Convolution Representation

We will now derive the convolution representation in its full generality. The first step is to express an arbitrary signal $x(\cdot)$ as a linear combination of shifted impulses, i.e.,

$$x(n)=\sum_{i=0}^n x(i)\delta(n-i) \isdef (x\ast \delta)(n) \protect$$ (6.4)

where ``$\ast$'' denotes the convolution operator. (See [83]^6.4 for an elementary introduction to convolution.)

If the above equation is not obvious, here is how it is built up intuitively. Imagine $\delta(\cdot)$ as a 1 in the midst of an infinite string of 0s. Now think of $\delta(\cdot -i)$ as the same pattern shifted over to the right by $i$ samples. Next multiply $\delta(\cdot -i)$ by $x(\cdot)$, which plucks out the sample $x(i)$ and surrounds it on both sides by 0's. An example collection of waveforms $x(i)\delta(\cdot -i)$ for the case $x(i) = i, i = -2, -1, 0, 1, 2$ is shown in Fig.5.4a. Now, sum over all $i$, bringing together all the samples of $x$ one at a time, to obtain $x(\cdot)$. Figure 5.4b shows the result of this addition for the sequences in Fig.5.4a. Thus, any signal $x(\cdot)$ may be expressed as a weighted sum of shifted impulses.

Equation (5.4) expresses a signal as a linear combination (or weighted sum) of impulses. That is, each sample may be viewed as an impulse at some amplitude and time. As we have already seen, each impulse (sample) arriving at the filter's input will cause the filter to produce an impulse response. If another impulse arrives at the filter's input before the first impulse response has died away, then the impulse response for both impulses will superimpose (add together sample by sample). More generally, since the input is a linear combination of impulses, the output is the same linear combination of impulse responses. This is a direct consequence of the superposition principle which holds for any LTI filter.

Figure 5.4: Any signal may be considered to be a sum of impulses. (a) Family of impulses at various amplitudes and time shifts. (b) Addition of impulses in (a), giving part of a ramp $x(i)=i$.

We repeat this in more precise terms. First linearity is used and then time-invariance is invoked. Using the form of the general linear filter in Eq. (4.2), and the definition of linearity, Eq. (4.3) and Eq. (4.5), we can express the output of any linear (and possibly time-varying) filter by

$$\begin{eqnarray*} y(n) &=& {\cal L}_n\{x(\cdot)\}\\ &=& {\cal L}_n\{(x \ast \... ...ght\}\\ &\isdef & \sum_{i=-\infty}^\infty x(i) h(n,i) \protect \end{eqnarray*}$$

where we have written $h(n, i) \isdef {\cal L}_n\{\delta(\cdot - i)\}$ to denote the filter response at time $n$ to an impulse which occurred at time $i$. If we are to be completely rigorous mathematically, certain ``smoothness'' restrictions must be placed on the linear operator ${\cal L}$ in order that it may be distributed inside the infinite summation [37]. However, essentially all practically useful filters of the form of Eq. (5.1) satisfy these restrictions. If in addition to being linear, the filter is time-invariant, then $h(n, i) = h(n - i)$, which allows us to write

$$y(n) =\sum_{i=-\infty}^\infty x(i) h(n - i) \isdef (x\ast h)(n) \protect.$$ (6.5)

This states that the filter output $y(n)$ is the convolution of the input $x(n)$ with the filter impulse response $h(n)$. The infinite sum in Eq. (5.5) can be replaced by more typical practical limits. By choosing time 0 as the beginning of the signal, we may define $x(n)$ to be 0 for $n < 0$ so that the lower summation limit of $-\infty$ can be replaced by 0. Also, if the filter is causal, we have $h(n) = 0$ for $n < 0$, so the upper summation limit can be written as $n$ instead of $\infty$. This gets us down to the ultimate convolution representation of a linear, time-invariant, causal digital filter:

$$\zbox {y(n) = \sum_{i=0}^n x(i) h(n - i) = (x \ast h)(n)}$$

Since the above equation is a convolution, and since convolution is commutative (i.e., $(x \ast h)(n) = (h \ast x)(n)$ [83]), we can rewrite it as

$$\zbox {y(n) = \sum_{i=0}^n h(i) x(n - i) = (h \ast x)(n)}$$

or

$$y(n) = h(0) x(n) + h(1) x(n-1) + h(2) x(n-2) + \cdots + h(n) x(0).$$

This latter form looks more like the general difference equation presented in Eq. (5.1). In this form one can see that $h(i)$ may be identified with the $b_i$ coefficients in Eq. (5.1). It is also evident that the filter operates by summing weighted echoes of the input signal together. At time $n$, the weight of the echo from $i$ samples ago [$x(n - i)$] is $h(i)$.

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, then the difference equation and the convolution formula are identical; in this case, $h(i) = b_i$ and there are no $a$ coefficients in Eq. (5.1). 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 simply truncate the response after an appropriate length of time, such as after it falls below the quantization noise level due to round-off error.