1.

What is the response of the system with impulse response h(n)=(1/2)^nu(n) and the input signalx(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 mentionedThe 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»

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


Discussion

No Comment Found

Related InterviewSolutions