A set of samples x[n] from a signal is converted to a set of numbers X[k] by using the DFT.

(a) If {x[0] , x[1] , x[2] , x[3] , x[4] } = {2, 8, ?3, 1, 9} , find the numerical value of X[0] .

(b) If x[n] consists of 24 samples taken from exactly three periods of a sinusoid at a sampling rate which is exactly eight times the frequency of the sinusoid, two values of X[k] in the range 0 ? k < 24 are not zero. Which ones are they?

(c) If x[n] consists of seven samples, all of which are the same, ?5, in the range 0 ? k < 7, which X[k] values are zero?

(d) If {x[0] , x[1] , x[2] , x[3] } = {a, b, 0, b} and
{X[0] , X [1] , X[2] , X[3] } = {A, B, 0, B * }
how are a and b related to each other? Express B in terms of a and b.




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