PCM System

• The quantization is used to reduce the effects of noise,

& the sampling allows for the TDM a no. of messages.
Hence the combined operations of sampling &
quantization generate a quantized PAM whose
amplitudes are restricted to a finite no. of discrete
magnitudes/levels.
• Hence the quantized amplitude can be binary coded for
the transmission of samples in binary form known as
PCM as shown in Fig.11.
• Hence in PCM each sample value falling to a
quantization level is coded to an appropriate binary
sequence which are then converted to electrical pulses
for transmission as shown in Fig.11.
Fig.-11(a) :- A complete PCM system
Fig.-11(b) :- A complete PCM modulation & Demodulation circuit.
Fig.-11© :- Electrical representation of binary digits is
known as line coding.
• The quantizer at the PCM receiver removes the
noise possessed by each binary digits to
estimate/decide/confirm it as a ‘0’ or ‘1’ for ease
of decoding to reconstitute the quantized PAM
signal. Then it is passed through a holding ckt (0
order or 1st order) & a reconstruction low pass
filter having message BW B Hz for getting back
the original message as shown in Fig.11.
• Pb.01:- Let us assume a message signal m(t) =
10cos(2π103t) is sampled & transmitted through
the channel using 4-bit PCM. If the sampling rate
is 50% higher than the Nyquist rate, then
calculate (i) step size (ii) Sampling rate ?
• Solution :- Here, N = 4, M (no. of quantized
levels) = 24 =16, amplitude of message signal =
±10v & fm = 103Hz =1KHz.
VH  VL
(i) step size =  10  (10)  20  1.25v
M 16 16

(ii) Nyquist rate = fs = 2 fm = 2KHz  Sampling

rate = 0.5x2KHz + 2KHz =3000 Samples/sec. (Ans.)
• Pb-2: A channel with bit rate Rb= 36kbps is
available for PCM voice transmission. Find
appropriate values of binary digits N, the no. of
quantization levels M & the sampling rate fs
assuming fm = 1.6 kHz.
• Ans:- fs ≥ 2 fm = 3.2 kHz & Nfs ≤ Rb =36kbps
So, N ≤ Rb /fs ≤ 36/3.2 = 5.6; Thus N=5 &
M = 2N = 32
Hence, fs = Rb /N =36/5 =7.2kHz. (Ans)
 Bandwidth of the PCM system
• Let us assume that, there are ‘n’ numbers of users,
each band limited to fm, required to be time division
multiplexed for transmission.
• Let N be the length of the PCM code, i.e., there are
2N = M quantization levels.
• Then the BW of the PCM system depends on the bit
duration (bit time slot, Tb).
• Considering sampling frequency, fs as 2fm, then the
sampling period, Ts will be (1/ 2 fm).
• As there are n channels in this TDM system with N
bits per sample & One synchronizing bit, then the
 total number of bits accommodated per frame
(i.e., within one sampling period) = nN +1.
• Therefore the bit duration
Sampling period 1 1
Tb    s ....( A)
tota l n umber of bits (nN  1) f s (nN  1)2 f m

• Hence, the required BW for this PCM system is :

1
BW   (nN  1) 2 f m  (nN  1) f s Hz
Tb
 nNf s Hz (if , nN 1) ...( B)
 Pb-3:- A voice signal band limited to 3.4 kHz is
sampled at 8 kHz & pulse code modulated using
64 quantization levels. 10 numbers of such
channels are time division multiplexed using a 2-
bit synchronizing word. Compute the minimum
BW required ?

• Solution:-
Here, no. of quantization levels, M = 64
so, length of PCM code N = log2 (64) = 6
n = 10, fs = 8kHz & Synchronizing word size = 2
Thus the min. BW required = (nN+2)8 = 496 kHz
(Ans.)
 Merits & Demerits of PCM
• Merits of PCM :- (1) It is highly immune to noise
due to two level of signal transmission (‘0’ & ‘1’).
(2) The repeaters used in PCM regenerate the original
received PCM signal eliminating all the acquired
noise during transmission. Again, the repeaters in
PCM system are extremely simple compared to
analog comn systems.
(3) It is possible to use various coding techniques so
that only the intended person can decode the
received signal, i. e., data security is high.
• Demerits of PCM :- (1) It’s sampling, quantizing,
encoding & decoding circuitry are more complex.
(2) It requires large BW in comparison to other
systems.
 Companding
• We know that quantization error depends upon the
step size, S & mean-square quantization noise power
S2
Nq  .
12
• When the steps are uniform in size, the small
amplitude signals will have a poor signal-to-
quantization noise power (SNR) than larger
amplitude signals as the quantization noise
power is same for all signals, since the signal/
sample power is not constant & proportional to
the square of signal amplitude.
• Companding (compressing + expanding) is a
process required to be implementated in PCM
system (as shown in Fig.-12(i)) in order to
improve the weak signal power (strength) to
quantization noise ratio & to maintain almost
same SNR irrespective of signal amplitudes.
• Thus companding is a process to keep the signal-
to-quantization noise ratio high by adjusting the
step size in such a way, i.e., the step size should
be small for small amplitude signals & larger for
larger amplitude signals (non-uniform quantizing)
as shown in Fig.-12(ii).
 y

