substituting (3.4)into(3.6) we may rewriteG ( f ) as
n
jnf
g(
) exp(
) ,  W  f  W (3.7)

2W
W
n  
n
g (t ) is uniquely determinedby g (
) for    n  
2W
n 

or  g (
)  containsall information of g (t )
 2W 
1
G( f ) 
2W

3
n 

T o reconstruct g (t ) from  g (
)  , we may have
 2W 

g (t )   G ( f ) exp( j 2ft )df

W

W

1
2W
n
j n f
g(
) exp(
) exp( j 2 f t)df

2W
W
n  

n 

W exp j 2 f (t  2W )df (3.8)

n sin(2 Wt  n )
  g(
)
2W
2 Wt  n
n  
n
1
  g(
)
2W 2W
n  
W

n
  g(
) sin c( 2Wt  n ) , -   t  
2W
n  
(3.9)is an interpolation formulaof g (t )
(3.9)
4
SamplingT heoremfor strictlyband - limitedsignals
1.a signal which is limited to  W  f  W , can be completely
n 

described by  g (
) .
 2W 
n 

2.T hesignal can be completelyrecoveredfrom  g (
)
 2W 
Nyquist rate  2W
Nyquist interval 1
2W
When thesignal is not band - limited(under sampling)
aliasing occurs.T oavoidaliasing, we may limit the
signal bandwidth or have higher samplingrate.
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
7
filter.
3.3 Pulse-Amplitude Modulation
Let s(t ) denotethesequence of flat - toppulses as
s (t ) 

 m(nT ) h(t  nT )
s
n  
(3.10)
s
0 t T
 1,
1
h (t )   ,
t  0, t  T
(3.11)
2
 0,
otherwise

T heinstantaneously sampled versionof m(t ) is
m ( t ) 

 m(nT ) (t  nT )
s
n  
(3.12)
s

m (t )  h(t )   m ( )h(t   )d






 m(nT ) (  nT )h(t   )d
s
s
n  


 m(nTs)  (  nTs)h(t   )d (3.13)

n  
Using thesiftingproperty, we have
m ( t )  h ( t ) 

 m(nT )h(t  nT )
s
n  
s
(3.14)
8
T he P AM signal s (t ) is
s ( t )  m ( t )  h ( t )
 S ( f )  Mδ ( f ) H ( f )
Recall (3.2) g (t )  fs
M ( f )  f s
S( f )  fs
(3.15)
(3.16)

 G( f  m f )
m  
s
(3.2)

M( f k f )
s
k  
(3.17)

 M ( f  k f )H ( f )
k  
s
(3.18)
9
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.
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling
Figure 11-3. Pulse amplitude modulator, natural sampling.
Pulse Amplitude Modulation –
Natural and Flat-Top 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
low but not zero)
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling

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.
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling

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 DT in Figure 11-4.
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling
Figure 11-4. Sample-and-hold circuit and flat-top sampling.
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling
Figure 11-5. Flat-top PAM signals.
Recovering the original message signal m(t) from PAM signal
Wherethefilter bandwidth is W
T hefilter outputis f s M ( f ) H ( f ) . Note that the
Fourier transformof h (t ) is given by
H ( f )  T sinc( f T ) exp( j f T )
amplitudedistortion delay  T
(3.19)
2
 apartureeffect
Let theequalizer responseis
1
1
f


(3.20)
H ( f ) T sinc( f T ) sin( f T )
Ideally the originalsignal m(t ) can be recoveredcompletely.
10
3.4 Other Forms of Pulse Modulation
a. Pulse-duration modulation (PDM)
b. Pulse-position modulation (PPM)
PPM has a similar noise performance as FM.
11
Pulse Width and Pulse Position Modulation

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.
Pulse Width and Pulse Position Modulation




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.
Pulse Width and Pulse Position Modulation
Figure 11-12. Pulse width modulator.
Pulse Width and Pulse Position Modulation

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 voltagecontrollable multivibrators.

One example is the popular 555 timer IC. Other (pulse
output) VCOs, like the 566 and that of the 565 phaselocked 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.
Demodulation

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 timeaveraged (integrated) to recover the analog signal.
Pulse Width and Pulse Position Modulation
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.
Demodulation

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 flipflop 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.
Pulse Code Modulation (PCM)

