# Answered Essay: Consider the LTI system with the following impulse response: h[n] = (1/2)^n u[n] + (3)^n u[-n -1] Where u[n] is the unit step function. D

Consider the LTI system with the following impulse response: h[n] = (1/2)^n u[n] + (3)^n u[-n -1] Where u[n] is the unit step function. Derive the Z-transform H(z) in closed form, indicating the ROC as an algebraic inequality. Sketch the pole-zero plot of H(z). Clearly indicate the ROC in your sketch. Is this system causal? Clearly explain why or why not. Is this system stable? Clearly explain why or why not. Is this system minimum phase? Clearly explain why or why not.

1). The Z transform of h(n) using standard Z-transform pair is given below

$H(z)=\frac{1}{1-\frac{1}{2}Z^{-1}}-\frac{1}{1-3Z^{-1}}$

and ROC: $\frac{1}{2}<\sigma<3$

2).The pole zero plot of the given transfer function is shown below where, yellow part represents ROC

3).The given system is not causal because for the causal system ROC should be outermost of the greatest pole. Here the greatest pole is 3 but ROC is inside 3 not outside

4). System is stable because ROC includes the unit circle. The black circle in solution 2 represents unit circle

5). System is non-minimum phase because it has one zero at infinity.

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