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(U.P. Tech., Sem. Examination, 2003-2001)
As discussed earlier, in uniform quantization, the quantizer has a linear characteristics. The step size also remains same throughout the range of quantizer. Therefore, over the complete range of inputs, the maximum quantization error also remains same.
We know that the quantization error is given as under:
Maximum quantization error =                                                        …(4.33)
Also, step size ” is expressed as,
If x(t) is normalized, its maximum value i.e, xmax= 1.
Therefore, we have,  D =                                                                                  …(4.34)
Let us consider an example of PCM system in which we take v = 4 bits.
Then number of levels q will be,
q = 24= 16 levels.
Thus, using equation (4.34) the step size D will be,
Hence, quantization error is given from equation (4.33) as,
Hence, note that here the quantization error is th part of the full voltage* range. For simplicity, we assume that full range voltage is 16 volts. Then maximum quantization error will be 1 volt. However, for the low signal amplitudes like 2 volts, 3 volts etc., the maximum quantization error of 1 volt is quite high i.e., about 30 to 50%. This means that for signal amplitudes which are dose to 15 volts, 16 volts etc., the maximum quantization error (which is same throughout the range) of 1 volt can be considered to be small. In fact, this problem arises because of uniform quantization. Therefore non-uniform quantization should be used in such cases.
* If we consider the input signal as a voltage signal.
NOTE: In other words, we can say that it is desirable that SNRQ should remain essentially constant over a wide range of input power levels. A quantizer that satisfies all these requirements is known as a Robust Quantizer. Infact, such a robust performance can be obtained by using a nonuniform quantization.
            We know that speech and music signals are characterized by large crest factor. This means that for such signals the ratio of peak to rms value is quite high.
We know that the signal to noise ratio is given by,
Expressing in decibles, the last expression becomes
log10 (3 x 22v x P)
If we normalize the signal power i.e., if P = 1, then above equation becomes,
Here, power P is defined as
v2signal =  mean square value of signal voltage = x2(t)
Hence, normalized power will be,
[with R = 1]
P = x2(t)                                                                         …(4.38)
From equation (4.35), crest factor is given as,
Crest factor =                                                        …(4.39)
or                     Crest factor =                               since P = x2(t)                 …(4.40)
When we normalize the signal x(t), then
xmax = 1                                                            …(4.41)
Substituting this value of xmax in equation (4.40), we get
Crest factor =                                                                      …(4.42)
For a large crest factor of voice (i.e., speech) and music signals, P should be very very les, than one in above equation.
ie..                                           P << 1             for large crest factor
Therefore, actual signal to noise ratio would be significantly less than the value which is given by equation (4.37) since in this equation P=1.
Again, consider equation (4.36).
x x P
(3 x x P                                   …(4.43)
This equation illustrates that the signal to noise ratio for large crest factor signal (P << 1) would be very very less than that of the calculated theoretical value. The theoretical value is obtained for normalized power (P = 1) by equation (4.37).
NOTE: Therefore, such large crest factor signals (i.e., speech and music) should use non-uniform quantization to overcome the problem just discussed. Signal to noise ratio reduces at low power levels (P << 1) just now we have observed by equation (4.43). This means that at low signal levels, signal to noise ratio reduces i.e., noise increases. However, the quantization noise is directly related to step size. Therefore, at low signal levels (P << 1), noise can be kept low by keeping step size low. This means that at low signal levels, signal to noise ratio can be increased by decreasing step size ‘D’. This means that step size ‘D’ should be varied according to the signal level to keep signal to noise ratio at the required value. This is nothing but nonuniform quantization. Now let’s see how nonuniform quantization is achieved through companding in the section to follow.
If the quantizer characteristics is nonlinear and the step size is not constant instead if it is variable, dependent on the amplitude of input signal then the quantization is known as nonuniform quantization. In non-uniform quantization, the step size is reduced with the reduction in signal level. For weak signals (P < < 1), the step size is small, therefore the quantization noise reduces, to improve the signal to quantization noise ratio for weak signals. The step size is thus varied according to the signal level to keep the signal to noise ratio adequately high. This is nonuniform quantization. The non-uniform quantization is practically achieved through a process called companding. We shall discuss companding in the next section.
