Digital Signal Processing Reference
In-Depth Information
1.5
1
0.5
0
−30
−20
−10
0
10
20
30
(a) Sample Index n
0.5
0
−0.5
−30
−20
−10
0
10
20
30
(b) Sample Index n
1.5
1
0.5
0
−30
−20
−10
0
10
20
30
(c) Sample Index n
Figure 2.9:
(a) Even component of a decaying exponential sequence; (b) Odd component of same;
(c) Reconstruction of original exponential sequence, obtained by summing the even and odd components
shown in (a) and (b).
2.4.10 GEOMETRIC SEQUENCE
The sum of a decreasing exponential sequence of numbers
a
n
, where
|
a
|
<
1, converges to the value 1/(1-
a
), i.e.,
∞
1
a
n
→
(2.2)
−
1
a
n
=
0
A more general statement of this proposition is that
∞
a
N
a
n
→
(2.3)
1
−
a
n
=
N
which allows computation of the sum starting from a value of
n
greater than 0.
Another way of thinking of this is that the sum of a geometric sequence
a
n
(we assume
|
a
|
<
1) is
its first term divided by one minus the convergence ratio
R
, where
