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 identities

Number 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