Causal Systems with Linear Phase

Linear phase is one condition a linear system must satisfy in order to be distortionless. In this problem, we will derive a simple condition that the impulse response of a linear system must meet to have linear phase. You may assume throughout the problem that the impulse response h(t) is real valued.

1.

Show that if the impulse response of a linear system is even, h(t)=h(-t), then the frequency response H(f) is real.

2.

Show the converse of the property in (a), i.e., if H(f) is real and h(t) is real then h(t) is even.

3.

Can a linear system be causal and have zero phase? Explain!

4.

Let h(t) be the impulse response of a linear system with zero phase and let h'(t) = h(t-t₀). Compute the Fourier transform of h'(t) in terms of H(f).

5.

Is it possible to have a causal system with linear phase? If yes, sketch an example impulse response h'(t). If not, explain why not.

6.

Formulate a set of conditions that the impulse response of a causal linear system must satisfy in order to have linear phase.