1 C o n t i n u o u s a n d D i s c r e t e S i g n a l s In this c h a p t e r we shall review several c o n c e p t s c o n c e r n i n g analog and digital sig› nals, n a m e l y the Fourier, Z, and L a p l a c e t r a n s f o r m s , the s a m p l i n g t h e o r e m , and the aliasing problem. These topics are p r e s e n t e d in order to e s t a b l i s h n o t a t i o n that we will use in m i x e d signal circuits. We will also p r e s e n t e x p o n e n t i a l , Euler, and bilin› ear m a p p i n g s from the s d o m a i n to the z domain, as well as t r a n s f e r functions de› scribing t w o - d i m e n s i o n a l s y s t e m s in both domains. Finally, we will d e s c r i b e the discrete cosine t r a n s f o r m , which is very i m p o r t a n t in image c o m p r e s s i o n and will be used in the s e c o n d part o f the book. 1.1 FOURIER, Z, AND LAPLACE T R A N S F O R M S A d i s c r e t e - t i m e signal is d e f i n e d as a s e q u e n c e { x ( k ) } r e s u l t i n g from s a m p l i n g a c o n t i n u o u s - t i m e signal x(t). The symbol x(k) denotes the e l e m e n t o f the s e q u e n c e that is equal to the value o f the function x(t) for t = kT, where T is the s a m p l i n g in› terval. The r e l a t i o n x(k) = r xk(t)dt - x d e s c r i b e s the s a m p l i n g o p e r a t i o n , where (1.1 ) (1.2) o(t) is the delta function or d i s t r i b u t i o n function. The function o b t a i n e d as a sum o f (1.2) for all indices k 3