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Chapter 3: Pulse Modulation
CHAPTER 3
PULSE MODULATION
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
•
3.1 Introduction
3.2 Sampling Process
3.3 Pulse Amplitude Modulation
3.4 Other Forms of Pulse Modulation
3.5 Bandwidth-Noise Trade-off
3.6 The Quantization Process
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Chapter 3: Pulse Modulation
3.1 Introduction
• This chapter is a transitional chapter between
analog and digital modulation techniques.
• In CW modulation, as we have studied in chapter
2, one parameter of the sinusoidal carrier wave is
continuously varied in accordance with a given
message signal.
• In the case of pulse modulation we have a pulse
train and some parameter of the pulse train is
varied in accordance with the message signal.
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Chapter 3: Pulse Modulation
CommSystems 1 – Analog
Communication Techniques
• In the first part of Communication Systems we
studied transmission techniques of analog
waveforms (analog sources) over analog
signals (lines).
– Why is modulation necessary?
– What types of modulation did we study?
– When we studied a specific modulation type what
were the specific subjects we discussed?
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Chapter 3: Pulse Modulation
CommSystems 2 – Digital
Communication Techniques
• In the second part we have two major topics
– analog waveforms (analog sources) transmission
using baseband signals
– digital waveforms (digital sources) transmission
using band-pass signals
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Chapter 3: Pulse Modulation
Why digital?
• Digital approximation of analog signals can be
made as precise as required
• Low cost of digital circuits
• Flexibility of digital approach – possibility of
combining analog and digital sources for
transmission over digital lines
• Increased efficiency – source coding/channel
coding separation
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Chapter 3: Pulse Modulation
Goals of this course:
• To study digital communication systems and their conceptual
basis in information theory
• To study how analog waveforms can be converted to digital
signals (PCM)
• Compute spectrum of digital signals
• Examine effects of filtering – how does filtering affect the
ability to recover digital information at the receiver.
– filtering produces ISI in the recovered signal
• Study how to multiplex data from several digital bit streams
into one high speed digital stream for transmission over a
digital system (TDM)
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Chapter 3: Pulse Modulation
Motivation and Development
• Digital transmission –1960s
• Real application – after 1970s
– developments in solid state electronics, micro-electronics, large scale
integration
– all common information sources are inherently analog
• Historical steps
– Sampled analog sources transmitted using analog pulse modulation
(PAM, PPM)
– Samples are quantized to discrete levels (PCM, DM)
– Conversion from analog and transmission were implemented as a single
step
• Today
– Layered approach – different steps are distinguished and separately
optimized (source coding and channel coding)
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Chapter 3: Pulse Modulation
• We distinguish between:
– analog pulse modulation
• a periodic pulse train is used as a carrier wave;
• a parameter of that train (amplitude, duration, position)
is varied in a continuous manner in accordance with the
corresponding sample value of the message signal;
• information is transmitted basically in analog form, but
at discrete times.
– digital pulse modulation
• message represented in a discrete way in both time and
amplitude;
• sequence of coded pulses is transmitted.
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Chapter 3: Pulse Modulation
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Chapter 3: Pulse Modulation
Source Coding
• Problem of coding: efficient representation of
source signals (speech waveforms, image
waveforms, text files) as a sequence of bits for
transmission over a digital network
• Paired problem of source decoding –
conversion of received bit sequence (possibly
corrupted) into a more-or-less faithful replica
of the original
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Chapter 3: Pulse Modulation
Channel Coding
• Problem of the efficient transmission of a
sequence of bits through a lower layer channel
– 4 KHz telephone channel, wireless channel
• Recovery at the channel output in the remote
receiver despite distortions
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Chapter 3: Pulse Modulation
Why separate source and channel
coding?
• Basic theorem of information theory:
– If a source signal can be communicated through a
given point-to-point channel within some level of
distortion (by any means) then the separate source
and channel coding can also be designed to stay
within the same limits of distortion.
• WHY then…(delay, complexity…)
• Pros and cons? Does it always hold true?
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Chapter 3: Pulse Modulation
Shannon and the Channel Coding
Theorem
• Channel coding can help reduce the error
probability without reducing the data rate
• Date rate depends on the channel itself –
channel capacity
• Channel bandwidth W, input power P, noise
power then the channel capacity in bits is:
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Chapter 3: Pulse Modulation
Digital Interface
• Interface between source coding/channel coding –
issues continuity, rate etc.
– continuous sources
– packet sources
– complex combinations
protocols discussed in
details in Data
Communications
course
Here: min number of bits from source and
max transmission speed over channel
source coder rate = channel encoder rate (source-channel
coding theorem)
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
•
3.1 Introduction
3.2 Sampling Process
3.3 Pulse Amplitude Modulation
3.4 Other Forms of Pulse Modulation
3.5 Bandwidth-Noise Trade-off
3.6 The Quantization Process
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Chapter 3: Pulse Modulation
3.2 Sampling Process
• Sampling converts an analog signal into a
corresponding sequence of samples that are
uniformly distributed in time.
• Proper selection of the sampling rate is very
important because it determines how uniquely
the samples would represent the original
signal.
• It is determined according to the so called
sampling theorem.
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Chapter 3: Pulse Modulation
• The model:
– we consider an arbitrary signal g(t) with finite energy,
specified for all time
– we sample the signal instantaneously and at an uniform
rate, once every Ts seconds
– we obtain an infinite sequence of samples spaced Ts
seconds apart; they are denoted by [g(nTs)], where n can
take all possible integer values
– Ts is referred to as the sampling period, and fs=1/Ts is the
sampling rate.
– let gδ(t) denote the signal obtained by individually
weighting the elements of a periodic sequence of delta
functions spaced Ts seconds apart by the numbers [g(nTs)].
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Chapter 3: Pulse Modulation
The sampling process.
(a) Analog signal.
(b) Instantaneously sampled version of the analog
signal.
Figure 3.1
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Chapter 3: Pulse Modulation
• For the signal gδ(t), called the ideal sampled signal,
we have the following expression :

