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 xi(n) is the sequence shown in Figure P3-21(b). Because xi(n) is exactly two cycles of a cosine sequence, the imaginary parts of Xl(m) are all zero-valued samples, making | Xi(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.