A Pilot Discussion on the Product of Impulse and the Signal which Is Discontinuous at the Singularity Point
Jiang Xiao-lei · Journal of Electrical & Electronic Education · 2008
It is well known that if x(t) is continuous at t0,then x(t)δ(t-t0) = x(t0)δ(t-t0).Here the case in which x(t) is discontinuous at t0 is explored.By convolution theorem in the frequency domain,it is shown that x(t)δ(t-t0) = 1/2[x(t-0)+x(t+0)]δ(t-t0).To avoid possible confusion and inconsistence,this equation is understood only as the shorthand in the time domain for its corresponding frequency-domain equation,and in the development of other formulas,only in the intermediate steps does it appear.This equation and its frequency-domain dual are useful to easily obtain solutions to two problems.First,in the sampling of a continuous-time signal,what value does the obtained discrete-time signal take on at the discontinuous point of the signal being sampled.Second,what is the output of an ideal frequency-selective filter to a signal whose spectrum contains impulse just at the edge of the passband of the filter.