digital transmission pulse modulation
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EE401B/EE3DTR/EE4DTRDigital Transmission
Pulse Modulation
Dr John A.R. Williams
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Overview
Pulse Modulation Schemes
Pulse Amplitude and Position Modulation
Time Division Multiplexing
Pulse Code Modulation
Quantisation and Companding
Differential and Delta Pulse Code Modulation
Adaptive Modulation Techniques
Subband Coding
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Pulse Modulation Schemes
Pulse PositionModulation(PPM) t
Pulse FrequencyModulation
(PFM) t
t
tPulse Carrier
Message Signal
t
Pulse AmplitudeModulation
(PAM)
t
Pulse WidthModulation
(PWM)
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Natural Sampling
0
t
(t)
W W0
|F( )|
W = 2B W = 2B0
t
(t)
W W0
|F( )|
W = 2B W = 2B
W W0
|F( )|
W = 2B W = 2B
p(t)
00
t
T 2T 3T2T T
1
T Pn
2/T/T
0 2
0
02
0
p(t)
00
t
T 2T 3T2T T
1
T PnPn
2/T/T
0
0 2
02
0
0
02
02
0
2/T
0 2 0 02 0
Fs()
0
t
(t)s
0 T 2T 3T2T T
T
2/T
0 2 0 02 0
2/T
0
0 2 02 0 0 02 02 0
Fs()Fs()Fs()
0
t
(t)s(t)s
0 T 2T 3T2T T
T
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Nyquist Sampling Theorem
1
T0 W
T=1/2W
W 1
T
1
T0 W
T=1/2W
W 1
T
1
T
1
T
1
TW+
1
TWW0 W +
1
TW
T1/2W
W +1
TW
1
TW W
1
TW
Undersampled
A band-limited signal of finite energy, which has no fre-
quency components higher than WHz is completely
described by sampling values of the signal at instantsof time separated by1/(2W)seconds
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Pulse Amplitude Modulation - Flat Topped Sampling
0
Fs( )
T
0
t
(t)s
0
Fs( )Fs( )
TT
0
t
(t)s (t)s
q(t)
0
t
1
0
Q()
-2/ 2/
q(t)
0
t
q(t)
0
t
1
0
Q()
-2/-2/ 2/
t
0
T(t)s
q(t)*
2/T
0 2 0 02 0 0
Fs ( ) Q( )
t
0
T(t)s
q(t)*(t)s
(t)s
q(t)* q(t)*
2/T
0 0 2 02 0 0 02 02 0 0
Fs ( ) Q( )Fs ( )Fs ( ) Q( )
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PAM Demodulation
Sample and Hold Circuit
C R
T
Rs
R sC > TC R
T
Rs
R sC > TC R
T
T
RsRs
R sC T
t
T
input
sample-and-holdoutput
low-pass filteredoutput
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Time Division Multiplexing (TDM)
single communication
channel carrying N
signals:
signal 1
signal 2
signal N
synchronised
commutatorsignal 1
signal 2
signal N
low-pass filters to
band-limit signals
low-pass filters to
recover signals
Time Division Multiplexing Of 2 PAM Signals
t
TT
sTT
sT
s
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TDM Example
A four channel time division multiplexed PAM system has input signals band-
limited as follows:
channel 1: 016 kHz
channel 2: 020 kHzchannel 3: 025 kHz
channel 4: 2530 kHz
If the four channels are sampled at equal time intervals using very short pulses at
the minimum frequency possible, and the TDM signal is low-pass filtered prior totransmission determine
1. The minimum clock frequency and the commutator recycling frequency
2. the minimum cut-off frequency of the LPF consistent with recovery of the
signals after transmission
3. What would be the minimum bandwidth required if the four channels were
frequency division multiplexed using
(a) DSB-AM
(b) SSB
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TDM Example
1. Channel 4 has a maximum frequency = 30 kHz and must,
therefore, be sampled at 60 kHz. This is the maximum frequency
in the set of channels. Hence for the four channels sampled at
equal time intervals, the clock rate=460=240kHz and thecommutator must cycle at 60 kHz
2. For the TDM signal we must have a filter cut-off at half the
maximum frequencyBc=1/(2Ts) =120kHz, andTs=4.17s
3. (a) DSB-AM requires2 (16 + 20 + 25 + 5) =132kHz
(b) SSB requires132/2=66kHz
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PPM and PWM Generation
inputsignal
Sample
and hold
Clock
Sawtooth
Generator
reference level
PWMComparator
reference level
PWMComparator
PPM
Pulse
Generator
sample-and-holdoutput
sample-and-holdoutput
inputsignalinputsignal
sawtoothsawtooth
PWMoutputPWMoutput
PPMoutputPPM
output
sumsum
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Pulse Code Modulation
0.0
0.2
0.4
0.6
0.8
1.0
Amplitude
123
45678
9101112131415
0
