Minimum Variance - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 186

Moreover, this implicit hypothesis of the narrowband filters, which is necessary
in order to write equation [7.35], is detrimental to the properties of this estimator.
This consequence is discussed in section 7.5.2.

Most frequently, the proposed normalization [IOA 94, KAY 88] consists of
dividing the power

()

Pf by the quantity:

MVc

1 1

H

B = EE

c

= [7.40]

e f f

T

c

T M

s s

This quantity B e is the equivalent bandwidth of the impulse response E of a

f

c

Fourier estimator and not that of the response

f A of a MV filter. It is thus about an
approximation. The statistic performances of this estimator are studied in [IOA 94].

c

An alternative normalization is proposed in section 7.6.

7.5.2. Spectral refinement of the NMV estimator

The NMV method is not designed to have a high resolution. Its objective is to
estimate a homogenous quantity to a power spectral density and not to a power. The
frequency resolution is not improved, but the visualization makes it possible to
better separate close frequencies. In Figure 7.10 we present the improvement
brought by the NMV estimator in relation to the MV estimator when two
frequencies are close. At a finite order M, the peaks sharpen, but they still remain
centered on the same frequency.

()

β NMV c f and it can be
explained starting from the MV filter study presented in section 7.2 ([DUR 00]). Let

This effect is due to the normalization by the factor

()
()

Pf
S

MV
NMV

us calculate the ratio at the exponential frequency f exp and at a close

f

frequency ( f exp + ε) with ε tending to 0.

When the filter frequency f c is equal to the frequency f exp , the frequency response
is given by equation [7.15]. In this case, the normalization factor defined by
equation [7.36] and which we will note

()

β NMV f is written:

c

exp