In this paper we discuss the use of cooperative game theory for analyzing interference channels. We extend our previous work, to games with N players as well as frequency selective channels and joint TDM/FDM strategies. We show that the Nash bargaining solution can be computed using convex optimization techniques. We also show that the same results are applicable to interference channels where only statistical knowledge of the channel is available. Moreover, for the special case of two players $2\times K$ frequency selective channel (with K frequency bins) we provide an $O(K \log_2 K)$ complexity algorithm for computing the Nash bargaining solution under mask constraint and using joint FDM/TDM strategies. Simulation results are also provided.
Computing the capacity region of the interference channel is an open problem in information theory [2]. A good overview of the results until 1985 is given by van der Meulen [3] and the references therein.
The capacity region of general interference case is not known yet. However, in the last forty five years of research some progress has been made. Ahslswede [4], derived a general formula for the capacity region of a discrete memoryless Interference Channel (IC) using a limiting expression which is computationally infeasible. Cheng, and Verdu [5] proved that the limiting expression cannot be written in general by a single-letter formula and the restriction to Gaussian inputs provides only an inner bound to the capacity region of the IC. The best known achievable region for the general interference channel is due to Han and Kobayashi [6]. However the computation of the Han and Kobayashi formula for a general discrete memoryless channel is in general too complex. Sason [7] describes certain improvement over the Han Kobayashi rate region in certain cases. A 2x2 Gaussian interference channel in standard form (after suitable School of Engineering, Bar-Ilan University, Ramat-Gan, 52900, Israel. Part of this work has been presented at ISIT 2006 [1]. This work was supported by Intel Corporation. e-mail: leshema@eng.biu.ac.il . normalization) is given by:
where, s = [s 1 , s 2 ] T , and x = [x 1 , x 2 ] T are sampled values of the input and output signals, respectively.
The noise vector n represents the additive Gaussian noises with zero mean and unit variance. The powers of the input signals are constrained to be less than P 1 , P 2 respectively. The off-diagonal elements of H, α, β represent the degree of interference present. The capacity region of the Gaussain interference channel with very strong interference (i.e., α ≥ 1 + P 1 , β ≥ 1 + P 2 ) was found by Carleial given by
This surprising result shows that very strong interference dose not reduce the capacity. A Gaussian interference channel is said to have strong interference if min{α, β} > 1. Sato [8] derived an achievable capacity region (inner bound) of Gaussian interference channel as intersection of two multiple access gaussian capacity regions embedded in the interference channel. The achievable region is the intersection of the rate pair of the rectangular region of the very strong interference (2) and the region
A recent progress for the case of Gaussian interference is described by Sason [7]. Sason derived an achievable rate region based on a modified time-(or frequency-) division multiplexing approach which was originated by Sato for the degraded Gaussian IC. The achievable rate region includes the rate region which is achieved by time/frequency division multiplexing (TDM/ FDM), and it also includes the rate region which is obtained by time sharing between the two rate pairs where one of the transmitters sends its data reliably at the maximal possible rate (i.e., the maximum rate it can achieve in the absence of interference), and the other transmitter decreases its data rate to the point where both receivers can reliably decode their messages.
While the two users fixed channel interference channel is a well studied problem, much less is known in the frequency selective case. An N × N frequency selective Gaussian interference channel is given by:
where, s k , and x k are sampled values of the input and output signal vectors at frequency k, respectively.
The noise vector n k represents the additive Gaussian noises with zero mean and unit variance. The power spectral density (PSD) of the input signals are constrained to be less than p 1 (k), p 2 (k) respectively. The offdiagonal elements of H k , represent the degree of interference present at frequency k. The main difference between interference channel and a multiple access channel (MAC) is that in the interference channel, each component of s k is coded independently, and each receiver has access to a single element of x k .
Therefore iterative decoding schemes are much more limited, and typically impractical.
One of the simplest ways to deal with interference channel is through orthogonal signaling. Two extremely simple orthogonal schemes are using FDM or TDM strategies. For frequency selective channels (also known as ISI channels) we can combine both strategies by allowing time varying allocation of the frequency bins to the different users. In this paper we limit ourselves to joint FDM and TDM scheme where an assignment of disjoint portions of the frequency band to the several transmitters is made at each time instance. This technique is widely used in practice because simple filtering can be used at the receivers to eliminate interference. In this paper we will assume a PSD mask limitation (peak power at each frequency) since this constraint is typically enforced by regulators.
While information theoretical considerations allow all points in the rate region, we argue that the interfe
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