Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 411

In other words, the quantized value of the input lying within the levels d m and
d m + 1 is given by r m , which equals 0.5( d m + d m + 1 ). The quantization levels
{ d 0 , d 1 ,..., d L } are referred to as the decision levels , while the output levels
{ r 0 , r 1 ,..., r L − 1 } are referred to as the reconstruction levels .
Equation (9.27) approximates the analog sample values by using a finite
number of quantization levels. The approximation introduces a distortion, which
is referred to as the quantization error. The peak value of the quantization error
is one-half of the quantile interval in the positive or negative direction.
The quantizer illustrated in Fig. 9.12(a) is called a uniform quantizer because
the quantization levels are uniformly distributed between the minimum and
maximum ranges of the input sequence. In most practical applications, the
distribution of the amplitude of the input sequence is skewed towards low
values. In speech communication, for example, low speech volumes dominate
the sequence most of the time. Large-amplitude values are extremely rare and
typically occupy only 15% to 25% of the communication time. A uniform
quantizer will be wasteful, with most of the quantization levels rarely used.
In such applications, we use non-uniform quantization, which provides fine
quantization at frequently occurring lower volumes and coarse quantization at
higher volumes. The input-output relationship of a non-uniform quantizer is
shown in Fig. 9.12(b). The quantile interval is small at low values of the input
sequence and large at high values of the sequence.

Example 9.3
Consider an audio recording system where the microphone generates a CT
voltage signal within the range [ − 1, 1] volts. Calculate the decision and recon-
struction levels for an eight-level uniform quantizer.

Solution
For an L = 8 level quantizer with peak-to-peak range of [ − 1 , 1] volts, the
quantile interval is given by

= 1 − ( − 1)
8

= 0 . 25 V .

Starting with the minimum voltage of − 1 V, the decision levels d m are uniformly
distributed between − 1 V and 1 V. In other words,

d m =− 1 + m for 0 ≤ m ≤ L .

Substituting different values of m , we obtain

d m =− 1V , − 0 . 75 V , − 0 . 5V , − 0 . 25 V , 0V , 0 . 25 V ,
0 . 50 V , 0 . 75 V , 1V .