g (t )
 g  nT    t  nT 
n 
s
s
(3.1)
• As the idealized delta function has unit area, the
multiplication factor g(nTs) can be considered as
“mass” assigned to it (samples are “weighted”);
• A delta function weighted in this manner is
approximated by a rectangular pulse of duration Δt
and amplitude g(nTs)/Δt.
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Chapter 3: Pulse Modulation
• Knowing that the uniform sampling of a continuoustime signal of finite energy results into a periodic
spectrum with a period equal to the sampling rate
using the FT gδ(t) can be expressed as:

g (t )


f
s
 G  f  mf 
(3.2)
s
m 
• So if we take FT on both sides of (3.1) we get:
G ( f ) 

 g nT exp  j 2nfT 
n  
s
s
(3.3)
discrete time Fourier transform
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Chapter 3: Pulse Modulation
• The relations derived up to here apply to any
continuous time signal g(t) of finite energy
and infinite duration.
• If the signal g(t) is strictly band-limited, with
no components above W Hz, then the FT G(f)
of g(t) will be zero for |f| ≥ W.
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Chapter 3: Pulse Modulation
(a) Spectrum of a strictly band-limited signal g(t).
(b) Spectrum of the sampled version of g(t) for a
sampling period Ts = 1/2 W.
Figure 3.2
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Chapter 3: Pulse Modulation
• For a sampling period Ts=1/2 W after substitution in
3.3 we get the following expression:
 n 
 jnf 
G ( f )   g 
 exp  

 W 
n    2W 

(3.4)
• and using 3.2 for the FT of gδ(t) we can also write:
G ( f )  f s G f   f s

 G f  mf 
m  
m0
s
(3.5)
m=0
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Chapter 3: Pulse Modulation
• and for the conditions specified about f we get:
1
G( f ) 
G  f ,
W  f  W
2W
(3.6)
• and when we substitute (3.4) and (3.6) we get:
1
G( f ) 
2W
 n 
 jnf 
g
 exp  
,

 W 
n    2W 
(3.7)

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Chapter 3: Pulse Modulation
Conclusion:
• 1. If the sample values g(n/2W) of an analog
signal g(t) are specified for all n, then the FT
G(f) of the signal is uniquely determined by
using the discrete-time FT of equation (3.7).
• 2. Because g(t) is related to G(f) by the inverse
FT, the signal g(t) is itself uniquely determined
by the sample values g(n/2W) for -∞ < n <+∞.
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Chapter 3: Pulse Modulation
Second part: reconstructing the signal
from the samples
• We substitute equation (3.7) in the inverse FT formula and after
some reorganizing we get:

 n  1
g (t )   g 

n    2W  2W

n 

exp
j
2

f
t

W   2W df
W
(3.8)
• which after integration ends to be:
 n  sin( 2Wt  n )
g (t )   g 

n    2W  ( 2Wt  n )

