BSA Flow Software Installation and User's Guide - CSI

Filters are implemented using FFT-techniques to reduce calculation time:
1. Switch to time-domain performing a Fourier transform of the spectrum.
2. Apply a so-called Lag-window to the resulting correlation estimate.
3. Fourier transform back to the frequency-domain.
The Lag-window multiplies each correlation estimate with a factor w(τ)
depending on the lag-time τ of the estimate in question. This factor is one for
a lag-time of zero, and then decreases until it reaches zero. The various
filters differ only in the way the factor varies in between:
Box:
w
( τ)
⎧1
= ⎨
⎩0
for τ ≤ T
otherwise
Hanning:
⎧1
⎡ ⎛ τ ⎞ ⎤
⎪ π τ T
w( τ)
= ⎢cos⎜
⎟ + 1
⎣ ⎝ T ⎠ ⎥ for ≤
⎨2
⎦
⎩
⎪ 0
otherwise
Papoulis:
⎧ 1 ⎛ τ ⎞ ⎛ τ⎞⎛τ ⎞
⎪ sin⎜ π⎟
+ ⎜1−
⎟ cos⎜ π⎟for τ≤T
w( τ)
= ⎨π
⎝ T ⎠ ⎝ T⎠⎝T ⎠
⎪
⎩ 0
otherwise
In the Box-window (also known as a Top-hat window) the weighting factor
does not decrease gradually as with most other window-functions. Instead it
jumps from 1 to 0 at a threshold lag T, determined by the Filter width W:
T = W⋅τmax
-where τmax is the maximum lag-time of the correlation estimate calculated
by Fourier-transformation of the original spectrum.
6-56 BSA Flow Software: Options and Add-ons