In Section 2 8, the conjugate symmetric and conjugate antisymmetric components of a sequence x n were defined, respectively, as x e n (x n x n ) X 0 n (x n x n ) In Section 8 6 4, we found it convenient to respectively define the periodic conjugate symmetric and periodic conjugate anti symmetric co...

In Section 2.8, the conjugate-symmetric and conjugate-antisymmetric components of a sequence x[n] were defined, respectively, as x

Question:

In Section 2.8, the conjugate-symmetric and conjugate-antisymmetric components of a sequence x[n] were defined, respectively, as

x_e[n] = ½ (x[n] + x*[– n]).

X₀[n] = ½ (x[n] – x*[– n]).

In Section 8.6.4, we found it convenient to respectively define the periodic conjugate symmetric and periodic conjugate-anti symmetric components of a sequence of finite duration N as

x_ep[n] = ½{x[((n))_N] + x*[((– n))_N]}, 0 ≤ n ≤ N – 1,

x_0p[n] = ½ {x[((n))_N] − x*[((– n))_N]}, 0 ≤ n ≤ N – 1,

(a) Show that x_ep[n] can be related to x_e[n] and that x_0p[n] can be related to x₀[n] by the relations

x_ep[n] = (x_e[n] + x_e[n – N]), 0 ≤ n ≤ N – 1,

x_0p[n] = (x₀[n] + x₀[n – N]), 0 ≤ n ≤ N – 1.

(b) x[n] is considered to be a sequence of length N, and in general, x_e[n] cannot be recovered from

X_ep[n], and x₀[n] cannot be recovered from x_0p[n]. Show that with x[n] considered as a sequence of length N, but with x[n] = 0, n > N/2, x_e[n] can be obtained from x_ep[n], and x₀[n] can be obtained from x_op[n].