DFT/IDFT of Complex Data

cfft(A) Returns the Discrete Fourier Transform of a vector or matrix. The result has the same number of rows and columns as A. If A is a vector, the result is given by:

where n is the number of elements in v. For matrix arguments, cfft returns the two-dimensional Fourier transform, scaled by 1/n. Calculated using the Singleton method.

icfft(u) Returns the inverse Fourier transform for u created with cfft.

CFFT(A) Returns the Discrete Fourier Transform of a vector or matrix. The formula is equivalent to cfft, but is scaled by 1/n instead of 1/√n (1/n² in the 2-D case), and uses a negative exponent going from the time to the frequency domain.

ICFFT(u) Returns the inverse Fourier transform for u created with CFFT. The formula is equivalent to icfft, but is scaled by 1/n instead of 1/√n, and uses a positive exponent going from the frequency to the time domain.

Arguments:

• A is a complex vector or matrix of any size, although the function works faster when the number of elements in the vector is 2^m. A vector whose length is a large prime number slows down the function.

Notes:

• If A is a real-valued vector that has 2 ^m elements (m > 2), use fft/FFT instead, for speed.

• The frequency spacing associated with the kth element in the calculated FFT spectrum is given by:

  

  where f_s is the sampling frequency of the original signal and n is the number of samples. As a consequence, since k must be an integer, spreading in the spectrum occurs unless the sampling frequency is chosen so that

  

  is always equal to an integer for any period 1/T in the signal.

• The complex transform is cyclic, from 0 (inclusive) to 2pi (noninclusive). This means that the first half of the values returned corresponds to the positive frequency spectrum, and the second half to the negative frequency spectrum. The first value in the returned sequence is always at zero frequency (DC), and the remaining values can be split in half and recentered, if desired. If the input, A, is real-valued, these remaining values are complex conjugate symmetric. See the QuickSheet.
• The sampling frequency is distinct from the frequency(ies) of the original time-domain signal.

QuickSheet