%%
clc;
clear;

N = 100;
eps = 1e-10;

% 生成傅里叶矩阵
F_n = dftmtx(N);

% 输出傅里叶矩阵
for i = 1: N
    for j = 1: N
        fprintf(string(F_n(i, j)) + ", ");
    end
    fprintf("\b\b\n");
end
fprintf("\n");

% 检查傅里叶矩阵
assert(abs(F_n(2, 2) ^ 2 - F_n(2, 3)) < eps);
assert(abs(F_n(2, 2) ^ N - 1) < eps);

%% 循环卷积
x = randn(N, 1);
z = randn(N, 1);
p = cconv(x, z, N);

% 检查循环卷积公式的正确性
for i = 1: N
    tmp = 0;
    for j = 1: N
        tmp = tmp + x(j) * z(mod(i-j+N, N)+1);
    end
    assert(abs(tmp - p(i)) < eps);
end

% 检查 x * z = A(x)z
A_x = toeplitz([x(1) fliplr(x(2:end)')], x);
assert(is_vector_equal(p, A_x' * z, eps));

% 检查卷积定理
x_hat = fft(x);
z_hat = fft(z);
xz_hat = fft(p);
assert(is_vector_equal(x_hat .* z_hat, xz_hat, eps));