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t his reason, we shall merely mention here some o f these insights without discussion
o r proof. T he interested reader should read Sec. 2.7 for more explanation.
T he s ystem's behavior is s trongly influenced by t he c haracteristic roots (or
modes) of t he system. T he s ystem responds strongly t o i nput signals similar to its
characteristic modes, a nd responds poorly to i nputs very different from its characteristic modes. In fact, when t he i nput is a c haracteristic mode of t he s ystem, the
response goes t o infinity provided t hat t he m ode is a nondecaying signal. This is
t he resonance phenomenon. T he w idth o f an impulse response h[k] i ndicates the
response time (time required t o r espond fully t o a n i nput) o f t he s ystem. I t is the
t ime c onstant o f t he system. t Discrete-time pulses are generally dispersed when f h'flu[k] = f + n] + an-18[k + n - 1] + ... + a18[k + 1] + ao8[kJ) = bn8[k + n]
+ bn_18[k + n - 1] + ... + b18[k + 1] + bo8[k] k ~ 0 I f we s et k = 0 in this equation a nd recognize t hat 8 = 1 a nd 8[m] = 0 when
m i ' 0, all b ut t he l ast terms vanish o n b oth sides, yielding
Aoao = bo a nd
A Special Case: ao = 0
I n t his case Ao = £IL becomes indeterminate. T he p rocedure has t o b e modified
ao tThis conclusion follows from the fact that (see Sec. B.7-4) k =-oo A o(8[k slightly. W hen ao = 0, Q [E] c an be expressed as E Q[E]' a nd Eq. (9.28) can be
expressed as l'Yilk = 1 _l/'Yi! k =O E Q[E]h[k] = P [E]8[k] Therefore, if /'Yi! < 1 for i = 1 ,2, . .. , n , ~
~ k =-oo Ih[kll:S ~_1_
~l-/'Yil < 00 i =l Although we have assumed all the roots to be distinct in this derivation, it is also valid for repeated
rO<lts proVIded that they lie inside the unit circle.
j:The converse is not true. See footnote on p. 152.
tThis part of the discussion applies to systems with impulse response h[k] that is a mostly positive
(or mostly negative) pulse. I f we recognize t hat i is a delay operator, this equation can be rearranged as j:Note that
Q[E] (Yn[k]) = 0 for all k. But if we restrict the mode terms to be causal, the equation is valid only for k
is, Q[E] (Yn[k]u[k]) = 0 k 2: 0 ~ 0; that 6 10 9 T ime-Domain Analysis o f D iscrete-Time Systems Q[EJh[kJ = P[EJ (~8[kJ) = P[EJ8[k - h[kJ = Ao8[kJ + A18[k - IJ c onditions exist: (i) a t l east one r oot is o utside t he u nit circle, (ii) t here a re
r epeated r oots o n t he u nit circle. IJ I n t his case t he i nput vanishes n ot for k 2: 1, b ut for k 2: 2. T herefore, t he response
c onsists n ot only of t he z ero-input t erm a nd a n i mpulse Ao8[kJ ( at k = 0), b ut also
of an impulse A 18[k - IJ ( at k = 1). Therefore + Yn[kJu[kJ W e c an determine t he u nknowns Ao, A I, a nd t he n coefficients in Yn[kJ from t he
n + 2 n umber o f initial values h , h , . .. , h [n + 1], d etermined as usual from t he
i terative s olution of t he e quation Q[EJh[kJ = P[EJ8[kJ. Similarly, if ao = a l = 0,
we need t o use t he form h[kJ = Ao8[kJ + A 18[k - IJ + A 2 8[k - 2J + Yn[kJu[kJ...
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