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Signals and Systems
Lecture 3:
Sinusoids
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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Fig. 2-6: x(t) = 20cos(2π(40)t - 0.4π)
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Sinusoidal signal : x(t) = 10cos(2π(440)t - 0.4π)
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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)
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Basic Properties of sine and cosine functions
Equivalence
Periodicity
Evenness of
cosine
Oddness of sine
sin = cos( - /2)
or
cos = sin( +/2)y
cos( + 2 k) = cos , k =
integer
cos(-) = cos 
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
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Some basic trigonometric identities
Number
Equation
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sin2 + cos2  = 1
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cos2 = cos2  - sin2
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sin2 = 2 sin  cos 
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sin (α + β) = sinα cosβ + cosα sinβ
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cos (α + β) = cosα cosβ + sinα sinβ
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Relation of Frequency to Period
X(t)=A cos(0t+ )
x(t + T0) = x(t)
A cos(0 (t + T0) +  )= A cos(0t+ )
cos(0 t + 0 T0 +  )= cos(0t+ )
Since cosine function has a period of 2π
0 T0 = 2π
2πf0 T0 = 2π
T0 = 1/ f0
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Fig 2-7: x(t) = 5cos(2πfot) for different values of fo
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Phase Shift and Time Shift
x0 (t - t1) = A cos(0 (t - t1) = A cos (0t + )
cos(0 t -0 t1 )= cos(0t + )
t1 = -/ 0 = -/ 2πf0
Phase Shift is negative when time-shift is
positive
 = - 2πf0 t1 = - 2πt1 /T0
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Phase Shift and Time Shift
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Phase Shift is Ambiguous
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
− X(t) =Acos(wt +Φ)
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Sinusoid from a Plot
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Represent following graph in form of
X(t) =Acos(wt +Φ)
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− 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)
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Sampling and Plotting Sinusoids
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Effect of Sampling Period
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Sample Spacing
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Complex Numbers
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Plot Complex Numbers
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Complex Addition = Vector Addition
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Polar Form
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Polar versus Rectangular
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Practice
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Solution
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Complex Conjugation
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