Question

What is the inverse of the system with impulse response h(n)=(1/2)^nu(n)?

(a) δ(n)+1/2 δ(n-1)

(b) δ(n)-1/2 δ(n-1)

(c) δ(n)-1/2 δ(n+1)

(d) δ(n)+1/2 δ(n+1)

The question was asked in an online interview.

Enquiry is from Inverse Systems and Deconvolution topic in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right option is (b) δ(n)-1/2 δ(n-1)

Explanation: Given impulse response is h(n)=(1/2)^nu(n)

The system function corresponding to h(n) is

H(z)=\(\frac{1}{1-\frac{1}{2} z^{-1}}\) ROC:|z|>1/2

This system is both stable and causal. Since H(z) is all pole system, its inverse is FIR and is given by the system function

HI(z)=\(1-\frac{1}{2} z^{-1}\)

Hence its impulse response is δ(n)-1/2 δ(n-1).