BSA Flow Software Installation and User's Guide - CSI

Estimator smoothing
As explained above the frequency resolution ∆f of the calculated spectrum
depend on the duration T of the sampling period. Consequently the
additional information acquired increasing T is used solely to produce
spectrum estimates at a higher number of discrete frequencies rather than
reducing the variance of any one particular spectrum estimate.
Analysis of LDA-measurements is based on discrete velocity samples, and
you might attempt to reduce the estimator variance by increasing the sample
rate, producing more samples within the same finite sampling period T.
Unfortunately this will not improve things either: According to the Nyquist
sampling criterion you will not be able to estimate the spectrum at
frequencies above half the sampling rate. The additional information
acquired increasing the sample rate will be used to produce additional
spectrum estimates at higher frequencies, and again this does not reduce the
variance of any one particular spectrum estimate.
In practice, the random errors of an estimate produced by (7-75) can be
reduced by smoothing the estimated spectrum.
–There are two ways of doing this:
• Ensemble smoothing: Averaging over an ensemble of estimates.
• Frequency smoothing: Averaging over neighboring frequencies.
Ensemble smoothing Ensemble smoothing is implemented by splitting the raw data into blocks of
equal duration. A separate spectrum estimate is calculated for each block,
and the final estimate is determined as the average of the calculated spectra:
i= q
S$ 1
f $ ∑ q
( ) = S′ ( f)
uv uv, i
i=
1
(7-76)
- where S’(f) is introduced to distinguish the raw and the smoothed estimate.
This will reduce the variance of the estimator by a factor q, where q is the
number of blocks used in the calculation. Consequently the standard
deviation and thus the normalized standard error is reduced by a factor √q:
ε
r
[ S$ uv(
f)
]
( )
σ 1
≡ =
(7-77)
S f q
uv
Since the duration of each block is only a fraction of the total sampling
period, the method has the drawback of reducing the frequency resolution of
the calculated spectrum. With a direct estimate based on Fourier transform of
the entire sampling period T the frequency resolution becomes ∆f=1/T, while
ensemble averaging reduces resolution to ∆f=q/T.
Apart from reducing estimator variance, the ensemble averaging will also
reduce calculation time slightly, since a separate Fourier transform of several
smaller blocks is faster than calculating the Fourier transform of one large
block.
Frequency smoothing Frequency smoothing is done by averaging together the results for l
neighboring spectral components in the estimate based on Fourier transform
of the entire sampling period:
( l )
j=
−1
2
1
( ) ∑ ( + )
S$ $
uv fi= S′ uv fij
l= 135 , , , K
l
( l)
j=
1− 2
(7-78)
7-132 BSA Flow Software: Reference guide