[Fluent Inc. Logo] return to home search
next up previous contents index

28.10.2 Windowing

The discrete FFT algorithm is based on the assumption that the time-sequence data passed to the FFT corresponds to a single period of a periodically repeating signal. Since, in most situations, the first and the last data points will not coincide, the repeating signal implied in the assumption can often have a large discontinuity. The large discontinuity produces high-frequency components in the resulting Fourier modes, causing an aliasing error. You can condition the input signal before the transform by "windowing" it, in order to avoid this problem.

Suppose that we have $N$ consecutive discrete (time-sequence) data sampled with a constant interval, $\Delta t$:


 \phi_k \equiv \phi(t_k), \hspace{0.3in} t_k \equiv k \, \Delta t, \hspace{0.5in} k = 0,1,2,...,(N-1) (28.10-1)

Windowing is done by multiplying the original input data ( $\phi_j$) by a window function, $W_j$:


 \tilde{\phi}_j = \phi_j \; W_j \hspace{0.5in} j = 0,1,2,...,(N-1) (28.10-2)

FLUENT offers four different window functions:

Hamming's window:

 W_j = \left\{\begin{array}{ll} 0.54 - 0.46 \cos \left(\frac... ... & \,\,\, \frac{N}{8} < j < \frac{7N}{8} \end{array}\right.\\ (28.10-3)

Hanning's window:

 W_j = \left\{\begin{array}{ll} 0.5 [1 - \cos (\frac{8 \pi j... ... & \,\,\, \frac{N}{8} < j < \frac{7N}{8} \end{array}\right.\\ (28.10-4)

Barlett's window:

 W_j = \left\{\begin{array}{ll} \frac{8 j}{N} & \,\,\, j \leq... ... & \,\,\, \frac{N}{8} < j < \frac{7N}{8} \end{array}\right.\\ (28.10-5)

Blackman's window:

 W_j = \left\{\begin{array}{ll} 0.42 - 0.5 \cos (\frac{8 \pi... ... & \,\,\, \frac{N}{8} < j < \frac{7N}{8} \end{array}\right.\\ (28.10-6)

These window functions preserve a large fraction ( $3/4$) of the original data, affecting only $1/4$ of the data on both ends.


next up previous contents index Previous: 28.10.1 Limitations of the
Up: 28.10 Fast Fourier Transform
Next: 28.10.3 Fast Fourier Transform
© Fluent Inc. 2006-09-20