Question

An LTI system S with impulse response h[n] and frequency response H(ejω) is known to have the property that, when −π≤ω0≤π, then S(cos(ω0n))=∣ω0∣cos(ω0n).

a) Determine H(ejω).

Answer 1

To determine the frequency response H(ejω), we need to find the Fourier transform of the impulse response h[n] and apply the given property. The frequency response H(ejω) is (|δ(ω - ω0)| + |δ(ω + ω0)|)/2.

Step-by-step explanation:

To determine the frequency response H(ejω), we need to find the Fourier transform of the impulse response h[n]. From the given property, we know that S(cos(ω0n)) = |ω0|cos(ω0n).

The Fourier transform of cos(ω0n) is given by (δ(ω - ω0) + δ(ω + ω0))/2. The frequency response H(ejω) is then the magnitude of the Fourier transform, which can be written as (|δ(ω - ω0)| + |δ(ω + ω0)|)/2.

Therefore, the frequency response H(ejω) is (|δ(ω - ω0)| + |δ(ω + ω0)|)/2.