Digital Signal Processing Reference
In-Depth Information
MV power
zoom in frequency
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Figure 7.6.
Illustration of the bias and of the MV estimator resolution. Superposition of
spectra estimated by the MV method at orders 3, 10, 20, and 70 on a signal made up of a
noisy sinusoid (0.25 Hz, amplitude 1, sampling frequency 1 Hz, signal to noise ratio of 20 dB,
128 time samples). Horizontal axis: zoomed frequency around the frequency of the sinusoid.
Vertical axis: linear spectrum
As regards the variance given by equations [7.21] and [7.22], in the particular
case of an exponential embedded in an additive white noise (see equation [7.7]), the
variance is written at the exponential frequency:
4
2
2
σ σ
=+⋅
C
(
)
()
var
Pf
2
for
f
=
f
MV
c
c
exp
2
M
M
This equation shows us again dependency of the variance in relation to the order
M. Actually, it will be more interesting to calculate this variance at another
frequency than the frequency of the exponential, or starting from the frequency
response given by [7.12]. The calculations are more complex there and we will
content ourselves with experimental observations.
[CAP 70] and [CAP 71] presented an experimental study in the array processing
domain in order to argue the fact that the MV estimator has a weaker variance than a
periodogram and an autoregressive estimator for long time-duration signals. This
study was extended in spectral analysis in [LAC 71] and taken again in
[KAY 88].
This weak variance is an important property of the MV estimator that we have
already illustrated in Figures 7.1, 7.2 and 7.3. Figure 7.7 illustrates the variance
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