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Explanation of Fourier Transform for cos + cos

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sonaiko

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FFT for x(t)=cos(2 pi f0 t) is simply a delta function at frequency of f0, with a certain amplitude (isnt this amplitude should equal 1/2?)

Does that mean FFT for y(t)=cos(2 pi 2f0 t) is a delta function at frequency of 2f0, with the amplitude half of the previous one? (Based on the rule of f(a)=>F(f/a)/a.

Based on that, since FFT is linear, then FFT for x(t)+y(t) should equal X(f)+Y(f), which means two delta functions one at freq f0 with amp c, and another at freq 2f0 with amp c/2.

is what I said above true?
 

TheArcane

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FFT for cos+cos

Hi,

First of all, to clarify, we are talking about the Fourier Transform, not FFT. FFT is a discrete-time transform that may yield entirely different results based on the frequency of the cosine function and the sampling frequency.

Now, regarding the FT of a cosine function, it is two delta functions, one at f0 and one at -f0, multiplied by 0.5 each. For cos(2*pi*2f0*t) the delta functions will be at 2f0 and -2f0 and they will still be multiplied by 0.5. The scale rule you pointed out is correct, but for integrable signals only. The cosine function is not integrable and the FT for it is generalized.

Finally, regarding linearity, you are correct. The FT of a sum of functions equals the sum of the individual FTs.
 

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