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 S_{0} 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.