《DSP using MATLAB》Problem 5.12

        1、从别的地方找的证明过程：

        2、代码

function x2 = circfold(x1, N)
%% Circular folding using DFT 
%% ---------------------------------------------
%% x2 = circfold(x1, N)
%% x2 = circulary folded output sequence
%% x1 = input sequence of length <= N
%%  N = circular buffer length 
%% 
           
X1k_DFT = dft(x1, N);
Xk = dft(X1k_DFT, N);
x2 = Xk/N;

　　

%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%%            Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf('        <DSP using MATLAB> Problem 5.12 \n\n');

banner();
%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

% --------------------------------------------------------------
%              1  x(n) = [1, 3, 5, 7, 9, -7, -5, -3, -1]                                      
% --------------------------------------------------------------

xx1 = [1, 3, 5, 7, 9, -7, -5, -3, -1];
NN1 = length(xx1);                              % length is 6
nn1 = [0 : NN1-1];


% ----------------------------------------------------
%             1st way to get Circulary folded
% ----------------------------------------------------
xx1_cf = xx1( mod(-nn1, NN1)+1 );


figure('NumberTitle', 'off', 'Name', 'P5.12 x(n)')
set(gcf,'Color','white'); 
subplot(2,1,1); stem(nn1, xx1);
xlabel('n'); ylabel('x(n)');
title('x(n) ori sequence');  grid on;
subplot(2,1,2); stem(nn1, xx1_cf);
xlabel('n'); ylabel('x(n)');
title('x((-n))_N Circulary folded by mod way');  grid on;


% ------------------------------------------------------
%         2nd way to get Circulary folded
%              DFT
% ------------------------------------------------------


xx2 = circfold(xx1, NN1);

figure('NumberTitle', 'off', 'Name', 'P5.12 x(n)')
set(gcf,'Color','white'); 
subplot(2,1,1); stem(nn1, xx1);
xlabel('n'); ylabel('x(n)');
title('x(n) ori sequence');  grid on;
subplot(2,1,2); stem(nn1, xx2);
xlabel('n'); ylabel('x(n)');
title('x((-n))_N Circulary folded by DFT way');  grid on;

　　运行结果：