Geoscience Reference
In-Depth Information
Fig. A6
Approximation of a function
x
=
x
(
τ
)
by a sequence of pulses of width
Δ
τ
.
In numerical applications of the convolution integral, the input function
x
(
t
) can be
represented by a histogram consisting of pulses of width
τ
, as illustrated in Figure A6.
The contribution to the total response by the input pulse at
τ
=
(
k
τ
)isgivenby
x
(
k
τ
)
u
(
τ
;
t
−
k
τ
)
τ
in which
u
(
τ
;
t
−
k
τ
) represents the response of the system to an input pulse of
width
τ
. The total response at
t
is the sum of the contributions by all input pulses, or
+∞
y
(
t
)
=
x
(
k
τ
)
u
(
τ
;
t
−
k
τ
)
τ
(A17)
k
=−∞
which is the discrete analog of the general convolution integral (A11). Again, if
t
repre-
sents time, and the system is non-anticipatory with an input starting at
t
=
0, one obtains
the discrete analog of (A14), or
n
y
(
t
)
=
x
(
k
τ
)
u
(
τ
;
t
−
k
τ
)
τ
(A18)
k
=
0
in which
τ
=
n
τ
(
≤
t
) is the time of the last input pulse prior to the designated response
time
τ
=
t
.
Relationships between the moments
The convolution integral provides also convenient relationships between the moments
of the three functions involved, namely the input function
x
(
t
), the output function
y
(
t
)
and the unit response function
u
(
t
). Denote the mean values of
t
(or centers of area) of
these three functions respectively as
m
y
1
,
m
x
1
and
m
u
1
and the
n
th moments about these
means of the three functions, respectively as
m
yn
,
m
xn
and
m
un
. Assume for convenience
that these functions are properly scaled, so that their zero-order moments
ydt
,
xdt
and
udt
are equal to unity. The center of area of the output function, which is the first
moment about the origin, can be calculated as
∞
m
y
1
=
ty
(
t
)
dt
(A19)
−∞
Making use of the convolution integral (A11) one obtains
