In this problem, we consider a procedure for computing the DFT of four real symmetric or...

In this problem, we consider a procedure for computing the DFT of four real symmetric or antisymmetric N-point sequences using only one N-point DFT computation. Since we are considering only finite-length sequences, by symmetric and antisymmetric, we explicitly mean periodic symmetric and periodic antisymmetric, as defined in Section 8.6.4. Let x₁[n], x₂[n], x₃[n], and x₄[n] denote the four real sequences of length N, and let X₁[k], X₂[k], X₃[k], and X₄[k] denote the corresponding DFTs. We assume first that x₁[n] and x₂[n] are symmetric and x₃[n] and x₄[n] are antisymmetric; i.e.,

(a) Define y₁[n] = x₁[n] + x₃[n] and let Y₁[k] denote the DFT of y₁[n]. Determine how X₁[k] and X₂[k] can be recovered from Y₁[k].

(b) y ₁[n] as defined in part (a) is a real sequence with symmetric part x₁[n] and antisymmetric part x₃[n]. Similarly, we define the real sequence y₂[n] = x₂[n] + x₄[n], and we let y₃[n] be the complex sequence

First, determine how Y₁[k] and Y₂[k] can be determined from Y₃[k], and then, using the results of part (a), show how to obtain X₁[k], X₂[k], X₃[k], and X₄[k] from Y₃[k].

The result of part (b) shows that we can compute the DFT of four real sequences simultaneously with only one N-point DFT computation if two sequences are symmetric and the other two are antisymmetric. Now consider the case when all four are symmetric; i.e.,

for n = 0, 1, . . . , N − 1. For parts (c)–(f), assume x₃[n] and x₄[n] are real and symmetric, not antisymmetric.

(c) Consider a real symmetric sequence x₃[n]. Show that the sequence

is an antisymmetric sequence; i.e., u₃[n] = −u₃[N − n] for n = 1, 2, . . . , N − 1 and u₃[0] = 0.

(d) Let U₃[k] denote the N-point DFT of u₃[n]. Determine an expression for U₃[k] in terms of X₃[k].

(e) By using the procedure of part (c), we can form the real sequence y₁[n] = x₁[n] + u₃[n], where x₁[n] is the symmetric part and u₃[n] is the antisymmetric part of y₁[n]. Determine how X₁[k] and X₃[k] can be recovered from Y₁[k].

(f) Now let y₃[n] = y₁[n] + jy₂[n], where

for n = 0, 1, . . . , N − 1. Determine how to obtain X₁[k], X₂[k], X₃[k], and X₄[k] from Y₃[k]. (Note that X₃[0] and X₄[0] cannot be recovered from Y₃[k], and if N is even, X₃[N/2] and X₄[N/2] also cannot be recovered from Y₃[k].)