Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 37

Coloured Gaussian noise has identical structure or shape to the input (u k ) amplitude
statistics, as in Figure 2.12(b). The output is said to be coloured, because spectral
energy is not constant across all the frequencies, as shown in Figure 2.13(a)
Taking linear systems having either an IIR or an FIR and exciting them with
Gaussian noise results in another Gaussian noise of different mean and variance.
This is an important characteristic exhibited by linear systems and is illustrated in
Figures 2.12 and 2.13. In this context, it is worth recollecting that a sine wave input
to a linear system results in another sine wave with the same frequency but different
amplitude and phase.

2.5.1 Linear System Driven by Arbitrary Noise
Consider an AR system defined by (2.23) and (2.24) and where the input u k is
uncorrelated and stationary. The output of this system is normal. As an example, let
us excite the system with u k having a Weibull distribution as in Figure 2.14 with
¼ 0 : 1 and ¼ 0 : 8:

(

f ð u Þ¼ ð= Þ u 1 e ð u =Þ

; u 0 ;

ð 2 : 29 Þ

0 ; u < 0 :

The output is normal, as shown in Figure 2.14. This can be understood by realising
that the output can be best approximated using an MA process as in (2.5) and (2.6)
and that the statistics of u k are the same as the statistics of u k þ i . Then use the three
rules in Section 2.3.1 to obtain the output as a normal distribution.

0.5

0.4

< Weibull distribution

0.3

0.2

< Normal distribution

0.1

0

− 150 − 100 − 50 0 50 100 150 200 250 300

Figure 2.14 Input and output statistics of u k and y k

2.6 Multivariate Functions
Understanding functions of several variables is important for estimating parameters
of systems. Often the problem is stated like this: Given a system, find out its response