Question

If x(n)=Ae^jωn is the input of an LTI system and h(n) is the response of the system, then what is the output y(n) of the system?

(a) H(-ω)x(n)

(b) -H(ω)x(n)

(c) H(ω)x(n)

(d) None of the mentioned

I had been asked this question in exam.

My doubt is from Frequency Domain Characteristics of LTI System in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right answer is (c) H(ω)x(n)

To explain I would say: If x(n)= Ae^jωn is the input and h(n) is the response o the system, then we know that

y(n)=\(\sum_{k=-∞}^∞ h(k)x(n-k)\)

=>y(n)=\(\sum_{k=-∞}^∞ h(k)Ae^{jω(n-k)}\)

= A \([\sum_{k=-∞}^∞ h(k) e^{-jωk}] e^{jωn}\)

= A. H(ω). e^jωn

= H(ω)x(n)