Question

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

(a) (1/2)^nu(n)

(b) -(1/2)^nu(-n-1)

(c) (1/2)^nu(n) & -(1/2)^nu(-n-1)

(d) None of the mentioned

The question was asked during an online exam.

The origin of the question is Inverse Systems and Deconvolution in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right answer is (c) (1/2)^nu(n) & -(1/2)^nu(-n-1)

Explanation: The system function of given system is H(z)=\(1-\frac{1}{2} z^{-1}\)

The inverse of the system has a system function as H(z)=\(\frac{1}{1-\frac{1}{2} z^{-1}}\)

Thus it has a zero at origin and a pole at z=1/2.So, two possible cases are |z|>1/2 and |z|<1/2

So, h(n)= (1/2)^nu(n) for causal and stable(|z|>1/2)

and h(n)= -(1/2)^nu(-n-1) for anti causal and unstable for |z|<1/2.