Convolution Filtering

Welcome to this section on convolution and filtering for signals. In the last section, we saw examples of different audio effects we can create in MATLAB. This included adding a noise to our signal to create certain sound effects. Typically, we want to do the opposite. Often we're only provided a noisy signal and we want to remove the noise and recover the original signal. This is widely applicable to signals in general, not just audio signals. So how do we remove noise? Let's consider a snippet of a signal. It could be any type of signal over any time frame. But here, this is a snippet of an audio signal that has been corrupted by noise displayed over a short 10 millisecond time frame. The simplest and most common way to reduce noise is to smooth the signal. We assume that the noise has mostly high frequency content. That is, the noise signal changes rapidly, whereas the true signal changes slowly. This is certainly the case for the signals and noises seen in our plots. We see our signals are relatively smooth and slowly changing, whereas our noise involves rapid, erratic changes. High-frequency noise can be smoothed out by averaging our signal over time. This brings us to signal filtering methods. The term filtering generally refers to the process of manipulating the frequency content of a signal. If we reduce the high frequency portion of the signal while letting the low-frequency portion remain, we are filtering the signal. This type of filter is called a smoothing filter. Because by reducing high frequencies, we reduced the rapid changes in the signal, which makes the signal change more slowly over time, which corresponds to a smoother looking shape. It is also called a low-pass filter because it allows the low frequencies to pass through the filter while the high frequencies are filtered out. Filtering can be performed by computing a new value for each sample that is equal to a weighted average of the signal over time. The two things we control are the width of the filter. That is, how many signal samples do we include in our averaging window and the weight that's applied to each of those samples in the weighted average. Here's an example of a potential averaging window. Our filter computes an average in a window over time. Let's consider first one specific time, t equals 0.245 seconds, shown with the green line here.