Convolution as a Filtering Operation

In a convolution of two signals $x\circledast y$ , where both $x$ and $y$ are signals of length $N$ (real or complex), we may interpret either $x$ or $y$ as a filter that operates on the other signal which is in turn interpreted as the filter's ``input signal''.^7.5 Let $h\in\mathbb{C}^N$ denote a length $N$ signal that is interpreted as a filter. Then given any input signal $x\in\mathbb{C}^N$ , the filter output signal $y\in\mathbb{C}^N$ may be defined as the cyclic convolution of $x$ and $h$ :

$$y = h\circledast x = x \circledast h$$

Because the convolution is cyclic, with $x$ and $h$ chosen from the set of (periodically extended) vectors of length $N$ , $h(n)$ is most precisely viewed as the impulse-train-response of the associated filter at time $n$ . Specifically, the impulse-train response $h\in\mathbb{C}^N$ is the response of the filter to the impulse-train signal $\delta\isdeftext [1,0,\ldots,0]\in\mathbb{R}^N$ , which, by periodic extension, is equal to

$$\delta(n) = \left\{\begin{array}{ll} 1, & n=0\;\mbox{(mod $N$)} \\ [5pt] 0, & n\ne 0\;\mbox{(mod $N$)}. \\ \end{array} \right.$$

Thus, $N$ is the period of the impulse-train in samples--there is an ``impulse'' (a `$1$ ') every $N$ samples. Neglecting the assumed periodic extension of all signals in $\mathbb{C}^N$ , we may refer to $\delta$ more simply as the impulse signal, and $h$ as the impulse response (as opposed to impulse-train response). In contrast, for the DTFT (§B.1), in which the discrete-time axis is infinitely long, the impulse signal $\delta(n)$ is defined as

$$\delta(n) \isdef \left\{\begin{array}{ll} 1, & n=0 \\ [5pt] 0, & n\ne 0 \\ \end{array} \right.$$

and no periodic extension arises.

As discussed below (§7.2.7), one may embed acyclic convolution within a larger cyclic convolution. In this way, real-world systems may be simulated using fast DFT convolutions (see Appendix A for more on fast convolution algorithms).

Note that only linear, time-invariant (LTI) filters can be completely represented by their impulse response (the filter output in response to an impulse at time 0 ). The convolution representation of LTI digital filters is fully discussed in Book II [71] of the music signal processing book series (in which this is Book I).