Sampling and Quantization - Digital Speech: Coding for Low Bit Rate Communication Systems - page 26

channel transmission bit rate is given by,

T c = Bf s bits/second (3.7)

Given a fixed sampling frequency, the only way to reduce the channel bit
rate T c is by reducing the length of the codeword c(n) . However, a reduced
length c(n) means a smaller set of discrete amplitudes separated by larger
and, hence, larger differences between the analogue and discrete amplitudes
after quantization, which reduces the quality of reconstructed signal. In order
to reduce the bit rate while maintaining good speech quality, various types
of scalar quantizer have been designed and used in practice. The main aim of
a specific quantizer is to match the input signal characteristics both in terms
of its dynamic range and probability density function.

3.3.1 QuantizationError
When estimating the quantization error, we cannot assume that i = i + n if
the quantizer is not uniform [2]. Therefore, the signal lying in the i th interval,

i
2

i
2

x i − ≤ s(n)<x i + (3.8)

is represented by the quantized amplitude x i and the difference between the
input and quantized values is a function of i . The instantaneous squared
error, for the signal lying in the i th interval is (s(n) −

x i ) 2 . The mean squared
error of the signal can then be written by including the likelihood of the signal
being in the i th interval as,

x i +

i
2

E i x i ) 2 p(x)dx = (x − (3.9)

i
2

x i −

where s(n) has been replaced by x for ease of notation and p(x) represents the
probability density function of x . Assuming the step size i is small, enabling
very fine quantization, we can assume that p(x) is flat within the interval
x i

2 . Representing the flat region of p(x) by its value at the centre,
p(x i ) , the above equation can be written as,

2

− to x i +

p(x i )

i
2

i

E i y 2 dy = =

12 p(x i )

(3.10)

i
2

−

The probability of the signal falling in the i th interval is,

x i +

i
2

i = p(x)dx = p(x i ) i (3.11)

2

x i −