Linear Systems

(40 points) Consider the following system in which H₁(f) and H₂(f) are transfer functions of linear systems with impulse response h₁(t) and h₂(t), respectively.

1.

Find the output y(t) when $x(t) = \delta(t)$ is the input. Repeat for $x(t) = \delta(t-\frac{1}{4f_0})$.

2.

Now, find y(t) for $x(t)=\delta(t-\tau)$, where $\tau \gt 0$ is arbitrary.

3.

For what condition on h₁(t) and h₂(t) is the above system time-invariant?

4.

Assume that the two linear systems are identical with frequency response $H_1(f) = H_2(f) = \Pi(\frac{f}{f_1})$, where f₁ < f₀ and $\Pi(\cdot)$ denotes the rectangular pulse function, i.e., $\Pi(x)=1$ for $\vert x\vert\leq\frac{1}{2}$ and $\Pi(x)=0$ for $\vert x\vert\gt\frac{1}{2}$.Find the frequency response H(f) of the overall system, i.e., the system with input x(t) and output y(t).

5.

Sketch the magnitude of H(f).

Hint: You may need $\cos(a-b) = \cos a \cos b + \sin a \sin b$.