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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 fq(t). The difference between fq(t) and f (t) is known as the quantization error εq(t),
εq (t) = fq (t) - f(t)
Theoretically, fq(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) εq(t) present. For a small δv with a large number of levels, it can be shown that the mean-squared value of εq(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 Nq 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.
This is at the heart of the analysis of electromagnetics. It is based on a series of experiments conducted by the French scientist Ampere (1775 - 1836), in which the force between
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