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1
CHAPTER 2
Syllabus:
1) Pulse amplitude modulation
2) TDM
3) Wave form coding techniques
4) PCM
5) Quantization noise and SNR
6) Robust quantization
Pulse amplitude modulation
In pulse amplitude modulation, the amplitude of a carrier consisting of a periodic train of
rectangular pulses is varied in proportion to sample values of the message signal. In this the
pulse duration is held constant, by making the amplitude of each rectangular pulse the same as
the value of the message signal at the leading edge of the pulse, which is exactly equal to flat
top sampling.
The important feature of PAM is a conservation of time. According to definition given
before in terms of rectangular pulse a wider bandwidth is required to transmit PAM, however if
we formulate the PAM in terms of standard pulse, we may then define the a PAM wave, s(t) as
( ) ∑
( ) ( )
( ) are sample values of the message signal g(t), is the sampling period.
Time division multiplexing
The block diagram for TDM is illustrated as shown in the figure
2
Each input message signal is restricted in bandwidth by a low pass filter to remove the
frequencies that are nonessential to an adequate signal representation, the pre-alias filter output
are then applied to a commutator.
Commutator is usually implemented using electronic switching circuitry. The function of
commutator is folds
1) To make a narrow sample of each of the input messages at a rate fs that is slightly
higher than 2W
2) To sequentially interleave these N samples inside a sampling interval Ts = 1/fs
Multiplexed signal is applied to pulse amplitude modulator the purpose of which is to
transform the multiplex signal into a form suitable for transmission over the channel.
Suppose ‘N’ message signal to be multiplexed having similar properties, then sampling
rate of each message signal is calculated. Let Ts denote sampling period and Tx denote the time
spacing between adjacent samples in multiplexed signal.
At receiving end, the received signal is applied to pulse amplitude demodulator which
performs the reverse operation of pulse amplitude modulator. The short pulses are applied to
LPF through decommutator, which operates in synchronism with the commutator in the
transmitter.
Example:
The waveform shown in the figure below illustrate the operation of a TDM system for N
= 2.The PAM waves g1 (t) and g2 (t) corresponding to message signal m1 (t) and m2 (t) are
depicted as sequence of uniformly spaced rectangular pulses. The PAM wave corresponding to
g1(t) is as shown in the shaded.
3
Pulse code modulation
The block diagram of pulse code modulation is as shown in the figure
Basic signaling elements of a PCM system are:
a) Transmitter
b) Transmission path
c) Receiver
The incoming message signal is passed through the low pass filter of cutoff frequency
‘W’ hertz, these filter blocks all the frequency components that are higher than ‘W’ hertz, and
hence m (t) is band limited signal.
a) The essential parts of transmitter are
i) Sampling:
The incoming message wave is sampled with train of narrow rectangular pulses
in order to ensure perfect reconstruction of the message signal at the receiver, the
sampling rate must be greater than twice the highest frequency component.
4
ii) Quantizing
Sampled signal is fed to quantizer, where the sampled signal approximated to
nearest preferred representation level.
The quantizing has two fold effects:
a) The peak – to – peak range of input sample values is subdivided into finite set
of decision levels that are aligned with the raisers of the staircase.
b) The output is assigned a discrete value selected from a finite set of
representation levels that are aligned with the treads of the staircase
iii) Encoding.
The combined process of sampling and quantizing will convert the continuous
baseband signal into its discrete set of values but not in the best suited form for
transmission.
Encoding process translate the discrete set of sample values to more appropriate
form of signal. A particular arrangement of symbol used in a code to represent a single
value of the discrete set is called a code – word or character.
b) Transmission path:
The regenerative repeaters are located at sufficiently close spacing along the
transmission path which has the ability to control the effects of distortion and noise
produced by transmitting a PCM wave through a channel.
c) Receiver
At the receiving end binary pulses are fed to the binary decoder which convert
the binary coded signals to a approximated pulses of discrete magnitudes these
approximated pulses are fed to reconstruction filters which reconstructs the original message,
the final output is a analog signal obtained from low pass filter.
