Digital Signal Processing Reference
In-Depth Information
Chapter 8
Subspace-based Estimators
8.1. Model, concept of subspace, definition of high resolution
8.1.1.
Model of signals
Let
x
a
(
t
) be an analogic signal made up of
P
complex sine waves in an additive
noise:
p
()
∑
j
2
π
f t
()
x
t
=
α
e
+
b
t
[8.1]
i
a
i
a
i
=
where the α
i
are centered complex random and independent variables of variances
2
σ
,
where
b
a
(
t
)
is a centered, complex, white noise of variance
E
= (|
b
a
(
t
)|
2
) = σ
2
independent from α
i
. We suppose the noise and the signals to be circular so that the
methods presented here will be based only on the first correlation functions of the
different signals. Moreover, we will further work on the sampled signals
() ( )
and
() ( )
bk b k
=
where
T
e
is a sampling period verifying the
Shannon theorem for the complex sine waves.
x kx T
=
a
e
a
e
Using the non-correlation hypotheses of the sine waves and of the noise, the
autocorrelation function of the observations is written:
pp
( )
(
)
(
)
j
2
π
f
−
f
kTj
2
π
f
−
mf
'
T
(
)
∑∑
*
2
[8.2]
γ
kmm
,, '
=
E
αα
e
i
l
e
e
i
l
e
+
σδ
xx
i
l
m
−
m
'
i
==
11
l
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