By Arthur B Baggeroer
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1 4. -k W()') = [ 2kP] k 1 -I (3. ) where b l! J2P k). = (kb ) cos 1 )'k - 1 t - A (A) : b sin{k b t) co s (kb t ) + + (3. 25, - b ) sin(kbt) ' • � [1 �] si n(kbT) = 2 b sin(kbt) si n(kbt) By substituting Equation 37 substituting Equa • The transition matrix for W O) i s ( EXAMPLES cos(kbT). 61) The eige nvalues are given b y the roots o f t h i s e q u a t i o n . 62 for J. gives us the }'i in terms o f the bj , A •. k 2P = - [I + (Mt) where }'; = is the a normal i zing facto r. ei genfu nctions.
T) f�o cO�:' 't") K,,(r, 't")CT(T)f(r) dK�:. t) ( OT('t". t)CT('t")fCr) to t, we obtain cit = + + We have used ' K,,(t. t) first term of the 2. l0b and canceled eaua! terms. 34) we • Note that we could just as well derive the results for the state vector covariance opcrator dircctly. In this approach we would operate on a function f'(r) with K,,(t, T) and then substitute C:r(T)f(T) for the operator of interest here. 14, we obtain d�(t) a - t = F(t} I' O(t, t)K,,(t, t)CT(t)f(t) To + [F(t)Kx(t, t) + G(t)QGT(t)] After rearranging d��t) At this = terms � � t T,.
6. 0 EXAMPLES 43 0. 2. The l l ighcr-Order Butte rworth Processes* We c a n a n alyze t he h i gher-order Butterworth processes just as we did the previous exa m p le. S i nce these processes a re often employed as a t heoret ical tool , it is usefu l to s u m marize lhe res ults of s u ch an analysis. (co : II) = 211P sin(1l/211) . 7 1 The comp l e x i ty of h i s a n a l y t i c results i n d icate the a ppropriateness of a numerical algori l h m . 5. The state matrices a re speci fied on the res pect i ve figures.