As we have seen, the techniques of Fourier analysis are of value in examining continuous...

As we have seen, the techniques of Fourier analysis are of value in examining continuous-time LTI systems because periodic complex exponentials are eigenfunctions for LTI systems. In this problem, we wish to substantiate the following statement: Although some LTI systems may have additional eigenfunctions, the complex exponentials are the only signals that are eigenfunctions of every LTI system.

(a) What are the eigenfunctions of the LTI system with unit impulse response What are the associated eigenvalues?

(b) Consider the LTI system with unit impulse response , but that is an eigenfunction of the system eigenvalue 1. Similarly, find the eigenfunctions with eigenvalues 1/2 and 2 that are not complex exponentials. (Hint: You can find impulse trains that meet requirements.)

(c) Consider a stable LTI system with impulse response h(t) that is real and even. Show that are eigenfunctions of this system.

(d) Consider the LTI system with impulse response h(t) = u(t). Suppose that must satisfy, and solve the equation. This result, together with those of parts (a) through (c), should prove the validity of the statement made at the beginning of the problem.