Signal Processing and Linear Systems-B.P.Lathi copy

A system can be conveniently illustrated by a black

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Unformatted text preview: systems consists of three major areas: mathematical modeling, analysis, a nd design. Although we shall b e dealing with mathematical modeling, our main concern is w ith analysis a nd design. T he m ajor portion of this book is devoted t o t he analysis p roblem-how t o d etermine t he s ystem o utputs for the given inputs a nd a given mathematical model of t he s ystem (or rules governing the system). To a lesser extent, we will also consider t he problem of design or s ynthesis-how t o c onstruct a system which will produce a desired set of o utputs for the given inputs. 78 1 I ntroduction t o Signals a nd S ystems a nd all capacitor voltagest are needed t o d etermine t he o utputs a t a ny i nstant t if t he i nputs a re given over t he i nterval [0, tl. + R + 1. 7 y (t) c ! (t) F ig. 1 .26 An e xample of a s imple e lectrical s ystem. Data Needed to Compute System Response I n o rder t o u nderstand w hat d ata we need t o c ompute a system response, consider a simple R C c ircuit with a current source / (t) as its i nput (Fig. 1.26). T he o utput voltage y (t) is given by y(t) = R /(t) 1 +C Jt - 00 / (r)dr (1.36a) T he limits of t he i ntegral on t he r ight-hand side are from - 00 t o t b ecause this integral r epresents t he c apacitor charge due t o t he c urrent / (t) flowing in t he capacitor, a nd t his charge is t he r esult of t he c urrent flowing in t he c apacitor from - 00. Now, Eq. (1.36a) can b e e xpressed as y(t) = R /(t) +.!. JO C / (r) dr +.!. C - 00 t /(r) dr Jo (1.36b) T he m iddle t erm o n t he r ight-hand side is vc(O), t he capacitor voltage a t t = O. T herefore 11t y(t) = vc(O) + R /(t) + - C T his equation c an b e readily generalized as y(t) = vc(to) 0 11t + R /(t) + - C /(r) dr 79 1.7 Classification of S ystems 1. 2. 3. 4. 5. 6. 7. ~ 0 Classification o f S ystems S ystems m ay b e classified broadly in t he following categories:t L inear a nd n onlinear systems; C onstant-parameter a nd t ime-varying-parameter systems; I nstantaneous (memoryless) a nd d ynamic (with memory) systems; C ausal a nd n oncausal systems; L umped-parameter a nd d istributed-parameter s ystems; C ontinuous-time a nd d iscrete-time systems; Analog a nd D igital systems; 1 .7-1 Linear and Nonlinear Systems The Concept o f Linearity A s ystem whose o utput is p roportional t o i ts i nput is a n e xample o f a linear system. B ut l inearity implies more t han this; it also implies a dditivity p roperty, i mplying t hat if several causes a re a cting o n a s ystem, t hen t he t otal effect on t he s ystem due t o all these causes c an b e d etermined by considering each cause separately while assuming all t he o ther causes t o b e zero. T he t otal effect is t hen t he s um o f all t he c omponent effects. T his p roperty m ay b e expressed as follows: for a linear system, if a cause q a cting alone has a n effect e l, a nd if a nother c ause C2, also acting alone, has a n effect e2, then, w ith b oth c auses acting on t he s ystem, t he t otal effect will b e e l + e2. Thus, if (1.37) a nd t hen for all Cl a nd C2 (1.38) (1.36c) I n a ddition, a linear system m ust s atisfy t he h omogeneity o r scaling property, which s tates t hat for a rbitrary real o r i maginary n umber k , if a cause is increased k-fold, t he e...
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