Question

A causal system produces the output sequence y(n)={1,0.7} when excited by the input sequence x(n)={1,-0.7,0.1}, then what is the impulse response of the system function?

(a) [3(0.5)^n+4(0.2)^n]u(n)

(b) [4(0.5)^n-3(0.2)^n]u(n)

(c) [4(0.5)^n+3(0.2)^n]u(n)

(d) None of the mentioned

This question was addressed to me by my school principal while I was bunking the class.

Question is taken from Inverse Systems and Deconvolution topic in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right option is (b) [4(0.5)^n-3(0.2)^n]u(n)

The best explanation: The system function is easily determined by taking the z-transforms of x(n) and y(n). Thus we have

H(z)=\(\frac{Y(z)}{X(z)} = \frac{1+0.7z^{-1}}{1-0.7z^{-1}+0.1z^{-2}} = \frac{1+0.7z^{-1}}{(1-0.2z^{-1})(1-0.5z^{-1})}\)

Upon applying partial fractions and applying the inverse z-transform, we get

[4(0.5)^n-3(0.2)^n]u(n).