Consider a finite-length sequence x[n] of length N as indicated in Figure. (The solid line is used to suggest the envelope of the sequence values between 0 and N ? 1.) Two finite-length sequence x₁[n] and x₂[n] of length 2N are constructed from x[n] as indicated in Figure, with x₁[n] and x₂[n] given as follows:?

The N-point DFT of x[n] is denoted by X[k], and the 2N-point DFTs of x₁[n] and x₂[n].

(a) Specify whether X₂[k] can be obtained if X[k] is given. Clearly indicate your reasoning.?

(b) Determine the simplest possible relationship whereby one can obtain X[k] X₁[k]. ?

Transcribed Image Text:

x[n]. 0s nsN-1. 0. x1[n]3D otherwise. 0sns N-1. -x[n - N]. Nsn< 2N-1. otherwise. x2[n] = 0. %3D x[n] N- 1 N - 1 (a) 2N - 1 x2{n] 2N - 1 N -1 (b)