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

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 idea being tested is why spectral analysis of a sampled sequence uses a periodic extension as a modeling step. The discrete-time Fourier transform naturally produces a spectrum that is 2π periodic in frequency. When we have a finite-length data record, we often treat it as if it were one period of a periodically repeating sequence. This periodic extension is what lets the discrete Fourier transform (and its fast implementation) work with a finite set of numbers: the N samples we have are viewed as one period, and the DFT samples the DTFT at N equally spaced frequencies. This viewpoint also makes circular convolution match linear convolution in a convenient way, so analysis and filtering become straightforward. It’s a mathematical tool, not a claim that the actual signal becomes periodic just by sampling. The other statements misstate what sampling does or what periodicity means in this context.