BIBO, Impulse R es p r i se arid Frequmcy Response Curve 385 ~ A discrete LTI-systcui is stable if and orily if its inipulsc response is itrid calculate the value The result is only fi
Trang 116.1 BIBO, Impulse R es p r i se arid Frequmcy Response Curve 385
~
A discrete LTI-systcui is stable if and orily if its inipulsc response is
itrid calculate the value
The result is only finite when thc inipulse response can be absolutely intcgrated, othcrwisc l g ( t ) I WOLIM riot Le I)onnckd
As we discovered in Chapter 9.2.2, if the impulsc response can be abwluLely iritegratetl, the Fouricr transform H ( j w ) - .F{h(t)}, ~ the frequcricy response of the stable systerri rmist exi Because the Foiirier integral is 1)oiindrd (SCT (9.5)),
it c m be analytically corrti d, arid is thv saim as thc L a p l a ~ e transform on the imaginary axis s = jw That mcaris that for slable systems, tlrp imagiiiarv axis
of the s-plane is part of the rcgiori of c o i i v q p i c e of the system fuwtion '12-ic
wnse of' a, stable system cannot h a w ariy siiigiilwrities or discontinu- itirs
iscrete Systems
For discrete LTI-systems:
(16.7) I
T h e prod is exactly as for continuou5 systems
the Foiiricr translorrri I I ( d 2 ) = F*{h[kj} of tlie iinpiiBe response h [ k ] agrees m i t h
the transfer function H ( z ) = Z { h [ k ] } on the iiiiit circlc of the :-planr Correspond- ingly, the fi cqucncy rtsporw o f R stnblc svstein m i s t riol h a w m y disc oritirriiities
or ringularitics,
TL can be slion7ii in the same m7ay liom thci cxistenre ot thc frequel~c.y rcspot
JVe will clarifv the use of the stahilitv criterin (1 S.3) tor contiriuous systems with
a k>w exsinplei Stdbility is showri in tlic s a n e way for discrete 5ybltims, using (16.7)
Example 16.1
Fhr the simplest clxaniplc we consider a, system with irqiilse rcspoiise
h ( t ) = ~ - " ~ ; ( f ) rx E IR (16.8)