Applying the Convolution Theorem

The convolution theorem is useful as a qualitative tool in predicting the effects of different operations in discrete linear time-invariant systems. For example, many authors use the convolution theorem to show why periodic sampling of continuous signals results in discrete samples whose spectra are periodic in the frequency domain. Consider the real continuous time-domain waveform in

Figure 5-44(a), with the one-sided spectrum of bandwidth B. Being a real signal, of course, its spectrum is symmetrical about 0 Hz. (In Figure 5-44, the large right-pointing arrows represent Fourier transform operations.) Sampling this waveform is equivalent to multiplying it by a sequence of periodically spaced impulses, Figure 5-44(b), whose values are unity. If we say that the sampling rate is fs samples/second, then the

sample period ts = 1/fs seconds. The result of this multiplication is the sequence of discrete time-domain

impulses shown in Figure 5-44(c). We can use the convolution theorem to help us predict what the frequency- domain effect is of this multiplication in the time domain. From our theorem, we now realize that the spectrum of the time-domain product must be the convolution of the original spectra. Well, we know what the spectrum of the original continuous waveform is. What about the spectrum of the time-domain impulses? It has been shown that the spectrum of periodic impulses, whose period is ts seconds, is also periodic impulses in the

frequency domain with a spacing of fs Hz as shown in Figure 5-44(b)[30].

Now, all we have to do is convolve the two spectra. In this case, convolution is straightforward because both of the frequency-domain functions are symmetrical about the zero-Hz point, and flipping one of them about zero Hz

is superfluous. So we merely slide one of the functions across the other and plot the product of the two. The convolution of the original waveform spectrum and the spectral impulses results in replications of the waveform spectrum every fs Hz, as shown in Figure 5-44(c). This discussion reiterates the fact that the DFT is

always periodic with a period of fs Hz.

Here’s another example of how the convolution theorem can come in handy when we try to understand digital signal processing operations. This author once used the theorem to resolve the puzzling result, at the time, of a triangular window function having its first frequency response null at twice the frequency of the first null of a rectangular window function. The question was “If a rectangular time-domain function of width T has its first spectral null at 1/T Hz, why does a triangular time-domain function of width T have its first spectral null at 2/T Hz?” We can answer this question by considering convolution in the time domain.

Look at the two rectangular time-domain functions shown in Figures 5-45(a) and 5-45(b). If their widths are each T seconds, their spectra are shown to have nulls at 1/T Hz as depicted in the frequency-domain functions in Figures 5-45(a) and 5-45(b). We know that the frequency magnitude responses will be the absolute value of the classic sin(x)/x function.† If we convolve those two rectangular time-domain functions of width T, we’ll get the triangular function shown in Figure 5-45(c). Again, in this case, flipping one rectangular function about the zero time axis is unnecessary. To convolve them, we need only scan one function across the other and determine the area of their overlap. The time shift where they overlap the most happens to be a zero time shift. Thus, our resultant convolution has a peak at a time shift of zero seconds because there’s 100 percent overlap. If we slide one of the rectangular functions in either direction, the convolution decreases linearly toward zero. When the time shift is T/2 seconds, the rectangular functions have a 50 percent overlap. The convolution is zero when the time shift is T seconds—that’s when the two rectangular functions cease to overlap.

† _{The sin(x)/x function was introduced in our discussion of window functions in Section 3.9 and is covered in greater detail in Section}

3.13.

Figure 5-45 Using convolution to show that the Fourier transform of a triangular function has its first null at

Notice that the triangular convolution result has a width of 2T, and that’s really the key to answering our question. Because convolution in the time domain is equivalent to multiplication in the frequency domain, the Fourier transform magnitude of our 2T-width triangular function is the |sin(x)/x| in

Figure 5-45(a) times the |sin(x)/x| in Figure 5-45(b), or the (sin(x)/x)2 function in Figure 5-45(c). If a triangular function of width 2T has its first frequency-domain null at 1/T Hz, then the same function of width T must have its first frequency null at 2/T Hz as shown in Figure 5-45(d), and that’s what we set out to show. Comparison of

Figures 5-45(c) and 5-45(d) illustrates a fundamental Fourier transform property that compressing a function in the time domain results in an expansion of its corresponding frequency-domain representation.

We cannot overemphasize the importance of the convolution theorem as an analysis tool. As an aside, for years I thought convolution was a process developed in the second half of the twentieth century to help us analyze discrete-time signal processing systems. Later I learned that statisticians had been using convolution since the late 1800s. In statistics the probability density function (PDF) of the sum of two random variables is the convolution of their individual PDFs.