Suppose that we represent an analog bandpass filter by the frequency response...

Suppose that we represent an analog bandpass filter by the frequency response

where C(f) is the frequency response of an equivalent lowpass filter, as shown in Fig. P9.13. (a) Show that the impulse response c(t) of the equivalent lowpass filter is related to the impulse response h(t) of the bandpass filter as follows:

(b) Suppose that the bandpass system with frequency response H(F) is excited by a bandpass signal of the form

where u(t) is the equivalent lowpass signal. Show that the filter output may be expressed as

(Hint: Use the frequency domain to prove this result.)