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

# 1 26 2 design a digital lowpass b utterworth filter

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: a re t he j s ystem i nputs. F or a linear system, these equations reduce t o a s impler linear form V3 = Xl - V4 = Xl - (13.2) 2X2 T his s et o f e quations is known a s t he o utput e quation o f t he s ystem. I t is clear from t his s et t hat e very possible o utput a t some i nstant t c an b e d etermined from a knowledge o f X I(t), X2(t), a nd f (t), t he s ystem s tate a nd t he i nput a t t he i nstant t. O nce we solve t he s tate e quations (13.1) t o o btain X I(t) a nd X2(t), we c an d etermine e very possible o utput f or a ny given i nput f (t) . • I f we a lready have a system equation in t he form o f a n n th-order differential e quation, we c an c onvert i t i nto a s tate e quation as follows. Consider t he s ystem e quation d ny d n - 1y +a - 1 - d tn n d tn-l dy + ... + al- +aoy = dt f (t) (13.3) k =I,2, . .. , n (13.6a) a nd t he o utput e quations a re o f t he form Ym = CmlXI + Cm2X2+" . +cmnx n + dmlfl + d m 2h+" · +dmjfj m = 1 ,2, . .. , k (13.6b) T he s et of E quations (13.6a) a nd (13.6b) is called a d ynamical e quation. W hen i t is used t o d escribe a system, i t is called t he d ynamical-equation d escription o r s tate-variable d escription of t he s ystem. T he n s imultaneous first-order s tate e quations a re also known as t he n ormal-form e quations. These equations c an b e w ritten m ore conveniently in m atrix form: .... < 13 S tate-Space A nalysis 788 13.2 A S ystematic P rocedure for D etermining S tate E quations HI a ll X2 a l2 al n a 21 Xl a 22 a 2n b ll X2 + . .................. a nI Xn an2 a nn b lj b 2j 12 ..... ..... b nl Xn A 10 h b 22 bn 2 , '-v-' x '-v-' x b l2 b 21 Xl IH 789 b nj f (13.7a) Ij ,'-v-' v F ig. 1 3.2 R LC network for Example 13.2. f B a nd 3. W rite t he l oop e quations a nd e liminate all variables o ther t han s tate variables ( and t heir first derivatives) from t he e quations d erived in S teps 2 a nd 3. C ll Y2 C i2 C in C21 YI C22 C 2n d ll X2 + . ................. C kl Yk C kn Ck2 '-v-' y c d l2 d lj h d 21 Xl d 22 d 2j 12 . ................. d kl Xn • , '-v-' x d kj d k2 (13.7b) Ij ,'-v-' D E xample 1 3.2 Write the state equations for the network shown in Fig. 13.2. S tep 1. There is one inductor and one capacitor in the network. Therefore, we shall choose the inductor current X l a nd the capacitor voltage X 2 a s t he state variables. S tep 2. T he relationship between t he loop currents and the s tate variables can be written by inspection: f (13.9a) or (13.9b) x= A x+Bf (13.8a) y = C x+Df (13.8b) S tep 3. The loop equations are (13.lOa) E quation (13.8a) is t he s tate e quation a nd E q. (13.8b) is t he o utput e quation. T he v ectors x , y , a nd f a re t he s tate vector, t he o utput v ector, a nd t he i nput v ector, respectively. For discrete-time systems, t he s tate e quations a re n s imultaneous f irst-order difference equations. Discrete-time systems a re d iscussed in Sec. 13.6. 1 3.2 A Systematic Procedure for Determining S tate Equations (13.10b) Now we eliminate...
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