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

As in t he case o f t he laplace transform with

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Unformatted text preview: lds (zn + an_IZ n - 1 + ... + a lz + ao)Y[zJ = (bnz n + bn_IZ n- 1 + ... + b tz + bo)F[zJ zn F [J z As i n t he case o f L TIC s ystems, t his r esult leads t o a n a lternative d efinition of t he L TID s ystem t ransfer f unction as t he r atio o f Y[zJ t o F[zJ ( assuming all initial conditions zero). H[zJ == Y[zJ = Z [zero-state responseJ F[zJ Z [inputJ (11.35) Because Y[z], t he z -transform o f t he z ero-state response y[k], is t he p roduct o f F[zJ a nd H[z], we c an r epresent a n L TID system in t he frequency d omain b y a block diagram, as i llustrated i n Fig. 11.5. J ust a s in continuous-time systems, we c an r epresent discrete-time systems in t he t ransformed m anner b y representing all signals by t heir z -transforms a nd all system components (or elements) by t heir t ransfer functions. O bserve t hat t he d enominator o f H[zJ is Q[z], t he c haracteristic polynomial of t he s ystem. Therefore t he poles o f H[zJ a re t he c haracteristic r oots o f t he s ystem. C onsequently, t he s ystem s tability c riterion c an b e s tated i n t erms o f t he poles o f t he t ransfer function o f a n L TID s ystem a s follows: 1. A n L TID s ystem is asymptotically s table if a nd o nly i f all t he poles o f i ts t ransfer f unction H[zJ lie inside a u nit circle (centered a t t he origin) in t he c omplex plane. T he p oles m ay b e r epeated o r unrepeated. 2. A n L TID s ystem is unstable if a nd o nly if either one or b oth o f t he following conditions exist: (i) a t l east one pole of H[zJ is outside t he u nit circle; (ii) t here a re r epeated p oles of H [z J o n t he u nit circle. 3. A n LTID s ystem is marginally stable if a nd only if t here a re no poles of H [zJ o utside t he u nit circle, a nd t here a re some u nrepeated poles o n t he u nit circle. 6 92 11 D iscrete- Time S ystems Analysis Using t he Z - Transform f [k I u [k I so t hat f [k-II u [ k-I] F[z I Y[zl L ..-_ _..:J =1. z 11.4 S ystem R ealization F [zl and y[k] F ig. 1 1.6 Ideal unit delay and its transfer function . • E xample 1 1.6: T he T ransfer F unction o f a U nit D elay Show t hat the transfer function of a unit delay is 1 / z. I f the input to the unit delay is f [k]u[k], then its output (Fig. 11.6) is given by o T he z-transform of this equation yields [see Eq. (11.15a)] 1 Y[z] = ; F[z] = H[z]F[z] 6 I t follows that the transfer function of the unit delay is 1 z • = f [k + 2( _0.5)k] u[k] • C omputer E xample C 11.3 Solve Example 11.7 using MATLAB. Plot y[k] for 0 :S k :S 10. 0 E xercise E l1.11 H[ ] z = (z (11.36) E xample 1 1.7 Find the response y[k] of an LTID system described by the difference equation + 2] + y[k + 1] + 0.16y[k] [~( _0.2)k - ~(-0.8)k (11.38) A d iscrete-time s ystem is described b y t he following t ransfer f unction: H[z] = - y[k = (z_Z ) _~ (z_Z ) + 2 (z_Z ) + 0.2 3 + 0.8 + 0.5 k =0:10; b =[O 1 0 .32]; a =[l 1 0 .16]; f =(-2). - (-k); y =fiiter(b,a,f) ; s tem(k,y) x label('k');ylabel('y[kl') y[k] = f [k - l]u[k - 1] • Y[z] _ ~ -3 693 + 1] + 0.32f[k] z - 0.5 +...
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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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