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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

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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.

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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.

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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

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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

( )

; ( )

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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

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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

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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.

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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

( ) {

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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

.

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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.

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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’

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( )

( ) ------------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

∑

[ ( )

]

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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.

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**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.