Pulse code modulation (PCM) is produced by analogto-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
Define partitioncell
J k : mk  m  mk 1 , k  1,2,, L
(3.21)
Wheremk is thedecision level or the decision threshold.
Amplitudequantization : T heprocessof transforming the
sampleamplitudem(nTs ) intoa discreteamplitude
 (nTs ) as shown in Fig 3.9
If m(t )  J k then thequantizer outputis νk where νk , k  1,2,, L
are therepresentationor reconstructionlevels, mk 1  mk is thestep size.
T hemapping  g( m)
(3.22)
is called thequantizercharacteristic, which is a staircasefunction.
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
Let thequantization error be denot edby therandom
variableQ of sample value q
q  m 
(3.23)
Q  M  V , ( E [ M ]  0)
(3.24)
Assuming a uniformquantizer of themidrise type
2m max
thestep - size is D 
L
 m max  m  m max , L : totalnumber of levels
D
D
 1
,  q
fQ ( q)   D
2
2
 0, otherwise
D
2
D

2
 Q2  E[Q 2 ]  
D2

12
(3.25)
(3.26)
D
1
q 2 fQ ( q)dq   2D q 2dq
D 2
(3.28)
When thequat ized sampleis expressedin binary form,
L  2R
(3.29)
where R is thenumber of bit s per sample
R  log2 L
(3.30)
2m max
D R
(3.31)
2
1 2 2 R
2
 Q  mmax 2
(3.32)
3
Let P denotetheaverage power of m(t )
 ( SNR) o 
P
 Q2
3P 2 R
 ( 2 )2
mmax
(3.33)
(SNR) o increasesexponentially wit hincreasingR (bandwidth).
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.
Find the two sets

L
k k 1
and J

L
k k 1
, thatminimize
theaverage distortion
L
D  
k 1
mJ k
d ( m, k ) f M ( m )dm
(3.37)
where f M ( m ) is thepdf
T hemean - square distortionis used commonly
d ( m,k )  ( m  k ) 2
(3.38)
T heoptimization is a nonlinearproblem which
may not haveclosed formsolution.However the
quantizer consistsof two components: an encoder
characterized by J k , anda decoder characterized by  k
Condition1 . Optimalityof theencoder for a given decoder
Given theset  k k 1 , find theset J k k 1 thatminimizesD .
L
L
T hatis to find theencoder defined by thenonlinearmapping
g( m)   k , k  1,2,, L
(3.40)
such that wehave
L
D   d (m, g (m)) f M (m)dm   
A
A
k 1
mJ k
min d (m,k )f M (m)dm (3.41)
T o attainthelower bound , if
d (m, k )  d ( m, j ) holds for all j  k
T hisis called nearest neighborcondition.
(3.42)
Condition2 .Optimality of thedecoder for a given encoder
Given theset J

L
k k 1
, find theset 

L
k k 1
thatminimizedD .
For mean - square dist ort ion
L
D  
k 1
2
mJ k
( m   k ) f M ( m )dm, f M ( m)
(3.43)
L
2
D
 2  ( m   k ) f M ( m )dm  0 (3.44)
mJ k
 k
k 1
 k , opt



mJ k
m f M ( m)dm
mJ k
f M ( m)dm
(3.45)
Probability Pk (given)
 E M mk  m  mk  1
Using iterat ion,until D reachesa minimum
(3.47)
Pulse Code Modulation
Figure 3.13 The basic elements of a PCM system.
Quantization (nonuniform quantizer)
 - law
log(1   m )
 
log(1   )
(3.48)
d m log(1   )

(1   m ) (3.49)
d

A - law
A( m )

1
 1  log A 0  m 
A
 
1  log( A m ) 1

 m 1
A
 1  log A
1
 1  log A
0 m 
dm 
A

A
1
d
(1  A) m
 m 1

A
(3.50)
(3.51)
Figure 3.14 Compression laws. (a)  -law. (b) A-law.
Encoding
Line codes:
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)
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.
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
Figure 3.19 Block diagram of TDM system.
Synchronization
Example 2.2 The T1 System
3.10 Digital Multiplexers
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)
Let mn  m(nTs ) , n  0,1,2, 
where Ts is thesamplingperiodand m(nTs ) is a sampleof m(t ).
T heerrorsignal is
en  mn  mq n  1
eq n  D sgn(en )
mq n  mq n  1  eq n
(3.52)
(3.53)
(3.54)
where mq n is thequant izeroutput , eq n is
thequant ized versionof en , and D is thest ep size
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
n
mq n  D sgn(ei )
i 1
n
  eq i 
(3.55)
i 1
Two types of quantization errors :
Slope overload distortion and granular noise
Slope Overload Distortion and Granular Noise
Denotethequantization error by qn  ,
mq n   mn   qn 
(3.56)
Recall (3.52), we have
en   mn   mn  1  qn  1 (3.57)
Exceptfor qn  1, thequantizer input is a first
backward differenceof theinput signal( differentiator )
T o avoidslope- overloaddistortion, we require
D
dm(t )
(slope)
 max
(3.58)
Ts
dt
On theotherhand, granular noise occurs when step size
D is toolarge relativeto thelocal slope of m(t ).
Delta-Sigma modulation (sigma-delta modulation)
The D   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 differentiator in the conventional DM receiver is cancelled by
the integrator )
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 (  )
T hefilter output(T helinear preditionof theinput ) is
p
xˆ n    wk x ( n  k )
(3.59)
T hepredictionerror is
en   xn   xˆ n 
Let theindex of performance be
(3.60)
k 1


