Digital Signal Processing Reference
In-Depth Information
4.4 The response or the Green function
A nonzero initial time t
s
= 0 in place of t
0
= 0 can also be employed in
spectral methods that rely upon the Green function, as in the statespace
formulation of the FPT. Here, the main quantity is the delayed Green operator
which represents the delayed resolvent
∞
R
(s)
(u)≡(1u−U)
−1
U
s
=
U
n+s
(τ)u
−n−1
.
(4.10)
n=0
The associated delayed Green function G
(s)
(u) is defined by the standard
matrix element (Φ
0
|R
(s)
(u)|Φ
0
) which together with (4.10) yields
∞
|R
(s)
(u)|Φ
0
) =
G
(s)
(u) = (Φ
0
c
n+s
u
−n−1
.
(4.11)
n=0
This result is a direct generalization of the customary nondelayed Green
function
∞
c
n
u
−n−1
G(u) =
(4.12)
n=0
where G
(0)
(u)≡G(u). In signal processing, the delayed counterpart of the
standard Green function can also be obtained by splitting the original infinite
time interval [0,∞] into two parts, as in [0,∞] = [0,s−1] + [s,∞]. This maps
(4.12) to
s−1
c
n
u
−n−1
+ u
−s
∞
c
n+s
u
−n−1
G(u) =
n=0
n=0
s−1
c
n
u
−n−1
+ u
−s
G
(s)
(u)
=
(4.13)
n=0
−1
n=0
where
≡0. This procedure yields a relationship between the two
exact Green functions G
(s)
(u) and G
(0)
(u) that are related to the case with
and without the delay, i.e., s = 0 and s = 0, respectively. It is seen from
(4.12) that the delayed spectrum G
(s)
(u) is multiplied by the term u
−s
which
compensates for the time evolution of the system from t
0
= 0 to t
s
= 0.
By substituting (4.7) into the rhs of (4.10), we have
∞
∞
K
d
(s
k
u
k
c
n+s
u
−n−1
=
u
−n−1
n=0
n=0
k=1
K
∞
K
d
(s)
d
(s)
k
u
−1
(u
k
/u)
n
k
u
−1
(1−u
k
/u)
−1
=
k=1
n=0
k=1
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