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1 Lnear Systems Revew Notes adapted from notes by Mchael Braun Typcally n electrcal engneerng, one s concerned wth functons of tme, such as a voltage waveform System descrpton s therefore defned n the domans of tme and temporal frequency In these notes, a more general approach s taken The generalzed ndependent varable may be tme, but t may also represent spatal poston n one dmenson (1D) If treated as a vector, may also stand for multdmensonal quanttes such as poston n 3D space Let f ( be a functon of the generalzed varable We wll assume that the functon satsfes the estence condtons for the Fourer transform We wll denote the Fourer transform of f ( by F (, { f ( } = f ( e 2π s d = F, where = 1 Note that f s tme, then s s the temporal frequency measured n cycles per second If, on the other hand, s the spatal poston, then s s the spatal frequency measured n cycles per unt length System A system s anythng we may care to eamne that can be characterzed by a black bo When the system receves a stmulus (nput functon f ( ), t produces a response (output functon g ( ), The system can be represented mathematcally by a system operator S whch maps the nput functons to output functons, g ( = S f ( { } Lnear System A system s sad to be lnear f the response to a sum of two dfferent nputs s a sum of the responses produced separately by each nput It also follows that scalng the nput scales the output by the same factor Thus a system s lnear f S s a lnear operator, that s, S{ α f1( + β f 2 ( } = α S{ f1( } + β S{ f 2 ( } where α, β are constants Ths naturally etends to any fnte sum of weghted functons, n n S α f ( = α S{ f ( } = 1 = 1 A system s sad to possess etended lnearty f the above holds when n s nfnte and when the summaton s replaced by ntegraton For the latter case, we have b S α ( ') f (, ') d' α( ' ) S f (, ') d' { } { } a = b a
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5 The shft nvarant mpulse response thus completely characterzes an LSI system Cascaded LSI system If two or more LSI systems are cascaded, ther combned response s gven by g ( = f ( * h1 ( * h2 ( Thus the two systems can be consdered as a sngle system wth mpulse response h = h * h 1 2 Response to a Comple Eponental Response of an LSI system to a comple eponental f ( = ep( s s a scaled verson of the nput, e { e } = a e S where a s a comple constant Ths s sometmes referred to as harmonc response to harmonc nput because the output has the same frequency as the nput (ampltude and phase may, however, dffer) In other words, no frequences can appear at the output that are not present at the nput To prove ths asserton, wrte g ( as the superposton ntegral ' = e h( ') d' Introducng a new varable ' ' = ', we obtan 2π '' = e e h( '') d' Snce the ntegrand s ndependent of, the whole ntegral s smply a comple constant, = a e = a f ( A comple eponental s sad to be an egenfuncton of a LSA system, and the comple constant a s ts egenvalue Transfer functon The comple egenvalue a s known as the transfer functon of the system and s usually denoted as a functon of frequency (temporal or spatal) H ( s { e } = H ( e S We wll now show that, f the nput and output functons have Fourer transforms = F { f ( } and G( = F { }, then for a lnear system wth transfer functon H (, G ( = H ( Wrtng the Fourer transform n full, f ( = e 2π ds
6 we see that ths s equvalent to epressng a functon as a superposton of comple eponentals Snce n a lnear system each eponental s passed through ndependently, we can wrte = S = { e ds} s S { e s } = H ( e ds = F { H ( } Therefore, G ( = H ( as stated above It follows drectly that the transfer functon s the Fourer transform of the mpulse response, H ( = F { h( } For a cascade of n LSI systems, f the overall mpulse response s h( = h1 ( * h2 ( ** hn (, then the overall transfer functon s smply the product of ndvdual transfer functons, n H ( = Π H ( Response of Physcal Systems to a Snusod In physcal systems, real-valued nputs lead to real-valued responses The mpulse response wll also be real-valued Consder a snusod nput functon of frequency s 0, s0 f ( = cos 2π s0 = Re{ e } The response to ths nput can be found as follows Let the transfer functon, φ ( s ) H ( s ) = A ( s ) e, be gven by ampltude and phase functons A ( and φ (, respectvely Then the response to the snusod nput s φ ( s0 ) s0 = Re{ A( s0 ) e e } = Acos(2π s0 + φ) Thus the response s a snusod of the same frequency as the nput but wth possbly dfferent ampltude and phase = 1 ds
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