Tables and Formulas - Digital Signal Processing: An Introduction with MATLAB and Applications - page 306

Natural logarithm: ln ð x Þ¼ log e ð x Þ; e 2 : 718281828459 ;
Common logarithm: log 10 (x).
12. Power series expansion: 1 =ð 1 x Þ¼ 1 þ x þ x 2 þ x 3 þ ð for j x j \1 Þ ;

sin ð x Þ¼ x x 3 = 3! þ x 5 = 5! ;
sin 1 ð x Þ¼ x þ x 3 = 6 þ 3x 5 = 40 þ 5x 7 = 112 þ 35x 9 = 1152 þ ;
cos ð x Þ¼ 1 x 2 = 2! þ x 4 = 4! ;
cos 1 ð x Þ¼ p = 2 x x 3 = 6 3x 5 = 40 5x 7 = 112 ð for j x j \1 Þ ;
tan ð x Þ¼ x þ x 3 = 3 þ 2x 5 = 15 þ 17x 7 = 315 þ 62x 9 = 2835 þ ð for j x j \p = 2 Þ
tan 1 ð x Þ¼ x x 3 = 3 þ x 5 = 5 ð for j x j \1 Þ ;
tan 1 ð 1 þ x Þ¼ p = 4 þ x = 2 x 2 = 4 þ x 3 = 12 þ x 5 = 40 þ
e x ¼ x þ x 2 = 2! þ x 3 = 3! þ ð for j x j \ 1Þ ;
ln ð 1 þ x Þ¼ x x 2 = 2 þ x 3 = 3 x 4 = 4 þ ð for j x j \1 Þ
13. The convolution integral: x ð t Þ h ð t Þ¼ R 1

1 x ð k Þ h ð t k Þ dk;

14. The correlation integral: R xy ð s Þ¼ R 1

1 x ð t Þ y ð s þ t Þ dt;

Autocorrelation: R x ð s Þ¼ R 1 1 x ð t Þ x ð s þ t Þ dt :

15. Discrete convolution: x ð n Þ h ð n Þ¼ P 1

k ¼1 x ð k Þ h ð n k Þ ;
16. Dirac delta function, d(t): It is defined by the integral

Z

1

x ð t Þ d ð t t o Þ dt ¼ x ð t o Þ;

1

where x(t) is any continuous function.
Note that: d ð at Þ¼ð 1 =j a jÞ d ð t Þ ; d ð t Þ¼ d ð t Þ ; R 1 1 d ð t Þ dt ¼ 1.

17 : x ð t Þ d ð t t o Þ¼ x ð t t o Þ ; x ð t Þ d ð t t o Þ¼ x ð t o Þ d ð t t o Þ

p e ð k m Þ 2 = 2r 2 ; m ¼ mean ; r 2 = variance
19. Schwarz's inequality: j R g 1 ð x Þ g 2 ð x Þ dx j 2 ð R j g 1 ð x Þj 2 dx Þð R j g 2 ð x Þj 2 dx Þ
(equality holds only when g 1 (x) = kg * (x), k being a constant.)
20. Roots of the quadratic equation: ax 2 + bx + c = 0 are given by
r 1 ; 2 ¼ b

18. Gaussian distribution: p ð k Þ¼ r

2p

p

b 2 4ac

2a