Consider a real-valued finite-duration sequence x[n] of length M. Specifically, x[n] = 0 for n < 0 and n > M – 1. Let X [k] denote the N-point DFT of x[n] with N ≥ M and N odd. The real part of X[k] is denoted X_R[k].

(a) Determine, in terms of M, the smallest value of N that will permit X[k] to be uniquely determined from X_R[k].

(b) With N satisfying the condition determined in part (a), X[k] can be expressed as the circular convolution of X_R[k] with a sequence U_N[k]. Determine U_N[k].