1. Engineering Background The firs t chapte r i s designed t o convince a mathematician tha t analyti c func - tions f(s) occu r naturall y i n describin g a "blac k box, " tha t i s s doe s indee d correspond t o frequency , an d tha t stabl e boxe s ( a ver y importan t class ) giv e f(s) wit h n o pole s i n th e righ t half-plan e (R.H.P.) . Thes e ar e modes t expecta - tions o f a lon g chapter an d fastidiou s attentio n t o detai l i s not essential . I n fac t beyond Chapte r 2 , the firs t chapte r i s not necessary . Much of engineering concerns boxes which take in inputs and put ou t outputs . In thes e talk s th e input s wil l be i n th e vecto r spac e C n an d th e output s i n C m , and they vary with time. Thu s a box is really a map B fro m a space of C n -valued functions t o C m -valued function s (se e Figure 1.1) . Common physica l propertie s o f a box ar e 1. Causality. I f tw o input s ii(t) an d %2{t) ar e identica l befor e tim e to, the n the output s ar e identica l before tim e to- 2. Time-invariance. I f a n inpu t functio n i s shifte d b y a the n th e outpu t i s shifted b y a. (A n experimen t a t noo n give s th e sam e answe r a s a n experimen t at 2 p.m.) 3. Linearity. B i s a linear map . All system s discusse d i n th e lecture s wil l enjo y thes e thre e properties . A n im - portant propert y whic h som e enjo y an d som e do no t is : 4. Stability. Ever y decayin g inpu t functio n produce s a decayin g outpu t func - tion. For concretenes s th e reade r shoul d thin k o f the bo x a s a n electrica l circuit . Conceptually th e bo x i s wha t i s given . I t act s o n som e spac e o f function s which we choose t o put i n the bo x and respond s b y puttin g ou t othe r functions . There i s n o a prior i reaso n tha t th e appropriat e clas s o f input s shoul d b e L z or L°° o r an y othe r class . I n fac t whic h spac e w e us e depend s o n th e proble m we ultimately wan t t o solv e and whic h quantit y w e designate a s th e inpu t o r a s the output . Thi s choic e i s usually somewha t arbitrary fo r example , o n a circui t output wit h tw o terminal s w e usuall y measur e th e voltag e v(t) acros s th e tw o terminals an d th e curren t c(t) throug h th e tw o wires . Whethe r w e cal l c(t) th e input an d v(t) th e output , o r vic e versa, i s arbitrary. However , onc e w e make a choice th e genera l input-outpu t theor y w e ar e describin g holds . 3 http://dx.doi.org/10.1090/cbms/068/01
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