TRANSMISSION BANDWIDTH IN A PCM SYSTEM , QUANTIZATION NOISE / ERROR IN PCM

what is TRANSMISSION BANDWIDTH IN A PCM SYSTEM ? and QUANTIZATION NOISE / ERROR IN PCM ?
In this section, we shall evalute the transmission bandwidth for PCM system. Let us assume that the quantizer use ‘v’ number of binary digits to represent each level.
q = 2^v                                                               …(4.3)
Here ‘q’ represents total number of digital levels of a q-level quantizer.
For example, if v=4 bits, the total number of levels will be,
q = 2⁴ = 16 levels
Each sample is converted to ‘v’ binary bits i.e.,
Number of bits per sample = v
We know that,
Number of samples per second = f_s
Therefore, Number of bits per second is expressed as
(Number of bits per second) = Number of bits per samples) x (Number of samples per second) = v bits per sample x f_s samples per second ….(4.4)
As a matter of fact, the number of bits per second is known as signaling rate of PCM and is denoted by ‘r’ i.e.,
Signaling rate in PCM, r = v f_s                                                               …(4.5)
Where                                      f_x ≥ f_m
Also, since bandwidth needed for PCM transmission is given by half of the signaling rate therefore, we have
Transmission Bandidth in PCM,
BW ≥                                                                      …(4.6)
But                                          r = vf_s
Therefore,                                BW ≥                                                                       …(4.7)
Again, since                            f_s ≥ 2f_m                                                                             …(4.8)
This is the required expression for bandwidth of a PCM system.
4.12     QUANTIZATION NOISE / ERROR IN PCM
            In this section, we shall derive an expression for quantization noise (i.e., error) in a PCM system for linear quantization or uniform quantization. Because of quantization, inherent errors are introduced in the signal. This error is called quantization error. As defined earlier, the quantization error is given as
…(4.9)
Let us assume that the input x(nT_s) to a linear or uniform quantizer has continuous amplitude in the range
Form figure 4.9(a), it may be observed that the total excursion of input x(nT_s) in mapped into ‘q’ levles on veritcal axis. This means that when input is 4D, output is  and when input is -4D, output is . Thus, + represents . Therefore, the total amplitude range becomes,
Total amplitude range = x_max – (-x_max) = 2 x_max
Now, if this total amplitude range is divided into ‘q’ levles of quantizer, then the step size ‘D’ will be,
‘step size’ D =                                       …(4.10)
Again, now if signal x(t) is normalized to minimum and maximum values equal to 1, then we have
…(4.11)
Therefore, step side would be,
(for normalized signal)                                    …(4.12)
Now, if step size ‘D’ is considered as sufficiently small, then it may be assumed that the quantization error ‘ will be an uniformly distributed random variable. We know that the maximum quantization error is given as,
…(4.13)
i.e.,
Hence, over the interval quantization error may be assumed as an uniformly distributed random variable.
DIAGRAM
FIGURE 4.11 (a) A Uniform distribution (b) A Uniform distribution for quantization error
Figure 4.11 (a) shows an uniformly distrubuted random variable ‘X’ over an interval (a,b). Receall that the PDF of uniformly distributed random variable ‘X’ is given as
EQUATION
Thus, with the help of equation (4.15), the probability density function (PDF) for quantization error ” has aero average value. In other words, the mean ‘m’ of the quantization error is zero.
Further, we know that the signal to quantization noise ratio of the quantizer is defined as,
…(4.17)
If type of signal at input i.e., x(t) is known, then it is possible to calculate signal power.
The noise power is expressed as,
Noise power =                                      …(4.18)
Here,  is taken as the mean square value of noise voltage. Since, here noise is defined by random variable ” and PDF  therefore, its mean squre value is given as,
mean square value =                          …(4.19)
We know that the mean squre value of a random variable ‘X’ is expressed as,
EQUATION
Here                                                      EQUATION
Using equation (4.16), above equation may be written as,
EQUATION
or                                                            EQUATION
Simplifying, we get
Or                                                                                                              …(4.22)
Now, using equation (4.19), then mean square value of noise voltage would be
*          Here, we are assuming that x(t) is a voltage signal.
