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Posted on 2013-02-04

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?

I know that

This is the fourier transform equation: Fourier Transform Equation

Both

2 Comments

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)

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
```

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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