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Which is the variable for amplitude in fourier transform equation?

Posted on 2013-02-04
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Last Modified: 2013-11-25
In this Sinusoidual signal equation: Equation

I know that A_0 is to shift the signal up or down, and the A_n is for the amplitude. What I don't understand however is when this equation is transformed into Fourier transform equation, the variable for the amplitude goes missing?

This is the fourier transform equation: Fourier Transform Equation
Both a_n and b_n are variables for the phase. What about the variable for the amplitude?
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Question by:xenonn
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2 Comments
 
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Accepted Solution

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aburr earned 250 total points
ID: 38851892
The a_n and b_n are the amplitudes of the sin and cos part of the final answer.
They combine to give the amplitude (and phase) of the periodic wave
a cos (wt) + b sine(wt) = c sin(wt + p)
where
c = (a^2 + b^2)^0.5            p = arctan(b/a)         b = c cos(p)      a = c sin(p)
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Assisted Solution

by:yuk99
yuk99 earned 250 total points
ID: 38852216
The answer provided by aburr is exactly right. I just get late a bit. :)

Since you put your question also to MATLAB zone let me give you some code to illustrate the function behavior depending on a_n and b_n;

x = linspace(-pi*2,pi*2,100);
clf
n = 3;
hold on
for a = 2:n
    plot(x,a*sin(x)+cos(x),'color','r')
end
for a = 2:n
    plot(x,sin(x)+a*cos(x),'color','b')
end
for a = 1:n
    plot(x,a*sin(x)+a*cos(x),'color','g')
end
hold off

Open in new window


On the resulting plot you will see red lines for a>b, blue - for a<b and green - for a==b. You can see that if a==b, the greater the coefficients, the greater the amplitude. And the greater the difference, the greater the phase shift.
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