EVALUATION OF THE PARSEVAL'S INTEGRALS IN THE ANALOG DOMAIN
A m e t h o d f o r t h e d i r e c t e v a l u a t i o n of 1 a n d 2 v a r i a b l e __Parseval's i n t e g r a l s i n t h e ana log a n d d i g l t a l d o m a i n s a n d u t i l i s e s t h e e x l s t i n q m e t h o d of p s e u d o p a r t l a l f r a c t i o n e x p a n s l o n p r e s e n t e d i n an e a r l i e r a r t ~ c l e [l;. a r e a l s o q i v e n . Some i l l u s t r a t i v e e x a m p l e s I . INTRODUCTION R e c e n t l y J u r y a n d R u r i d a n t [ z ] hdv.? a d d r e s s e d t h e p r o b l e m of e v a l u a t i n q t h e d o u b l e s q u a r e i n t e q r . 1 : t h a t is, H(s h ( t l , t 2 ) . m e t h o d f o r t h e e v a l u a t i o n o f (1) based on t h e u s e of t h e d o u b l e b i l i n e a r t r a n s f o r m (DOT) , s i ) I n t h i s a r t i c l e ih:?.y h a v e s u q q e s t e d a n l n t e r e s t i n q is t h e 2 v a r m b l e ( 2 V ) L a p l a c e t r a n s f o r m of 1 Z . ( 3 ) = k = 1,: T h e y c l e v e r l y reduce t h e proolrm of em?iuai ing ( 1 ) CO one o l e v a l u a i i n g t h e iollovlng double Parseval's I P r e g r a l I? rhe unit b i d i s k of the (zl, z z j domain: J = 1 [ a ( m , " ) I 2 m=o n=o where a(m n) i r e t h e Taylor c o e f f i c i e n t s of U z l . z 2 j and M.N.S Swamy, E. Plotkin E l e c t n c a l E n g i n e e r i n g Dept. C o n c o r d i a Universi ty Montreal , Canada I n t h i s a r t i c l e we develop an a l t e r n a t e method of e i a l u a t i n g (1) by us ing the isomorphism of the OBT and applying the method o f pseudo-partial f r a c t i o n enpansion. domain i t s e l f . 1 -V Parseval's j n t e g r a i i n the analog domaln, f a r which ).he au tho r s are not aware of any e x i ~ t i n g technique. This enables us t o eva iua te ( ? ) d i r r c t l y i n the analog Inc idenca l ly Chis provides a d i r e c t method of eva:uacing a 11. EV,ZLtiATIOV OF 2-1 PARSEVAL'S :NTECRAL I V THE iNALOG DOMAIN i f 1 s observed from ( 4 ) t h a t J does n o t e x i s t ( t h a t 15. is not -, f i n i t e ) i f t h e funcc lon . A ( z , , z z l has a o i a r singularities i n G-, t h e c l o s e d u n i t b i d i s k . Hence, in v i e w of ( 5 ) , a necessa ry ; > n a i t l a n f o r t h e e x i s c a n c e a i 3 i s t h a t t h e f u n c t i o n F(sl, s 2 ) i n t h e ( s l . 9.1 domain, c a r r e s p o n d r n g io t h e f u n c t i o n A ( z , , z 2 ) , F(Sl, s 2 ) = $(l+sl)(1+s2) H ( s l , s 2 ) ( 6 ) is devo id of poles i n E', t h e closed r igh t -ha l f of che ( L ~ , "* ) b i p l a n e . t h i s arricie We assume t h a t F(sl. s 2 ) has n e i t h e r p o l a r s i i n g u l a r i t i e s n o r n o n e s s e n t i e l s i n g u l a r i t i e s o f the second kind i n E 2 . A(zl , z 2 ) is devo id of singuiariries i n iz and t h a t J i s f i n i t e . can use t h e method given i n [ l ] f a r e v a l u a t i n g ( 4 ) . L e t :n T h i s assures t h a t Hence, we P(L1, L*) A('1, z 2 ) = Then t h e i n t e g r a n d of ( 4 ) i s m y be put i n t h e form W z l , z z ) P 1 ( z l , "2) k1 k2 --.-. Q ( Z i , Z 2 1 Q 1 ( L 1 . Z ? ! z1 L 2 where P , and Q Tlien ( 4 ) may be rearranged as are polynomials in (zl, z 2 ) , and i l and k2 are i n t e g e r s . ' 1 293 24ACSSC-1219010293 $1.00