Let us consider a system with a real and causal impulse response h(t) that does not have...

Let us consider a system with a real and causal impulse response h(t) that does not have any singularities at t = 0. In Problem 4.47, we saw that either the real or the imaginary part of H(jω) completely determines H(jω). In this problem we derive an explicit relationship between H_R(jω) and H_I(jω), the real and imaginary parts of H(jω).

(a) To begin, note that since h(t) is causal,

except perhaps at t = 0. Now, since h(t) contains no singularities at t = 0, the Fourier transforms of both sides of eq. (P4.48-1) must be identical. Use this fact, together with the multiplication property, to show that

Use eq. (P4.48-2) to determine an expression for H_R(jω) in terms of H_I(fω) and one for H_I (jω) in terms of H_R(jω).

(b) The operation

is called the Hilbert transform. We have just seen that the real and imaginary parts of the transform of a real, causal impulse response h(t) can be determined from one another using the Hilbert transform. Now consider eq. (P4.48-3), and regard y(t) as the output of an LTI system with input x(t). Show that the frequency response of this system is

(c) What is the Hilbert transform of the signal x(t) = cos 3t?