We have shown that the output
of any LTI filter may be calculated
by convolving the input
with the impulse response
. It is
instructive to compare this method of filter implementation to the use
of difference equations, Eq.
(5.1). If there is no feedback (no
coefficients in Eq.
(5.1)), then the difference equation and
the convolution formula are essentially *identical*, as shown in
the next section.
For recursive filters, we can convert the difference equation into a
convolution by calculating the filter impulse response. However, this
can be rather tedious, since with nonzero feedback coefficients the
impulse response generally lasts forever. Of course, for stable
filters the response is infinite only in theory; in practice, one may
truncate the response after an appropriate length of time, such as
after it falls below the quantization noise level due to round-off
error.

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