パラメータ

A

実数または複素数ベクトル, 実数または複素数配列(ベクトル, 行列またはN-D配列).

X

Aと同じ形状の実数または複素数配列

sign

-1 or 1 : sign of the ±2iπ factor in the exponential term inside the transform formula, setting the direct or inverse transform. The default value is -1 = Direct transform.

directions

a vector containing indices of A dimensions (in [1, ndims(A)]) along which the (multidirectional) FFT must be computed. Default directions=1:ndims(A): The "cumulated" FFT is computed for all directions. See the Description section.

symmetry

optional character string, helping fft() to choose the best algorithm:

• "symmetric": forces to consider A or all its "slices" as conjugate symmetric. This is useful when an exact symmetry of A or its "slices" is possibly altered only by round-off errors.
  A N-D array B of sizes [s1,s2,..,sN] is conjugate symmetric for the FFT if and only if B==conj(B([1 s1:-1:2],[1 s2:-1:2],...,[1 sN:-1:2])). In such a case, the result X is real, and an efficient specific algorithm can be used to compute it.

• "nonsymmetric": Then fft() does not take care of any symmetry.

• not specified: Then an automatic determination of symmetry is performed.

dims

整数値を有する正の数のベクトルまたは正の整数値のベクトル. 詳細は説明のパートを参照ください. 各要素はAの要素の総数の約数とする 必要があります. 要素の積はAの全要素数より 少ない必要があります.

incr

整数値を有する正の数のベクトルまたは正の整数値のベクトル. 詳細は説明のパートを参照ください. incr は, dimsと同じ要素数とする必要があります. 各要素は Aの全要素数の約数とする必要があります. incr要素は厳密に昇順とする必要があります.

説明

for i1 = 1:size(A,1)
    for i3 = 1:size(A,3)
        X(i1,:,i3) = fft(A(i1,:,i3), -1);
    end
end

for i2 = 1:size(A,2)
    X(:,i2,:) = fft(A(:,i2,:), -1);
end

例

1次元FFT

// Frequency components of a signal
// --------------------------------
// build a noised signal sampled at 1000hz  containing  pure frequencies
// at 50 and 70 Hz
sample_rate = 1000;
t = 0:1/sample_rate:0.6;
N = size(t,'*'); // number of samples
s = sin(2*%pi*50*t) + sin(2*%pi*70*t+%pi/4) + grand(1,N,'nor',0,1);

y = fft(s);

// s is real so the fft response is conjugate symmetric and we retain only the first N/2 points
f = sample_rate*(0:(N/2))/N;  // associated frequency vector
n = size(f,'*')
clf()
plot(f, abs(y(1:n)))

2次元FFT

//----------------------------------
A = zeros(256,256);
A(5:24,13:17) = 1;
X = fftshift(fft(A));
set(gcf(), "color_map",jet(128));
clf;grayplot(0:255,0:255,abs(X)')

N-Dimensional FFT:

// simple case, 3 1-D fft at a time
N = 2048;
t = linspace(0,10,2048);
A = [2*sin(2*%pi*3*t)+ sin(2*%pi*3.5*t)
     10*sin(2*%pi*8*t)
     sin(2*%pi*0.5*t) + 4*sin(2*%pi*0.8*t)];
X = fft(A,-1,2);

fs = 1/(t(2)-t(1));
f = fs*(0:(N/2))/N; // associated frequency vector
clf; plot(f(1:100)', abs(X(:,1:100))')
legend(["3 and 3.5 Hz","8 Hz","0.5 and 0.8 Hz"],"in_upper_left")

// 45  3-D fft at a time
Dims = [5 4 9 5 6];
A = matrix(rand(1,prod(Dims)),Dims);

y = fft(A,-1,[2 4 5]);

// equivalent (but less efficient code)
y1 = zeros(A);
for i1 = 1:Dims(1)
    for i3 = 1:Dims(3)
        ind = list(i1,:,i3,:,:);
        y1(ind(:)) = fft(A(ind(:)), -1);
  end
end

// Using explicit formula for  1-D discrete Fourier transform
// ----------------------------------------------------------
function xf=DFT(x, Sign);
    n = size(x,'*');
    // Compute the n by n Fourier matrix
    am = exp(Sign*2*%pi*%i*(0:n-1)'*(0:n-1)/n);
    xf = am*matrix(x,n,1);//dft
    xf = matrix(xf ,size(x)); // reshape
    if Sign==1 then
        xf = xf/n;
    end
endfunction

// Comparison with the fast Fourier algorithm
a = rand(1,1000);
norm(DFT(a,1) - fft(a,1))
norm(DFT(a,-1) - fft(a,-1))

tic(); DFT(a,-1); toc()
tic(); fft(a,-1); toc()

参照

• corr — 相関 , 共分散
• fftw_flags — fftプランナアルゴリズム選択用手法を設定する
• get_fftw_wisdom — fftw wisdomを返す
• set_fftw_wisdom — fftw wisdomを設定
• fftw_forget_wisdom — fftw wisdomをリセット