Problem 38

Consider the system in the block diagram below.

1.

Determine the Fourier transforms X₁(f) and X₂(f) of the signals x₁(t) and x₂(t) as a function of the Fourier transform V(f) of the input signal v(t).

2.

Find the Fourier transforms of the signals Z₁(f) and Z₂(f) of the signals z₁(t) and z₂(t) as a function V(f) and H(f).

3.

Which value of the angle $\phi$ allows W(f), the Fourier transform of w(t), to be written as

$$W(f) = G(f) \cdot V(f),$$

where G(f) is not a function of V(f)? Express G(f) as a function of H(f) and f₀.

4.

Assume now that H(f) is the transfer function of an ideal lowpass with cut-off frequency $f_c = \frac{f_0}{2}$. Sketch |G(f)| for this case.

Hint: You may need the following identities:

$$\sin(a+b) = \sin a \cos b + \cos a \sin b$$

$$\cos(a-b) = \cos a \cos b + \sin a \sin b$$