# proof that orthogonal signals not interrupting each other

I know that signals that are orthogonal do not disturb each other.

What I am curious is what is the proof behind why orthogonal signals in a single signal (i.e. a single signal can be broken down into signals that have their unique frequency.) do not disturb each other so that Fourier transform can be carried out.

Edit: By signals not disturbing each other, I mean that when Fourier transform into frequency contents is carried out, each signal's frequency content is not disturbed. For example, harmonic signals (waves). two signals, each with frequency $f$ and $2f$ are combined into one signal. When the signal is received, a receiver can figure out the two signals that were combined. Similar with OFDM. According to the text, it says that this is due to its orthogonal nature. I get what orthogonality mean, but I am not sure how being orthogonal leads to non-disturbance of signal components.

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Not getting an answer to such a question after about 5 hours is a good indication that people don't really get what you want. I, as an example, have no idea what exactly you might possibly mean by saying that signals 'do not disturb each other'. – user20266 May 18 '12 at 17:56
@Thomas I edited the question – user2346 May 18 '12 at 23:18
The Fourier transform is a linear operator in the sense that $FT\left(af(x)+bg(x)\right)=a\cdot FT(f(x))+b\cdot FT(g(x))$. This is simply a consequence of the linearity of integral transforms in general: the transform of a superposition is the superposition of the transforms. Since you mention orthogonality, I think you must have something more than this in mind, but I'm not seeing what it is. Can you write down in formulas the property you are trying to understand? – Will Orrick May 19 '12 at 1:21
@WillOrrick For example, harmonic signals. two signals, each with frequency $f$ and $2f$ are combined into one signal. When the signal is received, a receiver can figure out the two signals that were combined. Similar with OFDM. According to the text, it says that this is due to its orthogonal nature. I get what orthogonality mean, but I am not sure how being orthogonal leads to non-disturbance of signal components. – user2346 May 19 '12 at 2:56
You said it yourself (emphasis mine) When the signal is received, a receiver can figure out the two signals that were combined. A receiver will perform an orthogonal projection to the received signal. And we are able to tune the receiver in such a way that the orthogonal projection it performs is the desired one, i.e. it lets the desired signal component through and kills the others. – Jyrki Lahtonen May 19 '12 at 6:44