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

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

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

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