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TE312: Introduction to Digital
Telecommunications
Lecture #4Pulse Code Modulation (PCM)
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Introduction
Points to be discussed in this lecture
Quantization and Encoding
Uniform Quantization
Non-Uniform Quantization
Bandwidth Requirements for PCM
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Introduction
Reading Assignment
Simon Haykin, Digital Communications, JohnWiley & Sons,
Inc., 1988, Chapter 5, Sec. 5.1,Sec. 5.3 and Sec. 5.4.
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Quantization and EncodingThe sampled signal ( )sm nT of the
messagesignal ( ), band-limited to Hzm t B is discrete intime but
continuous in amplitude.
Digital representation of )m t requires: -(i) Quantization of
the amplitude of a
sampled signal ( ).sm nT
(ii) Encoding of each quantized samplevalue resulting in a pulse
codemodulation (PCM) system.
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Quantization and Encoding
Amplitude quantization transforms the sampleamplitude ( ) into
amplitudesm nT ) takenfrom a finite set of possible
sm nT
L amplitudes.
Definition: When quantization of a samplevalue ( ) is
independent of sample values
( ) , the quantization process is
memoryless and instantaneous .
sm nT ,sm kT k n
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Quantization and Encoding
( )sm nT ( ) sm nT ( )m t 0110001
Sampler Encoder Quantizer
DecoderReconstruction
LPF filter
( ) sm nT ( )m t 0110001
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Quantization and EncodingThe quantization process partitions
theamplitude range of the continuous-valued
samples into L intervals.
The interval is determined by decision
levels or threshold levels and
thl l I
l D 1l D + i.e.
{ }1: , 1,2,...,l l l I D m D l L+< < =
All sample values that fall in are represented
by a target level or representation level
l I
l lT I .
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Quantization and Encoding
The spacing between two adjacent targetlevels (or two adjacent
decision levels)
lT T = is the step size of the quantizer.1l l+ Definition: The
quantizer is called a uniformquantizer if 1l l lT T + = = for all l
. Otherwiseit is a non-uniform quantizer.
1lT +lT
l D 1l D + 2l D +
1l I +l I
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Quantization and EncodingIn PCM system, each quantized sample )
sm nT is encoded into an R-bit sequence of binary
digits (bits) called a code word where 2log R L= .
In binary coding, the target level isrepresented by the binary
equivalent
Rb b b b of its ordinal number 1l
thl
1 2 ... ...k .
With binary code, there may be a change ofmore than one bit for
two adjacent target levelsresulting in distorted receiver output
due tochannel noise and interference.
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Quantization and Encoding A binary code R can be converted to
aGray code R
1 2 ... ...k b b b b
1 2 ... ...k g g g g as follows:
1 1
1, 2k k k
g b
g b b k
==
With Gray code, there is a change of only onebit for two
adjacent target levels.
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Uniform Quantization
The quantizer input/output characteristic for auniform quantizer
is a staircase function which
can be a midtread or midriser .
Quantization process introduces quantizationerror .
The input to the quantizer is modeled as asample value m of a
zero mean randomvariable with pdf ) M f m and amplitude range
.max max
m m m
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Uniform Quantization
The quantization step size is given by
max2m L
=
Quantization noise mq m= is a sample valueof a zero mean random
variable Q with uniformpdf i.e.
( )1 -
2 2
0 otherwise
Q
q f q
