Tables and Formulas - Digital Signal Processing: An Introduction with MATLAB and Applications - page 316

Note that G 2 = 0 in this case.
Sinusoidal PLL: 2nd-order system equation:
/ ð k þ 2 Þ 2/ ð k þ 1 Þþ / ð k Þ¼ K 2 sin ½ / ð k Þ rK 2 sin ½ / ð k þ 1 Þ;
where r ¼ 1 þ G 2 = G 1 :

Fixed point analysis: g(x) = x has a solution x* only if j g 0 ð x Þj \1 :
Jacobian matrix: o ð u ; v Þ

where u, v are functions of x, y.

o ð x ; y Þ ¼ ou = ox

ou = oy

ov = ox ov = ou

Eigenvalues {k} of a matrix X:

j kI X j¼ 0; where I is the identity matrix :

Input Signal,

x ( k )

Sampler &
ADC

x ( t ) DF

N −bits

y ( k )

DCO

Output Pulses, v o ( k )

Integrator H(z) = 1/(1 - z -1 ); Differentiator H(z) = (1 - z -1 )/T s
Error functions:

p Z

x

erf ð x Þ¼ 2 e v 2 dv

0

Z ¼ 1

1

Q ð x Þ¼ 1 e v 2 = 2 dv ¼ 1

2

x
2

2 1

x

p

2p

erfc p erf p 2

2

x

For x [ 0 we have:

p e x 2 = 2 X

5

Q ð x Þ 1 b k g k ;

2p

where

k ¼ 1

g ¼ 1 =ð 1 þ 0 : 2316419x Þ;
b 1 ¼ 0 : 3198 ; b 2 ¼ 0 : 3565 ; b 3 ¼ 1 : 7814 ;

b 4 ¼ 1 : 8212 ; b 5 ¼ 1 : 3302

Direct forms of IIR implementation:

y ð n Þ¼ b o x ð n Þþ b 1 x ð n 1 Þþþ b M x ð n M Þ a 1 y ð n 1 Þ a N y ð n N Þ
H ð z Þ¼ P i ¼ 0 b i z i

! ¼ H 1 ð z Þ H 2 ð z Þ¼ H 2 ð z Þ H 1 ð z Þ:

1 þ P k ¼ 1 a k z k ¼ X M

1
1 þ P k ¼ 1 a k z k

i ¼ 0 b i z i