J  E e 2 n  (meansquare error) (3.61)
Find w1 , w2 ,, w p to minimizeJ
From (3.59)(3.60)and (3.61)we have


p
J  E x 2 n   2 wk E xn xn  k 
k 1
p
p
  w j wk E xn  j xn  k 
j 1 k 1
(3.62)
Assume X (t ) is stat ionaryprocesswit h zero mean ( E[ x[n ]]  0)
 X2  E x 2 n   ( E xn )2


 E x 2 n 
T heautocorrelation
RX (  kTs )  RX k   E xn xn  k 
We may simplify J as
p
p
p
J   X2  2 wk RX k    w j wk RX k  j 
k 1
(3.63)
j 1 k 1
p
J
 2 RX k   2 w j RX k  j   0
wk
j 1
p
 w R k  j   R k   R  k  , k  1,2, ,p (3.64)
j 1
j
X
X
X
(3.64)are called Wiener- Hopf equations
For convenience,
we may rewrite
the Wiener-Hopf equations
1
1
as , if R X exist s
where
w 0  R X rX
w 0  w1 , w2 ,, w p 
(3.66)
T
rX  [ RX [1], RX [2],...,RX [ p ]]T
R X 1

 RX 0
 R 1
RX 0

X

RX 




 RX  p  1 RX  p  2 
The filter coefficients are uniquely determined by
RX  p  1
RX  p  2




R X 0 
RX 0 , RX 1 ,, R X  p 
Subst it ut ing (3.64)int o(3.63)yields
p
p
k 1
k 1
J min   X2  2  wk RX k    wk R X k 

p
2
X
  wk RX k 
k 1
  X2  rXT w 0   X2  rXT R X1rX
 rXT R X1rX  0, J min is always less t han X2
(3.67)
Linear adaptive prediction (If RX k for varying k is not available)
T hepredictoris adaptivein thefollowsense
1. Computewk , k  1,2,, p, startingany initial values
2. Do iterationusing themethodof steepestdescent
Define thegradient vector
J
gk 
, k  1,2 , ,p
(3.68)
wk
wk n  denotesthe value at iterationn . T henupdate wk n  1
1
wk n  1  wk n   g k , k  1,2 , ,p (3.69)
2
1
where  is a step - size parameterand is for convenience
2
of presentation.
Differentiating (3.63), we have
P
J
gk 
 2 RX k   2 w j RX k  j 
wk
j 1
p
 2 E xn xn  k   2 w j E xn  j xn  k  , k  1,2,, p
(3.70)
j 1
T o simplify he
t comput ingwe use xn xn  k  for E[x[n]x[n - k]]
(ignore t heexpect at ion)
p
gˆ k n   2 xn xn  k   2 w j n xn  j xn  k  , k  1,2,, p
(3.71)
j 1
Substituting (3.71) into (3.69)
p


wˆ k n  1  wˆ k n   xn  k  xn    wˆ j n xn  j 
j 1


 wˆ k n   xn  k en  , k  1,2,, p
p
where en   xn    wˆ j n xn  j 
by (3.59) (3.60)
j 1
T heabove equat ionsare called lease - mean - square algorit hm
(3.72)
(3.73)
Figure 3.27
Block 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:
en   mn   mˆ n 
(3.74)
mˆ n  is a predictionvalue.
T hequantizer outputis
eq n   en   qn 
(3.75)
where qn  is quantization error.
T hepredictionfilter input is
mq n   mˆ n   en   qn  (3.77)
From (3.74)
mn 
 mq n   mn   qn 
(3.78)
Processing Gain
T he(SNR) o of t he DP CM syst emis
 M2
(SNR) o  2
Q
(3.79)
where  M2 and  Q2 are variancesof mn  ( E [m[n ]]  0) and qn 
 M2  E2
(SNR) o  ( 2 )( 2 )
 E Q
 G p (SNR )Q
(3.80)
where  E2 is t he varianceof t hepredict ions error
and t hesignal - t o - quant izat on
i noise rat iois
 E2
(SNR )Q  2
Q
(3.81)
 M2
P rocessingGain, G p  2 (3.82)
E
Design a predict ionfilt er t omaximizeG p (minimize E2 )
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).