Question

The impulse response of a discrete-time system is given by 

\(h(n)=\frac{1}{2}(δ[n]+δ[n-2])\)

the magnitude of the response can be expressed as

1. |cos Ω|
2. cos Ω
3. |sin Ω|
4. sin Ω

Answer 1

Correct Answer - Option 1 : |cos Ω|

Concept:

The Discrete-Time Fourier transform of a signal of infinite duration x[n] is given by

\(X\left( ω \right) = DTFT\left\{ {x\left[ n \right]} \right\} = \mathop \sum \limits_{n = - \infty }^\infty x\left[ n \right]{e^{ - jω n}}\)

Analysis:

\(h(n)=\frac{1}{2}(δ[n]+δ[n-2])\)

H(e^jω) = 0.5 + 0.5 e^-j2ω 

= e^-jω [0.5 e^jω + 0.5 e^-jω]

= e^-jω cosω 

The magnitude of the frequency response is: |cos ω|