BSA Flow Software Installation and User's Guide - CSI

-again S’(f) is used to distinguish between the raw and the smoothed
estimate.
Similar to ensemble averaging, the variance of the estimator is reduced by a
factor l, and standard deviation and normalized standard error are thus
reduced accordingly:
ε
r
[ S$ uv(
f)
]
S ( f)
σ
≡ =
uv
1
l
As opposed to ensemble smoothing this method preserves frequency
resolution, but frequency smoothing introduces the problem of peak
smearing instead.
(7-79)
The problem arises from the fact that the smoothed spectrum estimate at any
one particular frequency is actually the result of calculations performed over
a range of neighboring frequencies.
If there is a peak in the true spectrum, the energy contained in this peak will
“leak” to neighboring frequencies, while at the same time the true peak is
“contaminated” by the lower spectrum estimates from neighboring
frequencies. This leads to artificial broadening of peak width and reduction
of peak height.
In severe cases a peak in the true spectrum may not be recognized at all.
In practice frequency smoothing is implemented as filter functions, some of
which are more advanced than simple mean values. To some extent these
advanced filter functions may reduce the above mentioned problem, but it
cannot be removed completely. The filter functions are described in more
detail later in section 0 of this manual.
Smoothing bias Both frequency and ensemble smoothing increase bias according to the
mechanism illustrated in Figure 7-88. As explained above, ensemble
smoothing reduce frequency resolution by increasing ∆f, thereby directly
increasing bias as explained on page 7-131. Frequency smoothing preserves
resolution, but introduce bias by including spectrum estimates from
neighboring frequencies. The effect is exactly the same as for ensemble
averaging: The resulting smoothed spectrum estimate is an average value
estimated over a wider frequency range, and bias is introduced accordingly if
the second order derivative of the true spectrum with respect to frequency is
nonzero within the range in question.
End effects
Correlations and spectra are calculated using Fast Fourier Transformation
(FFT). This approach gives much higher calculation speed than a direct
implementation, but the method is based on the assumption that the input
signal is cyclic with period corresponding to the sampling period T. In most
cases this assumption does not hold, and consequently errors are introduced.
The error is present both in the frequency and the time-domain. In the
frequency-domain the error is known as cyclic noise, but the problem is
easier explained and understood in the time-domain, where it is known as
circular correlation (see Figure 7-89).
BSA Flow Software:Reference guide 7-133