Quantisationlevel
Binarycode
1111111011011100101110101001
10000111011001010100
0011001000010000
0111 1101 1110 1010 1001 0101 0011 0101 0111
Samplinginstants
sampled signal
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PCM Transmission
PCM Transmitter
analogueto
digital converter
PCM
signal
input(analogue)
signalSampler EncoderQuantiser
Parallelto
serial
converter
NRZ coded PCM signal with 7-bit samples
1011001 1110011 0101010
framing bit
PCM Receiver
Output MessageSignal
DecoderRegenerator Reconstruction
Filter
PCM signal+noise
+distortion
Output MessageSignal
DecoderRegenerator Reconstruction
Filter
PCM signal+noise
+distortion
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PCM Advantages
Digital transmission providesrobustnessagainst noise and
interference
PCM signals can beregeneratedat intermediate repeaters. Modulating and demodulating circuits areentirely digitaloffering
compatibility with VLSI: high reliability and low cost
Signals can be stored in memory;digital signal processingoperationssuch as time scaling can be easily performed
Encryptionusing special codes can allow secure communication
Source codingmay be used to avoid unnecessary repetitions offrequent message
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Quantisation
(t)=f(t)-f (t)
f(t)
q
t
/2
/2
f (t)q
0
0
Error
Magni
tude
0
1
2
3
4
M-3
M-2
M-1
M
M
2
+1M2
V+
2
V
2
True SignalLevel
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Quantisation Noise
Assume all values in the range /2< < /2are equally probable. p() =1/.Then
2
=
1
Z /2
/2
2
d=
1
3
3/2/2 =
2
12
If a signal occupies thefull quantiser range. Peak signal amplitude isM/2giving
(SNRQ)peak=(M/2)2
2/12 =3M2 =322n
(SNRQ)peak=10{log10 3 + 2n log10 2} =4.8 + 6n[dB]
If we have asinusoidal signal, mean signal power is(M/2)2/2and
(SNRQ)mean=10{log10 1.5 + 2n log10 2} =1.8 + 6n[dB]
TheSNRQincreases by 6 dB for each additional bit used in quantisation
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Quantisation Example
What is the quantisation noise for the case where the quantiser always rounds down to
the next lowest level (i.e. the maximum error is).
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Quantisation Example
What is the quantisation noise for the case where the quantiser always rounds down to
the next lowest level (i.e. the maximum error is).If we round down the quantisation error is equiprobable in the range 0
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Companding
A-law compression characteristic
1/A1-1 0
-1/A
Vout
Vin1/A
1-1 0
-1/A
Vout
Vin
Vout= AVin
1 + logeA , 0 |Vin|
1
A (linear region)
=1 + loge(AVin)
1 + logeA,
1
A |Vin| 1, (logarithmic region)
A=87.7in Europe
Vout
Vin
Increasing A
Vout
Vin
Increasing A
0
1/A
2dB {
Vin
SNRQ
WithoutCompanding
0
1/A
2dB {
Vin
SNRQ
WithoutCompanding
0
1/A
2dB {
Vin
SNRQ
WithoutCompanding
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Segmented Companding
640 128 256
512 1024 2048 4096
32
48
64
80
96
112
128
0
Input Level
128 16
64 32
64 16
256 16512 16
1024 16
2048 166 bi ts equiv.
10 bits equiv.
12 bits equiv.
11 bits equiv.
9 bi ts equiv.
8 bi ts equiv.
7 bi ts equiv.
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NICAM
Nearly Instantaneous Companded Audio Multiplex
BitsRange 0
MSB LSB
Range 1Range 2Range 3Range 4
1 2 3 4 5 6 7 8 91011121314
Transmit ted bits
14 bit resolution with 5 ranges represented in 3 bits
1 range setting per block of 32 samples (1 ms)
Ranges packed into 7 bits every 3 blocks (3ms)
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Differential Pulse Code Modulation
If we oversample changes in signal amplitude are small and we can use
fewer bits to quantise.
SampledInput Signal
InputSignal
Range
DifferentialSignalRange
t
t
Ts
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Differential Pulse Code Modulation
DPCM Encoder
Sampler
Predictor
+ Quantiserx(t) xn
xn
+
en en
{ai}
en=xn
p
i=1
aixni
DPCM Decoder
+
Predictor
en
xn
xn
{ai}
nx= en+
p
i=1
ai xni
xnxn =
en+
p
i=1 ai xni
en+
p
i=1 aixni
= qn+p
i=1
ai( xnixni)
i.e. there is an accumulation of quantisation errors at the receiver.
