We can visualize the shape of the filter by plotting the weights the filter uses in the weighted average when the filter is applied at time 0.245 seconds. First, let's consider using equal weights. Next, we control the weights that are used in the weighted average. Our filter will compute a weighted average of all of the entries in our noisy red curve within this range to determine what value is assigned to our new filtered signal at the specific time, 0.245 seconds. Thus, the range of time bracketed by the blue lines is defined as our filter width, which we call w. This time range is indicated by our blue lines. Our filter will average all of the values of the red curve contained in a certain time range surrounding this green line. Let's consider first one specific time, t equals 0.245 seconds, shown with the green line here. Our filter computes an average in a window over time. Here's an example of a potential averaging window. 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. The two things we control are the width of the filter. Filtering can be performed by computing a new value for each sample that is equal to a weighted average of the signal over time. 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. 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. This type of filter is called a smoothing filter. If we reduce the high frequency portion of the signal while letting the low-frequency portion remain, we are filtering the signal. The term filtering generally refers to the process of manipulating the frequency content of a signal. This brings us to signal filtering methods. High-frequency noise can be smoothed out by averaging our signal over time. We see our signals are relatively smooth and slowly changing, whereas our noise involves rapid, erratic changes. This is certainly the case for the signals and noises seen in our plots. That is, the noise signal changes rapidly, whereas the true signal changes slowly. We assume that the noise has mostly high frequency content. The simplest and most common way to reduce noise is to smooth the signal. But here, this is a snippet of an audio signal that has been corrupted by noise displayed over a short 10 millisecond time frame. It could be any type of signal over any time frame. So how do we remove noise? Let's consider a snippet of a signal. This is widely applicable to signals in general, not just audio signals. Often we're only provided a noisy signal and we want to remove the noise and recover the original signal. This included adding a noise to our signal to create certain sound effects. In the last section, we saw examples of different audio effects we can create in MATLAB. Welcome to this section on convolution and filtering for signals.
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