In Section 2.5, we characterized the unit doublet through the equation for any...

In Section 2.5, we characterized the unit doublet through the equation

for any signal. x(t). From this equation, we derived the relationship

(a) Show that eq. (P2.69-2) is an equivalent characterization of u₁ (t) by showing that eq. (P2.69-2) implies eq. (P2.69-1). [Hint: Fix t, and define the signal

Thus, we have seen that characterizing the unit impulse or unit doublet by how it behaves under convolution is equivalent to characterizing how it haves under integration when multiplied by an arbitrary signal g(t). In fact, as indicated in Section 2.5, the equivalence of these operational definitions holds for all signals and, in particular, for all singularity functions.

(b) Let f (t) be a given signal. Show that

by showing that both functions have the same operational definitions.

(c) What is the value of

Find an expression for