The frequency response \(H\left( \Omega \right)\) of a system for impulse sequence response

\(h\left[ n \right] = \delta \left[ n \right] + \delta \left[ {n - 1} \right]\) is

Question

Question

This question was previously asked in

ESE Electronics 2010 Paper 1: Official Paper

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\(H\left[ \Omega \right] = 2\cos \left( {\frac{\Omega }{2}} \right)\angle - \frac{\Omega }{2}\)
\(H\left[ \Omega \right] = \cos \Omega \angle - \Omega\)
\(H\left[ \Omega \right] = 2\cos \Omega \angle - \frac{\Omega }{2}\)
\(H\left[ \Omega \right] = 2\angle - \frac{\Omega }{2}\)

Answer 1

Analysis:

h(n) = δ(n) + δ(n – 1)

H(Ω) = 1 + e^-jΩ

H(Ω) = e^-jΩ/2 (e^jΩ/2+ e^-jΩ/2)

\(= 2{e^{ - \frac{{j{\rm{\Omega }}}}{2}}}\cos \left( {\frac{{\rm{\Omega }}}{2}} \right)\)

∴ \(H\left( {\rm{\Omega }} \right) = 2\cos \left( {\frac{{\rm{\Omega }}}{2}} \right)\angle - \frac{{\rm{\Omega }}}{2}\)