Convolution

Abstract

Having covered the theory and basics of spectral techniques in the frequency domain we next turn to the parallel path of convolution (in the time domain). After defining convolution and applying it to causal signals we set out on a marathon of convolution examples. Convolution is an elaborate process, since unlike integration which yields a number, convolution yields a function. It takes two functions, flips one, offsets it and for each offset point calculates the corresponding area of the product of both functions. Convolution can be made easier by the use of graphics, and that was used heavily for all presented examples. For each case we plot the initial functions, flip one of them, incrementally offset it, identify the overlap product area, and plot the resulting area. From rectangles, triangles, unit step function, negative exponential, ramp, sine, cosine, and ending in the periodic pulse, we mix and match what we can fit and describe the intricate details of convolution. We wrap the chapter with the important topic of convolution with the delta function.