Digital Signal Processing Reference
In-Depth Information
to improve the performance of the estimator. A brief discussion about the fast
implementation of APES appears in Section 2.6.
2.2 PROBLEM FORMULATION
Consider the problem of estimating the amplitude spectrum of a complex-valued
uniformly sampled discrete-time signal
N
−
1
{
y
n
}
0
.For a frequency
ω
of interest, the
n
=
signal
y
n
is modeled as
)
e
j
ω
n
y
n
=
α
(
ω
+
e
n
(
ω
)
,
n
=
0
,...,
N
−
1
,ω
∈
[0
,
2
π
)
,
(2.1)
where
α
(
ω
) denotes the complex amplitude of the sinusoidal component at fre-
quency
) denotes the residual term (assumed zero-mean), which in-
cludes the unmodeled noise and interference from frequencies other than
ω
, and
e
n
(
ω
ω
. The
N
−
1
problem of interest is to estimate
α
(
ω
)from
{
y
n
}
for any given frequency
ω
.
n
=
0
2.3
FORWARD-ONLY APES ESTIMATOR
Let
h
(
) denote the impulse response of an
M
-tap finite impulse response (FIR)
filter-bank
ω
)]
T
h
(
ω
)
=
[
h
0
(
ω
)
h
1
(
ω
)
···
h
M
−
1
(
ω
,
(2.2)
)
T
denotes the transpose. Then the filter output can be written as
h
H
(
where (
·
ω
)
y
l
,
where
y
l
+
M
−
1
]
T
y
l
=
[
y
l
y
l
+
1
···
,
l
=
0
,...,
L
−
1
(2.3)
are the
M
×
1 overlapping forward data subvectors (snapshots) and
L
=
N
−
M
+
1.
)
H
denotes the conjugate transpose.
For each
·
Here (
ω
of interest, we consider the following design objective:
L
−
1
h
H
(
)
e
j
ω
l
2
h
H
(
ω
)
y
l
−
α
ω
ω
ω
=
,
min
(
s.t.
)
a
(
)
1
(2.4)
α
(
ω
)
,
h
(
ω
)
l
=
0
where
a
(
ω
)isan
M
×
1vector given by
[1
e
j
ω
e
j
ω
(
M
−
1)
]
T
a
(
ω
)
=
···
.
(2.5)
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