Complex Exponential signal as an Eigensignal:

Consider an LSI system with impulse response h(t). We will verify that the complex exponential is an eigensignal of the LSI system. The output y(t) of the LSI system, corresponding to input x(t) = can be obtained by convolving the x(t) with the impulse response h(t).

Noting that can be moved outside the integral, we get

Note that stability of the LSI system guarantees convergence of y(t). Thus we have:

Hence, we have shown that complex exponential signals are eigensignals for the LSI systems (when they do produce a convergent output). The constant , for each fixed , is the eigenvalue corresponding to the exponential signal.