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

ij 2fkj 2 fk ij 3 fk c yk 3j 2yk 2j

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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

Ask a homework question - tutors are online