x[n] is a real-valued, nonnegative, finite-length sequence of length N; i.e., x[n] is real and nonnegative for 0 ≤ n ≤ N – 1 and is zero otherwise. The N-point DFT of x[n] is X[k], and the Fourier transform of x[n] is X(e^jω). 

Determine whether each of the following statements is true or false. For each statement, if you indicate that it is true, clearly show your reasoning. If you state that it is false, construct a counterexample.

(a) If X(e^jω) is expressible in the form 

X(e^jω) = B(ω)e^jaω,

where B(ω) is real and α is a real constant, then X[k] can be expressed in the form 

X[k] = A[k]e^jyk,

where A[k] is real and γ is a real constant,

(b) If X[k] is expressible in the form 

X[k] = A[k]e^jyk,

where A[k] is real and y is a real constant, then X(e^jω) can be expressed in the form 

X(e^jω) = B(ω)e^jaω,

where B(ω) is real and a is a real constant.