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

434j note t hat a is restricted to only positive

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: which uses zero-input/zero-state separation of t he i nput) t o t he electrical engineering community. 8 +1 (8 2 + 58 + 6) Y(8) - (28 + 11) = - 4 8+ so t hat (8 2 + 58 + 6) Y (8) = (28 + 11) + '-...-' i nitial c ondition t erms 8+1 8 +4 :r " '--..--' i nput t erms T herefore Y (8) = 28 + 11 82 + 58 + 6 "-v-"' z ero-input c omponent 8 +1 (8 + 4)(8 + 58 + 6) + -:-----:-;-;-;;---::--:-:2 , ... # z ero-state c omponent 6 7 5] [ -1/2 2 3 /2 ] = [ 8 +2-8+3 + 8 +2+8+3-8+4 Exercise E 6.6 S olve d2 d dt~ + 4 ft + 3y(t) = T aking t he i nverse transform of this equation yields z ero-input r esponse 393 df 2 di + I (t) for t he i nput I (t) = u (t). T he i nitial conditions are y (O-) = 1 a nd y (O-) = 2. Answer: y (t) = ~(l + g e- t - 7e- 3t )u(t) \ 7 z ero-state r esponse lH 20 lH 20 Comments on Initial Conditions at 0 - and at 0+ T he initial conditions in Example 6.9 are y(O-) = 2 a nd y(O-) = 1. I f we let t = 0 in t he t otal response in Eq. (6.44), we find y(O) = 2 a nd y(O) = 2, which is a t o dds with t he given initial conditions. Why? T he reason is t hat t he initial conditions are given a t t = 0 - (just before t he i nput is applied), when only t he zeroinput response is present. T he z ero-state response is t he result of t he i nput f (t) applied a t t = o. Hence, this component does not exist a t t = 0 -. Consequently, t he initial conditions a t t = 0 - are satisfied by t he zero-input response, not b y t he t otal response. We can readily verify in this example t hat t he z ero-input response does indeed satisfy the given initial conditions a t t = 0 -. I t is t he t otal response t hat satisfies t he initial conditions a t t = 0+, which are generally different from the initial conditions a t 0 -. T here also exists a L:+ version of t he Laplace transform, which uses t he initial conditions a t t = 0+ r ather t han a t 0 - (as in o ur present L:_ version). T he L:+ version, which was in vogue till t he early sixties, is identical t o t he L:_ version except t he l imits of Laplace integral [Eq. (6.18)] are from 0+ t o 0 0. Hence, by definition, t he o rigin t = 0 is excluded from t he domain. This version, still used in some m ath b ooks, has some serious difficulties. For instance, t he Laplace transform of 6(t) is zero b ecause 8(t) = 0 for t 2: 0+. Moreover, this approach is p ractically useless in t he t heoretical s tudy of linear systems because t he response obtained by this method c annot be separated into zero-input a nd zero-state components. As we know, the zero-state component represents t he system response as a n explicit function of t he i nput, and without knowing this component, it is n ot possible to assess t he effect of t he i nput on t he s ystem response. The L:+ version can separate t he response i n t erms of t he n atural a nd t he forced components, which are not as interesting a s t he zero-input a nd t he z ero-state components. Note t hat we can + .! 5 (a) F ( b) ( c) o 1-- F ig. 6 .8 Analysis of a networ...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online