Digital Signal Processing Reference
In-Depth Information
R
=
a
n
/a
n
−
1
For example, we can determine the sum of the following sequence using Eq. (2.3), where
N
=0:
+
...
Here we see that
a
= 1/2 since
a
0
=1,
a
1
=1/2, etc., so the sum is
1
+
1
/
2
+
1
/
4
+
1
/
8
1
1
1
/
2
=
1
/
2
=
2
1
−
We can determine the sum of the following geometric sequence, for example,
1
/
3
+
1
/
9
+
1
/
27
+
...
using Eq. (2.3) as
1
/
3
1
/
3
=
1
/
2
1
−
We can verify this result with the simple MathScript call
format long; n = 1:1:50; ans = sum((1/3).ˆn)
which yields
ans
= 0.500000000000000.
Note that it was only necessary to use the first 50 terms of the infinite sequence to obtain a value
close (in this case equal within the limitations of accuracy imposed by the computer) to the theoretical
value. As
a
approaches unity in value, more terms are needed to obtain a sum close to the theoretical
value.
Sometimes the sum of a finite number of terms of such a sequence is needed. Supposing that the
sum of the first
N
- 1 terms is needed; we can subtract the sum for terms
N
to
∞
from the sum for all
terms, i.e.,
N
−
1
∞
∞
a
N
a
N
1
1
−
a
n
a
n
a
n
=
−
=
a
−
a
=
1
−
1
−
1
−
a
n
=
0
n
=
0
n
=
N
2.4.11 RANDOM OR NOISE SEQUENCES
Noise is an ever-present background signal in communications systems. It is generated by many natural
sources such as the Sun and Jupiter, lightning, many man-made sources, by active devices in electronic
systems, etc. Noise assumes random values over time (rather than predictable values such as those of a sine
wave, for example) which are described using statistics such as the probability density function, mean,
standard deviation, etc.
It is often necessary to simulate noise in signals, and MathScript can be used to generate random
sequence values using the functions
rand(m, n)
or
randn(m, n)
where
m
and
n
are dimensions of the matrix of random numbers to be created.
