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The degrees of freedom (DoF) regions are characterized for the multiple-input multiple-output (MIMO) broadcast channel (BC), interference channels (IC) (including X and multi-hop interference channels) and the cognitive radio channel (CRC), when there is perfect and no channel state information at the receivers and the transmitter(s) (CSIR and CSIT), respectively. For the K-user MIMO BC, the exact characterization of the DoF region is obtained, which shows that a simple time-division-based transmission scheme is DoF-region optimal. Using the techniques developed for the MIMO BC, the corresponding problems for the two-user MIMO IC and the two-user MIMO CRC with an arbitrary number of antennas at each of the four terminals are addressed. For both of these channels, inner and outer bounds to the DoF region are obtained and are seen to coincide for a vast majority of the relative numbers of antennas at the four terminals, thereby characterizing DoF regions for all but a few cases.Finally, the DoF regions of the K-user MIMO IC, the CRC, and X networks are derived for certain classes of these networks, including the one where all transmitters have an equal number of antennas and so do all receivers. The results of this paper are derived for distributions of fading channel matrices and additive noises that are more general than those considered in other simultaneous related works. The DoF regions with and without CSIT are compared and conditions on the relative numbers of antennas at the terminals under which a lack of CSIT does, or does not, result in the loss of DoF are identified, thereby providing, on the one hand, simple and robust communication schemes that don't require CSIT but have the same DoF performance as their previously found CSIT counterparts, and on the other hand, identifying situations where CSI feedback to transmitters would provide gains that are significant enough that even the DoF performance could be improved. ).The material in this paper was presented in part at the IEEE Intl. Symp. of Inform. Th., Austin, TX, Jun. 2010. 1 Broadcast channel, cognitive radio channel, degrees of freedom region, interference channel, multihop interference network, multiple-input multiple-output systems, X network. I. INTRODUCTIONM Ultiple-input multiple-output (MIMO) systems are of great interest because they can provide a significantly higher capacity as compared to their single-input single-output (SISO) counterparts by exploiting the spatial dimension. One way of measuring this benefit at high signal-to-noise ratio (SNR) is via the spatial multiplexing gain or the degrees of freedom (DoF), which is defined as the limit of the ratio of the capacity to the logarithm of the SNR. For example, the point-to-point MIMO channel with M transmit and N receive antennas has min(M, N) DoF whereas its SISO counterpart has only 1 DoF [1]. Interestingly, min(M, N) DoF are achievable over the MIMO channel even if there is perfect channel state information (CSI) just at the receiver (CSIR). In other words, the presence or absence...

This paper presents an in-depth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multi-input multi-output (MIMO) systems with no fewer receive than transmit antennas. In spite of much prior work on this subject, we reveal several new and surprising analytical results in terms of the well-known performance metrics of output signal-to-noise ratio (SNR), uncoded error and outage probabilities, diversity-multiplexing (D-M) gain tradeoff, and coding gain. Contrary to the common perception that ZF and MMSE are asymptotically equivalent at high SNR, we show that the output SNR of the MMSE equalizer (conditioned on the channel realization) is ρ mmse = ρ zf + η snr , where ρ zf is the output SNR of the ZF equalizer, and that the gap η snr is statistically independent of ρ zf and is a non-decreasing function of input SNR. Furthermore, as snr → ∞, η snr converges with probability one to a scaled F random variable. It is also shown that at the output of the MMSE equalizer, the interference-to-noise ratio (INR) is tightly upper bounded whereas for the MMSE-V-BLAST architecture, the SNR gain due to ordered detection is even better, and significantly so. KeywordsThis work was supported in part by the National Science Foundation Grant CCF-0423842 and CCF-0434410. Zero forcing, minimum mean squared error, MIMO, error probability, V-BLAST, diversity gain, spatial multiplexing gain, tradeoff, outage capacity, outage probability.

Abstract-A general, asymptotic (high signal-to-noise (SNR)) error analysis is introduced for quadratic receivers in frequency-flat and multipath Rayleigh-fading channels with multiple transmit and receive antennas. Asymptotically tight expressions for the pairwise error probabilities are obtained for coherent, noncoherent, and differentially coherent space-time receivers. Not only is our unified analysis applicable to more general modulation schemes and/or channel models than previously considered, but it also reveals a hitherto unrecognized eigenvalue structure that is common to all of these problems. In addition to providing an easy recipe for computing the asymptotic pairwise error rates, we make some conclusions regarding criteria for the design of signal constellations and codes such as a) the same design criteria apply for both correlated and independent and identically distributed (i.i.d.) fading processes and b) for noncoherent communications, unitary signals are optimal in the sense that they minimize the asymptotic union bound.

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The degrees of freedom (DoF) region of the 2-user multiple-antenna or MIMO (multiple-input, multiple-output) interference channel (IC) is studied under fast fading and the assumption of delayed channel state information (CSI) wherein all terminals know all (or certain) channel matrices perfectly, but with a delay, and each receiver in addition knows its own incoming channels instantaneously. The general MIMO IC is considered with an arbitrary number of antennas at each of the four terminals.Dividing it into several classes depending on the relation between the numbers of antennas at the four terminals, the fundamental DoF regions are characterized under the delayed CSI assumption for all possible values of number of antennas at the four terminals. In particular, an outer bound on the DoF region of the general MIMO IC is derived. This bound is then shown to be tight for all MIMO ICs by developing interference alignment based achievability schemes for each class. A comparison of these DoF regions under the delayed CSI assumption is made with those of the idealistic 'perfect CSI' assumption where perfect and instantaneous CSI is available at all terminals on the one hand and with the DoF regions of the conservative 'no CSI' assumption on the other, where CSI is available at the receivers but not at all at the transmitters.

Abstract-In addition to optimality in symmetric energy, it is also shown that under certain conditions, the optimum DFD achieves the AEE performance of the exponentially complex maximum-likelihood detector for all users simultaneously. None of the results of this paper make the perfect feedback assumption. The implications of our work on power control for multiuser detection are also discussed.

The degrees of freedom (DoF) region of the fast-fading MIMO (multiple-input multiple-output) Gaussian broadcast channel (BC) is studied when there is delayed channel state information at the transmitter (CSIT). In this setting, the channel matrices are assumed to vary independently across time and the transmitter is assumed to know the channel matrices with some arbitrary finite delay. An outerbound to the DoF region of the general K-user MIMO BC (with an arbitrary number of antennas at each terminal) is derived. This outer-bound is then shown to be tight for two classes of MIMO BCs, namely, (a) the two-user MIMO BC with arbitrary number of antennas at all terminals, and (b) for certain three-user MIMO BCs where all three receivers have an equal number of antennas and the transmitter has no more than twice the number of antennas present at each receivers. The achievability results are obtained by developing an interference alignment scheme that optimally accounts for multiple, and possibly distinct, number of antennas at the receivers. Index TermsBroadcast channel, degrees of freedom, delayed CSIT, interference alignment, outer bound.