FE Electrical and Computer Signal Processing Practice Problems

The current FE Electrical and Computer specification assigns 5-8 of the 110 exam questions to Signal Processing. Form A contains 6 original problems in this area. The sample below is published in full, with the same worked-solution format used throughout the book.

A causal discrete-time linear time-invariant system has the transfer function $H (z) = \frac{z}{( z - 0.5 ) ( z - 1.25 )}$ . The statement that correctly describes the region of convergence and the stability of this system is:

(A)Region of convergence is $∣ z ∣ > 0.5$ ; the system is stable.
(B)Region of convergence is $0.5 < ∣ z ∣ < 1.25$ ; the system is stable.
(C)Region of convergence is $∣ z ∣ < 0.5$ ; the system is unstable.
(D)Region of convergence is $∣ z ∣ > 1.25$ ; the system is unstable.

Show worked solution

Answer: (D)

Causality forces the region of convergence outside the outermost pole at 1.25, and that region excludes the unit circle, so the response grows without bound.

Causality admits only right-sided sequences, whose z-transforms converge in the exterior of a circle passing through the pole of largest magnitude.

∣ z ∣ > max (0.5, 1.25) = 1.25

Bounded-input bounded-output stability additionally demands that this exterior region contain the unit circle, which is impossible with a pole at radius 1.25.

∣ p_{2} ∣ = 1.25 > 1

FE Reference Handbook — Electrical and Computer Engineering: z-Transforms, Region of Convergence and Stability

Why the other choices appear

(A)Using the innermost pole to bound the exterior region and then judging stability from that pole alone, ignoring the pole at 1.25.
(B)Selecting the annular region, which belongs to the two-sided noncausal inverse transform; it is stable but contradicts the stated causality.
(C)Reading the region as the interior of the smallest pole, the fully anticausal choice, which is inconsistent with a causal system.