Fig.-12 (i) A Model of a Compander ckt.

• Fig. 12(iii) shows the location of a compander

ckt.
• Thus compressor 1st compress the i/p signal &
then uniform quantized as shown in Fig.-12(ii) (b).
• m/mp is the
normalized i/p & y
is the normalized
o/p which is
uniformly
quantized. Non-uniform quantization
• The compressor
maps i/p signal
increments ∆m into
larger increments
∆y for small i/p
signals & vice
versa for large i/p
signals.
• Hence small Fig.-12 (ii) A non-uniform Quantization process
 Compression Ckt
Compander

X X X X

Expansion Ckt
Fig.-12(iii) :- A PCM system with Compander
 m/mp contains large numbers of steps
(quantization levels) & the quantization noise is
lower for smaller i/p signal power.
• an appropriate logarithmic compression
characteristic yields a quantization noise nearly
proportional to the signal power, thus making the
SNR practically independent of the i/p signal
power over a large dynamic range.
• As a result the loud talkers & stronger signals are
penalized with higher noise/quantization steps to
compensate the soft talkers & weaker signals.
• Thus before applying the signal to the quantizer
we must pass it through a network (known as
Compressor) having an i/p-o/p characteristic as
shown in Fig.-13(a). Here at low amplitudes the
slope is larger than at large amplitudes.
• As a result, the larger amplitudes are compressed
whereas lower amplitudes are expanded while
passing through the compressing ckt, hence
giving rise to signal distortion.
• To undo the distortion, at the receiver we pass the
recovered signal through an expander network
which has an inverse i/p-o/p characteristic to the
 characteristic of the compressor as shown in Fig.-
13(a).
• Thus both the inverse distortions of compressor
& expander generate a final o/p signal without
distortion.
• Fig. 13(a) shows the characteristics of a
compander which is the combination of
compressor & expander characteristics .
• Thus the original signals are first compressed
before the quantization process at the PCM
transmitter using a compression (μ or A) law as
shown in Fig.-14.
Fig. 13(a) - Characteristics of compandor.
• Hence to maintain the quality of SNR companding
(both Compression & expansion) is done.
• At the transmitter, the weak signals are amplified &
strong signals are attenuated by a compressor ckt.
Then the signal uniformly quantized for
transmission.
• At the receiver the reverse process is followed by an
expander circuit to recover the original
signal/samples against distortion by compressor.
• Due to the inverse nature of compressor & expander,
the overall characteristics of the compandar is a
straight line (as shown in solid line in Fig.-13(a)),
Fig. 13 (b) - Characteristics of compressor only.
 hence the original signals are brought back at the
receiver before holding circuit to avoid original
signal distortion as shown in Fig.-12(iii).
• Hence by the use of a large no. of steps for small
message samples the quantization noise power
can be lowered in order to improve the SNR
regardless of the signal amplitudes.
• Among several choices, 2 compression laws have
been accepted as desirable standards by the ITU-
T : the μ-law (eqn-1) used in North America &
Japan, & the A-law (eqn-2) used in Europe & rest
of the world.
• Both the μ-law & the A-law curves have odd
symmetry about the vertical axis. Here, m is the
i/p voltage whereas y is the normalized o/p
voltage.
• These characteristics are shown in Fig.-14.
• The μ-law (for +ve amplitude) is given by
1  m  m
y ln 1   ; 0  1 ....(1)

ln(1   )  m p  mp
The A  law ( for  ve amplitude) is
 A  m  m 1
   ; 0  
 1  ln A  m p  mp A
y ...(2)
 1  Am  1 m
1  ln A 1  ln m  ; A  m  1
  p  p

