Signals and Systems - Electrical Engineering

144 C H A P T E R 2: Continuous-Time Systems which can be written as y(t)= 1 2 T ∫t t−T x(τ)dτ+ 1 2 T ∫t+T t x(τ)dτ At the prese ...

2.3 LTI Continuous-Time Systems 145 n If the input signalx(t)is causal (i.e.,x(t)=0 fort<0), we can simplify further the abov ...

146 C H A P T E R 2: Continuous-Time Systems FIGURE 2.11 Graphical convolution for a unit-step input into an averager withT= 1. ...

2.3 LTI Continuous-Time Systems 147 Solution In this case,x(t)=h(t)=u(t)−u(t− 1 ). Again we plotx(τ)andh(t−τ)both as functions o ...

148 C H A P T E R 2: Continuous-Time Systems FIGURE 2.13 Block diagrams for connecting two LTI systems with impulse responsesh 1 ...

2.3 LTI Continuous-Time Systems 149 Parallel Connection If we connect inparalleltwo LTI systems with impulse responsesh 1 (t)and ...

150 C H A P T E R 2: Continuous-Time Systems nExample 2.14 Consider the block diagram in Figure 2.14 with input a unit-step sign ...

2.3 LTI Continuous-Time Systems 151 If we then let 1 →0 we have that (recall thatds(t)/dt=h(t)is the relation between the unit-s ...

152 C H A P T E R 2: Continuous-Time Systems The input resistance of this circuit isRin=∞and the output resistance isRout=0 so t ...

2.3 LTI Continuous-Time Systems 153 2.3.9 Bounded-Input Bounded-Output Stability Stability characterizes useful systems. A stabl ...

154 C H A P T E R 2: Continuous-Time Systems is transformed into heat, and so RLC circuits spend the energy given to them. This ...

2.3 LTI Continuous-Time Systems 155 FIGURE 2.17 Echo system with two paths. × × + x(t) Delay τ 1 Delay τ 2 α 2 α 1 y(t) wherex(t ...

156 C H A P T E R 2: Continuous-Time Systems FIGURE 2.18 Positive feedback system: the microphone picks up input signalx(t)and t ...

Problems 157 to the theory of differential equations and shown some features that will come back when we apply transforms. Our n ...

158 C H A P T E R 2: Continuous-Time Systems FIGURE 2.20 Problem 2.2. − 4 − 3 − 2 − 1 0 1 2 3 − 1 −0.5 0 0.5 1 t(sec) x (t ) 2.3 ...

Problems 159 FIGURE 2.21 Problem 2.5: p-n diode and i-v characteristic. − + 0 i(t) i(t) Is v(t) v(t) (a) Consider the voltagev(t ...

160 C H A P T E R 2: Continuous-Time Systems FIGURE 2.23 Problem 2.6. 1F + − + − is(t) vs(t) 1H iL(t) 2.7. Time-varying capacito ...

Problems 161 (c)Letf(t)=u(t)−u(t− 2 )and the inputx(t)=u(t). Find the corresponding outputy(t). Suppose you shift the input so t ...

162 C H A P T E R 2: Continuous-Time Systems 2.14. Steady-state response of averager—MATLAB An analog averager is given by y(t)= ...

Problems 163 2.17. Response to unbounded inputs versus BIBO stability The BIBO stability assumes that the input is always bounde ...