Minimum Variance - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 192

7.6. The CAPNORM estimator

The strong idea of the normalized minimum variance estimator presented in
section 7.5 is to define a spectral density estimator starting from the power estimator
given by the minimum variance. This idea is based on equation [7.35]. However,
this equation makes the implicit hypothesis of the MV narrow band filters, a
hypothesis that is not verified. It is of interest to propose another writing of this
equation.

A solution consists of adapting the integration support to the filter shape. [DUR
00] proposes such an adaptation, which leads to a new estimator called CAPNORM.

The significant part of the frequency response is located around the filter
frequency f c . Let ∆ B be the width of the lobe containing f c , the contribution to the
integral of equation [7.2] is close to 0 outside this band, either S x ( f ) ≈ 0, or
|A f c ( f )| ≈ 0. We can then write:

2

() () ()

≈ ∫

Pf Af Sf f [7.51]

MV c f x

c

∆ B

If we make the hypothesis that S x ( f ) is constant in the band ∆ B , such as
S x ( f ) ≈ S x ( f c ), then equation [7.51] is written:

() 2 () ()

∫

Pf ≈ Sf Af f [7.52]

MV c x c f

c

∆ B

By substituting [7.6] into [7.52], we obtain the definition of a new estimator
noted by

()

S f [DUR 00]:

CNorm c

1

()

S f = [7.53]

CNorm c

H
fc

− 1

()

ERE β f

x fc CNorm c

()

with β CNorm f defined by:

c

() 2

()

≈ ∫

β f A f df [7.54]

CNorm c f

c

∆ B

It remains to evaluate this local integral that we will numerically approximate

by:

() 2 ()

β f ≈∆ B sup A f [7.55]

CNorm c f

c

f ∈∆ B

with the band ∆ B = [ f - , f + ], which can be called an equivalent band of the filter at the
frequency f c , defined by: