1.

What is the output sequence of the system with impulse response h(n)=(1/2)^nu(n) when the input of the system is the complex exponential sequence x(n)=Ae^jnπ/2?(a) \(Ae^{j(\frac{nπ}{2}-26.6°)}\)(b) \(\frac{2}{\sqrt{5}} Ae^{j(\frac{nπ}{2}-26.6°)}\)(c) \(\frac{2}{\sqrt{5}} Ae^{j({nπ}{2}+26.6°)}\)(d) \(Ae^{j(\frac{nπ}{2}+26.6°)}\)I got this question during an interview for a job.The above asked question is from Frequency Domain Characteristics of LTI System topic in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer»

The correct answer is (b) \(\frac{2}{\sqrt{5}} Ae^{j(\frac{nπ}{2}-26.6°)}\)

To elaborate: First we evaluate the FOURIER transform of the impulse response of the system h(n)

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

At ω=π/2, the above equation yields,

H(π/2)=\(\frac{1}{1+j 1/2}=\frac{2}{\sqrt{5}} e^{-j26.6°}\)

We KNOW that if the input signal is a complex exponential signal, then y(n)=x(n) . H(ω)

=>y(n)=\(\frac{2}{\sqrt{5}} Ae^{j(\frac{nπ}{2}-26.6°)}\)


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