Test on Spectral Analysis and Fourier Transforms

Question 1: The Discrete Fourier Transform (DFT) solely relies on time domain sampling for its operation, rather than frequency domain sampling.

A. Ano

B. Ne

Explanation: The study materials state: "DFT uses sampling in frequency domain." This contradicts the statement that it solely relies on time domain sampling.

Question 2: The spectral density derived for the discrete time rectangular signal in the study materials is 1 + 2cos(Ω) + 2cos(2Ω).

A. Ano

B. Ne

Explanation: The derivation in 'DTFT Example #2' shows that the spectral density of the discrete time rectangular signal s[n] is calculated as 1 + 2cos(Ω) + 2cos(2Ω).

Question 3: According to the study materials, when analyzing the Discrete-Time Fourier Transform (DTFT) of a rectangular signal, what occurs as the length of the impulse (number of samples) increases?

A. The spectral density increasingly resembles a sinc() function.

B. The spectral density becomes a constant value, similar to a unit impulse.

C. The height of the spectral density plot becomes equal to the number of samples.

D. The spectral density completely loses its periodicity.

Explanation: The study materials state that when the impulse length (number of samples) of a rectangular signal increases, 'The spectral density more approaches the sinc() function'. It also notes that 'The height is equal to the length of the impulse (number of samples)'. The spectral density remains a periodic extension and does not become a constant or lose its periodicity.

Question 4: According to the provided study materials, which condition ensures the convergence of the Discrete Time Fourier Transform (DTFT) for a signal s[n]?

A. The signal must be a periodic sequence.

B. The signal must have limited energy.

C. The spectral density must be a continuous function of frequency.

D. The signal energy must be infinite.

Explanation: The study materials state that 'Limited signal “energy” is the condition of the sum convergence' for the DTFT. It also clarifies that this condition is 'not met by periodic sequences' and the spectral density being a continuous function is a consequence, not a condition for convergence, and is 'not suitable for DSP'.

Question 5: The DFT coefficient X_hat_N(N-k) is equal to the DFT coefficient X_hat_N(k).

A. Ano

B. Ne

Explanation: According to the study materials, the symmetry property states that X_hat_N(N-k) = (X_hat_N(k))*, meaning the DFT coefficient X_hat_N(N-k) is the complex conjugate of X_hat_N(k), not necessarily equal to it.