Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 26

significance in linear systems and is given as

r k ¼ X

n ¼ N

h n þ 1 h k þ n ¼ h k h k :

ð 2 : 8 Þ

n ¼ 0

The DFT of r k results in the power spectrum of the system. The DFT of the given
autocorrelation function r k of a linear system is shown in Figure 2.4(b). Note that r k
of a linear system and its power spectrum j H ð e j ! Þj are synonymous and can be used
interchangeably. In fact, the Wiener-Khinchin theorem states the same.

2.1.6 Decomposing h k
The impulse response of a given linear system can be considered as a sum of
impulse responses of groups of first- and second-order systems. This is because the
system transfer function H ð z Þ¼ B ð z Þ= A ð z Þ can be decomposed into partial fractions

of first order (complex or real) or a sum of partial fractions of all-real second-order
and first-order systems. This results in the given impulse response h k being
expressed as a sum of impulse responses of several basic responses:

h k ¼ X

M
i ¼ 1 i h i k

where i is a constant ; ð 2 : 9 Þ

where h i k is the unit sample response of either a first-order system or a second-order
system.

2.2 Linear System Representation
We would like to avoid continuous systems in this textbook, but we are compelled
to mention them, although this is probably the last time. Consider the popular
second-order system

2 2

€

x ¼ 2

! n _

x !

n x þ ! n u ð t Þ

2 f n x þ 4 2 f n u ð t Þ:

¼ 4 f n _ x 4 ð 2 : 10 Þ

Here

d 2 x
dt 2

dx
dt :

€ x ¼ and x ¼ _

Taking numerical ¼ 0 : 1 and f n ¼ 100 Hz we get

x ¼ 125 : 7 x 394 784x þ 394 784u ð t Þ: ð 2 : 11 Þ