Processing of Signals - Digital Signal Processing: A Practitioner's Approach - page 11

1.4.2.1 Parametric Spectrum
Given the parameter vector p (1.26) we obtain the parametric spectrum H e j ðÞ j j as

He j !

¼

Be j ðÞ
Ae j ðÞ

ð 1 : 27 Þ

P m ¼ 0 b m þ 1 e j ! m :

1 P n ¼ 1 a n e j ! n

¼ ð 1 : 28 Þ

Using the given value of p , we obtain the parametric spectrum via (1.28). The

function j H ð e j ! Þj

is periodic (with period 2 Þ in ! .

is real-valued, He j ðÞ

When the parameter vector p j j is also symmetric in ! , and we

need to evaluate (1.28) only for 0 . Figure 1.6 is the parametric spectrum of

the narrowband signal. We have obtained this spectrum by first obtaining the
parameter vector

<!<

p (1.26) of the given time series and substituting this value in

(1.28).

1

0.8

0.6

0.4

0.2

0

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Frequency (Hz)

Figure 1.6 Parametric spectrum jj H ð e jw Þjj using (1.28)

1.5 Converting Analogue Signals to Digital
An analogue signal y ð t Þ is in reality a mapping function from the real line, to the
real line, defined as R!R , where R denotes the set of real numbers. When
converted to a digital signal, this function goes through transformations and
becomes modified into another signal which mostly preserves the information,
depending on how the conversion is done. The process of modification or morphing
has three distinct stages:

Windowing
Digitisation: sampling
Digitisation: quantisatioin