FFT/IFFT of Real Data

fft(v) Returns the Fast Fourier Transform of an n-element vector v, where n = 2^m. Result is a 1 + 2^m-1 element vector whose jth element is given by:

where n is the number of elements in v and i is the imaginary unit. Calculated using the Cooley-Tukey algorithm.

ifft(u) Returns the inverse Fourier transform for u created with fft. Result is a 2^m element vector whose jth element is given by:

FFT(v) Returns the Fast Fourier Transform of a vector v. The formula is equivalent to fft, but is scaled by 1/n instead of 1/√n, and uses a negative exponent going from the time to the frequency domain.

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

Arguments:

• v is a real-valued vector with 2^m elements (m > 2), representing samples at regular time intervals.
• u is a complex-valued vector with 1 + 2^m-1 elements (m > 2), representing samples at frequencies.

Notes:

• The frequency 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 sampling frequency is distinct from the frequency(ies) of the original time-domain signal.
• The fft/ifft functions take advantage of the complex conjugate symmetry of the Fourier transform, which only applies for real input data. To save time and memory, Mathcad does not calculate the second half of the frequency spectrum. For vectors with complex values, or with an arbitrary number of elements, use cfft or CFFT instead. Likewise, for 2-dimensional Fourier transforms, use cfft/CFFT instead.
• You can also evaluate Fourier transforms symbolically.

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