Convolution

Convolution is an important operation on two functions which produces a third ( $\mathrm{f}_3 = \mathrm{f}_1 \ast \mathrm{f}_2$, $\ast$ is the symbol for convolution) in the following way

$$(\mathrm{f}_1 \ast \mathrm{f}_2)(x) = \int_{-\infty}^{\infty} \mathrm{f}_1(x') \mathrm{f}_2(x - x') \textrm{d}x'$$ (3.23)

Intuitively, this means, multiplying two functions and integrating them, then plotting the result at zero, then shift one function by x, again multiply and integrate, plot the result at x, and so on. Convolution is, loosely spoken, a sliding weighted average of one function with another function providing the weights. An example of a convolution is depicted in Fig. 3.3. Two box functions convolved result in a tent function.

Figure 3.3: An example for a convolution. Two box functions convolved result in a tent function.

As another example consider an arbitrary function convolved with an impulse function at a certain position k, i.e., $\mathrm{f}_2(x) = \delta(x-k)$. We simply plug this into the definition of the convolution

$$\mathrm{f}_C(x) = \int_{-\infty}^{\infty} \delta(x - k) \mathrm{f}(x - x') \textrm{d}x = \mathrm{f}(x - k)$$ (3.24)

The convolution of a function with an impulse function shifted by k is the function itself shifted by k.

Important for our purposes is the convolution of a set of samples (located at integer positions for the sake of simplicity) with a continuous (finite) function (which we can regard as a reconstruction filter). First we note that the result will be continuous and second, if the continuous function has the value one at x=0 and zero at all other integer positions, the resulting function will interpolate the samples. In Fig. 3.4 a set of samples is convolved with a tent function which results in a linear interpolation of the samples.

Figure 3.4: Convolution of a set of samples and a tent function of width of the sampling distance results in linear interpolation of the samples.

Convolution itself has a few nice properties. The first, we used it without explanation in Eq. 3.24, is commutativity, i.e.,

$$\mathrm{f}_1 \ast \mathrm{f}_2 = \mathrm{f}_2 \ast \mathrm{f}_1$$ (3.25)

Furthermore, it is distributive and associative, i.e.,

$$\mathrm{f}_1 \ast (\mathrm{f}_2 + \mathrm{f}_3) = \mathrm{f}_1 \ast \mathrm{f}_2 + \mathrm{f}_1 \ast \mathrm{f}_3$$ (3.26)

$$\mathrm{f}_1 \ast (\mathrm{f}_2 \ast \mathrm{f}_3) = (\mathrm{f}_1 \ast \mathrm{f}_2) \ast \mathrm{f}_3$$ (3.27)

In fact, convolution possesses a whole algebra, which will not be discussed here in detail.