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206 Chapter 5 THE DISCRETE FOURIER TRANSFORM
P5.22 A 512-point DFT X(k) ofareal-valued sequence x(n) has the following DFT values:
X(0)=20+jα; X(5)=20+j30; X(k 1)=−10 + j15; X(152) = 17 + j23;
X(k 2)=20− j30; X(k 3)=17− j23; X(480) = −10 − j15; X(256) = 30 + jβ
and all other values are known to be zero.
1. Determine the real-valued coefficients α and β.
2. Determine the values of the integers k 1, k 2, and k 3.
3. Determine the energy of the signal x(n).
4. Express the sequence x(n) inaclosed form.
P5.23 Let x(n) beafinite length sequence given by
x(n) =
{
...,0, 0, 0, 1, 2, −3, 4, −5, 0,...
↑
Determine and sketch the sequence x((−8 − n)) 7R 7 (n) where
R 7 (n) =
{
1, 0 ≤ n ≤ 6
0, elsewhere
P5.24 The circonvt function developed in this chapter implements the circular convolution as a
matrix-vector multiplication. The matrix corresponding to the circular shifts {x((n − m)) N ;
0 ≤ n ≤ N − 1} has an interesting structure. This matrix is called a circulant matrix, which
is a special case of Toeplitz matrix introduced in Chapter 2.
1. Consider the sequences given in Example 5.13. Express x 1(n) asacolumn vector x 1 and
x 2((n − m)) N as a circulant matrix X 2 with rows corresponding to n =0, 1, 2, 3.
Characterize this matrix X 2. Can it completely be described by its first row (or column)?
2. Determine the circular convolution as X 2x 1 and verify your calculations.
P5.25 Develop a MATLAB function to construct a circulant matrix C given an N-point sequence
x(n). Use the toeplitz function to implement matrix C. Your subroutine function should
have the following format:
}
function [C] = circulnt(x,N)
% Circulant Matrix from an N-point sequence
% [C] = circulnt(x,N)
% C = circulant matrix of size NxN
% x = sequence of length