Signals and Systems Lecture 3: Sinusoids

2 Today's lecture −Sinusoidal signals −Review of the Sine and Cosine Functions  Examples −Basic Trigonometric Identities −Relation of Frequency to Period −Phase Shift to Time Shift  Example Sampling and Plotting Sinusoids −Complex Exponentials and Phasors −Complex Number Representation −Addition of Complex Numbers  Mathematical Addition  Graphical Addition

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4 Fig. 2-6: x(t) = 20cos(2π(40)t - 0.4π)

5 Sinusoidal signal : x(t) = 10cos(2π(440)t - 0.4π)

6 MATLAB Demo of Tuning Fork −% TuningFork −t = 0:.0001:.01; −y = 10*cos(2*pi*440*t-0.4*pi); −plot(t,y) −grid −pause; −t = 0:.0001:1; −y = 10*cos(2*pi*440*t-0.4*pi); −sound (y)

7 Basic Properties of sine and cosine functions Equivalence sin  = cos(  -  /2) or cos  = sin(  +  /2)y Periodicity cos(  + 2  k) = cos , k = integer Evenness of cosine cos(-  ) = cos  Oddness of sine sin(-  ) = - sin  Zeros of sine sin (  k) = 0, k = integer Ones of cosine cos (2  k) = 1, k = integer Minus ones of cosine cos [2  (k + ½)) = -1, k = integer

8 Some basic trigonometric identities NumberEquation 1 sin 2  + cos 2  = 1 2 cos2  = cos 2  - sin 2  3 sin2  = 2 sin  cos  4sin (α + β) = sinα cosβ + cosα sinβ 5cos (α + β) = cosα cosβ + sinα sinβ

9 Relation of Frequency to Period X(t)=A cos(  0 t+  ) x(t + T 0 ) = x(t) A cos(  0 (t + T 0 ) +  )= A cos(  0 t+  ) cos(  0 t +  0 T 0 +  )= cos(  0 t+  ) Since cosine function has a period of 2π  0 T 0 = 2π 2πf 0 T 0 = 2π T 0 = 1/ f 0

10 Fig 2-7: x(t) = 5cos(2πf o t) for different values of f o

11 Phase Shift and Time Shift x 0 (t - t 1 ) = A cos(  0 (t - t 1 ) = A cos (  0 t +  ) cos(  0 t -  0 t 1 )= cos(  0 t +  ) t 1 = -  /  0 = -  / 2πf 0 Phase Shift is negative when time-shift is positive  = - 2πf 0 t 1 = - 2πt 1 /T 0

12 Phase Shift and Time Shift

13 Phase Shift is Ambiguous

14 −X(t) =Acos(wt +Φ)

15 Sinusoid from a Plot

16 Represent following graph in form of X(t) =Acos(wt +Φ)

17 −A=6 −T =6 −f=1/6 −tm=2; −Φ=-wtm −Φ=-2*pi*f*tm −-2pi/3; − X(t)=6cos(pi/3 -2pi/3)

18 Sampling and Plotting Sinusoids

19 Effect of Sampling Period

20 Sample Spacing

21 Complex Numbers

22 Plot Complex Numbers

23 Complex Addition = Vector Addition

24 Polar Form

25 Polar versus Rectangular

26 Practice

27 Solution

28 Complex Conjugation