Question

An LTI system with impulse response h1 [n] = -2(\(\frac{1}{4})^n\)  u[n] is connected in parallel with another causal LTI system with impulse response h2 [n]. The resulting interconnection has frequency response H (e^jω) = \(\frac{-12+5e^{-jω}}{12+7e^{-jω}+e^{-j2ω}}\). Then h2[n] is ___________

(a) (\(\frac{1}{3}\))^n u[n-1]

(b) (\(\frac{1}{3}\))^n u[n]

(c) (\(\frac{1}{4}\))^n u[n]

(d) (\(\frac{1}{4}\))^n u[n-1]

I have been asked this question in homework.

My question is taken from Fourier Analysis topic in portion Fourier Analysis on Complex Spaces of Signals and Systems

Answer 1

The correct answer is (b) (\(\frac{1}{3}\))^n u[n]

The best explanation: H (e^jω) = H1 (e^jω) + H2 (e^jω)

Or, \(\frac{-12+5e^{-jω}}{12+7e^{-jω}+e^{-j2ω}} = \frac{1}{1-\frac{1}{3} e^{-jω}} + \frac{(-2)}{1-\frac{1}{4} e^{-jω}}\)

∴ H2(e^jω) = \(\frac{1}{1-\frac{1}{3} e^{-jω}}\)

So, h2 [n] = (\(\frac{1}{3}\))^n u[n].