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

# Such cases c an b e analyzed by t he b ilateral or

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: showed t hat d iscrete-time systems c an b e a nalyzed by t he L aplace transform as if t hey were continuous-time systems. I n fact, we showed t hat t he z -transform is t he Laplace transform with a change in variable. I n practice, we o ften have t o d eal w ith h ybrid systems consisting of discretetime a nd c ontinuous-time subsystems. Feedback hybrid systems are also called s ampled-data systems. I n such systems, we can relate t he s amples of t he o utput t o t hose of the input. However, t he o utput is generally a continuous-time signal. T he o utput values during t he successive sampling intervals c an b e found by using t he m odified z-transform. T he m ajority of t he i nput signals a nd p ractical systems are causal. Consequently, we a re required t o deal with causal signals m ost o f t he time. W hen all 1 1.1-1 Using the definition of the z-transform, show t hat [J ( ab k -1 u k-l ( b) u[k - mJ 1 1.1-2 = k 1 = -z-, ( e) ; 'u[kJ k. z = ( d) (In 7)k u[kJ z m(z _ 1) k e"'l/z = ( Fz Using only the z-transform Table 11.1, show t hat ( a) 2k+1u[k - IJ + e k - 1 u[kJ z~2 + e (z"-e) = = (z -, [TkCOS(~k)] = z:,o-~5g=~~)25 ( b) k ,ku[k - IJ , z )2 Hint: Express u[k - IJ in terms of u[kJ. ( e) u[k - IJ Hint: See the hint for p art b. ( d) k (k - 1)(k - 2)2 k- 3u[k - mJ (z~~)' for m =O, 1, 2, or 3. Hint: Examine what happens to the function if u[k - mJ is replaced by u[kJ. = 11.1-3 Find the inverse z-transform of ( a) ( b) ( e) z (z - 4) z2 - 5z + 6 z -4 z2 - 5z + 6 ( e- 2 - 2)z (z - e- 2)(z - 2) z (z - 2) z2 - z+ 1 ( h) 2z2 - 0.3z + 0.25 z2 + 0.6z + 0.25 z (2z + 3) (z - 1)(z2 - 5z ( e) 2z(3z - 23) (z - 1)(z2 - 6z + 25) ( j) z (-5z + 22) (z + l )(z - 2)2 ( f) (i) z(3.S3z + 11.34) (z - 2)(z2 - 5z + 25) ( k) ( d) 1 1.1-4 ( g) z2( _2z2 + Sz - 7) (z - l)(z - 2)3 z (1.4z + O.OS) (z - 0.2)(z - 0.S)2 + 6) Find the first three terms of f[kJ if F[zJ 3 2 = 2z + 13z + Z z3 + 7z2 + 2z + 1 ., 712 11 D iscrete-Time S ystems A nalysis U sing t he Z -Transform .... if y [-I] f [kJ 1 o 713 P roblems 1 1.3-7 4 II I m -l y[k + 2 ]- 2y[k f [rkJ _j_ rlIII! k --- = 2, y [-2] = 3, a nd I [k] = (3)k u [k]. Solve Fig. P ll.2-1 + 1] + 2y[k] = I [k] w ith y [-I] = 1, y [-2] = 0, a nd J[k] = u[k]. 1 1.3-8 Solve + 2y[k - 1] + 2y[k - 2] = I [k - 1] + 2 /[k - 2] 1, a nd I [k] = eku[k]. y[k] w ith y[O] Fig. P ll.2-2 1 1.3-9 = 0, y[l] = ( a) F ind t he z ero-state r esponse o f a n L TID s ystem w ith t ransfer f unction F ind y our answer by e xpanding F[z] a s a p ower series in z -l. 1 1.1-5 H [z]- (z B y e xpanding F[z] = ( as a power series in z -l, show t hat I [k] 1 1.2-1 Z -"( a nd t he i nput J[k] = e(k+l)u[k]. ( b) W rite t he difference e quation r elating t he o utput y[k] t o i nput I [k]. = k"(ku[k]. For a discrete-time signal shown in Fig. P l1.2-1 s how t hat 1 1.3-10 1 1.3-11 1 1.2-3 R epeat P rob. 11.3-9 i f I [k] = u[k] a nd = z:-, k ( a) e"(ku[k] ( c) a [ u[k]- u[k - mll ( b) 1 1.2-4 H[z] a nd p roperties o f t he z -transform, find t he k3 u...
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