Tables and Formulas - Digital Signal Processing: An Introduction with MATLAB and Applications - page 303

Appendix C: Tables and Formulas

Basic Definitions

The rectangular pulse: P ð t = T Þ¼ P T ð t Þ¼ 1 ; j t j T = 2

0 ; elsewhere

The triangular pulse: K ð t = T Þ¼ K T ð t Þ¼ 1 2 j t j= T ; j t j T = 2
0 ; elsewhere

,

The unit step function: u ð t Þ¼ 1 ;

t 0

0 ; t\0

The sinc function: sinc ð t Þ¼ sin ð pt Þ
ð pt Þ

1 ; t 0

The signum function: sgn ð t Þ¼

1 ; t\0

Useful Formulas

e j/ ¼ cos ð / Þ jsin ð / Þ (Euler);
j e j/ j¼ 1;
cos ð x Þ¼ cosh ð jx Þ¼ð e jx þ e jx Þ= 2;
sin ð x Þ¼ð 1 = j Þ sinh ð jx Þ¼ð e jx e jx Þ=ð 2j Þ ;

1 :

cos 2 ð / Þþ sin 2 ð / Þ¼ 1;
tan ð x Þ¼ sin ð x Þ= cos ð x Þ ;
cos 2 ð / Þ¼ð 1 = 2 Þþð 1 = 2 Þ cos ð 2/ Þ ;
sin 2 ð / Þ¼ð 1 = 2 Þð 1 = 2 Þ cos ð 2/ Þ

2 :