The wideband sum slope of K-User interference channels

In this work we consider a K-user interference channel in the low power regime. It is well known that the minimum received energy per bit of -1.59dB is achieved in the low power regime, that is when the spectral efficiency R converges to zero, which can equivalently be obtained by letting the bandwidth B go to infinity, or by letting the time to transmit a bit go towards infinity (with fixed and finite bandwidth). For an interference channel, the minimum received energy per bit is also -1.59dB; thus interference does not affect performance. This is only true in the limit, though. When the spectral efficiency is small but finite, performance is affected. One way this can be quantified is through the wideband slope developed by Verdu. For example, if the interference channel uses TDMA, the minimum energy per bit of -1.59dB is achieved, but the wideband slope per user is 2/K, i.e., K times the bandwidth of a single user system is required. We show that if the bandwidth tends to infinity, a wideband slope of at least 1 per user is achievable with probability 1 by using a type of interference alignment, i.e., the bandwidth requirement is only twice that of a single user system. On the other hand, if the transmission time tends to infinity (but bandwidth is fixed), we show (at present for K = 3) that the wideband slope per user is 2/K+ε with non-zero probability. Thus, for the interference channel it does matter how the low power limit is approached. While the wideband slope for the finite bandwidth model can be as poor as TDMA, we show that for most cases it is possible to obtain a wideband slope larger than TDMA.