Signal Construction in Terms of Convolution Integrals

Abstract

Two arbitrary signals can be convolved together to get a third signal; but a very relevant and pertinent case is when one of the signals contributes to the input and it is also the output! In other words, how can we generate a signal by convolving it (or a variant thereof) with another? We show in the text how this can be done by various means. For example we can generate the signal by convolving it with the delta function. Or we can generate it by convolving its derivative with the unit step function. Or we can generate it by convolving its second derivative with the ramp function. Or even more we can generate it by convolving its third derivative (divided by 2) with the quadratic function; and so on. But why would we want to go to all this trouble? The premise is, if we know the system (circuit) response to any of the generating functions (the impulse, unit step, …) and if we know how to generate our stimulus of those generating functions (which is the topic of this chapter), then we can know the response of the system due to stimulus by using the same convolution steps—not on the stimulus, but on the system response. For each case we do an example showing all the intermediate details, especially graphically. We wrap the chapter by tying the Fourier transform to the convolution integral using the complex exponential and the delta function.