Continuous and Discrete Modeling for IBIS-AMI
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1 Cotios ad Discrete Modelig for IBIS-AMI Bob Ross Eropea IBIS Smmit Naples, Italy May, Page Teraspeed Cosltig Grop LLC
2 Problem ad Traditioal Methods Page Teraspeed Cosltig Grop LLC
3 Refereces B. Ross, Taylor Series Dality, Proc. 7 th IEEE Worshop of Sigal Propagatio o Itercoects, Siea, Italy, May -,, pp Fll eatios i paper (presetatio temporary) Page Teraspeed Cosltig Grop LLC
4 Special Notio - Eatios ()-() Laplace Trasform X (s) s a s b s a b, Differetial Eatio ( t) b ( t) b ( t) iitial coditios, (),, (), Differece Eatio ( t) d ( t) d ( t) iitial coditios, ),, (), ( Z Trasform Z( z) z( c z d z c) z d. Page Teraspeed Cosltig Grop LLC
5 Coversios ad Resposes ()-(6) Nmerator Coefficiets Iitial Coditios (Traditioal methods) Laplace Trasform Z Trasform B D A, M(T) = ep(at) Differetial Eatio Differece Eatio I-M E I-L A E, L(T) = l(e)/t M(T), E, Taylor Series Time Resposes L(T), A, Dal of T.S. Differetial Resposes Page Teraspeed Cosltig Grop LLC
6 , Differetial Differece E s ()-(8) E s (9)-() (t) = z(t) = [ ( t), ( t),, ( t)] T a = c = B = D = a = B() Page 6 c = Dz() Teraspeed Cosltig Grop LLC
7 Differetial Differece E s ()-(6) E s (8)-() d(t)/dt = A(t) z(t+t) = Ez(t) A = E = (t+t) = M(t) dz(t)/dt = Lz(t) M = ep(at) Page 7 E = ep(lt) L = l(e)/t Teraspeed Cosltig Grop LLC
8 Differetial Differece E (7) E () I-M = E = A = I-L = Page 8 Teraspeed Cosltig Grop LLC
9 Characteristic Eatio Cayley-Hamilto Theorem a matri satisfies its ow characteristic eatio Comptatio of characteristic eatios: Based o bilt i mathematical fctios Or based o calclatig traces (sm of diagoal terms) of powers of M or L Frther simplificatios possible (otside scope of this presetatio) Page 9 Teraspeed Cosltig Grop LLC
10 Taylor Series Dal of T.S. E s ()-() E ()-(6) Page Teraspeed Cosltig Grop LLC
11 Recrsive Taylor Series (Repeat b ad c) a) Iitialize: i =,, - (B) b) Eted: i =,, p (A) c) Net time step: i =,, - (Taylor series) R. I. Ross, Evalatig the Trasiet Respose of a Networ Fctio, Proc. IEEE, vol., pp. 6-66, May 967 Page Teraspeed Cosltig Grop LLC
12 Ecel T.S. Implemetatio Page Teraspeed Cosltig Grop LLC
13 Coversios ad Resposes ()-(6) Nmerator Coefficiets Iitial Coditios (Traditioal methods) Laplace Trasform Z Trasform B D A, M(T) = ep(at) Differetial Eatio Differece Eatio I-M E I-L A E, L(T) = l(e)/t M(T), E, Taylor Series Time Resposes L(T), A, Dal of T.S. Differetial Resposes Page Teraspeed Cosltig Grop LLC
14 Coclsios Recrsive Taylor Series I place, if eeded No special hadig for mltiple or comple poles as with partial fractio epasios Embedded software ad spread sheets Good covergece with Taylor Series M or L ca be calclated by fctios of matrices ca be sed for eact cotios ad discrete system coversios Page Teraspeed Cosltig Grop LLC
15 Page Teraspeed Cosltig Grop LLC Bacgrod - Symmetry (State Trasitio/Logarithmic Matrices) Differece (Time) Differetial z(t), L(T) (t), M(T)
16 Page 6 Teraspeed Cosltig Grop LLC Traspose Symmetry (Differece/Differetial Eatio) Differece (Time) Differetial A E
17 Page 7 Teraspeed Cosltig Grop LLC Traspose Symmetry (Taylor Series/Dal Resposes) Differece (Time) Differetial Dal of T.S. Taylor Series A E
18 Taylor Series ad Dal of Taylor Series Derivatios Taylor Series from State Trasitio Matri (t+t) = M(T)(t) = (I + AT + A T /! )(t) A (t) is -th derivative of (t) Dal of T.S. from Natral Logarithm Matri dz(t)/dt = L(T)z(t) = - [(I - E) + (I - E) / + (I - E) / + ]z(t)/t E z(t) is the -th shifted sample of z(t) Collect the terms for each power of E Page 8 Teraspeed Cosltig Grop LLC
19 Page 9 Teraspeed Cosltig Grop LLC Logarithmic Epasio: Biomial Series & Pascal Triagle Redctio 6, i factor i , i factor i i i i T ) (, > T ) (, > Correctio: replace with i
20 Eample: si( t), Some Origial & Scaled Terms Scalig by weightig samples: y i = ep(- t) i T Scaled T ( terms) e+7.9 (maimm vale) -6.89e+9 -. (set by scalig) -.8e e e- Page Teraspeed Cosltig Grop LLC
21 Eample: si( t), Last / Cycle, 8-th Derivative. <= t <=., T =., samples Fctio Eact Dal T.S. (error bold) si[]. -.87e- si[6] si[7] si[8] si[9] si[]..87e- (All other iterative methods are Eact ) Page Teraspeed Cosltig Grop LLC
22 Eample: si( t), Last / Cycle, 9-th Derivative. <= t <=., T =., samples Fctio Eact Dal T.S. (error bold) cos[] cos[6] cos[7] cos[8].97.9 cos[9] cos[]..9 (All other iterative methods are Eact ) Page Teraspeed Cosltig Grop LLC
23 Coclsios Eact trasformatios Practical modelig applicatios Commo roties for both domais Taylor Series & Biomial Series dality Accrate with scalig Bt ot as accrate ad stable as other iterative methods Page Teraspeed Cosltig Grop LLC
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