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Quantization is a highly nonlinear process and is very difficult to analyze precisely. Approximations and assumptions are madeto make analysis tractable.
The roundoff or truncation errors at any point in a system at each time are random , stationary , and statistically independent (white and independent of all other quantizers in a system).
That is, the error autocorrelation function is . Intuitively, and confirmed experimentally in some (but notall!) cases, one expects the quantization error to have a uniform distribution over the interval for rounding, or for truncation.
In this case, rounding has zero mean and variance and truncation has the statistics
Please note that the independence assumption may be very
bad (for example, when quantizing a sinusoid with an integerperiod
). There is another
quantizing scheme called
dithering , in which
the values are randomly assigned to nearby quantizationlevels. This can be (and often is) implemented by adding a
small (one- or two-bit) random input to the signal before atruncation or rounding quantizer.
Pretend that the quantization error is really additive
Gaussian noise with the same mean and
variance as the uniform quantizer. That is, model
As
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