Signals and Systems - Electrical Engineering

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CHAPTER 1 Continuous-Time Signals......................................................................... A journey of a thousa ...

66 C H A P T E R 1: Continuous-Time Signals One more realization could be that the processing of signals requires us to consider ...

1.3 Continuous-Time Signals 67 1.3 CONTINUOUS-TIME SIGNALS That signals are functions of time-carrying information is easily ill ...

68 C H A P T E R 1: Continuous-Time Signals As a result of the sampling, we obtain the discrete-time signal v(nTs)=v(t)|t=nTs 0 ...

1.3 Continuous-Time Signals 69 0 2 4 6 8 10 − 1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (a) t(sec) v( t) 1.45 1.5 1.55 −0.5 −0.4 ...

70 C H A P T E R 1: Continuous-Time Signals nExample 1.1 Characterize the sinusoidal signal x(t)= √ 2 cos(πt/ 2 +π/ 4 ) −∞<t& ...

1.3 Continuous-Time Signals 71 n Complex, composed of two sinusoids of frequency=π/2 rad/sec, phaseπ/4 in rad, and amplitude √ ...

72 C H A P T E R 1: Continuous-Time Signals FIGURE 1.3 Diagrams of basic signal operations: (a) adder, (b) constant multiplier, ...

1.3 Continuous-Time Signals 73 FIGURE 1.4 (a) Continuous-time signal, and its (b) delayed, (c) advanced, and (d) reflected versi ...

74 C H A P T E R 1: Continuous-Time Signals Likewise, we have that x(t+ 2 )= { 1 0≤t+ 2 ≤1 or− 2 ≤t≤− 1 0 otherwise The signalx( ...

1.3 Continuous-Time Signals 75 1.3.2 Even and Odd Signals Symmetry with respect to the origin differentiates signals and will be ...

76 C H A P T E R 1: Continuous-Time Signals Solution The reflection ofx(t)isx(−t)=cos(− 2 πt+θ). Then: x(t)is even ifx(t)=x(−t) ...

1.3 Continuous-Time Signals 77 Ifx( 0 )=2, we have xe(t)=0.5[x(t)+x(−t)] =    cos( 4 t) t> 0 cos( 4 t) t< 0 2 t= 0 whil ...

78 C H A P T E R 1: Continuous-Time Signals Solution The analog frequency is 0 = 2 π/T 0 soT 0 = 2 π/ 0 is the period. Wheneve ...

1.3 Continuous-Time Signals 79 whereNandMare positive integers not divisible by each other so thatMT 1 =NT 0 becomes the period ...

80 C H A P T E R 1: Continuous-Time Signals wherei(t)andv(t)are the current and voltage in the resistor. Theenergyin the resisto ...

1.3 Continuous-Time Signals 81 n An analog signal x(t)is said to beabsolutely integrableif x(t)satisfies the condition ∫∞ −∞ |x( ...

82 C H A P T E R 1: Continuous-Time Signals we have that Px= 1 8 ∫^4 0 cos(πt+π/ 2 )dt+ 1 8 ∫^4 0 dt= 0 +0.5=0.5 The first integ ...

1.3 Continuous-Time Signals 83 Although this signal has infinite energy, it has finite power. LettingT=NT 0 whereT 0 is the peri ...