Parallel Gaussian interference channels are not always separable

It is known that the capacity of parallel (multicarrier) Gaussian point-to-point, multiple access and broadcast channels can be achieved by separate encoding for each subchannel (carrier) subject to a power allocation across carriers. In this paper we show that such a separation does not apply to parallel Gaussian interference channels in general. A counterexample is provided in the form of a 3 user interference channel where separate encoding can only achieve a sum capacity of 2 log(1 + 3 SNR) while the actual capacity, achieved only by joint encoding across carriers, is 3 log(1 + 2 SNR). As a byproduct of our analysis, we propose a class of multiple-access-outer bounds on the capacity of the 3 user interference channel.