Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 32

2.2.8 Solution of Linear Difference Equations
There are many methods available for solving difference equations along the lines
of differential equations, by solving for complementary function and particular
integral. The advent of powerful software packages such as MATLAB and
SIMULINK means that we no longer need to know the details, but if we do
understand them, it will help us get the most out of our software.
It is always better to visualise the systems in the form of (2.16) since this best
matches the software packages in the commercial market. The transfer function
H ð z Þ¼ B ð z Þ= A ð z Þ can be converted into a sum of second- and/or first-order partial

fractions, depending on whether the roots of the polynomial A ð z Þ

are real or

complex:

A ð z Þ ¼ X ð order 2 systems Þþ X ð order 1 systems Þ:

B ð z Þ

ð 2 : 25 Þ

A second-order system has a solution of the form h k ¼ Ar k cos ðÞþ Br k sin ðÞ ,
whereas a first-order system is of the form h k ¼ Cr k ; the constants can be
evaluated from the initial conditions. Equations (2.25) and (2.9) are the same
except the domains are different. The overall solution for a specific input u k can be
obtained using (2.6).

2.3 Random Variables
We will introduce two important ideas:

Continuous random variable (CRV)
Discrete random variable (DRV)

Note that not every discrete system is a DRV; discrete systems and discrete random
variables are separate concepts. Sampling or converting into a discrete signal show
its effect on the autocorrelation function r k but not on its probability density
function f x ð x Þ . For completeness, we recollect that a random variable (rv) is a
function mapping from a sample space to the real line.
Consider a roulette wheel commonly used in casinos. The sample space is
{0 2 } and the function is ¼ 1 = 2 . In the case of a CRV, suppose we

¼ 20 . The spontaneous answer would be zero. But if
we want P ð 10 20 Þ , then it is 1

ask, What is the probability

36. The idea of presenting a CRV this way is
to illustrate the continuous nature of its pdf .
Consider another experiment of throwing a die or picking a card from a pack of
52. This is a DRV experiment for the simple reason that the sample space is finite
and discrete. The pdf is discrete and exists only where an event occurs. If the earlier
roulette wheel

=

is mechanically redesigned so it will stop only at ¼

f 0 1 2 359 g , then ; ; ; ...; becomes a DRV.