Syntax Supported by the IML Procedure and the iml Action

IFFTC (x) ;

This function is supported by the IML procedure and the iml action.

The IFFTC function computes the inverse finite Fourier transform of a complex-valued vector, where x is an (or ) numeric matrix that represents a complex vector. The first column (or row) of x specifies the real components, and the second column (or row) specifies the imaginary components. (If x is a matrix, the first row contains the real components, and the second row contains the imaginary components.)

The IFFTC function returns an matrix that contains the complex Fourier coefficients that correspond to x. The first column of the matrix contains the real components of the Fourier coefficients, and the second column contains the imaginary components.

The inverse Fourier transform is related to the Fourier transform. The expression IFFTC(FFTC(x)) is equal to nx for every vector x. Similarly, FFTC(IFFTC(x)) equals nx.

The Fourier coefficients can be expressed in terms of a complex matrix multiplication. If the input x represents a complex column vector z of dimension n and if , where , then define a matrix W as

Then the complex vector of Fourier coefficients is simply the product .

The following example demonstrates the IFFTC function:

/* sample every millisecond from 0 to 10 seconds */
t = do(0,10,0.001);
pi = constant("pi");
/* complex signal takes values on the unit circle and makes
   three full rotations per second in the *clockwise* direction */
freq = 3;
x = cos(2*pi*freq*t); y = sin(-2*pi*freq*t);
Signal = t(x) || t(y);
f = fftc(Signal);

/* reconstruct the original signal from the Fourier coefficients */
z = ifftc(f) / nrow(f);
maxDiff = max(abs(Signal - z));  /* this should be essentially zero */
print maxDiff;