Unformatted text preview: . T he
n + 3 u nknown constants are d etermined from t he n + 3 values h[O], h [l], . .. , h[n +
1], h[n + 2], d etermined iteratively, a nd so on. 9 .8 Summary
T his c hapter discusses time-domain analysis of LTID (linear, time-invariant,
discrete-time) systems. T he a nalysis is parallel t o t hat o f LTIC systems, w ith some
m inor differences. Discrete-time systems are described by difference equations. For
a n n th-order s ystem, n a uxiliary conditions must b e specified for a unique solution
t o a n i nput s tarting a t k = O. C haracteristic modes are discrete-time exponentials
of t he form " (k c orresponding t o a n u nrepeated r oot "(, a nd t he m odes a re of t he
f orm k'''(k c orresponding t o a r epeated r oot "(.
T he u nit impulse function 8[kJ is a sequence o f a single n umber o f u nit value
a t k = O. T he u nit impulse response h[kJ o f a d iscrete-time s ystem is a linear
c ombination o f i ts characteristic modes. t
T he z ero-state response (response d ue to e xternal i nput) o f a linear system
is o btained b y breaking t he i nput i nto impulse components a nd t hen a dding t he
s ystem responses t o all t he i mpulse components. T he s um o f t he s ystem responses
t o t he i mpulse components is in t he form o f a s um, k nown as t he c onvolution sum,
whose s tructure a nd p roperties are similar t o t he c onvolution integral. T he s ystem
r esponse is obtained as t he c onvolution s um o f t he i nput f[kJ w ith t he s ystem's
i mpulse r esponse h[kJ. T herefore, t he knowledge o f t he s ystem's i mpulse response
allows us t o d etermine t he s ystem response to a ny a rbitrary i nput.
LTID systems have a very special relationship t o t he e verlasting exponential
signal zk b ecause t he response of a n L TID system t o s uch a n i nput s ignal is t he
s ame s ignal within a multiplicative constant. T he r esponse o f a n L TID system t o
t he e verlasting exponential i nput zk is H[zJzk, w here H[zJ is t he t ransfer function
of t he s ystem. 611 P roblems 3. A n L TID s ystem is marginally s table i f a nd o nly if t here a re n o r oots o utside
t he u nit circle, a nd t here a re s ome u nrepeated r oots o n t he u nit circle.
According t o a n a lternate d efinition o f s tability-the b ounded-input b oundedoutput ( BIBO) s tability-a s ystem is s table if a nd only if every b ounded i nput
p roduces a b ounded o utput. O therwise t he s ystem is unstable. A n a symptotically
stable s ystem is always BIBO-stable. T he converse is n ot n ecessarily t rue.
Difference e quations o f LTID systems c an also b e solved by t he classical m ethod,
where t he r esponse is o btained a s a s um o f n atural a nd forced responses. These are
n ot t he s ame as t he z ero-input a nd z ero-state components, a lthough t hey s atisfy t he
s ame equations, respectively. Although simple, t his m ethod suffers from t he fact
t hat i t is applicable t o a r estricted c lass of i nput signals, a nd t he s ystem respons...
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