Consider the sequence x[n] = aⁿ u[n] + bⁿ u[n], where u[n] denotes the unit-step sequence and 0 < |a| < |b| < 1. The region of convergence (ROC) of the z-transform of x[n] is

(A) |z| > |a|
(B) |z| > |b|
(C) |z| < |a|
(D) |a| < |z| < |b|

Correct answer: (B) |z| > |b|

Solution

The correct option is |z| > |b| because the z-transform converges for values of z outside the larger pole, which in this case is determined by the term bⁿ, as |b| is greater than |a| and both are less than 1.

Related GATE Technical questions

Which one of the following is an eigen function of the class of all continuous-time, linear, time-invariant systems (u(t) denotes the unit-step function)?
A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period Tₛ. In which one of the following cases is the sampled signal periodic?
Consider a two-port network with the transmission matrix: T = [[A, B],[C, D]]. If the network is reciprocal, then
Consider the following statements for a metal oxide semiconductor field effect transistor (MOSFET): P: As channel length reduces, OFF-state current increases. Q: As channel length reduces, output resistance increases. R: As channel length reduces, threshold voltage remains constant. S: As channel length reduces, ON current increases. Which of the above statements are INCORRECT?
The block diagram of a frequency synthesizer consisting of a Phase Locked Loop (PLL) and a divide-by-N counter (comprising ÷2, ÷4, ÷8, ÷16 outputs) is sketched below. The synthesizer is excited with a 5 kHz signal (input). The free-running frequency of the PLL is set to 20 kHz. Assume that the commutator switch makes contacts repeatedly in the order 1-2-3-4. The corresponding frequencies synthesized are:
The three-dimensional state of stress at a point is given by σ = [[10, 0, 0],[0, 40, 0],[0, 0, 0]] MPa. The maximum shear stress at the point is

⚔️ Practice GATE Technical free + battle 1v1 →