Which statement correctly describes the discrete-time representation of a finite-duration, non-periodic signal using the DFT?

Which statement correctly describes the discrete-time representation of a finite-duration, non-periodic signal using the DFT?

• A. The DFT yields a discrete spectrum corresponding to the finite time window.
• B. The DFT requires the signal to be strictly periodic.
• C. The DFT is identical to the CT Fourier transform.
• D. The DFT is unrelated to the time window.

Answer: (A)

Finite-length data lead to a spectrum that comes in as discrete samples. The DFT treats the N-point time sequence as one period of a periodic discrete-time signal with period N, so it represents the frequency content using N specific complex exponentials. This creates N frequency bins, i.e., a discrete spectrum, effectively sampling the underlying continuous spectrum of the windowed signal. If the signal inside the window is a sum of tones, those tones show up as lines at particular bins; if not, you see spectral leakage due to the window shape, but the spectrum remains discrete because of the finite time window.

The DFT does not require the original signal to be strictly periodic in reality; it analyzes a finite window as the data and interprets them as one period of a periodic sequence. It is not identical to the continuous-time Fourier transform, which yields a continuous spectrum; the DFT provides a discrete set of spectral values that are samples of the DTFT. And the time window is essential: it determines the number of bins and the exact spectral content you obtain.