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To determine the roundoff noise at the output of a digital filter we will assume that the noise due to a quantization is stationary, white, and uncorrelated with the filter input, output, and internal variables.

**Roundoff Noise:**

To
determine the roundoff noise at the output of a digital filter we will assume
that the noise due to a quantization is stationary, white, and uncorrelated
with the filter input, output, and internal variables. This assumption is good
if the filter input changes from sample to sample in a sufficiently complex
manner. It is not valid for zero or constant inputs for which the effects of
rounding are analyzed from a limit cycle perspective.

To
satisfy the assumption of a sufficiently complex input, roundoff noise in
digital filters is often calculated for the case of a zero- mean white noise
filter input signal *x(n)* of variance *a* ^{1}. This simplifies
calculation of the output roundoff noise because expected values of the form *E{x(n)x(n — k)}* are zero for *k =* 0 and give a^{2} when *k =* 0.

Another
assumption that will be made in calculating roundoff noise is that the product
of two quantization errors is zero. To justify this assumption, consider the
case of a 16-b fixed-point processor. In this case a quantization error is of
the order 2^{—1 5} , while the product of two quantization errors is of
the order 2^{—3 0} , which is negligible by comparison.

If a
linear system with impulse response *g(n)*
is excited by white noise with mean *m _{x}*
and variance a

Therefore,
if *g(n)* is the impulse response from
the point where a roundoff takes place to the filter output, the contribution
of that roundoff to the variance (mean-square value) of the output roundoff
noise is given by (3.25) with *a ^{2}*
replaced with the variance of the roundoff. If there is more than one source of
roundoff error in the filter, it is assumed that the errors are uncorrelated so
the output noise variance is simply the sum of the contributions from each
source.

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