Digital Signal Processing Reference
In-Depth Information
M
a
N,n
≈
(
ωτ
0
)
n
n
!
(
m −
1)
n
H
m
(
ω
)
.
(2.26)
m=1
We observe from (2.25) that as long as
e
(
m −
1)
ωτ
0
cos
θ
can be approx-
imated by a MacLaurin's series of order
N
(that is why the microphone
spacing should be small), which includes derivatives up to the order
N
, we
can build
N
th-order differential arrays. We also observe from (2.26) that the
gains
H
m
(
ω
)
,m
=1
,
2
,...,M
, can be determined given the coe-cients
a
N,n
,n
=0
,
1
,...,N
. The least-squares solution (
N
+1
>M
) is not appro-
priate since not only the beampatterns will be highly frequency dependent
(it is very hard, if not impossible, to numerically approximate a derivative
of order
N
with a smaller number of points,
M
) but it is also very hard
to have exact nulls in some specific directions and a one at
θ
=0
◦
. The
minimum-norm solution (
N
+1
<M
) is a good choice from both theoretical
and practical viewpoints; this concept will be elaborated in Chapter 6. But
for all the other chapters, it will always be assumed that the design of an
N
th-order differential array requires
N
+ 1 microphones.
2.3 Gain in Signal-to-Noise Ratio (SNR)
The first microphone serves as the reference and we recall that the desired
signal comes from the angle
θ
=0
◦
. In this case, the
m
th microphone signal
is given by
−
(
m −
1)
ωτ
0
X
(
ω
)+
V
m
(
ω
)
,m
=1
,
2
,...,M,
Y
m
(
ω
)=
e
(2.27)
where
X
(
ω
) is the desired signal and
V
m
(
ω
) is the additive noise at the
m
th
microphone. In a vector form, (2.27) becomes
T
y
(
ω
)=
Y
1
(
ω
)
Y
2
(
ω
)
···Y
M
(
ω
)
=
d
(
ω,
cos0
◦
)
X
(
ω
)+
v
(
ω
)
,
(2.28)
where the noise signal vector,
v
(
ω
), is defined similarly to
y
(
ω
).
The beamformer output is simply
M
∗
Z
(
ω
)=
H
m
(
ω
)
Y
m
(
ω
)
m=1
=
h
H
(
ω
)
y
(
ω
)
=
h
H
(
ω
)
d
(
ω,
cos0
◦
)
X
(
ω
)+
h
H
(
ω
)
v
(
ω
)
,
(2.29)
where
Z
(
ω
) is supposed to be the estimate of the desired signal,
X
(
ω
).
We define the input signal-to-noise ratio (SNR) as
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