ENGINEERING BACKGROUN D 5 where th e integra l i s take n ove r an y contou r Re s = K fo r whic h / £ #°°(fc +R.H.P.) . Basic propertie s o f the Laplac e transfor m ar e (l-2a) (?'/)(* ) = /-») , (1.2b) / a W = e - M / W , a0. The Paley-Wiene r theore m tell s us that th e Laplac e transfor m / o f any functio n / i n £ 2 [0, oo] extend s analyticall y t o al l o f th e R.H.P. , ha s value s a.e . o n th e imaginary axis , an d tha t th e resultin g functio n f(ioj) i s in L 2 [—ioc,ioo}. I n fac t Z2[0,oo] = H 2 (R.H.P. ) wher e u 2 / p u p \ A ! ' 1 ^ 2 (imaginar y axis) : analyti c i n th e R.H.P. , H (R.H.P. ) = | joo ^ ^ i(j + fl) |2 duJ M OQ f o r a i l a 0 The Laplac e transfor m o f L 2 [0, oc K] i s exactly H 2 o f th e half-plan e Re s if , the shif t o f R.H.P. b y K. Thi s i s easily derive d fro m th e basi c fact s (1.2) . Intuitively f(iu) i s the contributio n o f elu}t t o the functio n f(t). Fo r example , if f(t) i s a soun d wave , the n |/(iu )| 2 measure s ho w muc h powe r o f th e wav e is concentrate d i n th e e l(jjt componen t o f / . A devic e t o measur e f(iuj) fo r a signal f(t) i s calle d a spectru m analyzer . The y ar e commonplac e an d man y millions o f dollar s chang e hand s eac h yea r o n spectru m analyzer s fo r electri c signals, mechanica l vibrations , an d eve n sound . T o b e convincin g I brough t one alon g fo r analyzin g sound . I t i s cheap s o i t onl y measure s |/| 2 , th e "powe r spectrum," an d no t th e phas e o f / . Th e devic e displays th e grap h o f log \f\2 o n the fron t wit h lights . Sinc e I'l l leav e th e gizm o on , thos e o f you whos e eye s ca n follow rapidl y bouncin g light s will be abl e to follow th e res t o f the lectur e via it s (instantaneous) Fourie r transform . Laplace transform s greatl y facilitat e th e treatmen t o f boxes . Defin e th e "Laplace-transformed" B b y Bi = (Si) fo r al l i i n L 2 [0, oc]. Our discussio n o f Laplace transform s indicate s tha t B: F 2 (R.H.P.) - * H 2 (K + R.H.P. ) and tha t tim e invarianc e i s equivalent t o (1.3) B(e- a9 i) = e~ as Bi fo r ali i L 2 [0,oo] for a 0 . Thi s hold s fo r al l R e a 0 b y analyti c continuation . Tha t is , B commutes wit h multiplicatio n b y e^ lw+ °)s fo r al l u an d a 0 . Man y operato r theorists woul d quickl y gues s th e followin g fundamenta l fact : THEOREM 1. 1 (VARIAN T O F [FS]) . Suppose m = n = 1 . If B is linear, time invariant, and causal, i.e., satisfies (l.T) and (l.C), then there is a function b(s) defined and analytic for R e s K such that [Bi](s) = b(s)i(s) for R e s K.
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