BSA Flow Software Installation and User's Guide - CSI

Estimator bias
Estimator variance
Sk = S$ 1 *
uv( fk)
= Uk V
(7-75)
k
T
In principle the FFT-analysis also yield spectrum estimates at negative
frequencies, but these are not shown, since the spectrum is always
symmetrical around f=0.
The estimator in (7-75) will produce spectrum estimates at discrete
frequencies fk=k/T (k=0, 1, 2, … ,N/2) and the frequency resolution of the
estimated spectrum will thus be ∆f=1/T. Each estimate Sk represents an
average power spectral density in the range fk±∆f/2, and will be bias free if
the true spectrum is constant or changing at a constant rate within this range.
In other words the second order derivative of the true spectrum with respect
to frequency should be zero:
2
dS
⎡ ∆f ∆f⎤
S′′ ( f)
= = 0 for f ∈ − +
2
df
⎣
⎢
fk ; fk
2 2 ⎦
⎥
If this is not fulfilled the estimator will be biased as shown in Figure 7-88.
S
fk-1
∆f
fk
True value
Estimate
Bias
fk+1
Figure 7-88: Bias of the estimator due to curvature of true spectrum.
Obviously the bias increases with increasing ∆f and with increasing S”(f).
∆f decrease with increasing sampling time, so in the limit T→∞ the spectrum
estimator (7-75) will be bias free.
Intuitively you would expect the estimate to approach the true value as T
approaches infinity. This is true with respect to the mean-value of the
estimate (bias approaches zero), but unfortunately it doesn’t hold for the
variance.
On the contrary the normalized standard error εr is always a 100%:
ε
r
[ S$ uv(
f)
]
S ( f)
σ
≡ = 1
uv
[ ]
-where σ ( ) $ Suv f is the standard deviation of the spectrum estimate $Suv ( f)
and S ( f)
is the true value (see [Bendat & Piersol (1971)])
uv
BSA Flow Software:Reference guide 7-131
f
,