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IEEE SIGNAL PROCESSING MAGAZINE [23] MARCH 2015
variance (LCMV) beamformer [20], [21], where the power of the
output signal is minimized subject to a single constraint assuring
an undistorted response for the target source (or a filtered version
of it). Different versions of the MVDR beamformer exist, either
using the complete target ATF, the direct path of the ATF, or the
relative transfer functions (RTFs). In practice, the MVDR beam-
former is often implemented using a so-called generalized sidelobe
canceler (GSC) structure [22]–[25].
DERIVATION OF THE MVDR BEAMFORMER
The power spectral density (PSD) of the filter-and-sum beam-
former output signal
y is given by
{} { } ,Ey Ewxxw w w
HH H
2
xx
U== (5)
where {}E xx
H
xx
U =
9
denotes the crosspower spectral density
matrix of the observed microphone signals. The distortionless
response constraint requires that the desired component in the
output signal
y
s
0
is equal to the target signal ,s
0
i.e.,
.ysswh
!
s
H
00 0
0
== (6)
Hence, by solving the constrained minimization problem
,,min 1subject toww wh
HH
0
w
xx
U = (7)
we obtain the MVDR filter [20], [21]
.w
hh
h
H
0
1
0
1
0
MVDR
xx
xx
U
U
=
-
-
(8)
By assuming the target source, the interfering sources and the
noise to be mutually uncorrelated and of zero mean, the
crosspower spectral density matrix
xx
U can be written using (3) as
,hh
ss
H
0
0
xx vv
00
z UU =+ (9)
where {}E vv
H
vv
U =
9
denotes the crosspower spectral density
matrix of the interference and noise components and
.{| | }Es
ss 0
2
00
z = Using (9), it can be shown that the MVDR filter
in (8) can be written as [20]
.w
hh
h
H
0
1
0
1
0
MVDR
vv
vv
U
U
=
-
-
(10)
As can be seen, the MVDR filter is solely determined by the
crosspower spectral density matrix of the observations and the
ATFs
.h
0
However, due to the high order and the typically time-
varying nature of the corresponding RIRs (),ht
,m0
blindly iden-
tifying these impulse responses is generally difficult if at all
possible. Hence, instead of using the complete RIRs, one can
consider only the direct path of the RIRs (corresponding to the
free-field HRIR for the estimated or assumed target DOA),
which may, however, lead to target signal distortion, or one can
use the so-called RTFs.
+
−
x
1
(k, )
x
2
(k, )
u
2
(k, )
y (k,
)
g
2
(k, )
u
3
(k, )
g
3
(k, )
x
3
(k, )
x
M
(k, )
u
M
(k, )
g
M
(k, )
FB
BM
Σ
Σ
. . .
. . .
. . .
[FIG5] The GSC implementation of an MVDR beamformer
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