 n 
  g
 sin c(2Wt  n),
n    2W 

  t  
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Chapter 3: Pulse Modulation
• This is an interpolation formula for
reconstructing the original signal g(t) from a
sequence of sample values [g(n/2W)].
• The sinc function sinc(2Wt) is playing the role
of interpolation function.
• Each sample is multiplied by a suitably
delayed version of the interpolation function
and all the resulting waveforms are summed
up to obtain g(t).
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Chapter 3: Pulse Modulation
Sampling Theorem
• 1. A band-limited signal of finite energy, which has
no frequency components higher than W Hz, is
completely described by specifying the values of the
signal at instants of time separated by 1/2W (means
that sampling has to be done at a rate twice the
highest frequency of the original signal).
• 2. A band-limited signal of finite energy, which has
no frequency components higher than W Hz, may be
completely recovered from a knowledge of its
samples taken at the rate of 2W samples per second.
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Chapter 3: Pulse Modulation
Note:
• The sampling rate of 2W for a signal of
bandwidth W Hz, is called the Nyquist rate;
• Its reciprocal 1/2W (seconds) is called the
Nyquist interval;
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Chapter 3: Pulse Modulation
• The derivations of the sampling theorem so far
were based on the assumption that the signal
g(t) is strictly band limited.
• Practically – not strictly band-limited; the
result is under sampling so some aliasing is
produced by the sampling process.
• Aliasing is the phenomenon of a highfrequency component in the spectrum of the
signal taking on the identity of a lower
frequency in the spectrum of its sampled
version.
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Chapter 3: Pulse Modulation
(a) Spectrum of a signal.
(b) Spectrum of an undersampled version of the signal exhibiting
the aliasing phenomenon.
Figure 3.3
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Chapter 3: Pulse Modulation
• Practically there are two possible engineering
solutions:
– prior to sampling, a low-pass anti-aliasing filter is
used to attenuate the high-frequency components
that are not essential to the information baring
signal.
– the filtered signal is sampled at a rate slightly
higher than the Nyquist rate.
• Note: This also makes the design of the
reconstructing filter easier.
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Chapter 3: Pulse Modulation
(a) Anti-alias filtered
spectrum of an
information-bearing
signal.
(b) (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.
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Chapter 3: Pulse Modulation
• The reconstruction filter is low-pass, pass-band
–W to +W.
• The transition band of the filter is fs- W where
fs is the sampling rate.
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
•
3.1 Introduction
3.2 Sampling Process
3.3 Pulse Amplitude Modulation
3.4 Other Forms of Pulse Modulation
3.5 Bandwidth-Noise Trade-off
3.6 The Quantization Process
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Chapter 3: Pulse Modulation
3.3 Pulse Amplitude Modulation
• Definition: In Pulse Amplitude Modulation
(PAM) the amplitudes of regularly spaced
pulses are varied in accordance with the
corresponding sample values of the continuous
message signal;
• Note: Pulses can be rectangular or some other
form.
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Chapter 3: Pulse Modulation
Flat-top samples, representing an analog signal.
Figure 3.5
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Chapter 3: Pulse Modulation
PAM
•
Steps in realizing PAM:
1. Instantaneous sampling of the message signal
every Ts seconds, with sampling rate fs chosen
according to the sampling theorem.
2. Lengthening the duration of each sample to
obtain a constant value of T (duration of pulses).
3. These two are known as “sample and hold”.
4. Question is: how long should be the pulses (T)?
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Chapter 3: Pulse Modulation
• Assume:
– s(t) sequence of flat-top pulses generated as described.
s (t ) 

 mnT ht  nT 
n  
s
(3.10)
s
– where Ts is the sampling period, m(nTs) is the sample value
at time t=nTs
0t T
 1,
– standard rectangular
1
h(t )   ,
t  0, t  T
(3.11)
pulse is represented by:
2
0,
otherwise
– by definition the instantaneously sampled version of m(t) is:
m (t ) 

 mnT  t  nT 
n  
s
(3.12)
s
time-shifted delta function
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Chapter 3: Pulse Modulation
• after convolution and applying the sifting property of
the delta function we get:
m (t )  h(t )  

m ( )h(t   )d






 mnT    nT h(t   )d
s
n  

s

 mnT     nT h(t   )d
n  
s
m (t )  h(t ) 
s

(3.13)

 mnT ht  nT 
n  
s
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Chapter 3: Pulse Modulation
• The result in the previous slide means that (compare
3.10 and 3.14) the PAM signal s(t) is mathematically
represented by 3.15:


s (t ) 
 mnT ht  nT 
s
s
m (t )  h(t ) 
n  
 mnT ht  nT 
s
s
n  
(3.14)
(3.10)
s (t )  m (t )  h(t )
(3.15)
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Chapter 3: Pulse Modulation
• After taking FT on both sides we get:
S ( f )  M ( f ) H ( f )
(3.16)
• Using formula 3.2 for the relation between Mδ(f) and M(f), the
FT of the original message m(t) we can write:
M ( f )  fs

