Minimum Variance - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 167

The aim of this decomposition is to set apart the spectral information at
frequency f c . When ε tends to 0, the integral of the spectrum ()

o

x Sf

tends to the

searched quantity () Pf . By applying the first constraint defined by equation [7.1],

when ε tends to 0, equation [7.2] becomes:

c

+ f /2

c

2

() ()

i

( )

∫

P = A f S f df + P f [7.3]

f x c

c

− f /2

c

with:

+ f /2

c

()

o

()

∫

Pf = lim S fdf

c x

ε → 0

− f /2

c

The second constraint consists of minimizing the integral of equation [7.3].
Given the fact that the added term P ( f c ) is a constant, it is equivalent to minimizing
the total output power P. By applying the Parseval relation, the dual expression is
expressed according to the covariance matrix R x of the input signal x ( k ) and of the
vector A of the coefficients of a FIR filter impulse response at frequency f c :

f

c

H
f

P = ARA [7.4]

xf

c c

( )

( )

e π

2 jM − 1 fTc

()

H T 2 π jfTc

Af = EA E = 1, e , … ,

f f f f

c c c

with:

( )

( )

H T

( ) ( )

RXX = E ⋅ matrix M × M , X = x k − M , … , x k − 1

x

The exponents ( ) T and ( ) H represent transpose and hermitic transpose
respectively.

The minimization of equation [7.4] under the constraint [7.1] is achieved by the
technique of Lagrange multipliers [CAP 70]. It makes it possible to obtain the filter
impulse response at frequency f c :

− 1

RE

xf

A =

c

[7.5]

f

H
f

− 1

c

ERE

xf

c c