How to use convolution to implement filters: part 2

A moving-average filter can address white noise in the time domain but performs poorly in the frequency domain.

In part 1 of this series, we defined convolution, denoted by the * symbol, and looked at a simple geometrical example of how it operates to produce a new function y(t) from two given functions, f(t) and g(t).

Q: What’s a practical application of convolution?
A: In electrical engineering, we often have an input signal f(t) that we want to apply to a filter with the impulse response g(t). We can calculate the filter’s output y(t) = f(t) * g(t).

Q: Can you provide a specific numerical example?
A: The moving-average filter is a commonly used form of convolution filter. It averages m data samples to create a new y(t_n). For a 5-point moving-average filter:

Note that this equation represents the symmetrical form of a moving-average filter: to calculate y(t_n), we average f(t_n) with the two values to its left and the two values to its right. Alternatively, we can average f(t_n) with the four preceding or following samples. Figure 1a shows how the symmetrical form works. The convolution engine calculates y(t₆) as one-fifth of the sum of f(t₄) through f(t₈). When that calculation completes, the engine increments n, shifting one sample to the right, and calculates y(t₇) as one-fifth of the sum of f(t₅) through f(t₉), as shown in Figure 1b. The process continues until the entire dataset has been processed.

Figure 1. The convolution engine calculates y(tn) for n=6 (a) and then goes on to calculate y(tn) for n=7 (b).

Q: After the processing in Figure 1, what does the complete y(t) look like?
A: Figure 2a shows an unfiltered 4-Hz sawtooth wave (blue) and the output of our 5-point moving-average filter (red). The fall time of the filtered version is noticeably longer, as you would expect with a low-pass filter. If you look at the Fourier transform of the filtered signal (Figure 2b), you see that the higher frequency components are noticeably attenuated.

Figure 2. A filtered sawtooth wave shows longer fall times in the time domain (a) and attenuated high-frequency components in the frequency domain (b).

Q: Can I use the moving-average filter the same way I would a first-order low-pass analog filter?
A: No. The moving-average filter does not have the first-order analog filter’s smooth 20-dB per decade slope following a clean 3-dB cutoff frequency. One 15-point moving-average filter has the response shown in Figure 3. As you can see, this filter is not well-behaved in the frequency domain; the response periodically goes to zero and then rebounds.

convolution — Figure 3. The moving-average filter performs poorly in the frequency domain.

Q: What is it good for?
A: Look at the black trace in Figure 4, representing a simulated signal with considerable Gaussian white noise. The signal appears periodic, but whether it’s a sine wave, a triangle wave, or something else is hard to tell. The moving-average filter is very effective at removing such noise. A 5-point (orange trace), 15-point (blue trace), and 21-point (red trace) moving-average filter provide successively better approximations of the original sine wave.

Q: What’s the overall conclusion here?
A: In general, if you can most easily express a problem in the frequency domain—that is, you want to remove specific frequencies, above a cutoff frequency or within a stopband, for example—the moving-average filter is a poor choice. But if you want to smooth a signal — for example, remove wideband noise — in the time domain, then the moving-average filter is a good, simple choice.

Q: What else should I know about convolution?
A: There are many different types of convolution filters. We’ll discuss some in upcoming parts of this series. For example, we have looked at 1-dimensional filters, but 2-dimensional convolution filters are particularly applicable to image processing.