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STransform

# STransform - S — Transform N N =:aktk k=O hn/°° tnh(t...

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Unformatted text preview: S — Transform N N) = :aktk k=O hn /°° tnh(t) dt. 00 O,1,2,---N. |hn| < 00 for n m /: h(t) f(T — t) dt N (_t)k SH{f}(T) E FH(7') m—t) = z M Mr) k=0 ﬂew a ﬁrm. Forward 5 - Transform 00 N _ k / h(t) Z ( kt.) f’°('r) dt. — =0 ( _k1!)kfk(7) f: tkh(t) dt (—1)k k! hkfkcr). N Z k=0 N Z k=0 Deconvolution N _ k FHm = 2‘ 1,) hkfkm k=0 k. N (—1)k k FH(7') = hof(’r) + k, hkf (T) k=1 ' __ 1 N (*Dk k =>f(T) — h_O(FH(T)—k§::1 k! hkf (7))- Differentiation m _ 1 m N_m(—1)k k+m f (T)—h—O(FH(T)*k§1 k! hkf (T) m, d’” Where FH (’7') _— ———-d 7' Inverse S-Transform f (T) where 'wo wk N k 2 kaH(T) 1:20 1 — and ho k E Z p=1|Ip|=k (_1)p+k P h z'q 3.1 ‘ Example Inverse S' transform is veriﬁed here for a simple case. Let f(t) = a0 + a1t+ a2]!2 + (13t3 and (35) 1 t h(t) — T- rect (36) where, 1 for |t| 5 % rect(t) = (37) 0 otherwise. We have FH(r) = /_ h(t)f(r — t) dt (38) = /°° -11:rect (a0 + a1(r — t) + a2(1' — t)2 + (13(7' —- t)3) dt (39) = -1- (00 + (11(1’ - + (12( — t)2 + (13(7' -' T -§~ 2 T2 = ((10 + —a2) + ((11 + —-03> T + (127'2 + 037'? (41) 12 4 1;- 1 k = _ 42 hr 1; T t dt ( ) h = LN“ 2 for I: even (43) 0 for 1: odd. Therefore, we get T2 wo=1, 101:0, w2=—-(§Z),andw3= (44) Hence the inverse S transform of F H is f0) = woFﬁm + 101330) + wzFﬂt) + £03530) (45) T2 = Firm + (~53) out) (46) T2 T2 T2 = ((10 + Bag) + ((11 + 1—03) t + (122:2 + a3t3 —‘ (202 + 603i) a0 + a1t+ a2t2 + a3t3. Thus the inverse S transform has been veriﬁed for this case a 10 (48) S Transform Two-dimensional N-th order polynomial: N—m N f(93ay) = Z Z am,n\$myn m=O n=0 Z am’nwmyn OSm+nSN Iff 6 pg“, then fm’n=0 for m+n>N. Two-dimensional moments: OO 00 hmm E / / mm y" h(:I:, y) da: dy —oo -—00 for m,n = o,1,2,3,-~-. Forward S - Transform Mm, y) * f(:v, y) FH(5’3: y) H M N ﬂaw) = Z Elvis—1,1 Fig—1’1 II M20 .M 1.3 £3: Z(..1)19+z+j 1131 (hmqmq) w.. = 2,] qzl mq!nq| Summation over all mq,nq,p, for q = 1,2,3, o - o ,p. Conditions: m1+m2+~.+mp=z' n1+n2+~~+np=3 m1+n1217m2+n221)"'7mp+np21 ...
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STransform - S — Transform N N =:aktk k=O hn/°° tnh(t...

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