signals and systems lecture 3: sinusoids. 2 today's lecture −sinusoidal signals −review of...
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Signals and Systems
Lecture 3: Sinusoids
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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
ExampleSampling 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 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 (2k) = 1, k = integer
Minus ones of cosine
cos [2(k + ½)) = -1, k = integer
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Some basic trigonometric identitiesNumber Equation
1 sin2 + cos2 = 1
2 cos2 = cos2 - sin2
3 sin2 = 2 sin cos
4 sin (α + β) = sinα cosβ + cosα sinβ
5 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