1. Assume that an N-point DFT, performed on an N-sample x(n) time-domain sequence, results in a DFT frequency-domain sample spacing of 100 Hz. What would be the DFT frequency-domain sample spacing in Hz if the N-sample X(n) time sequence was padded with 4N zero-valued samples and we performed a DPT on that extended-time sequence?

2. This problem tests your understanding of the DFT's frequency-domain axis. Consider sampling exactly two cycles of an analog x(t) cosine wave resulting in the 8-point x₂(n) time sequence in Figure P3-21(a). The real part of the DFT of x_i(n) is the sequence shown in Figure P3-21(b). Because x_i(n) is exactly two cycles of a cosine sequence, the imaginary parts of X_l(m) are all zero-valued samples, making | X_i(m)| equal to the real part of X₂(m). (Note that no leak­age is apparent in |X₁(m)|.) Think, now, of a new frequency-domain se­quence X₂(m) that is equal to X₁(m) with eight zero-valued samples, the white squares in Figures P3-21(c) and P3-21(d), inserted in the center of the real and imaginary parts of X₁(m).

(a)  Draw the x₂(n) time sequence that is the inverse DFT of X₂(m).

(b)  Comment on how the x₂(n) time sequence is related to The original analog x(t) signal and the x₁(n) sequence.