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

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ESE Electronics 2012 Paper 1: Official Paper

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Answer 1

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|