An important property of the frequency response H(jω) of a continuous-time LTI system wi...

An important property of the frequency response H(jω) of a continuous-time LTI system with a real, causal impulse response h(t) is that H(jω) is completely specified by its real part, . The current problem is concerned with deriving and examining some of the implications of this property, which is generally referred to as real-part sufficiency.

(a) Prove the property of real-part sufficiency by examining the signal h_e(t), which is the even part of h(t). What is the Fourier transform of h_e(t)? Indicate ho h(t) can be recovered from h_e(t).

(b) If the real part of the frequency response of a causal system is

what is h(t)?

(c) Show that h(t) can be recovered from h₀(t), the odd part of h(t), for every value of t except t = 0. Note that if h(t) does not contain any singularities then the frequency response

will not change if h(t) is set to some arbitrary finite value at the single point t = 0. Thus, in this case, show that H(jω) is also completely specified by its imaginary part.