Geology Reference
In-Depth Information
Writing l
=
j
k , this becomes
−∞
h j
=
f j l g l
=
g l f j l
=
g k f j k .
(2.11)
l
=∞
l
=−∞
k
=−∞
Thus, the convolution of f j with g j is the same as the convolution of g j with f j .
The convolution of two energy signals produces an energy signal , and the con-
volution of an energy signal with a power signal results in a power signal , while
the convolution of two power signals does not exist .
Convolution, and many other operations with time sequences, are facilitated by
the use of the z-transform . For the time sequence
..., f 1 , f 0
,..., f j ,...,
(2.12)
the z-transform is defined as
f 1
z +
f j z j
F ( z )
=···+
f 0 +
f 1 z
+···+
+···
(2.13)
If h j is the sequence resulting from the convolution of the sequence f j with the
sequence g j , it has the z-transform
H ( z )
=
F ( z )
·
G ( z ).
(2.14)
Hence, the z-transform of the convolution of f j with g j is the product of the
z-transforms of f j and g j . The product of the z-transforms is
f 1
z + f 0 + f 1 z +···
···+ g 1
F ( z )
· G ( z )
=
···+
z + g 0 + g 1 z +···
···+ ··· f 1 g 1 ···
z 2
+ ··· f 1 g 0 + f 0 g 1 ···
z
=
+ ···
f 1 g 1
+
f 0 g 0
+
f 1 g 1
···
) z 2
+
(
···
f 0 g 1
+
f 1 g 0
···
) z
+
(
···
f 1 g 1
···
+···
,
(2.15)
where we have included only terms involving time indices
1, 0 and 1 in the
cient of the term in z j
sequences f j and g j . If all terms are included, the coe
takes the form
···+ f 1 g j + 1 + f 0 g j + f 1 g j 1 +···+ f k g j k +··· ,
(2.16)
in conformity with the general term in expression (2.10) for the convolution.
 
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