Image Processing Reference
In-Depth Information
9.2.2 Estimating the Power Spectrum of a Signal Source
Using Sensor Network Data
Again, let
x
denote a discrete version of the signal produced by the source and assume that it
is a zero-mean Gaussian WSS random process. The sampling frequency
f
s
associated with
x
(
n
)
(
n
)
is
arbitrary and depends on the frequency resolution desired in the spectrum estimation process.
We denote by
v
i
(
n
)
thesignalproducedatthefrontendofthe
i
th sensor node. We assume that
v
i
(
n
)
arerelatedtotheoriginalsourcesignal
x
(
n
)
bythemodelshowninFigure..helinearil-
ter
H
i
in this figure models the combined effect of room reverberations, microphone's frequency
response, and an additional filter which the system designer might want to include. The decimator
block which follows the filer represents the (potential) difference between the sampling frequency
f
s
associated with
x
(
z
)
and the actual sampling frequency of the Mote's sampling device. Here, it
is assumed that the sampling frequency associated with
v
i
(
n
)
(
n
)
is
f
s
/
N
i
where
N
i
is a fixed natural
number.
It is straightforward to show that the signal
v
i
(
n
)
in Figure . is also a WSS processes. The
autocorrelation coefficients
R
v
i
(
k
)
associated with
v
i
(
n
)
are given by
R
v
i
(
k
)=
R
x
i
(
N
i
k
)
(.)
and
R
x
i
(
k
)=(
h
i
(
k
)⋆
h
i
(−
k
))⋆
R
x
(
k
)
,
(.)
where
h
i
(
k
)
denotes the impulse response of
H
i
(
z
)
. We can express
R
v
i
(
k
)
as a function of the source
)=
z
−
signal's power spectrum as well. To do this, we deine
G
i
(
z
H
i
(
z
)
H
i
(
)
andthenuseittowrite
Equation . in the frequency domain:
π
∫
π
e
j
ω
e
j
ω
e
jk
ω
dω.
R
x
i
(
k
)=
P
x
(
)
G
i
(
)
(.)
−
π
Combining Equations . and ., we then get
π
π
e
j
ω
e
j
ω
e
jN
i
k
ω
dω.
R
v
i
(
k
)=
P
x
(
)
G
i
(
)
(.)
∫
−
π
x
i
(
z
)
v
i
(
n
)
x
(
n
-
D
)
N
i
H
i
(
z
)
Processor
Speech
source
x
(
n
)
FIGURE .
Relation between the signal
v
i
(
n
)
produced by the front end of the
i
ith sensor and the original source
signal
x
(
n
)
.
