Question

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 1

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