Digital Signal Processing Reference
In-Depth Information
where the criteria
()
N
J
θ
depends either on the data or on the correlation function.
In general, the approach followed consists of expanding the derivative of the criteria
into a Taylor series and using the asymptotic arguments presented earlier. This is the
case in the MUSIC algorithm presented in Chapter 8 for which the frequencies are
determined as arguments which minimize the square norm of the projection of
a
(
ω
)
on the estimated noise subspace
ˆ
V
:
2
1
H
H
()
()
N
J
ω
=
Π
a
ω
,
Π
=
V V
[3.28]
b
b
bb
2
We intend to give the main steps of the methodology of the analysis in the mono-
component case:
j n
ω
()
()
x n
=
e
+
bn
For a complete expansion we can refer to [STO 91], for example. Let
ˆ
ω
be the
frequency that minimizes the criteria [3.28]. The convergence of
ˆ
ω
towards
ω
0
helps use a Taylor expansion of
()
ˆ
J
'
ω
around
ω
:
N
0
() ( ( )
0'
=
J
ω
+"
J
ω ω ω
ˆ
−
+
N
0
N
0
0
Limiting ourselves to the first order, the error made asymptotically on the
estimation of
ω
0
is thus:
() ()
ωω
ˆ
−
−
J
'
ω
/
J
"
ω
[3.29]
0
N
0
N
0
with:
{
}
()
()
H
()
J
'
ω
=
Re
a
ω
Π
d
ω
b
N
0
{
}
()
() ()
()
H
H
()
J
"
ω
=
Re
a
ω
Π
d
'
ω
+
d
ω
Π
d
ω
b
b
N
0
Asymptotically, the projection
b
Π
converges towards
Π
which allows us to
write:
{
}
H
H
()
()
() ()
()
ω
ω
Π
ω
+
ω
Π
ω
J
"
Re
a
d
'
d
d
N
0
b
b
{
}
{
}
()
(
)
()
H
H
H
()
()
=
Re
d
ω
Π
d
ω
=
2Re
d
ω
I
−
V V
d
ω
[3.30]
b
s
s
(
)
2
=
MM
−
1/12
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