案例信號與系統(tǒng)chapter9-5the laplace transform

1、Chapter 9The Laplace Transform- Analysis and Characterization of LTI systems Using LT 9.7 Analysis and Characterization of LTI systems Using LT h(t)H(s)x(t)y(t)=x(t)*h(t)X(s)Y(s)=X(s)H(s)Consider a LTI system:system function or transfer function9.7.1 CausilityCausal: h(t)=0 for t0 (left sided)ROC is

2、 the left-half plane to the left of the rightmost pole.Note: the converse of this statement is not necessarily true. And its just right only for H(S) is rational.Inverse case? Examples(1) A causal system(2) An uncausal system(3) An uncausal system9.7.2 Stability A stable systemH(s), ROC include jw-a

3、xis(i.e. Res=0)Note: this means H(jw)=H(s)|s=jw; in another words, an unstable system has no H(jw) existed. A rational causal and stable systemAll poles of H(s) lie in the left-half of the s-plane(i.e. all poles have negative real parts)uncausalstableExample 9.20ImReS-plane-112ImReS-plane-112ImReS-p

4、lane-112ImReS-plane-112Causalunstableanticausalunstablethe system function: Usually, a practical system is causal and stable.LT9.7.3 LTI Systems Characterized by Linear Constant-Coefficient Differential EquationsExample =9.7.4 Examples Relating System Behavior to the System FunctionExample 9.25We ha

5、ve knownThen Example 9.26Determine the system function as the following information:1. causal2. H(s) is rational ,has only 2 poles, at s=-2 and s=-43.If x(t)=1, then y(t)=04.The value of the impulse response at t=0+ is 4.From 1 and 2,we can deduce Solution:From 3,From 4,H(s) has a zero at s=0suppose

6、Example 9.27Consider a stable and causal system with impulse response h(t) and system function H(s).Suppose H(s) is rational,contains a pole at s=-2, and does not have a zero at the origin. The location of all other poles and zeros is unknown.Determine each of the following statements： true, false,o

7、r uncertain.1. FTh(t)e3t converge.False2. False3. th(t) is the impulse response of a causal and stable systemTrue4.dh(t)/dt contain at least one pole in its LT.True5.h(t) has finite duration.False6.H(s)=H(-s).False7.UncertainExamplea real,causal and stable LTI system with H(s) and frequency response

8、Show that Proof:SupposeImpulse response h(t) is real,causalwe can deduce at t=0+Because h(t) is real, Re(w) is even9.7.5 Butterworth FiltersAn N th-order lowpass Butterworth filterN=5Frequency response of Butterworth filtersAs more poles are added (higher-order Butterworth filter), the LPF gets better.Relationships between poles l