Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 166

3.6.1 Quantization

Sampling is the first step to preparing an analog signal for processing with digital
techniques. After a signal is sampled it is often referred to as a pulse amplitude
modulation (PAM) signal. Sampling alone is not enough to fully prepare an analog
signal for digital processing, since the amplitude can take an infinite number of
possible values; quantization therefore needs to be subsequently applied. This
process involves representing the analog amplitudes using a finite number of
levels.
After quantization, data is typically encoded into a multibit binary sequence
before it is transmitted. This process is known as pulse code modulation (PCM),
and the resulting signal is known as PCM signal. In PCM, each sample is repre-
sented by one of 2 M code words, each of length M bits. If f s is the sampling rate,
the data rate is Mf s bps.
Normally multibit analog-to-digital converters (ADCs) perform all of the
above-mentioned operations: sampling, quantization and binary encoding. The
accuracy (resolution) of multibit ADC depends on the number of bits, M.This
accuracy can be increased by increasing the number of bits, M, but this obviously
requires more sophisticated hardware.
Quantization can be uniform or nonuniform, as discussed below.

3.6.1.1 Uniform Quantization

In uniform quantization, the quantization levels are uniformly distributed over the
full allowable range. For example, if the input voltage is limited to ±V volts, then
the whole range 2V is divided into N = 2 M steps that represent any analog voltage
between -V and +V. The approximation accuracy is dependent on the quantization
step D ¼ 2V = N : The quantization error e is given by:

D

2 \e\ D 2 :

If it is assumed that the analog signal takes all values between -V and +V with
equal probability, then the quantization error e is uniformly distributed over
ð D = 2 ; D = 2 Þ: Hence, the pdf of e is:

p ð e Þ¼ 1

D :

The mean of this error is zero since:

m e ¼ Z

D = 2

e p ð e Þ de ¼ Z

D = 2

e ð 1 = D Þ de ¼ 0 ð 3 : 53 Þ

D = 2 D = 2