BSA Flow Software Installation and User's Guide - CSI

The mean values of u(t) and v(t) are subtracted to improve calculation
accuracy, ensuring that large DC-components do not drown small
fluctuations. This will produce U(0,T)=V(0,T)=0, and consequently
Suv(0)=0 corresponding to (7-65) without the delta function δ(f).
In principle the Fourier transforms should be calculated integrating from -∞
to +∞, but in practice you always sample the signals over a limited period of
time, as indicated by the finite limits of the integrations in (7-67). Provided
the sampling period is much longer than the largest time scale you wish to
investigate, this will be of little significance.
Spectrum estimator Since both u(t) and v(t) are real, U(-f, T) equals U*(f, T), with U*
representing the complex conjugate of U. Exploiting this and omitting the
limiting and expectation operations in (7-66), the spectral density function
Suv(f) can be estimated from U(f, T) and V(f, T) directly:
$ *
Suv ( f)
= U ( f, T) V( f, T)
T 1
(7-68)
Correlation estimator Once the spectral density has been estimated, inverse Fourier transformation
yield an estimate of the covariance:
∞
∫
C ( τ)
= S ( f) e
uv uv
−∞
i2πf τ
df (7-69)
-from which the correlation can be calculated using (7-57):
R ( τ) = C ( τ)
+ uv
uv uv
Digital calculations based on discrete samples
The definitions of correlations and spectra are based on the assumed
knowledge of the true continuous signals u(t) and v(t), but in a real LDAexperiment
time history records are not continuous. We get a velocitysample
whenever a seeding particle passes through the measuring volume.
This happens with random time intervals, providing a discrete representation
of the time history with arrival times in sequence, but with varying
interarrival times:
{ ut ( i), vt ( i)| ( i= 0, , N−1) ∧( 0≤ ti ≤ ti+ 1 ≤T)
}
K (7-70)
Without knowledge of the true continuous signals u(t) and v(t) we can only
estimate the spectra Suv(f). The estimates are usually fairly good at low
frequencies, but tends to deviate randomly at frequencies significantly above
the average sample rate.
In theory the random sampling allow us to determine spectra Suv(f) at very
high frequencies, since there is a nonzero probability that samples will occur
at time intervals smaller than the mean, thereby providing information on the
high frequency components of the signal.
In practice it proves difficult to get satisfactory results at frequencies
significantly higher than the mean data rate, because the variance of the
estimated spectra becomes very large unless extremely long sampling times
are used.
BSA Flow Software:Reference guide 7-125