Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 121

Legend

x ( n ) y ( n )

α = 0.9

1 − α α = 0.5

1

α

z −1

Ω = ω T s

0 < α < 1

− π π

0

Im ( z )

π /2

P − Z
Diagram

Ω = ω T s

X Re ( z )

0

− π /2

− π π

Legend: Alpha Filter Averaging ( = 0.8 )

Moving Average (10 taps, 0.1 each)

α

15

10

5

0 10 20 Time (days)

Fig. 2.37 Alpha filter. Above Circuit and its frequency response. Below data averaging

To design a digital sinusoidal oscillator, one essentially needs a filter whose
impulse response is sinusoidal. From Tables- z Transform Pairs and Theorems,
such a filter transfer function is specified by:

sin ð b Þ z
z 2 2 cos ð b Þ z þ 1

h ð n Þ¼ sin ð bn Þ u ð n Þ ZT H ð z Þ¼ ð 2 : 38 Þ

To build H(z), one can write it as H ð z Þ¼ sin ð b Þ z 1 = 1 2 cos ð b Þ z 1 þ z ½ ; the
implementation of which is shown in Fig. ( 2.38 ). In this expression b corresponds
to the normalized radian frequency X o ¼ x o T s ; and hence the frequency of
oscillation is:

f o ¼ x o = 2p ¼ð b = T s Þ= 2p ¼ð b = 2p Þ f s Hz ;

provided that |b| \ p. The two poles of this system are the roots of the equation
z 2 - 2cos(b)z ? 1 = 0, which are given by:

p 1 ; 2 ¼ cos ð b Þ

p

¼ cos ð b Þ j sin ð b Þ¼ e jb :

cos 2 ð b Þ 1

ð 2 : 39 Þ

Hence, the poles are exactly on the circumference of the unit circle, as shown in
Fig. ( 2.39 ), and the system is strictly speaking, neither stable nor unstable.