Multiplexing:
In PCM, it is natural to multiplex different message sources by time division, as the
number of independent message sources is increased, the time interval that may be allocated to
each sources has to be reduced, since all of them must be accommodated into a time interval
equal to reciprocal of the sampling rate.
Synchronization:
For a PCM system with time division multiplexing to operate satisfactorily, it is
necessary that the timing operations at the receiver, except for the time lost in transmission and
5
regenerative repeaters, follow closely the corresponding operations at the transmitter, in general
way, this indicates that a clock is needed at the receiver to maintain same time as that of a
transmitter
One possible procedure to synchronize the transmitter and receiver is to set a code –
word derived from each independent message and to transmit this pulse every other frame, in
such a case, the receivers includes a circuit that would search for the pattern of 1s and 0s
alternating at half the frame rate, and there by establish synchronization between the transmitter
and the receiver.
Quantization
Sampled signal is fed to quantizer, where the sampled signal approximated to nearest
preferred representation level.
The quantizing has two fold effects:
a) The peak – to – peak range of input sample values is subdivided into finite set of
decision levels that are aligned with the raisers of the staircase.
b) The output is assigned a discrete value selected from a finite set of representation
levels that are aligned with the treads of the staircase
The difference between the two adjacent values is called “quantum” or a “step size”
indicated as ‘∆’.
Mid tread quantizer
In mid - tread quantizer the decision threshold of the quantizers are located at
and the representation levels are located at where ‘∆’ is
the step size. A uniform quantizer characterized in this way is referred as mid tread type,
because the origin lies in the middle tread of the staircase.
Suppose input lies between
( )
then the quantizer output is zero.
i.e.
( )
; ( )
For
( )
; ( )
6
Quantization error is given as: ( ) ( )
When ( ) , then ( ) , when ( )
, the quantizer output is zero
just before this level. Hence error is
near this level, hence maximum quantization error is
.
Mid riser quantizer
7
In mid - riser quantizer the decision threshold of the quantizers are located at
and the representation levels are located at
where ‘∆’ is
the step size. A uniform quantizer characterized in this way is referred as mid riser type, because
the origin lies in the middle riser of the staircase.
Suppose if the input is between 0 and ∆ or o and -∆, then the output is ∆/2 and -∆/2
respectively.
i.e. ( ) ; ( )
,
( ) ; ( )
Quantization error is given as:
( ) ( )
, when ( )
Maximum quantization error is
.
Encoding
The combined process of sampling and quantizing will convert the continuous baseband
signal into its discrete set of values but not in the best suited form for transmission.
Encoding process translate the discrete set of sample values to more appropriate form of
signal. A particular arrangement of symbol used in a code to represent a single value of the
discrete set is called a code – word or character.
In binary code, each word consists of ‘n’ bits then such a code,we may represent a total
of ‘2n’ distinct numbers. Ex: A sample quantized into one of 2
4 = 16 levels may be represented
by a 4 bit codeword.
There are several formats for the representations of binary sequence produced analog to
digital converter the below figure depicts two such formats first one is non return to zero
unipolar, where binary symbol is represented by a pulse of constant amplitude for a duration of
one bit, and ‘0’ is represented by switching off the pulse. Second one refers to non return to zero
polar, where binary symbol 1 and 0 are represented by pulse of positive and negative amplitude
respectively with each pulse occupying one bit duration.
Two binary wave forms a) nonreturn-to-
zero unipolar b) nonreturn-to-zero polar
8
Regeneration
The regenerative repeaters are located at sufficiently close spacing along the
transmission path which has the ability to control the effects of distortion and noise produced by
transmitting a PCM wave through a channel.
The three basic functions
a) Equalization:
The equalizer shapes the received pulse so as to accomplish the effect of
amplitude and phase distortion.
b) Timing circuit:
The timing circuit provides a periodic pulse train, derived from the received
pulse.
c) Decision making
In a PCM system with on – off signaling, the repeater makes decision in each bit
interval as to whether or not a pulse is present. If the decision is ‘yes’ a clean new pulse
is transmitted to the next repeater, if ‘no’ a clean base line is transmitted.
In this way, the accumulation of distortion and noise can be completely removed,
provided that the disturbance is not too large to cause an error in the decision making process.
