Why is it common to treat the discrete-time sequence obtained after sampling as if it were periodic for analysis?

• A. Because periodicity is required by the sampling theorem.
• B. Because discrete-time Fourier transform assumes periodic extension, allowing analysis.
• C. Because sampling guarantees periodicity.
• D. Because the signal is always periodic after sampling.

Answer: (B)

The key idea is that the discrete-time Fourier transform is inherently 2π-periodic in frequency. This happens because the DTFT is defined as X(ω) = Σ x[n] e^{-jωn}, and for any integer n we have e^{-j(ω+2π)n} = e^{-jωn} e^{-j2πn} = e^{-jωn} since e^{-j2πn} = 1. So X(ω) repeats every 2π regardless of what the sequence x[n] is. Because of this, analysts naturally treat the discrete-time sequence as if its spectrum is periodically extended in frequency, focusing on one 2π interval and using the periodic repeats to understand the whole spectrum. This isn’t saying the time-domain signal is truly periodic; it’s a consequence of how the DTFT is defined and how sampling affects the spectrum, which makes the periodic viewpoint the standard and convenient framework for analysis.