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)

For a discrete-time sequence, the natural way to understand its frequency content is to look at how the signal projects onto complex exponentials that lie on the unit circle, i.e., X(e^{jω}) = sum x[n] e^{-jωn}. When the sequence is finite, the Discrete Fourier Transform computes exactly these projections at N equally spaced points on the unit circle, with frequencies ω_k = 2πk/N. This gives a practical, finite spectrum that reveals which frequency components are present in the sequence. The other transforms don’t align with this unit-circle frequency view: the Laplace transform is for continuous-time signals in the s-plane; the Z-transform is a general discrete-time transform in the complex plane but not specifically framed as a frequency spectrum on the unit circle; the continuous-time Fourier transform applies to continuous-time signals. Thus, the Discrete Fourier Transform is the appropriate choice for analyzing the frequency content of a discrete-time sequence on the unit circle.

For a discrete-time sequence, the natural way to understand its frequency content is to look at how the signal projects onto complex exponentials that lie on the unit circle, i.e., X(e^{jω}) = sum x[n] e^{-jωn}. When the sequence is finite, the Discrete Fourier Transform computes exactly these projections at N equally spaced points on the unit circle, with frequencies ω_k = 2πk/N. This gives a practical, finite spectrum that reveals which frequency components are present in the sequence.

The other transforms don’t align with this unit-circle frequency view: the Laplace transform is for continuous-time signals in the s-plane; the Z-transform is a general discrete-time transform in the complex plane but not specifically framed as a frequency spectrum on the unit circle; the continuous-time Fourier transform applies to continuous-time signals. Thus, the Discrete Fourier Transform is the appropriate choice for analyzing the frequency content of a discrete-time sequence on the unit circle.