Question

A digital filter with impulse response $ h[n] = 2^n u[n] $ will have a transfer function with a region of convergence.

• A. includes unit circle
• B. excludes unit circle
• C. bounded by rings with circles of radius 0.5 and 2
• D. entire z-plane excluding origin and infinity

Hint:

The region of convergence (ROC) of a z-transform determines the stability and causality of a system. For a causal system, the ROC is outside a circle in the z-plane.

Answer 1

The Correct Option is B

The given impulse response is \( h[n]=2^nu[n] \) which is a right sided sequence. The z transform of \( a^n u[n] \) is given by \( \frac{1}{1-az^{-1}} \). Here \( a = 2 \) Therefore, \(H(z) = \frac{1}{1-2z^{-1}}\) or \( H(z) = \frac{z}{z-2} \). For the sequence to be stable the ROC (region of convergence) must be outside the circle with radius 2. Therefore, it excludes the unit circle.