Discrete Fourier Transform (DFT) is used to represent a finite-duration, non-periodic signal by sampling its spectrum. Which statement is accurate about this representation?

Discrete Fourier Transform (DFT) is used to represent a finite-duration, non-periodic signal by sampling its spectrum. Which statement is accurate about this representation?

• A. The DFT uses a finite sum over N samples to approximate the spectrum.
• B. The DFT yields a continuous spectrum.
• C. The DFT requires the signal to be strictly periodic.
• D. The DFT does not apply to finite-duration signals.

Answer: (A)

The DFT represents a finite-duration signal by a discrete set of spectral samples, obtained from a finite sum over the available time samples. When you have N time samples x[0] through x[N−1], the DFT computes N complex values X[k] = sum_{n=0}^{N−1} x[n] e^{-j 2π kn / N}, which correspond to N equally spaced frequency bins ω_k = 2π k / N. This means you are sampling the continuous spectrum of the signal (which would exist for all frequencies if you looked at the full Fourier transform) at N discrete points, not producing a continuous spectrum. The input need not be strictly periodic in time; the finite sequence is treated as one period of a periodic extension to justify the discrete spectrum, and the result is a finite, discrete set of spectral components. That’s why the statement about using a finite sum over N samples to approximate the spectrum is the accurate description.

The DFT represents a finite-duration signal by a discrete set of spectral samples, obtained from a finite sum over the available time samples. When you have N time samples x[0] through x[N−1], the DFT computes N complex values X[k] = sum_{n=0}^{N−1} x[n] e^{-j 2π kn / N}, which correspond to N equally spaced frequency bins ω_k = 2π k / N. This means you are sampling the continuous spectrum of the signal (which would exist for all frequencies if you looked at the full Fourier transform) at N discrete points, not producing a continuous spectrum. The input need not be strictly periodic in time; the finite sequence is treated as one period of a periodic extension to justify the discrete spectrum, and the result is a finite, discrete set of spectral components. That’s why the statement about using a finite sum over N samples to approximate the spectrum is the accurate description.