• The compression parameter μ (or A) determines

 μ-Law A-Law

Fig. 14 - Characteristics of compressor only.

 the degree of compression.
• To obtain a nearly constant S0/N0 over a dynamic
range of for i/p signal power 40 dB, μ should be
greater than 100.
• An optimum value of μ = 255 has been used for
all 8-bit (256-level) digital terminals. For the A-
law, a value of A = 87.6 gives comparable results
& has been standardized by the ITU-T.
• Thus by the use of compander along with a
uniform quantizer the non-uniform quantization is
implemented for improving o/p SNR.
• The compressor can be realized by a
semiconductor diode along with adjustable series
Resistors.
• The o/p SNR for the cases μ = 255 & μ = 0 (no
compression) as function of message signal power is
shown in Fig.-15.

Fig. 15 – Signal to quantization noise ratio in PCM with & without

compression.
 TDM of PCM signals
• The basic TDM scheme, called the T1 digital
system, accommodates 24 analog signals (s1-s24)
is shown in Fig-16.
• Each signal is limited to 3.3 KHz & sampled at
the rate of 8 KHz, i.e., it transmits 8000 no.s of
frames per sec or 8000 revolution per sec by the
commutator. Each frame contains 1+24x8 = 193
nos of bits (1st bit is the frame-synchronizing bit).
• Thus the bit rate on a T1 channel is 193
bits/125μs = 1.544 Mb/s as Ts = 125 μs.
Fig.-16 :- A T1 digital system.
 Multiplexing T1 lines-The T2 ,T3, T4, & T5
lines
• To take further advantage of the merits of TDM
& digital transmission, the common carriers
employ a hierarchy of further multiplexing as
shown in Fig.-17.
• 4 no.s of T1 lines are multiplexed in an M12
multiplexer to generate a T2 transmission system
having 193 x 4 +17 = 789 bits/frame.
• Thus the bit rate on a T2 channel is 789
bits/frame x 8000 frames/s = 6.312 Mb/s.
 T1

T2

T3

T4

Fig.-17 :- TDM hierarchy.

• Similarly 7 no.s of T2 lines are multiplexed in an
M23 multiplexer to generate a T3 transmission
system having 789 x 7 + 69 = 5592 bits/frame.
• Thus the bit rate on a T3 channel is 5592
bits/frame x 8000 frames/s = 44.736 Mb/s.
• Similarly 6 no.s of T3 lines are multiplexed in an
M34 multiplexer to generate a T4 transmission
system having 5592 x 6 + 720 = 34272
bits/frame.
• Thus the bit rate on a T4 channel is 34272
bits/frame x 8000 frames/s = 274.176 Mb/s.
• Similarly 2 no.s of T4 lines are multiplexed in an
M45 multiplexer to generate a T5 transmission system
having 34272 x 2 + 1476 = 70020 bits/frame.
•Thus the bit rate on a T5 channel is 70020 bits/frame
x 8000 frames/s = 560.16 Mbps.
 Line coding
• During the transmission of signal along the
channel from transmiiter to receiver, it may be
affected by several channel impirements which
may cause distortion to the transmitted signal.
• The different types of channel impirements that
affect the signal during transmission are :-
• (1) Signal distortion:- It distorts the signal
spectrum. As a result, the signal constellation
points are shifted from original location & get
distorted both in amplitude & phase.
• (2) Interference :- It is an unwanted disturbance
transmitted within the frequency band of the
modulated signal, for example :- ISI disrupts the
binary signal transmission severely.
• (3) Noise:- It also causes disturbances to the
signal. The noise either may be man made or
natural like thermal, mechanical, etc.
• (4) Fading :- is defined as the rapid fluctuation in
the strength of the received signal during short
periods of time of transmission due to variation of
refractive index of the medium or travelling
through multi-path.
• Hence to combat against these channel
impairments qualitative/efficient line coding
must be used.
• Also, quality line coding should care for the
following factors: (i) Minimum power & BW for
transmission, (ii) ability to extract/recover
timing/clock information, (iii) ability to reduce
low frequency or dc component as it is unsuitable
for ac coupled circuits, (iv) Favorable PSD for
matching to the available channel frequency
response, (v) ease of signal detection & decoding,
(vi) improve error detection & correction
capability, (vii) transparency to correctly transmit
a digital signal regardless of 0’s & 1’s, etc.