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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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    Say, if you want to find out the $xyz$-coordinates of a point $P$ in 3-space, you can project it to the three axes, and read the coordinates from the projections. This works, because the axes are orthogonal to each other in the sense that if you move $P$ by changing a single one of its coordinates, only one of the projections changes. Similarly, you can figure out the coefficients $a,b,c$ of a signal $r(t)=a\cos \omega t+ b\cos 2\omega t + c \cos 3\omega t$ by calculating the three projections $P_j(r)=\frac{\omega}{\pi}\int_0^{2\pi/\omega}r(t)\cos j\omega t\,dt$ for $j=1,2,3.$2012-05-19

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