Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 35

Water tub graph

0.7

0.6 Infantile mortality

Aging

0.5

Random failures

0.4

0.3

0.2

0.1

0

0 50 100 150 200 250 300 350 400

Progression of time

Reliability graph for software ends here

Figure 2.10 System reliability is a function of random variables

2.4 Noise
All textbooks of communication engineering and topics on other disciplines define
noise as an unwanted signal. Noise is characterised by two parameters: its
amplitude statistics (probability density function) and its autocorrelation function
r k . Both of them are important and independent.

2.4.1 Noise Generation
On many occasions we need to generate specific types of noise. The most frequently
used noise is white Gaussian noise (WGN). We cannot appreciate any noise in the
time domain. The only way to get some information is by constructing the pdf of
the given signal. There are many methods for generating WGN. The primary source
is a uniformly distributed noise (UDN), which can be generated using a
linear congruential generator (LCG), well explained by Knuth [10]. Figure 2.11
depicts the LCG relation given as k ¼ð a k 1 þ c Þ mod m, where the constants a

and c can be chosen to suit the purpose.

Figure 2.11 Linear congruential generator for UDF

WGN is generated by invoking the central limit theorem and also recollecting
that the variance of UDN is A 2

k ¼ P 12

= 12. Now WGN k is given as k .

i ¼ 1

Choosing 12 in the summation cancels the denominator 12 in the UDN k .