Minimum Variance - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 166

c c

c

Figure 7.4. Construction of a FIR filter bank for the analysis of the signal x ( k )
at all frequencies f c between 0 and f c /2

The first constraint implies that the filter frequency response, noted ()
c

, is

A f

f c

equal to 1 at the frequency f c , given:

() 1

Af = [7.1]

f c

c

In order take the second constraint into account, the power P of the filter output
y ( k ) is expressed as the spectral density of the input using the relation:

+ f /2

( ( )

c

2

2

() ()

= ∫ PEyk = A f S f f [7.2]

f x

c

− f /2

c

In order to set apart the frequency f c in which we take interest, the power spectral
density of the signal x ( k ) is decomposed in two parts:

() () ()

o i

Sf = S f + Sf

x x x

with:

() for

⎧
= ⎨ ⎩

Sf f −≤ ≤ + ε f f ε

o

()

x c c

Sf

x

0 other ise

() for

⎧
= ⎨ ⎩

Sf f <− f ε and f >+ f ε

i

()

x c c

Sf

x

0 other ise

knowing that ε . f

c