In Section 2.8, the conjugate-symmetric and conjugate-antisymmetric components of a sequ...

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

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

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

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