How are eigen-functions involved in OFDM and why are they important ?

Question

On theory of OFDM operation,it is said that the sinusoidal waveforms making up the tones in OFDM have the very special property of being the only EIGEN-FUNCTIONS of a linear channel.

Answer 1

If you model the air channel as a linear transformation matrix, then the eigenfunctions of that matrix are the sinusoidal tones that can pass the channel with only a multipath set of scale factors but without spectral distortions. The sinusoidal tones are eigenfunction and the scale factors are eigenvalues of the linear channel. Each of the eigenvalues represents one path through the multipath channel that the linear transformation matrix approximates.

In simple words, when you send a sinusoidal wave (with is an eigenfunction of the air channel) of a given frequency though the channel you receive a sinusoidal of the same frequency. The channel creates no new frequencies, it only delays and scales the amplitude of the input sinusoidal tone. The effects of passing through the multidimensional channel is thus simplified to the applying of just a scale factor to the sinusoidal tone. In contrast, for example, when you send a triangular wave you will not get only a scaled version of the triangular wave of the same frequency because the triangular wave is not an eigenfunction of the channel.

Therefore we should limit the component tones of an OFDM waveform to sinusoidal waves, so that they (being eigenfunctions of the air channel) can pass the channel without intermodulation and hence the orthogonality between each and every pair of frequencies is preserved by the air channel.