Question

What is the response of the system with impulse response h(n)=(1/2)^nu(n) and the input signal   x(n)=10-5sinπn/2+20cosπn?

(a) 20-\(\frac{10}{\sqrt{5}} sin(π/2n-26.60)+ \frac{40}{3}cosπn\)

(b) 20-\(\frac{10}{\sqrt{5}} sin(π/2n-26.60)+ 40cosπn\)

(c) 20-\(\frac{10}{\sqrt{5}} sin(π/2n+26.60)+ \frac{40}{3cosπn}\)

(d) None of the mentioned

The question was asked in an interview for job.

I would like to ask this question from Frequency Domain Characteristics of LTI System topic in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

The correct answer is (a) 20-\(\frac{10}{\sqrt{5}} sin(π/2n-26.60)+ \frac{40}{3}cosπn\)

The best I can explain: The frequency response of the system is

H(ω)=\(\sum_{n=-∞}^∞ h(n) e^{-jωn} = \frac{1}{1-\frac{1}{2} e^{-jω}}\)

For first term, ω=0=>H(0)=2

For second term, ω=π/2=>H(π/2)=\(\frac{1}{1+j\frac{1}{2}} = \frac{2}{\sqrt{5}} e^{-j26.6°}\)

For third term, ω=π=> H(π)=\(\frac{1}{1+\frac{1}{2}}\) = 2/3

Hence the response of the system to x(n) is

y(n)=20-\(\frac{10}{\sqrt{5}} sin(π/2n-26.6^0)+ \frac{40}{3}cosπn\)