Real-valued Systems

Throughout this problem, h(t) is a real-valued function which denotes the impulse response of a system.

1.

Which condition must h(t) satisfy to be the impulse response of a causal system?

2.

Show that the Fourier transform of an even, real-valued signal x(t) is purely real, i.e., X(f) = X^*(f).

3.

Show that the Fourier transform of an odd, real-valued signal y(t) is purely imaginary, i.e., Y(f) = -Y^*(f).

4.

Confirm that an arbitrary real-valued impulse response h(t) can always be decomposed into an even part h_e(t) and an odd part h_o(t), both real-valued:

h(t) = h_e(t) + h_o(t),

with

$$h_e(t) = \frac{1}{2}(h(t)+h(-t))$$

and

$$h_o(t)=\frac{1}{2}(h(t)-h(-t)).$$

Note: You have to show that h_e(t) and h_o(t) are even and odd, respectively and that h(t) = h_e(t) + h_o(t).

5.

Combining the results in parts (b)-(d), formulate simple rules for computing only the real part or only the imaginary part of the Fourier transform of a real-valued impulse response.

6.

Verify your rules by considering the example h(t)=1 for $0 \leq t\leq 1$.