 M  f  kf 
s
k  
(3.17)
• Finally, after substitution of 3.16 into 3.17 we get
S ( f )  fs

 M  f  kf H ( f )
k  
s
(3.18)
• which represents the FT of the PAM signal s(t).
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Chapter 3: Pulse Modulation
• Second part: recovery procedure
•assume that the message is limited to bandwidth W and
the sampling rate is fs which is higher than the Nyquist
rate.
•pass s(t) through a low-pass filter whose frequency
response is defined in 3.4c
•the result, according to 3.18 is
M(f)H(f), which is equal to
passing the original signal m(t)
through another low-pass filter
with frequency response H(f).
Fig. 3.4
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Chapter 3: Pulse Modulation
• To determine H(f) we use the FT of a rectangular pulse, plotted
on fig. 3.6a and 3.6b:
H ( f )  T sin c( fT ) exp(  jfT )
(3.19)
• By using flat-top samples to generate a PAM signal we
introduce amplitude distortion and delay of T/2
• This distortion is known as the aperture effect.
• This distortion is corrected by the use of an equalizer after the
low-pass filters to compensate for the aperture effect. The
magnitude response of the equalizer is ideally:
1
1
f


| H ( f ) | T sin c( fT ) sin( fT )
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Chapter 3: Pulse Modulation
(a)
Rectangular pulse h(t). (b) Spectrum H(f), made up
of the magnitude |H(f)|, and phase arg[H(f)]
Figure 3.6
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Chapter 3: Pulse Modulation
Conclusion on PAM:
• 1. Transmission of a PAM signal imposes strict
requirements on the magnitude and phase responses
of the channel, because of the relatively short
duration of the transmitted pulses.
• 2. Noise performance can never be better than a baseband signal transmission.
• 3. PAM is used only for time division multiplexing.
Later on for long distance transmission another
subsequent pulse modulation is used.
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
•
3.1 Introduction
3.2 Sampling Process
3.3 Pulse Amplitude Modulation
3.4 Other Forms of Pulse Modulation
3.5 Bandwidth-Noise Trade-off
3.6 The Quantization Process
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Chapter 3: Pulse Modulation
3.4 Other Forms of Pulse Modulation
• Rough comparison between CW modulation and
pulse modulation shows that latter inherently needs
more bandwidth. This bandwidth can be used for
improving noise performance.
• Such additional improvement is achieved by
representing the sample values of the message signal
by some other parameter of the pulse (different than
amplitude):
– pulse duration (width) modulation (PDM) – samples are
used to vary the duration of the individual pulses.
– pulse-position modulation (PPM) – position of the pulse,
relative to its un-modulated time of occurrence in
accordance with the message signal.
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Chapter 3: Pulse Modulation
Illustrating two
different forms of
pulse-time modulation
for the case of a
sinusoidal modulating
wave.
(a) Modulating wave.
(b) Pulse carrier.
(c) PDM wave.
(d) PPM wave.
Figure 3.8
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Chapter 3: Pulse Modulation
Comparison:
• 1. In PDM long pulses require more power, so PPM is more
effective.
• 2. Additive noise has no effect on the position of the pulse if it
is perfectly rectangular (ideal) but in reality pulses are not so
PPM is affected by channel noise.
• 3. As in CW systems the noise performance and comparison
can be done using the output signal-to-noise ratio or the figure
of merit.
• 4. Assuming the average power of the channel noise is small
compared to the peak pulse power, the figure of merit for a
PPM system is proportional to the square of the transmission
bandwidth BT, normalized with respect to the message
bandwidth W.
• 5. In bad noise conditions the PPM systems suffer a threshold
of its own – loss of wanted message signal.
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
•
3.1 Introduction
3.2 Sampling Process
3.3 Pulse Amplitude Modulation
3.4 Other Forms of Pulse Modulation
3.5 Bandwidth-Noise Trade-off
3.6 The Quantization Process
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Chapter 3: Pulse Modulation
3.5 Bandwidth-Noise Trade-Off
• As far as the analog pulse modulation schemes are
concerned the pulse position modulation exhibits
optimum noise performance.
• In comparison with CW modulation schemes it is
close to the FM systems.
– both systems have a figure of merit proportional to the
square of the transmission bandwidth BT normalized with
respect to the message bandwidth.
– both systems exhibit a threshold effect as the signal-tonoise ratio is reduced.
– can we do better – yes but not with analog methods….
