Biomedical Engineering Reference
In-Depth Information
R
−
1
total
(
f
)
l
(
r
)
w
(
r
,
f
)
=
,
(3.68)
)
R
−
1
l
T
(
r
total
(
f
)
l
(
r
)
where
R
total
(
)
=
R
T
(
)
+
R
C
(
f
f
f
)
. Then, the
F
-ratio image,
F
(
r
,
f
)
, is obtained
such that
)
R
T
(
)
R
C
(
T
T
)
=
w
(
r
,
f
f
)
w
(
r
,
f
)
−
w
(
r
,
f
f
)
w
(
r
,
f
)
F
(
r
,
f
.
(3.69)
)
R
C
(
T
w
(
r
,
f
f
)
w
(
r
,
f
)
On the right-hand side of Eq. (
3.69
), the first term in the numerator represents the
reconstruction of the source power in the target period and the second represents that
in the control period. Thus,
F
represents the ratio of the reconstructed source
power change to the power of baseline activities.
(
r
,
f
)
3.7.3 Frequency-Domain Implementation
The narrow-band beamformer can also be implemented in the frequency domain. We
first define the Fourier transform of the raw-trial vector
b
n
(
t
)
as
g
n
(
f
)
.Thesample
cross-spectrum matrix
R
(
f
)
is computed using
g
n
(
f
)
, such that
N
E
1
N
E
R
H
(
f
)
=
1
g
n
(
f
)
g
n
(
f
)
,
(3.70)
n
=
where the superscript
H
indicates the Hermitian transpose (complex conjugation
plus matrix transpose). Using Eq. (
3.70
), we compute the frequency-specific cross-
spectral matrix for the target period,
R
T
(
, and for the control period,
R
C
(
f
)
f
)
.The
frequency-selective weight
w
(
r
,
f
)
is obtained such that
R
−
1
total
(
f
)
l
(
r
)
w
(
r
,
f
)
=
,
(3.71)
)
R
−
1
l
T
(
r
total
(
f
)
l
(
r
)
where
R
total
(
)
=
R
T
(
)
+
R
C
(
f
f
f
)
. The pseudo
F
-ratio image is computed using
)
R
T
(
)
R
C
(
H
H
)
=
w
(
r
,
f
f
)
w
(
r
,
f
)
−
w
(
r
,
f
f
)
w
(
r
,
f
)
F
(
r
,
f
.
(3.72)
)
R
C
(
H
w
(
r
,
f
f
)
w
(
r
,
f
)
