Processing of Signals - Digital Signal Processing: A Practitioner's Approach - page 9

0.1

0.08

0.06

0.04

0.02

0

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Frequency (Hz)

Figure 1.5 Fourier power spectrum s ð k Þð 1 : 17 Þ

where W ¼ e j 2 = N . In general g k is a complex quantity. However, we restrict our
interest to the power spectrum of the signal y k , a real non-negative quantity given by

2

s k ¼ g k g k ¼j g k j ; ð 1 : 17 Þ

where ðÞ represents the complex conjugate and jj represents the absolute value
(Figure 1.5). The series S N , given by

S N ¼ f s 1 ; ...; s N g; ð 1 : 18 Þ

is another way of representing the signal. Even though the signal in the power
spectral domain is very convenient to handle, other considerations such as resolu-
tion, limit its use for online application. Also, in this representation, the phase
information of the signal is lost.
To preserve the total information, the complex quantity g k is represented as an
ordered pair of time series, one representing the in-phase component and the other
representing the quadrature component:

f g k g¼f g i k gþ j f g k g:

1.4.2 Parametric Representation
In a parametric representation the signal y k is modelled as the output of a linear
system which may be an all-pole or a pole-zero system [2]. The input is assumed to
be white noise, but this input to the system is not accessible and is only conceptual
in nature. Let the system under consideration be

x k ¼ X a i x k i þ X

p q 1

b j þ 1 k j ; ð 1 : 19 Þ

i ¼ 1 j ¼ 0

y k ¼ x k þ noise ; ð 1 : 20 Þ