(a) Show that if the response of an LTI system to x(t) is the output y(t), then the resp...

(a) Show that if the response of an LTI system to x(t) is the output y(t), then the response of the system to

is . Do this problem in three different ways:

(i) Directly from the properties of linearity and time invariance and the fact that

(ii) By differentiating the convolution integral.

(iii) By examining the system in Figure P2.45.

(b) Demonstrate the validity of the following relationships:

[Hint: These are easily done using block diagrams as in (iii) of part (a) and the fact that u₁ (t) × u_-1 (t) = δ(t).]

(c) An LTI system has the response Use the result of part (a) to aid in determining the impulse response of this system.

(d) Let s (t) be the unit step response of a continuous-time LTI system. Use part (b) to deduce that the response y (t) to the input x (t) is

(e) Use eq. (P2.45-1) to determine the response of an LTI system with step response

to the input

(f) Let s[n] be the unit step response of a discrete-time LTI system. What are the discrete-time counterparts of eqs. (P2.45-1) and (P2.45-2)?