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1、Chapter 5 The S-Domain Transform For Continuous Signals and Systems5.1 The Laplace Transform 5.1.1 Definef(t)F(j)f(t)e-t F(+j)-+|f(t)|e-t dt-+|f(t)|dt5.1.2 Region of Convergence(ROC)-+|f(t)|e-t dt-c.f(t)=et(t)-et(-t)ROC:If ROC include j axisF(j)=F(s)|s=j Else F(j)not existb.f(t)=-e-t(-t)ROC:-(t)1/s
2、0e-at(t)1/(s+a)-aeat(t)1/(s-a)aRule?cos0t s/(s2+02)0sin0t 0/(s2+02)05.2 Properties of The Unilateral Laplace Transformf(t)=F(s)-1F(s)=f(t)ROC=R (0)5.2.1 LinearityResult:F(s)=(e-2(s+2)-e-4(s+2)/(s+2)ROC:-2Ex5.2.2:f(t)=e-2t(t-2)-(t-4)F(s)=?5.2.2 Time ShiftingEx5.2.1:Proof:Ex5.2.3:Proof:5.2.3 Shifting
3、in the s-DomainEx5.2.4:f(t)=e-tcos(0t)(t)F(s)=?Result:F(s)=(s+)/(s+)2+02)5.2.4 Time ScalingResult:155.2.5 Conjugation5.2.6 Convolution PropertyEx5.2.6:h(t)=e-t(t)f(t)=(t)yf(t)=?5.2.7 Differentiation in the Time Domain Ex5.2.7:f(t)=(t)-(t-2)F(s)=?Result:yf(t)=(1-e-t)(t)Result:F(s)=(1-e-2s)/sROC:05.2.
4、8 Differentiation in the s-DomainEx5.2.8:f(t)=te-t(t)F(s)=?Result:F(s)=1/(s+)25.2.9 Integration in the Time DomainEx5.2.9:f(t)=t(t)F(s)=?Result:F(s)=1/s25.2.10 The Initial-and Final-Value Theorems if ROC include s=0 if lims-+sF(s)existEx5.2.10:F(s)=1/(s+1)f(0+)=?f(+)=?Result:f(0+)=1 f(+)=05.3 The Un
5、ilateral Laplace Inverse TransformF(s)f(t)1(t)s(1)(t)1/s(t)1/(s+a)e-at(t)1/(s+a)2te-at(t)s/(s2+02)cos(0t)(t)0/(s2+02)sin(0t)(t)m-1 Causal?Result:Causal Rocxx h(t)=(k1e-t+k2e-2t)(t)Ex5.8.3:Res-1 Causal?Result:h(t)=(k1e-(t+1)+k2e-2(t+1)(t+1)h(-1)=(k1+k2)0 No Causal 1.h(t)=0 t05.8.3 Stability H(s)LTI1.
6、if|f(t)|+then|y(t)|2 2)2 Res-1 Stable?Result:xxxx1)Res2 No Stable h(t)=(k1e-t+k2e2t)(t)2)2 Res-1 Stable h(t)=k1e-t(t)+k2e2t(-t)Ex5.8.5:Result:1)The system is causal(LTI)2)H(S)rational,Only 2 poles S1=-2,S2=43)If f(t)=1 then y(t)=04)h(0+)=4H(s)=?y(t)=estH(s)f(t)=1=e0t0=y(t)=e0tH(0)H(0)=0p(s)=sq(s)q(s)=4Ex5.8.6:Result:The system is causal and linearIf stable then k=?F(s)+-Y(s)K2&K2(K-2)02K-30Ex5.8.7:If stable then k=?Routh Array:Result:1331+K(8-k)/31+K(8-k)/301+K08K-118