Signals and Systems - Electrical Engineering

124 C H A P T E R 2: Continuous-Time Systems according to Faraday’s induction law. Solving this differential equation for the cu ...

2.3 LTI Continuous-Time Systems 125 be a line through the origin, its slopeAis very large. If|vd(t)|> 1Vthe output voltage is ...

126 C H A P T E R 2: Continuous-Time Systems n In many cases time invariance can be determined by identifying—if possible—the in ...

2.3 LTI Continuous-Time Systems 127 The FM system is nonlinear. Suppose that we scale the message toγm(t), for some constantγ, t ...

128 C H A P T E R 2: Continuous-Time Systems Nose Lips Nasal cavity Tongue Vellum Epiglottis Vocal chords Lungs (a) e LTI system ...

2.3 LTI Continuous-Time Systems 129 nExample 2.5 Consider constant linear capacitors and inductors, represented by differential ...

130 C H A P T E R 2: Continuous-Time Systems FIGURE 2.5 RLC circuit. v(t) i(t) + − t= (^0) RC L that theR,L, andCvalues are cons ...

2.3 LTI Continuous-Time Systems 131 On the other hand, if the initial conditions are different from zero, when checking linearit ...

132 C H A P T E R 2: Continuous-Time Systems complete response is their sum, y(t)=yzi(t)+yzs(t) Indeed,yzi(t)andyzs(t)satisfy (a ...

2.3 LTI Continuous-Time Systems 133 The solution of this differential equation is given by i(t)=[I 0 e−t+B( 1 −e−t)]u(t) (2.15) ...

134 C H A P T E R 2: Continuous-Time Systems FIGURE 2.6 Nonlinear behavior of RL circuit: (top)I 0 = 1 , B= 1 , v(t)=u(t),i 1 (t ...

2.3 LTI Continuous-Time Systems 135 Analog mechanical systems Making the analogy shown in Table 2.1 between the different variab ...

136 C H A P T E R 2: Continuous-Time Systems IfSis the transformation corresponding to an LTI system, so that the response of th ...

2.3 LTI Continuous-Time Systems 137 FIGURE 2.8 Response of an RL circuit to a pulse v(t)=u(t)−u(t− 2 ) using superposition and t ...

138 C H A P T E R 2: Continuous-Time Systems Next we define the impulse response of an LTI and find the response due tox(t). The ...

2.3 LTI Continuous-Time Systems 139 Remarks n We will see that the impulse response is fundamental in the characterization of li ...

140 C H A P T E R 2: Continuous-Time Systems The relation between the impulse response and the unit-step and the ramp responses ...

2.3 LTI Continuous-Time Systems 141 which corresponds to the accumulation of values ofx(t)in a segment [t−T,t] divided by its le ...

142 C H A P T E R 2: Continuous-Time Systems Ift−T<0 andt≥0, the above integral becomes ρ(t)= 1 T ∫t 0 σdσ= t^2 2 T 0 ≤t<T ...

2.3 LTI Continuous-Time Systems 143 where we used the sifting property of the impulse and that its area is unity. If we letT=T 0 ...