Quantization Error

Sampling followed by quantization is equivalent to quantization followed by sampling. Figure illustrates a message signal f (t) and its quantized version denoted by f_q(t). The difference between fq(t) and f (t) is known as the quantization error ε_q(t),

ε_q (t) = f_q (t) - f(t)

Theoretically, f_q(t) can be recovered in the receiver without error. The recovery of fq(t) can be viewed as the recovery of f (t) with an error (or noise) ε_q(t) present. For a small δv with a large number of levels, it can be shown that the mean-squared value of ε_q(t) is given by

When a digital communication system transmits an analog signal processed by a uniform quantizer, the best SNR that can be attained is given by

where S₀ and N_q represent the average powers in f (t) and εq(t), respectively. When f (t) fluctuates symmetrically between equal magnitude extremes, i.e., -|f(t)|_max ≤ f(t) ≤ |f(t)|_max, choosing a sufficiently large number of levels L, the step size δv comes out as

It can be seen that messages with large crest factors will lead to poor performance.