Also, if load resistance, R=1 ohm, then the noise power is normalized i.e.,
Noise power (normalized)   (putting R = 1 in equation (4.18)
=
Hence, finally, we write
Normalized noise power
or         Quantization noise power                               = , for linear quantization.
or         Quantization error (in terms of power)
4.13     SIGNAL TO QUANTIZATION NOISE RATIO FOR LINEAR QUANTIZATION
We know that in a PCM system for linear quantization the signal to quantization noise ratio is given as,
=
But, normalized noise power has been calculated as
Therefore,                     =                              …(4.24)
We know that the number of bits ‘v’ and quantization levels are related as,
q = 2^v                                       …(4.25)
Let us assume that input x(nT_s) to a linear quantizer has continuous amplitude in the range
Therefore, total amplitude range
= x_max – (-x_max) = 2x_max
Now, the step size will be
…(4.26)
Here, substituting the value of q from equation (4.25) in equation (4.26), we get

Now substituting this value in equation (4.24) we get,
Let normalized signal power be denoted as ‘P’.
Then,                                                   EQUATION
This is the required relation for signal to quantization noise ratio for linear quantization in a PCM system.
Hence, signal to quantization noise ratio:
…(4.27)
This expression shows that signal to noise power ratio of quantizer increase exponentially with increasing bits per sample.
Now if we assume that input x(t) is normalized, i.e.,
…(4.28)
Then, signal to quantization noise ratio will be,
…(4.29)
Also, if the destination signal power ‘P’ is normalized, i.e.,
…(4.30)
Then, the signal to noise ration will be given as
…(4.31)
Because x_max = 1 and p ≤ 1, the sigal to noise ratio given by equation (4.31) is said to be normalized.
Expressing the signal to noise ratio in decibels, we get
EQUATION
Thus, signal to quantization noise ratio for normazlied values of power ‘P’ and amplitude of input x(t), will be
…(4.32)
4.14     INFLUENCE OF NOISE ON THE PCM SYSTEM
To illustrate the influence of noise on the transmitted pulses, let us consider figure 4.12.
First, let us look at figure 4.12(a). Due to the noise superimposed on the pulse, only the PAM system shall be affected. However, the PWM, PPM and PCM systems will remain unaffected. The regeneration of the pulses is achieved by suing a clipper circuit with reference levels A and B. Now, let us observe figure 4.12(b). Here, the sides of the transmitted pulse are not perfectly vertical. In practice, the transmitted pulses usually have slightly sloping sides (edges). As the noise is superimposed on them, the width and the position of the regenerated pulses is changed. Now, this is going to distrot the information contents in the PWM and PPM signals.
DIAGRAM
FIGURE 4.12 Effect of noise on PCM.
NOTE: However, PCM is still unaffected as it does not contain any information in the width or the position of the pulses. Hence, PCM has much better noise immunity as compared to PAM, PWM and PPM systems.
4.15 VARIOUS IMPORTANT ASPECTS RELATED TO PCM
1.15.1. Advantages of Digital Representation of a Signal : Salient Features
The digital representation of a signal has following advantages :
(i)         Immunity to transmission noise and interference.
(ii)        It is possible to regenerate the coded signal along the transmission path.
(iii)       Communication can be kept private and secured by the use of encryption technique.
(iv)       The possibility of uniform format for different kinds of baseband signals.
(v)        It is possible to store the signal and process it whenever required.
4.15.2. Drawbacks
            The advantages listed above are attained at the cost of following factors:
(i)         Increased transmission bandwidth.
(ii)        Increased system complexity.
(iii)       PCM belongs to a class of signal coders known as waveform coders.
(iv)       The name waveform coders is given since in PCM, an analog signal is usually approximated by mimicking the amplitude – versus – time waveform.
4.15.3. Difference Between Waveform Coding and Source Coding
            (i)         Basically, the waveform coders are generally designed to be signal independent.
(ii)        The waveform coders are different from the source coders (i.e., linear predictive coders). The source coders depend on parameterization of the analog signal in accordance with an appropriate model for the generation of the signal.