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DPCM Processing Gain
Sampler
Predictor
+ Quantiserx(t) xn
xn
+
en en
{ai}
SNRO=2x
2
E
2E
2
Q
=GPSNRQ
Minimise2Eusing Yule-Walker equations
p
i=1 ai(i j) =(j), j=1,2, . . . ,p
where (n)is the autocorrelation function of the sampled signal sequence xn, whichmay be estimated from the finite set of samples {xn} by
(n) = 1
N
Nn
i=1
xixi+n,n=0,1, . . . ,p
(See Proakis Page 128)
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Practical DPCM
DPCM Encoder
Sampler + Quantiser
Predictor +
x(t) xn +
en en
xnxn
{ai}
en=xnp
i=
1
ai xni
DPCM Decoder
+
Predictor
en
xn
xn= xn+ en
{ai}
To lowpass
Filter
xn= en+p
i=1
ai xni
xnxn =
en+
p
i=1
ai xni
en+
p
i=1
ai xni
= qn
Error at receiver is only the quantisation error.c 20012008 Dr John A.R. Williams Pulse Modulation p. 24
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Improved DPCM
Encoder
Sampler + Quantiser
+Linear Filter
{ai}
Linear Filter{bi}
+
x(t) xn +
en en
xnxn
Decoder
+
+Linear Filter
{ai}Linear Filter
{bi}
en xn To lowpass
Filter
Improve estimate us-
ing linearly filtered past
values of the quantised
error.
xn =
p
i=1
ai xni
+m
i=1
bieni
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Delta Modulation (DM)
DM Encoder
Unit Delayz-1
QuantiserSamplerx t( ) xn en
~xn
~$ ~x xn n= 1
+
-
~en = 1 DM Decoder
Z-1
~en ~ ~$x x en n n= + To lowpassfilter
Ts
t
1 1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 1
Input Signal ( )f t
StepOverload
A simple practical approach
+
f t( )Comparator
BinaryOutputSignal
Clock
Flip-flop
(Output v)
R
C
R
C
RC Integrator
+V
-V
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Variable Step Size DM
DM Distortion
t
Slope-
overloaddistortion
Granular Noise
x t( )
t
Slope-
overloaddistortion
Granular Noise
x t( )
With Variable Step Size
t
Slope-overloaddistortion
Granular Noisex t( )
Adaptive DM Encoder
QuantiserSampler
Accumulator
x t( ) xn en+
-
~en = 1
~en1
z-1
z-1
To transmitter
n
n1
Adaptive DM Decoder
AccumulatorOutput
LowpassFilter
~en
~en1
z-1
z-1
n
n1
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Slope Overload Distortion Example
Consider a sine wave of frequency fmwith amplitudeAmapplied to a
delta modulator of step size. Show that the slope-overload distortionwill occur if
Am>
2fmTs
whereTsis the sampling period. What is the maximum power that may
be transmitted without slope overload distortion?
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Slope Overload Distortion Example
Consider a sine wave of frequency fmwith amplitudeAmapplied to a
delta modulator of step size. Show that the slope-overload distortionwill occur if
Am>
2fmTs
whereTsis the sampling period. What is the maximum power that may
be transmitted without slope overload distortion?Modulated Wavem(t) =Am cos2fmthas slopedm(t)
dt = 2fmAm sin2fmt.
Maximum average slope reproduced by the delta modulator is/Ts.Therefore require2fm>
Ts
orAm>
2fmTs.
Maximum power is A2m
2 =
2
8f2m
T2
s
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Adaptive Quantisation
000
M(4)
001
M(3)
010
M(2)
011
M(1)
100
M(1)
101M(2)
110
M(3)
111
M(4)
0
-3 -2 -1
1 2 3Input
Output Previous
output
Multiplier
7
2
12
52
32
32
52
12
72
EncoderInput
EncoderOutput
LevelEstimator
LevelEstimator
Quantiser with an adaptive step
N.S. Jayant, Digital Coding of
Speech Waveforms: PCM, DPCM,
and DM Quantizers, Proc IEE, vol62, pp. 611632.
PCM DPCM
2 3 4 2 3 4
M(1) 0.60 0.85 0.80 0.80 0.90 0.90
M(2) 2.20 1.00 0.80 1.60 0.90 0.90
M(3) 1.00 0.80 1.25 0.90
M(4) 1.50 0.80 1.70 0.90
M(5) 1.20 1.20
M(6) 1.60 1.60
M(7) 2.00 2.00
M(8) 2.40 2.40
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Adaptive Prediction
Calculate predictor coefficients from short term estimate of the autocorrelation
function ofxn.
Receiver predictor may compute its own predictor coefficients from signal esti-
mates provided quantisation error is small.
Adaptive Prediction withbackward estimation (APB)
Predictor
Logic foradaptive
Prediction
QuantiserSamplerx t( ) xn en
~en
~xn~$
xn
+
-
Adaptive Differential Pulse-Code Modulation (ADCPM) combines adaptive
quantisation and Prediction.
A standard for speech encoded transmission at 32 kb/s. Standard PCM requires
64 kb/s for speech encoded transmission.
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Adaptive Subband Coding
exploit quasi-periodic nature
of voiced speech permitting
pitch prediction reducing thelevel of prediction error
requiring quantisation
human ear cannot hear noise
below 15 dB below thesignal level in same band
can digitise speech at 16 kb/s
with a quality comparable to
64 kb/s PCM.
FilterBank f or
SubbandAnalysis
Adaptivebi t
assignmentcircuit
Multiplexer
Speech
Signal
To
channel
ADPCM Encoders
FilterBank f orSubbandAnalysis
De-multiplexer
Fromchannel
Output
ADPCM Decoders
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