BSA Flow Software Installation and User's Guide - CSI

Rxy(τ)
True correlation
Circular contribution
T
Figure 7-92: Separating true and circular correlation using zero padding.
With zero padding the correlation function can thus be estimated with lag
times as big as τ=T. For large lags you should remember however that
within the original data set used for the calculation only a very limited
number of samples have been available with this long time between them.
Consequently the correlations estimated at large lag times may not be very
accurate.
Improved resolution Apart from removing cyclic noise, zero padding has the benefit of improving
the frequency resolution of the calculated spectra. By adding a number of
zeroes after your true samples, the sampling period T is artificially increased.
Without zero padding the frequency resolution is ∆f=1/T, but increasing the
sampling period to 2T will change the resolution to ∆f=1/2T. In principle
more zeroes can be added to improve the resolution further, but since no new
information is really added, you should think of this technique as nothing but
an intelligent form of interpolation between raw estimates.
Calculation speed Since the amount of data going into the FFT-analysis increases, zero padding
will obviously slow down the calculations. With the number of samples
included in a typical LDA-experiment, the FFT-approach will however
remain superior to a direct calculation of correlations and spectra.
Filters
As explained in section 0, the “raw” spectrum estimates may deviate
considerably from the true value. Dividing the resampled data into blocks to
perform ensemble smoothing will improve this, and further smoothing of the
calculated spectrum can be achieved by averaging neighboring spectrum
estimates. In practice this so-called frequency smoothing is implemented as
filter-functions, sometimes referred to as lag-windows due to the way they
are implemented. Frequency-smoothing could be programmed as a sweep
over the frequency-range for which estimates have been calculated, but again
FFT-algorithms are used to increase speed by swapping back and forth
between frequency- and time-domain:
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 frequency-domain.
The lag-window is implemented by multiplying each correlation estimate
with a factor wL(τ) depending on the lag-time τ of the estimate itself.
7-136 BSA Flow Software: Reference guide
2T
τ