Convolution as a Filtering Operation

We may interpret either of the signals $x$ or $y$ as the impulse-train response of a linear, time-invariant, digital filter (see §8.3 for an introduction to the digital-filter point of view). To emphasize this interpretation, we use the notation $h(n)$ to denote the impulse-train-response signal at time $n$. More specifically, the impulse-train response $h\in{\bf C}^N$ may be defined as the response of the filter to the impulse-train signal $\delta \isdeftext [1,0,\ldots,0]\in{\bf R}^N$, 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 ${\bf C}^N$ for purposes of DFT analysis, we may refer to $\delta$ more simply as the impulse signal, and $h$ as the impulse response (as opposed to impulse-train response). However, because convolution as defined here (for DFT signals) is cyclic, the corresponding filter interpretation requires periodic extension of all input signals $x\in{\bf C}^N$. In contrast, for the DTFT (§B.1), in which the discrete-time axis is infinitely long, the impulse signal $\delta(n)$ is defined by

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

and no periodic extension is needed.

For any input signal $x\in{\bf C}^N$, we define the filter output signal $y\in{\bf C}^N$ as the cyclic convolution of $x$ and $h\in{\bf C}^N$:

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

As discussed below in §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).

The convolution representation of linear, time-invariant, digital filters is fully discussed in Book II [66] of the music signal processing book series (in which this is Book I).