            As a matter of fact, companding is nonuniform quantization. It is required to be implemented to improve the signal to quantization noise ratio of weak signals. We know that the quantization noise is given by
Nq =
This shows that in the uniform quantization, once the step size is fixed, the quantization noise power remains constant. However, the signal power is not constant. It is proportional to the square of signal amplitude. Hence signal power will be small for weak signals, but quantizatior noise power is constant. Therefore, the signal to quantization noise for the weak signals is very
poor. This will affect the quality of signal. The remedy is to use companding. Companding is a term derived from two words i.e., compression and expansion as under:
Companding = Compressing + Expanding
In pratice, it is difficult to implement the non-uniform quantization because it is not known in advance about the changes in the signal level. Therefore, a particular method is used. The weak signals are amplified and strong signals are attenuated before applying them to a uniform quantizer. This process is called as compression and the block that provides it is called as a compressor.
FIGURE 4.17 A companding model.
At the receiver exactly opposite is followed which is called expansion. The circuit used for providing expansion is called as an expander. The compression of signal at the transmitter and expansion at the receiver is combined to be called Compressor companding. The process of companding has been shown output in the form of a block diagram in figure 4.17.
4.21.1. Compressor Characteristic
Figure 4.18 shows the compressor characteristics. As shown in figure 4.18, the compressor provides a higher gain to the weak signals and smaller gain to the strong input signals. Thus, weak signals are artificially boosted to improve the signal to quantization noise ratio. It may be noted that this compressor characteristics has been shown only for the positive input signal but we can draw it even for the negative input signals using the some principle. In fact, the compressor is included at the PCM transmitter.
FIGURE 4.18 Compressor Characteristics.
4.21.2. Expander Characteristics
Figure 4.19 shows the expander characteristics. This characteristics is exactly the inverse the compressor characteristics. This ensures that all the artificially boosted signals by the compressor are brought back to their original amplitudes at the receiver end.
figure 4.19 Expander characteristics
Figure 4.17 shows the compander characteristics which is the combaination of the compressor and expander characteristics. Due to the inverse nature of compressor and expander, the overally characteristics of the compander is a straight line (dotted line in figure 4.17. This indicates that all the boosted signals are brought back to their original amplitudes.
FIGURE 4.20 Companding curves for PCM system
Ideally, we need a linear compressor characteristics for small amplitudes of the input signal and a logarithmic characteristic elsewhere. In pratice, this is achieved by using following two methods:
(ii)        A-law comapding
4.23.1   Companding
In the -law companding, the compressor characteristic is conntinouous. It is approximately linear for smaller values of input levels and logarithmic for high input levels. The -law compressor characteristic is mathematically expressed as under:
where                                      0 ≤  ≤ 1.
Here, z(x) represents the output and x is the input to the compressor. Also,  represents the normalized value of input with respect to the maximum value xmax. Further, (sgn x) term represents ± 1 i.e., positive and negative values of input and output. The -law compressor characteristics for different values of  have been shown in figure 4.18 (a). The practically used value of  is 255.
It may be noted that the characteristic corresponding to  = 0 corresponds to the uniform quantization. The -law comanding is used for speech and music signals. It is used for PCM telephone systems in United States, Canada and Japan. Figure 4.18(b) shows the variation of signal to quantiztion noise ratio with respect to signal level, with and without companding. It is obvious that SNR is almost constant at all the signal levels when companding is used.
4.23.2 A-law Comanding                                    (U.P. Tech, Sem. Exam., 2004-2005)
In the A-law companding, the compressor characteristic is piecewise, made up of a linear segment for low level inputs and a logarithmic segment for high level inputs. Figure 4.19 shows the A-law compressor characteristics for different values of A. Corresponding to A = 1, we observe that the characteristic is linear which corresponds to a uniform quantization. The practically used value of A is 87.56. The A-law companding is used for PCM telephone systems in Eurpoe. The linear segment of the characteristics is for low level inputs whereas the logarithmic segments is for high level input. It is mathematically expressed as under:
FIGURE 4.22 Compressor characteristic of A-law compressor.
DIAGRAM                                                    …(4.44a)