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28.10.3 Fast Fourier Transform (FFT)

The Fourier transform utility in FLUENT allows you to compute the Fourier transform of a signal, $\phi(t)$, a real-valued function, from a finite number of its sampled points.

The discrete Fourier transform of $\phi_k$ is defined by

 \phi_k = \sum_{n=0}^{N-1} \hat{\phi}_n e^{2 \pi i k n /N} \hspace{0.5in}k = 0,1,2,...(N-1) (28.10-7)

where $\hat{\phi}_n$ are the discrete Fourier coefficients, which can be obtained from

 \hat{\phi}_n = \frac{1}{N} \sum_{k=0}^{N-1} \phi_k e^{- 2 \pi i k n /N} \hspace{0.5in}n = 0,1,2,...(N-1) (28.10-8)

Equation  28.10-7 and Equation  28.10-8 form a Fourier transform pair that allows us to determine one from the other.

Note that when we follow the convention of varying $n$ from 0 to $N-1$ in Equation  28.10-7 or Equation  28.10-8 instead of from $-N/2$ to $N/2$, the range of index $1 \leq n \leq N/2 -1$ corresponds to positive frequencies, and the range of index $N/2 + 1 \leq n \leq N-1$ corresponds to negative frequencies. $n=0$ still corresponds to zero frequency.

For the actual calculation of the transforms, FLUENT adopts the so-called fast Fourier transform (FFT) algorithm which significantly reduces operation counts in comparison to the direct transform. Furthermore, unlike most FFT algorithms in which the number of data should be a power of 2, the FFT utility in FLUENT employs a prime-factor algorithm [ 372]. The number of data points permissible in the prime-factor FFT algorithm is any products of mutually prime factors from the set 2,3,4,5,7,8,9,11,13,16, with a maximum value of $720720 = 5 \times 7 \times 9 \times 11 \times 13 \times 16$. Thus, the prime-factor FFT preserves the original data better than the conventional FFT.

Just prior to computing the transform, FLUENT determines the largest permissible number of data points based on the prime factors, discarding the rest of the data.

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