# FFT Frequency bin relationship

I guess this is a stupid question or it is not clear. I will try again

I have a situation where I take the fft of a combined source, both at the same frequency but at different locations. I am trying to figure out what the relationship between the source and the combined.

Maybe I am using the wrong terminology here. A combined source in my application is at least two instances of the same frequency from different locations in the focal plane, combined in the same waveform.

Is the phase of the combined complex value the sum of the phase of the source? so one could write something like:

tanCombined = tan(combinedIm / combinedRe) = tan(s1Im/s1Re + s2Im/s2Re + ...);

Second Attempt.

Say I have a waveform waveOne that is a sin wave at a specific frequency that has an origin at a point in the focal plane relative to a single measuring microphone.

If I take the fft of this waveform I can calculate the phase angle and amplitude from the fft data, phaseOne and amplitudeOne.

Say I have another waveform waveTwo that is a sin wave of the same frequency that has its origin at a different point in the focal plane.

If I take the fft of this wave form I can calculate the phase angle and amplitude from the fft, phaseTwo and amplitudeTwo.

If I combine these waveforms into a single wave, waveThree and take its fft the same bin is excited as in the above two samples and both the amplitude and phase angle are different, as one would expect.

The question is, for the combined experiment, is there a relationship between the amplitude and phase of the combined fft value with those of the individual waveform.

• what's a "combined source"? – Marcus Müller Mar 20 '18 at 21:15
• hm, so combined physically means "added", right? You're thinking of physical signals, not only mathematical abstraction, as far as I can tell. Are you dealing with any special medium (especially: nonlinear)? – Marcus Müller Mar 20 '18 at 21:22
• No I just adjust the phase of a sin wave and add them together. I then take the fft and then wondered if I coul;d figure anything out from the fft's of the two individual waves and the fft of the combined wave given I knew the phase of each of the individual waveforms. – Nefarious Mar 20 '18 at 21:29
• Please don’t use code formatting on text that is not code. – Peter K. Mar 21 '18 at 16:21

Yes, simply use a trigonometric addition formula. The results can vary from complete constructive interference to most destructive interference. Depending on the relative phases at your microphone.

Here is the math:

$$W_1 = A_1 \cos( F t + P_1 )$$ $$W_2 = A_2 \cos( F t + P_2 )$$

$$W_1 = A_1 \cos( F t ) \cos( P_1 ) - A_1 \sin( F t ) \sin( P_1 )$$ $$W_2 = A_2 \cos( F t ) \cos( P_2 ) - A_2 \sin( F t ) \sin( P_2 )$$

$$W_3 = W_1 + W_2 = [ A_1 \cos( P_1 ) + A_2 \cos( P_2 ) ] \cos( F t ) - [ A_1 \sin( P_1 ) + A_2 \sin( P_2 ) ] \sin( F t )$$

Now

$$W_3 = A_3 \cos( F t ) \cos( P_3 ) - A_3 \sin( F t ) \sin( P_3 )$$

so

$$A_3 \cos( P_3 ) = A_1 \cos( P_1 ) + A_2 \cos( P_2 ) = C$$ $$A_3 \sin( P_3 ) = A_1 \sin( P_1 ) + A_2 \sin( P_2 ) = S$$

Calculate $C$ and $S$.

$$A_3 = \sqrt{ C^2 + S^2 }$$ $$P_3 = \operatorname{atan2}( S, C )$$

If they are in phase $(P_1 = P_2)$ then $A_3 = A_1 + A_2$.

If they are completely out of phase $( P_3 = P_1 = P_2 + \pi )$ then $A_3 = A_1 - A_2$.

Also, the FFT is a linear operator, so the FFT of your sum is the sum of your FFTs.

Hope this helps.

Ced

• "Also, the FFT is a linear operator, so the FFT of your sum is the sum of your FFTs." This is relative to the phase of the two signals? You don't mean I can add the real and imaginary fft values to get the combined? – Nefarious Mar 21 '18 at 18:03
• That's what it means. It is easy to see if your wave has a whole number of cycles in your sampling frame. Then the bracketed expressions in the W1+W2 equation represent your real and imaginary parts of the bin corresponding to the frequency. There is also a bin on the other side of the DC bin with the complex conjugate values. I recommend that you read my blog articles (link on my profile page). They are mostly dedicated to explaining the Discrete Fourier Transform from a fresh perspective. – Cedron Dawg Mar 21 '18 at 18:11