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 [zerostate responseJ
F[zJ
Z [inputJ (11.35) Because Y[z], t he z transform o f t he z erostate 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 continuoustime systems,
we c an r epresent discretetime 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 [kII u [ kI] 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 ztransform 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 iscretetime 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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