Minimum Variance - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 179

shows a zoom of it at the central frequency. It is clearly in evidence that the quantity
−

∆ d B of the estimated peak diminishes with the order, hence an improvement of
the frequency resolution with the order of the MV estimator.

It is interesting to know the capacity of a spectral estimator to separate two close
frequencies. Figure 7.8 illustrates this property for a signal whose noise to ratio
signal is of 10 dB. The three cases that were presented are compared to the
periodogram in its more resolutive version, that is, non-averaged and non-weighted.
Two peaks are separated if the amplitudes of the emerging peaks are higher than the
estimator standard deviation. Thus, in the top part of Figure 7.8, the difference
between the two frequencies is 0.02 Hz in reduced frequency and the MV estimator
as well as the periodogram separate the two peaks. In the middle of Figure 7.8, the
difference between the two frequencies is only 0.01 Hz. The MV estimator still
separates the two peaks while the periodogram presents only one peak. However, a
more detailed study of the periodogram could conclude on the possible presence of
two non-resolved peaks by comparing the obtained peak with the spectral window of
the estimator. Actually, the peak obtained by the periodogram has a bandwidth
greater than the frequency resolution of the periodogram (bandwidth at -3 dB for
128 samples equal to 0.008 Hz) [DUR 99]. In Figure 7.8 at the bottom, where the
frequency difference is only of 0.006 Hz, the estimators do not separate the peaks
even if two maxima are observed on the MV estimator. These maxima do not
emerge relatively to the estimator standard deviation.

7.3. Link with the Fourier estimators

The comparison with the Fourier-based methods is immediate if the estimator is
written in the form of a filtering. A periodogram or a non-averaged correlogram
FOU

( f ) can be written:

S

() 1

H

S f =

ERE

[7.25]

FOU f x f

M

This result is well known. The Fourier estimator is the output power of a filter
bank whose impulse response of each filter at the frequency f is an exponential
function at this frequency, a function limited in time by a rectangular window of M
samples, the length of the vector E f . This vector is the counterpart of

f A for the
Capon method (see equation [7.4]) but is totally independent of the signal. The
power is divided by 1 /M so that the result is the counterpart to a spectrum (see
section 7.5.1).

c

The statistic performances of a periodogram are well known. We transcribe
them in this section by using the same notations as the statistic study of the MV
estimator in section 7.2.2. The periodogram ( f ) is a non-central χ 2 , with two S

FOU