The convex geometry of subchannel attenuation coefficients in linearly precoded OFDM systems

Channels with spectral nulls are sometimes dubbed bad channels because they can cause poor performance in communication systems. This article investigates the validity of this intuition by studying the geometry of an orthogonal frequency-division multiplex (OFDM) system. It is shown that the subchannel attenuation coefficients form a natural coordinate system for describing finite-impulse response (FIR) channels in an OFDM framework. It is also shown that channels with spectral nulls are geometrically significant; they form the faces of the convex set of all subchannel attenuation coefficients. This novel perspective makes it immediately clear why the worst performance of a linearly precoded OFDM system is achieved over a channel having the greatest number of spectral nulls. The practical implications of these results are discussed.