The regenerative signal departs from the original signal from two main reasons:
a) The presence of channel noise and interference causes the repeater to make wrong
decisions occasionally, there by introducing bit error.
b) If spacing between the received pulses deviates from its original value, a jitter is
introduced into the regenerated pulse position thereby causing distortion.
9
Quantization noise and signal to noise ratio
Quantization noise is produced in the transmitter end of a PCM system by rounding off
sample values of analog baseband signal amplitude to nearest permissible representation level of
the quantizer.
Consider a memoryless quantizer that is both uniform and symmetric, with a total of ‘L’
representation levels. Let ‘x’ denotes the quantizer input and ‘y’ denote the quantizer output
these two are related by transfer characteristics of the quantizer
( )
Suppose that input ‘x’ lies inside the interval
{ } k= 1, 2…., n
Where ‘xk’ and ‘xk+1’ are decision threshold of the interval ‘Ik’ as depicted in figure
Correspondingly the quantizer output ‘y’ takes on discrete value ‘yk’, k = 1,2,3….. L that is
----------------- if ‘x’ lies in the interval ‘ik’
Let ‘q’ denote the quantization error, with values in the range
We may write
--------------------- if ‘x’ lies in the interval ‘ik’
Assume that quantizer input ‘x’ is the sample value of a random variable ‘x’ of zero mean and
variance .
When quantization is fine enough, the distortion produced by quantization noise affects
the performance of a PCM system as though it were it were an additive independent source of
noise with zero mean – square value determined by the quantizer step size ‘∆’.
It is found that the power spectral density of the quantization noise has a lager bandwidth
when compared with signal bandwidth. Thus quantization noise distributed throughout the
signal band.
The probability density function of quantization error is given by
( ) {
10
Q --------- Quantization error
q ---------- Denotes its sample
The mean of the quantization error is zero, and its variance is same as the mean square value.
[ ]
=∫
( )
=∫
(
)
=
Thus the variance of the quantization error produced by a uniform quantizer grows as the
square of the step size.
Let the variance of baseband signal x(t) at the quantizer input be denoted by when
baseband signal is reconstructed at the receiver outptut, we obtain the original signal plus
quantization noise, we may therefore define an output signal to quantization noise ration as
( )
=
Clearly the smaller the step size ‘∆’, the larger will be the SNR.
Channel noise
The ideal channel noise is the coding noise measured at receiver output with zero
transmitter input. The zero condition arises, for example silence in speech. The average power
depends on the quantizer used.
If quantizer is of midriser type zero input amplitude is encoded into one of the two inner
most level of representation
assuming that these two representation levels are equiprobable
the idle channel noise has zero mean and average power
.
It the quantizer is of midtread type, the output is zero for zero input and the ideal channel
noise is correspondingly zero. In practice the ideal channel noise is never be exactly zero due to
the inevitable presence of background noise or interferences. Accordingly the average power of
idle channel noise in a midtread quantizer is also in the order of or less than
.
11
Robust quantization
For a uniform quantizer with step size ‘∆’ the variance of the quantization noise is
provided that input signal does not overload the quantizer. Hence the variance of
quantization noise is independent of the variance of input signal.
Notably in the use of PCM where we transmission of speech signals, the same quantizer
has to accommodate input signals with widely varying power levels. It would therefore be
highly desirable from a practical viewpoint for the SNR to remain essentially constant, for a
wide range of input power levels. A quantizer that satisfies this requirement is said to be ‘robust’
The provision for such a robust performance necessitates the use of a “non uniform
quantizer” characterized by a step size that increases as the separation from the origin of transfer
characteristics is increased.
The desired form of non uniform quantization can be achieved by using a compressor
followed by a uniform quantizer, by cascading this combination with expander complementary
to the compressor the original signal samples are restored to their correct value except
quantization error
The figure depicts the transfer characteristics of the compressor, quantizer and expander
thus all the sample values of the compressor input, which lie in interval ‘Ik’ are assigned the
discrete value defined by the ‘kth
’ representation level at expander output.
