Convolution and Time Domain Filtering

Cuthbert Nyack

The above diagram shows the equivalence between the time and frequency domains. Passing a signal through a filter is equivalent to taking the convolution of the input signal with the impulse response of the filter. This approach is mostly used in digital signal processing but the principle is worth mentioning here.

In the diagram above, the red line shows an AM signal and the blue line shows the same signal with some added "noise". The AM signal has a finite bandwidth so that it is possible to remove some of the noise by passing the noisy signal through a bandpass filter or alternatively, taking the convolution of the noisy signal with the impulse response of a bandpass filter.

The impulse response of an ideal bandpass filter consists of a sinusoid within a sinc envelope and extending from -inf to +inf. An impulse response which approximates that of an ideal bandpass filter is shown above and consists of a sinusoid within the main lobe of a sinc function.

In the above plot, the red line shows the input without noise, the blue line shows the noisy input and the green line shows the convolution of the noisy signal with the impulse response of the bandpass filter. As can be seen, most of the noise has been removed.

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