Frequency Domain Representation of Sinusoids: Continuous Time
Linear Combinations of Sinusoids · 2014. 11. 3. · The Linear Combination of Sine and Cosine...
Transcript of Linear Combinations of Sinusoids · 2014. 11. 3. · The Linear Combination of Sine and Cosine...
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Linear Combinations of Sinusoids
Dr. Shildneck
Fall 2014
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LINEAR COMBINATIONS OF SINUSOIDAL FUNCTIONS
Part I
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The Sum of Two Sinusoids
If you add two sinusoidal functions (wave functions) that are not in phase, the result will be another sinusoid with the same period and a phase shift.
The amplitude of the result will be less than the sum of the amplitudes of the composed functions.
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The linear combination of function,
can be written as a single cosine function with a phase displacement (shift) in the form
Where A = amplitude of the new wave, and
D = the phase displacement.
cos siny a x b x
cos( )y A x D
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D
A
Graphically, D is the shift of the sinusoidal curve , while A is the amplitude of the new curve.
cosy x
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a = 1
b = 2
D
A
Q1. How can we use this information to find D?
Q2. How can we use it to find A? Use the Pythagorean Theorem.
Use arctan(b/a).
On the unit circle, D is an angle, in standard position whose horizontal component is a and vertical component is b (which are the coefficients in the original combination). A (the amplitude) is the distance of the horizontal and vertical components of the combination.
cos 2sin
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Example
Write in terms of a single cosine function.
3cos 4siny
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PROPERTY The Linear Combination of Sine and Cosine functions with equal periods, can be written as a single cosine function with phase displacement.
cos sin cos( )a x b x A x D
Where and . 2 2A a b arctanb
Da
Note: The signs of a and b specify the appropriate quadrant for D. A should be written in exact terms when possible. D can be rounded to 3 decimals.
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SUM AND DIFFERENCES OF PERIODIC FUNCTIONS
Part II
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DERIVE THE COSINE OF A DIFFERENCE
Using the Unit Circle to
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θ = u - v
(cos ,sin )B v v
(cos ,sin )A u u
u v
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'(1,0)B
'(cos ,sin )A
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'(1,0)B
'(cos ,sin )A
θ
θ = u - v
(cos ,sin )B v v
(cos ,sin )A u u
Since , we can write an equivalence relation for the lengths of the segments. ' 'AB A B
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DERIVE THE COSINE OF A SUM
Use the previous identity and even/odd identities to
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DERIVE THE SINE OF A SUM AND THE SINE OF A DIFFERENCE
You can use the previous identities, co-function identities, and even/odd identities to
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sin( ) sin cos sin cos
cos( ) cos cos sin sin
sin( ) sin cos sin cos tan tantan( )
cos( ) cos cos sin sin 1 tan tan
x y x y y x
x y x y x y
x y x y y x x yx y
x y x y x y x y
SUM and DIFFERENCE IDENTITIES
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Example 1
Find the exact value of cos75
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Example 2
Find the exact value of 7
sin12
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Example 3
Find the exact value of if ,
sin( )u v4
sin5
u
in Quadrant 1 and in Quadrant 2. 5
tan12
v
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Example 4
Write as an expression of x. cos(arctan1 arccos )x
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Example 5
Solve on sin sin 14 4
x x
[0,2 )
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ASSIGNMENT
Alternate Text
P. 395 #63-84(m3), 85, 87, 93-108(m3)
Foerster
P. 394 #1-9 (odd), 17, 23, 25