Read each part of this problem carefully to note the differences among parts.

(a) Consider the signal

which can be represented by the IDFT equation as

where X8[k] is the eight-point DFT of x[n]. Plot X8[k] for 0 ? k ? 7.

(b) Determine V16[k], the sixteen-point DFT of the sixteen-point sequence plot V16[k]for 0 ? k ? 15.

(c) Finally, consider |X16[k]|, the magnitude of the sixteen-point DFT of the eight-point Sketch |X16[k]| for 0 ? k ? 15 without explicitly evaluating the DFT expression. You will not be able to find all values of |X16[k]| by inspection as in parts (a) and (b), but you should be able to find some of the values exactly. Plot all the values you know exactly with a solid circle, and plot estimates of the other values with an open circle.

(a) + cos(an/4)-0.5 cos(37n/4). 0sns7. otherwise, x|n} (). x[n] = k=0 otherwise, 0. (b) I+ cos(7n/4) (0.5 cos(3:rn/4). 0sns 15. r[n| = otherwise. (c) JI+ cos(7n/4) - 0.5 cos(37n/4). 0