A modified discrete Fourier transform (MDFT) was proposed (Vernet, 1971) that computes samples of the z-transform on the unit circle offset from those computed by the DFT. In particular, with X_M[k] denoting the MDFT of x (n), 

Assume that N is even.

(a) The N-point MDFT of a sequence x[n] corresponds to the N-point DFT of a sequence x_M[n], which is easily constructed from x[n]. Determine x_M[n] in terms of x[n].

(b) If x[n] is real, not all the points in the DFT are independent, since the DFT is conjugate symmetric; i.e., X[k] = X*[((– k))_N] for 0 ≤ k ≤ N – 1. Similarly, if x[n] is real, not all the points in the MDFT are independent. Determine, for x[n] real, the relationship between points in X_M[k].

(c) (i) Let R[k] = X_M[2k]; that is, R[k] contains the even-numbered points in X _M[k]. From your answer in part (b), show that X_M[k] can be recovered from R[k]. 

(ii) R[k] can be considered to be the N/2-point MDFT of an N/2-point sequence r[n]. Determine a simple expression relating r[n] directly to x[n]. According to parts (b) and (c), the N-point MDFT of a real sequence x[n] can be computed by forming r[n] from x[n] and then computing the N/2-point MDRT of r[n]. The next two parts are directed at showing that the MDFT can be used to implement a linear convolution.

(d) Consider three sequences x₁[n], x₂[n], and x₃[n], all of length N. Let X_1M[k], X_2M[k], and X_3M[k], respectively, denote the MDFTs of the three sequences. If 

X_3M[k] = X_1M[k] X_2M[k],

 express x₃[n] in terms of x₁[n] ans x₂[n]. Your expression must be of the form of a single summation over a “combination” of x₁[n] x₂[n] in the same style as (but not identical to ) a circular convolution.

(e) It is convenient to refer to the result in part (d) as a modified circular convolution. If the sequences x₁[n] and x₂[n] are both zero for n ≥ N/2, show that the modified circular convolution of x₁[n] and x₂[n] is identical to the linear convolution of x₁[n] and x₂[n].

X 4[k] = X(z)| k = 0, 1. 2. .... N 1. z=e{?+k;Ni+;N