Signals and Systems - Electrical Engineering

84 C H A P T E R 1: Continuous-Time Signals which is again a sum of harmonically related frequency sinusoids, so thatx^2 (t)is p ...

1.4 Representation Using Basic Signals 85 The power of a sum of sinusoids, x(t)= ∑ k Akcos(kt)= ∑ k xk(t) (1.15) with harmonica ...

86 C H A P T E R 1: Continuous-Time Signals n If A is real, but a=j 0 , then we have x(t)=Aej^0 t =Acos( 0 t)+jAsin( 0 t) wh ...

1.4 Representation Using Basic Signals 87 Sinusoids Sinusoids are of the general form Acos( 0 t+θ)=Asin( 0 t+θ+π/ 2 ) −∞<t& ...

88 C H A P T E R 1: Continuous-Time Signals 1.4.2 Unit-Step, Unit-Impulse, and Ram Signals Unit-Step and Unit-Impulse Signals Co ...

1.4 Representation Using Basic Signals 89 Ignoring the value att=0 we define theunit-stepsignal as u(t)= { 1 t> 0 0 t< 0 Y ...

90 C H A P T E R 1: Continuous-Time Signals n The impulseδ(t)is impossible to generate physically, but characterizes very brief ...

1.4 Representation Using Basic Signals 91 where the first termx 1 a(t)is continuous and the secondx 1 b(t)is discontinuous. The ...

92 C H A P T E R 1: Continuous-Time Signals The signalx 2 (t)has jump discontinuities att=0,t=1, andt=2, and we can think of it ...

1.4 Representation Using Basic Signals 93 % t: time support % m: slope of ramp % ad : advance (positive), delay (negative) facto ...

94 C H A P T E R 1: Continuous-Time Signals FIGURE 1.8 Generation of y(t)= 3 r(t+ 3 )− 6 r(t+ 1 )+ 3 r(t)− 3 u(t− 3 ),− 5 ≤t≤ 5 ...

1.4 Representation Using Basic Signals 95 − 5 05 − 3 − 2 − 1 0 1 2 3 4 5 (a) (c) t (sec) y( t) − 5 05 − 2 0 2 4 ye (t ) − 5 05 0 ...

96 C H A P T E R 1: Continuous-Time Signals grid axis([min(t) max(t) -1 5]) function [ye,yo] = evenodd(t,y) % even/odd decomposi ...

1.4 Representation Using Basic Signals 97 − 10 − (^50510) − 1 −0.5 0 0.5 1 1.5 2 t x( t) − 2 − 1 0 1 2 3 4 − 8 − 6 − 4 − 2 0 2 4 ...

98 C H A P T E R 1: Continuous-Time Signals FIGURE 1.11 (a) The triangular signal3(t)and (b) its derivative. dt 0 0 (^11) − 1 1 ...

1.4 Representation Using Basic Signals 99 FIGURE 1.12 Eight periods of full-wave rectified signal x(t)=|cos( 2 πt)|, −∞<t< ...

100 C H A P T E R 1: Continuous-Time Signals ρ(t) dρ(t) dt 1 1 1 − 1 223 3 0 0 t t (1) (2) (−2) (−2) · · · · · · FIGURE 1.13 Cau ...

1.4 Representation Using Basic Signals 101 signalx(t)by a train of very narrow pulses of the sampling periodTs. For simplicity, ...

102 C H A P T E R 1: Continuous-Time Signals n A property that is not obvious and that requires the frequency representation ofS ...

1.4 Representation Using Basic Signals 103 Notice that time scaling also changes the frequency content of the signal. For instan ...