Question

If h(n) is the real valued impulse response sequence of an LTI system, then what is the imaginary part of Fourier transform of the impulse response?

(a) –\(\sum_{k=-∞}^∞ h(k) sin⁡ωk\)

(b) \(\sum_{k=-∞}^∞ h(k) sin⁡ωk\)

(c) –\(\sum_{k=-∞}^∞ h(k) cos⁡ωk\)

(d) \(\sum_{k=-∞}^∞ h(k) cos⁡ωk\)

This question was addressed to me in semester exam.

My question is based upon Frequency Domain Characteristics of LTI System in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right choice is (a) –\(\sum_{k=-∞}^∞ h(k) sin⁡ωk\)

To explain I would say: From the definition of H(ω), we have

H(ω)=\(\sum_{k=-∞}^∞h(k) e^{-jωk}\)

=\(\sum_{k=-∞}^∞h(k) cos⁡ωk-j\sum_{k=-∞}^∞h(k) sin⁡ωk\)

= HR(ω)+j HI(ω)

=> HI(ω)=-\(\sum_{k=-∞}^∞h(k) sin⁡ωk\)