A discrete-time system has poles at z = 0.5 and z = 1.5. Can its DTFT be computed?
AYes — the DTFT is always defined for any discrete-time system by evaluating X(z) at z = e^(jω)
BNo — the pole at z = 1.5 places it outside the unit circle, so the ROC of the Z-transform excludes the unit circle and the DTFT does not exist
CYes — the DTFT averages the contributions of stable and unstable poles, converging to a finite result
DNo — the DTFT cannot be computed for systems with more than one pole, regardless of their location
The DTFT is the Z-transform evaluated on the unit circle |z| = 1. For this evaluation to be valid, the unit circle must lie inside the Z-transform's region of convergence (ROC). A pole at z = 1.5 means the ROC is |z| > 1.5 (for a causal system), which does not include the unit circle. Physically, this corresponds to an unstable system whose impulse response grows without bound — no frequency-domain representation exists via the DTFT. The pole at z = 0.5 alone would give ROC |z| > 0.5, which includes the unit circle and allows the DTFT.
Question 2 Multiple Choice
What is the fundamental reason why the DTFT X(e^(jω)) is exactly periodic with period 2π in ω?
ADigital frequency must be normalized to fit within a bounded range, so periodicity is a mathematical convention
BMultiplying a discrete sequence x[n] by e^(j(ω+2π)n) produces exactly the same result as multiplying by e^(jωn), because e^(j2πn) = 1 for all integers n
CThe Z-transform is defined on a circle, making all evaluations naturally periodic in the angular variable
DThe DFT (which approximates the DTFT) is computed over N discrete points, introducing periodicity as a numerical artifact
The 2π-periodicity is not a convention or approximation — it is an exact algebraic identity. For any integer n, e^(j2πn) = cos(2πn) + j·sin(2πn) = 1. Therefore e^(j(ω+2π)n) = e^(jωn)·e^(j2πn) = e^(jωn)·1 = e^(jωn). Substituting into the DTFT sum, X(e^(j(ω+2π)}) = Σx[n]e^(-j(ω+2π)n) = Σx[n]e^(-jωn) = X(e^(jω)). The periodicity is baked into integer-indexed sampling: a discrete-time signal simply cannot distinguish frequencies that differ by 2π.
Question 3 True / False
The DTFT of any stable discrete-time system (all poles strictly inside the unit circle) always exists and is a continuous, periodic function of digital frequency ω.
TTrue
FFalse
Answer: True
For a stable system, all poles are inside the unit circle, which means the ROC of the Z-transform includes the unit circle. Evaluating X(z) on the unit circle (z = e^(jω)) is therefore valid, and the result X(e^(jω)) exists for all ω. It is continuous in ω (since the sum defining the DTFT is a sum of continuous functions of ω) and exactly 2π-periodic. This is the theoretical foundation for frequency-domain analysis of stable digital filters.
Question 4 True / False
The 2π-periodicity of the DTFT is a limitation of discrete sampling that can be overcome by using a higher sampling rate, which extends the usable frequency range beyond 2π radians.
TTrue
FFalse
Answer: False
The 2π-periodicity is a fundamental, exact property of discrete-time representation — not a limitation of the sampling rate. Every discrete-time system has a DTFT that is 2π-periodic, regardless of sampling rate. Increasing the sampling rate f_s changes the mapping between analog frequency Ω and digital frequency ω = 2πΩ/f_s (allowing higher analog frequencies to be represented before aliasing), but the digital DTFT itself remains 2π-periodic with exactly the same structure. The periodicity reflects the irreducible ambiguity of integer-indexed sequences: samples alone cannot distinguish frequencies that differ by integer multiples of the sampling rate.
Question 5 Short Answer
Explain how the 2π-periodicity of the DTFT is the frequency-domain expression of aliasing in sampled signals.
Think about your answer, then reveal below.
Model answer: When a continuous signal is sampled at rate f_s, any continuous frequency Ω and any frequency Ω + k·f_s (for integer k) produce identical sample sequences — they are aliased together and cannot be distinguished from the samples alone. In the digital frequency domain, these aliased frequencies map to the same digital frequency ω = 2πΩ/f_s. Aliasing is not a defect that appears only when the sampling rate is too low — it is structurally built into the discrete representation: the samples carry no information that distinguishes ω from ω + 2π. The DTFT's 2π-periodicity directly expresses this: X(e^(jω)) = X(e^(j(ω+2π)}) exactly, because both evaluate the spectrum of the same sample sequence at frequencies that are by definition indistinguishable.
The Nyquist theorem tells us how to avoid aliasing of the desired signal, but the periodicity is always present in the DTFT regardless. It is not a problem to be solved but a property to be understood when designing and analyzing discrete-time systems.