function y = circonvt(x1,x2,N)

	%% N-point Circular convolution between x1 and x2: (time domain)

	%% ------------------------------------------------------------------

	%% [y] = circonvt(x1,x2,N)

	%% y  = output sequence containning the circular convolution

	%% x1 = input sequence of length N1 <= N

	%% x2 = input sequence of length N2 <= N

	%%

	%% N = size of circular buffer

	%% Method: y(n) = sum( x1(m)*x2((n-m) mod N) )

	%% Check for length of x1

	if length(x1) > N

		error('N must be >= the length of x1 !')

	end

	%% Check for length of x2

	if length(x2) > N

		error('N must be >= the length of x2 !')

	end

	x1 = [x1 zeros(1,N-length(x1))];

	x2 = [x2 zeros(1,N-length(x2))];

	m = [0:1:N-1]; x2 = x2(mod_1(-m, N)+1); H = zeros(N,N);

	for n = 1:1:N

		H(n,:) = cirshftt(x2,n-1,N);

	end

	y = x1*conj(H');         % x1---row vector

%    H

%    y = H*x1';               % x1---column vector

%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

%%            Output Info about this m-file

fprintf('\n***********************************************************\n');

fprintf('        <DSP using MATLAB> Problem 5.24 \n\n');

banner();

%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

% -------------------------------------------------------------------

%

% -------------------------------------------------------------------

N = 4;

n1 = [0:3];

x1 = [1, 2, 2];

x2 = [1, 2, 3, 4];

y1 = circonvt(x1, x2, N)

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 <= N

	%%

	%%  N  = size of circulant matrix

	if length(x) > N

		error('N must be >= the length of x !')

	end

	x = [x zeros(1, N-length(x))];

    for i = 1 : N

	    c(i) = x(i);

    end

    m = [0:1:N-1]; x_fold = x(mod_1(-m, N)+1);

    r = x_fold;

    C = toeplitz(c,r);

function y = circonvt_v3(x1,x2,N)

	%% N-point Circular convolution between x1 and x2: (time domain)

	%% ------------------------------------------------------------------

	%% [y] = circonvt(x1,x2,N)

	%% y  = output sequence containning the circular convolution

	%% x1 = input sequence of length N1 <= N

	%% x2 = input sequence of length N2 <= N

	%%

	%% N = size of circular buffer

	%% Method: y(n) = sum( x1(m)*x2((n-m) mod N) )

	%% Check for length of x1

	if length(x1) > N

		error('N must be >= the length of x1 !')

	end

	%% Check for length of x2

	if length(x2) > N

		error('N must be >= the length of x2 !')

	end

	x1 = [x1 zeros(1,N-length(x1))];

	x2 = [x2 zeros(1,N-length(x2))];

	C = circulnt(x2, N);

%	m = [0:1:N-1]; x2 = x2(mod_1(-m, N)+1); H = zeros(N,N);

%	for n = 1:1:N

%		H(n,:) = cirshftt(x2,n-1,N);

%	end

%	y = x1*conj(H');         % x1---row vector

    y = C*x1';               % x1---column vector

%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

%%            Output Info about this m-file

fprintf('\n***********************************************************\n');

fprintf('        <DSP using MATLAB> Problem 5.25 \n\n');

banner();

%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

% -------------------------------------------------------------------

%

% -------------------------------------------------------------------

N = 4;

n1 = [0:3];

x1 = [1, 2, 2];

x2 = [1, 2, 3, 4];

%C = circulnt(x2, 4);

y1 = circonvt_v3(x1, x2, N)

function x3 = circonvf(x1, x2, N)

	%% N-point Circular convolution between x1 and x2: (frequency domain)

	%% ------------------------------------------------------------------

	%% [x3] = circonvf(x1,x2,N)

	%% x3 = output sequence containning the circular convolution

	%% x1 = input sequence of length N1 <= N

	%% x2 = input sequence of length N2 <= N

	%%

	%% N = size of circular buffer

	%% Method: x3(n) = IDFT[X1(k)X2(k)]  

	%% Check for length of x1

	if length(x1) > N

		error('N must be >= the length of x1 !')

	end

	%% Check for length of x2

	if length(x2) > N

		error('N must be >= the length of x2 !')

	end

	x1 = [x1 zeros(1,N-length(x1))];

	x2 = [x2 zeros(1,N-length(x2))];

    X1k_DFT = dft(x1, N);

    X2k_DFT = dft(x2, N);

    x3 = real(idft( X1k_DFT.* X2k_DFT, N));

%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

%%            Output Info about this m-file

fprintf('\n***********************************************************\n');

fprintf('        <DSP using MATLAB> Problem 5.26 \n\n');

banner();

%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

% -------------------------------------------------------------------

%

% -------------------------------------------------------------------

N = 4;

n1 = [0:3];

x1 = [4,3,2,1];

%x1 = [1,2,2];

n2 = [0:3];

x2 = [1, 2, 3, 4];

%C = circulnt(x2, 4);

y1 = circonvf(x1, x2, N)