The combination of compressor and expander is called a compander. Naturally in an actual
PCM system, the combination of compressor and uniform quantizer is located in the transmitter
while the expander is located at the receiver.
12
Variance of quantization error
The transfer characteristics of the compressor is represented by a memoryless nonlinear
c(x), where ‘x’ is the sample value of a random variable ‘X’ denoting the compressor input.
The characteristics c(x) is a monotonically increasing function that has odd symmetry
c(-x) = - c(x)
With sample value ‘x’ bounded in the range –xmax to xmax, the function c(x) similarly ranges from
–xmax to xmax, as shown by
{
c(x) ensures that it is completely invertible. Thus the sample value x of the compressor
input is reproduced exactly at the expander output. The compressor characteristics c(x) relates
nonuniform intervals at the compressor input to uniform intervals at the compressor output
Hence
( ) ( )
The uniform intervals are of width each, where ‘L’ is the number of
representation levels. The compressor characteristics c(x) in interval ‘Ik’ may then be
approximated by a straight line segment with slope equal to where is the width of
interval ‘Ik’.
( )
Where ( )
is the derivative of ‘c(x)’
Assumption:
1) The probability density function ( ) is symmetric
2) In each interval ‘Ik’, k=1,2,…..L-1 the probability density function ( ) is constant.
Hence, from second assumption we have
( ) ( )
Where the representation level ‘ ’ lies in the middle of interval ‘Ik’ i.e.
( )
Width of interval
( )
Accordingly, the probability that the random variable ‘X’ lies in interval ‘IK’
13
( )
( ) ------------1
With constraint
∑
Let the random variable ‘Q’ denotes quantization error
Variance of Q as:
[ ]
[( ) ]
∫ ( )
∫ ( )
( ) ---------------2
Using equation ‘1’ and dividing up the region of integration into ‘L’ intervals
∑
∫ ( )
∑
[ ( )
( ) ]
∑
{[(
( )]
[(
( ]]
}
∑
∑
In above formula, we have
, as the variance of quantization error conditional on
interval ‘Ik’ for a uniform quantizer of step size .
We have
∑
[ ( )
]
14
Substituting 4 in 3 we obtain,
∑ [
( )
]
We may equivalently write,
∑ ( ) [
( )
]
The output SNR of a nonuniform quantization is based on two assumptions:
1) The number of representation levels is large
2) The overload distortion is negligible.
Hence we have ( )
∫ ( )
** Hence
( ) ∫ ( )
∑ ( ) [ ( )
]
For Robust performance, the output signal – to-noise ratio should ideally be independent
of the probability density function of the input random variable ‘X’.
In nonuniform type of quantization the compressor characteristics ‘c(x)’ satisfy the first
– order differential equation.
( )
Where ‘k’ is a constant. Integrating above equation with respect to ‘x’ and using the boundary
condition that ( )
( ) (
)
If the ‘c(x)’tends to . Hence the c(x) is not realizable in practice.
15
**If the question is “Explain companding” start from this point.
Two widely used solutions to the problem are as follows:
1) µ law companding
In the ‘µ’law companding, the compressor characteristics c(x) is continuous,
approximating a linear dependency on ‘x’ for low input levels and logarithmic one for
high input levels.
The compression function ‘c(x)’ for µ law companding is
( )
(
) ( )
Where ‘µ’ is a constant .the typical value of ‘µ’ lies between 0 and 255. µ=0
corresponds to linear quantization. The ‘µ’ law is used for PCM telephone systems in the
US, Japan, Canada. The compressor characteristics are as shown in the figure.
2) A law companding:
In the ‘A’ law companding, the compressor characteristics c(x) is piecewise,
made up of linear segments for low input levels and logarithmic one for high input levels.
The compression function c(x) for A- law companding is
( )
{
(
)
‘A’ is constant here, practical value for ‘A’ is 87.56. The ‘A’ law is used for
PCM telephone systems in the Europe. The compressor characteristics are as shown in
the figure.
Note: As approximately logarithmic compression function is used for linear quantization, a
PCM scheme with non-uniform quantization scheme is also referred as “Log PCM” or
“Logarithmic PCM” scheme.