Digital Signal Processing Reference
In-Depth Information
ployed (f|g) = (g|f) without complex conjugation of either function. With
the special choice t
0
= 0, the first element of the data or Hankel matrix
H
n
(c
0
)≡{c
i+j
|Φ
0
) = 0. The general element
c
i,j
= c
i+j
of the matrix H
n
(c
0
) is the overlap between the two Schrodinger or
Krylov states c
i+j
= (Φ
j
|Φ
i
) = (Φ
0
|U
i+j
|Φ
0
)≡U
(0)
}is the signal point c
0
= (Φ
0
i,j
. The matrix which gives
the sought spectrum is not H
n
(c
0
), but rather the evolution matrix H
n
(c
1
)
with the general elements c
i+j+1
= (Φ
j
|U|Φ
i
)≡U
(1)
i,j
.
The matrices H
n
(c
0
) and H
n
(c
1
) are the particular cases of the general
delayed Hankel matrix H
n
(c
s
) ={c
n+m+s
}≡U
(s)
of the form
n
0
@
1
A
c
s
c
s+1
c
s+2
c
s+n−1
c
s+1
c
s+2
c
s+3
c
s+n
c
s+n+1
. . .
.
.
.
.
c
s+n−1
c
s+n
c
s+n+1
c
s+2n−2
c
s+2
c
s+3
c
s+4
H
n
(c
s
) = U
(s)
(4.6)
n
where U
(s)
={U
(s)
i,j
}
n−1
i,j=0
. Here, the signal point c
s
appearing in the small
parentheses in H
n
(c
s
) represents the leading element from the first row and
first column in the matrix (4.6). The corresponding delayed Hankel determi
nant H
n
(c
s
))≡det H
n
(c
s
) is defined by (2.200). In the special cases, s = 0
and s = 1 the Hankel matrices H
n
(c
0
) and H
n
(c
1
) coincide with the over
lap matrix S
n
= U
(0)
n
and the evolution or relaxation matrix U
n
= U
(1)
n
n
introduced in terms of the Schrodinger basis set{|Φ
n
)}.
In practice, many applications necessitate the usage of the nonzero initial
time, t
0
= 0. In this chapter in addition to t
0
= 0, we will also consider
the delayed time signals with the nonzero initial time t
s
= sτ(0≤s <
N−1). The corresponding mathematical model for the delayed time signal
{c
n+s
}(0≤n≤N−1 and 0≤s < N−1) is given by a sum of attenuated
complex exponentials
K
K
d
(s
k
u
k
d
k
e
−i(n+s)ω
k
τ
c
n+s
=
=
(4.7)
k=1
k=1
d
(s)
k
d
k
≡d
(0)
= d
k
u
k
u = e
−iωτ
u
k
= e
−iω
k
τ
(4.8)
k
where ω
k
and d
(s
k
are the fundamental angular frequencies and amplitudes,
respectively. This should be compared to the nondelayed time signal
K
K
d
k
e
−inω
k
τ
d
k
u
k
.
c
n
=
=
(4.9)
k=1
k=1
Evidently, spectral analysis of nondelayed{c
n
}and delayed time signals
{c
n+s
}(s = 0) can be accomplished on the same footing.
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