Convolution Representation Summary

We have shown that the output $ y$ of any LTI filter may be calculated by convolving the input $ x$ with the impulse response $ h$. It is instructive to compare this method of filter implementation to the use of difference equations, Eq.$ \,$(5.1). If there is no feedback (no $ a_j$ 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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