Example 6.36 Obtain the convolution product of two unit pulses of duration , one starting at and one starting at . Moreover, by converting to a piecewise format we can see the convolution product defined by subintervals. By the Convolution Integral Evaluating the convolution integral requires knowing where, in the -plane, both and are simultaneously nonzero. Hence, the inequalities must all be satisfied. This region is shaded in Figure 6.35, generated below.
Horizontal lines represent typical paths of integration along lines .
Any such line for which or yields a zero integrand, and a value of zero for the convolution. Any such line between and also yields an integrand of 1 and a value for which will be determined by the integral These values are consistent with the results from the convolution theorem. The red parallelogram is the region where the integrand has the value 1. Outside this region, the integrand has the value 0. We then embellish the figure to create Figure 6.35, as follows.