Which transform is most appropriate for analyzing the frequency content of a discrete-time sequence on the unit circle?

Which transform is most appropriate for analyzing the frequency content of a discrete-time sequence on the unit circle?

• A. Discrete Fourier Transform
• B. Laplace Transform
• C. Z-transform
• D. Continuous-time Fourier Transform

Answer: (A)

When analyzing the frequency content of a discrete-time sequence, we look at how the signal can be expressed as a sum of complex exponentials whose frequencies correspond to angles on the unit circle in the complex plane (z = e^{jω}). This connects directly to the discrete-time Fourier transform, which computes X(ω) = Σ x[n] e^{−jωn} and describes the spectrum over continuous ω. For a finite sequence, we don’t typically sweep all ω, but instead sample that spectrum at N evenly spaced frequencies using the Discrete Fourier Transform. Those DFT samples give a clear, practical view of which frequencies are present in the data, and they align with the unit-circle viewpoint since the spectrum is effectively taken on z = e^{jω}.

The Laplace transform targets continuous-time signals with complex frequency s and isn’t suited to discrete-time data. The Z-transform is the broader framework in the complex plane; evaluating it on the unit circle yields the DTFT, which the DFT approximates for finite sequences. The Continuous-time Fourier Transform is for continuous signals, not discrete-time ones.