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Chapter 3: Pulse Modulation
• what is required is discrete representation in
both time and amplitude.
• discrete in time – sampling
• discrete in amplitude – quantization
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
•
3.1 Introduction
3.2 Sampling Process
3.3 Pulse Amplitude Modulation
3.4 Other Forms of Pulse Modulation
3.5 Bandwidth-Noise Trade-off
3.6 The Quantization Process
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Chapter 3: Pulse Modulation
3.6 Quantization Process
• For a continuous signal (voice, music) the samples
have a continuous amplitude range.
• But humans can detect only finite intensity
differences
• So an original signal can be approximated, without
loss of perception, by a signal constructed of discrete
amplitudes selected on a min error basis.
• This is the basic condition for the existence of pulse
code modulation.
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Chapter 3: Pulse Modulation
• Definition: Amplitude quantization is defined
as the process of transforming the sample
amplitude m(nTs) of a message signal m(t) at
time t=nTs into a discrete amplitude of v(nTs)
taken from a finite set of possible amplitudes.
• We assume that the quantization process is
memoryless and instantaneous. (This means
that the transformation at time t is not affected
by earlier or later sample values.)
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Chapter 3: Pulse Modulation
Description of a memoryless quantizer.
Figure 3.9
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Types of quantizers
• based on the way representation values are
distributed and positioned around the origin:
–
–
–
–
unifrom – equally spaced representation levels
non-uniform – non-equally; considered later
mid-read – origin lies in the middle of a read;
mid-rise – origin lies in the middle of the rising
part of the staircase graph
– symmetric about the origin
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Two types of quantization: (a) midtread and (b) midrise.
Figure 3.10
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Quantization Noise
• Definition:
The error caused by the difference between the
input signal m and the output signal v is
referred to as quantization noise.
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Illustration of the quantization process.
Figure 3.11
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The model
• Assume:
– input value m, which is the sample value of a zero-mean
RV M; output value v which is the sample value of a RV V;
– quantizer g(*) that maps the continuous RV M into a
discrete RV V;
– respective samples of m and v are connected with the
following relation:
q  mv
(3.23)
or
Q  M V
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• We are trying to evaluate the quantization error Q.
– zero mean because the input is zero mean
– for the output signal-to-noise (quantization) ratio we need the mean
square value of the quantization error Q.
– the amplitude of m varies (-mmax, mmax); then for uniform quantizer the
step size is given by:
2m

max
L
(3.25)
with L being the total number of representation levels;
– for uniform quantizer the error is bounded by –Δ/2≤q≤Δ/2
– if step size is small Q is uniformly distributed (L large)


1
  ,  2  q  2
f Q (q)  

0,
otherwise
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• as mean is zero, variance is:
  E[Q ]
2
Q
2

 /2
 /2
2
1  /2 2
   q dq
  /2
2


12
2
Q
q fQ (q)dq
(3.28)
(3.27)
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• hidden in Δ is the number of levels used, which
directly influences the error.
• typically an L-ary number k, denoting the kth
representation level of the quantizer is transmitted to
the receiver in binary form.
• Let R denote the number of bits per sample used in
the binary code.
L2
R
R  log 2 L
(3.29)
(3.30)
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2mmax

2R
1 2 2 R
2
 Q  mmax 2
3
• so for the step size we get
(3.31)
• and for the variance
(3.32)
• If P denotes the average power of the message signal
m(t) we can find the output signal-to-noise ratio as:
(SNR)O 
P

2
Q
 3P
  2
 mmax
 2R
2

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Conclusion:
• The output SNR of the quantizer increases
exponentially with increasing the number of bits per
sample, R.
• Increasing R means increase in BT.
• So, using binary code for the representation of a
message signal provides a more efficient method for
the trade-off of increased bandwidth for improved
noise performance than either FM or PPM.
• Note: FM and PPM are limited by receiver noise,
while quantization is limited by quantization noise.
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