Adaptive Beamformers - Electromagnetic Brain Imaging - page 33

( )

This weight vector again does not depend on the norm of the lead field l r .The

output power of this beamformer is given by

l T R − 1 l ( r ) ( r )

2

s ( r , t ) =

) ] .

(3.18)

l T R − 2 l [ ( r ) ( r

This beamformer was first proposed by Borgiotti and Kaplan [ 8 ] and it is referred to
as the unit-noise-gain (constraint) minimum-variance beamformer, 2 or the weight-

normalized minimum-variance beamformer.

3.3 Semi-Bayesian Derivation of Adaptive Beamformers

The adaptive beamformer can be derived based on a Bayesian formulation [ 9 ]. Let
the relationship between the sensor data y ( t ) and the voxel source distribution x ( t )

be expressed in Eq. ( 2.15 ) , rewritten here as:

y ( t ) = Hx ( t ) + ʵ ,

where H is the lead field matrix defined in Eq. ( 2.11 ) . We also assume that the noise
in the sensor data is assumed such that

2 I

ʵ ∼ N( ʵ | 0 , ˃ ). (3.19)

We assume that the prior distribution for the source vector x ( t ) is the zero mean

Gaussian with a diagonal precision matrix, i.e.,

, ʦ − 1 p ( x ( t )) = N( x ( t ) | 0 ), (3.20)

where the precision matrix is expressed as

⊡
⊣

⊤
⊦ .

ʱ 1 ··· 0

. .

. . .

ʦ =

0 ··· ʱ N

An entirely rigorous Bayesian treatment using this prior distribution leads to the
Champagne algorithm, which is the topic of Chap. 4 . In the following, we show that
adaptive beamformer algorithm can also be derived using this prior distribution.

2 This name comes from the fact that spatial filter's noise gain is equal to the squared weight norm
w (

2 .

r )