Explore key aspects of the Z-transform in discrete-time signal analysis with this focused quiz. Assess your grasp on region of convergence, properties, and applications related to Z-transform techniques in signal processing.
Which expression correctly represents the unilateral Z-transform of a discrete-time sequence x[n]?
Explanation: The unilateral Z-transform for sequence x[n] is defined as the sum from n=0 to infinity of x[n] times z to the power of negative n. Option B uses a two-sided (bilateral) sum and changes the sign of the exponent, which is incorrect for the unilateral case. Option C uses an integral and exponential notation more appropriate for the Fourier transform. Option D uses z^n instead of z^(-n), which is not standard for Z-transform definitions.
For the finite-length sequence x[n] = {1, 2, 3} defined for n = 0, 1, 2, what is the region of convergence (ROC) of its Z-transform?
Explanation: The Z-transform of a finite-length sequence converges everywhere in the complex z-plane except possibly at z=0, because the sum has a finite number of terms. Option A is typically for specific right-sided or causal sequences with infinite length. Option C incorrectly gives a region typically associated with anti-causal infinite sequences, which does not apply here. Option D indicates the unit circle only, which is overly restrictive for this case.
If the Z-transform of x[n] is X(z), what is the Z-transform of a^n x[n]?
Explanation: Multiplying x[n] by a^n in the time domain results in X(z/a) in the Z-domain since the scaling affects the argument of the Z-transform. Option A simply multiplies by a instead of scaling the argument. Option B suggests substitution with az, which would not align with the correct transform property. Option D writes a^n X(z), which is not valid for linear time-invariant systems.
Given the Z-transform X(z) = z/(z-2) with ROC |z| u003E 2, what is the corresponding time-domain sequence x[n]?
Explanation: The expression z/(z-2) with ROC |z| u003E 2 matches the Z-transform of 2^n u[n], a right-sided sequence. Option B gives a sequence with alternating sign, which is not produced by the pole at 2. Option C includes an 'n' term, indicating a ramp or derivative, which is incorrect. Option D represents a constant sequence, not an exponential.
For a causal discrete-time system with system function H(z) = 1/(1-0.5z^-1), for what values of z is the system stable?
Explanation: Stability for a causal system requires the ROC to include the unit circle and be outside the outermost pole. The system has a pole at z = 0.5, so the ROC is |z| u003E 0.5 for causality and stability. Option B represents an anti-causal ROC, not compatible with a causal system. Option C sets the boundary too far out and is only stable if all poles are inside the unit circle. Option D describes the pole's location rather than a region.