Pulse Code Modulation
 PCM is a method of converting an analog signal into a
digital signal. (A/D conversion)
 The amplitude of Analog signal can take any value over
a continuous range i.e. it can take on an infinite values.
 Digital signal amplitude can take on finite values.
 Analog signal can be converted into digital by
sampling and quantizing.
Cont.
Cont.
 The amplitude of analog signal m(t) lie in the range
(-mp, mp) and is partitioned into L sub-intervals each
of magnitude 2mp/L
Binary pulse codes
Quantizing
 Digital signals come from variety of sources e.g.
computer
 Some sources are analog but are converted into digital
form by variety of techniques such as PCM and DM
 For quantizing , we limit the amplitude of m(t) to a
range(-mp, mp) as shown in the previous slides
 This amplitude is uniformly divided into L
subintervals and each interval is ,
Cont.
 A sample value is approximated by the mid point of
the interval
 The quantized samples are coded and transmitted as
binary pulses
 At the receiver some pulses will be detected incorrectly
 There are two types of errors
 Quantization error
 Pulse detection error
Cont.
 In almost all practical schemes, the pulse detection
error is very small compared to the quantization error
and can be ignored
 Now we analyze the quantization error
Cont.
Cont.
Cont.
 The integral of the cross product terms is zero and we
obtain,
 Because the sampling rate is 2B, hence the total
number of samples over the averaging interval is 2BT
 This is called the mean of the quantization error
Cont.
 The quantized levels are separated by 2mp/L
 Since sample value is approximated by the midpoint of
the subinterval in which the sample falls
 The maximum quantization error is
 The mean square quantizing error is
Cont.
Cont.
Non-uniform quantization
 SNR is an indication of the quality of the received signal
 Ideally we would like to have constant SNR
 Unfortunately, the SNR is directly proportional to the




signal power, which varies from talker to talker
The signal power can also vary because of the connecting
circuits
SNR vary even for the same talker, when the person speaks
softly
Smaller amplitudes pre-dominate in speech and larger
amplitude much less frequent.
This means the SNR will be low most of the time
Cont.
 The root of this difficulty is that the quantization steps
are of uniform value
 The quantization noise is directly proportional to the
square of the step size.
 The problem can be solved by using smaller steps for
smaller amplitudes as shown in fig. on the next slide
Cont.
Cont.
 The same result can be obtained by first compressing a
signal and then using uniform quantization
 The input-output characteristics of compressor are
shown in fig.
Cont.
 The horizontal axis is normalized input signal and the
vertical axis is the output signal y.
 The compressor maps the input signal into larger
increments
 Hence the interval delta(m) contains large number of
steps when m is small
 The quantization noise is small for smaller input signal
 Thus loud talker and stronger signals are penalized
with higher noise steps in order to compensate the soft
talker and weak signals
Compression Laws
 There are two laws regarding compressions
 (1)
 This law is used in North America and Japan
 (2) A-Law
 This law is used in Europe and the rest of the word
Cont.
 The compressed samples are restored to their original
values at receiver by using an expander
 The compressor and expander together are called
compandor.
 Compression of a signal increases its bandwidth but in
PCM, we are not compressing the signal but its
samples the number of samples does not change,
therefore bandwidth does not rise
 When meu-law compandor is used then output SNR is
Transmission BW and output SNR
 For binary PCM, we assign distinct group of n binary
digits to each of the L quantization levels
 Each quantized level is encoded into n-bits
 Minimum channel BW is
 This is the theoretical minimum transmission
bandwidth required to transmit the PCM signal
Example 6.2
 A signal m(t) band-limited to 3kHz is sampled at a rate
33.33% higher than Nyquist rate, a maximum
acceptable error in the sample amplitude is 0.5% of the
peak amplitude. The quantized samples are binary
coded. Find the minimum channel BW required to
transmit the coded signal. If 24 such channels are
time-division multiplexed, determine the minimum
transmission BW required to transmit the multiplexed
signal
Solution
Exponential Increase of output SNR
SNR in decibel scale
Cont.
Example 6.3
Comments on Logarithmic Units
 Very small and very large values are expressed in
logarithmic units
T1 carrier system
 A schematic of T1-system is shown in fig.
Cont.
Cont.
Differential Pulse Code Modulation
Taylor's series
Cont.
Analysis of DPCM
Cont.
Delta Modulation
Cont.
Cont.
Delta Modulator
Delta Demodulator
Delta Modulator output
Working of DM