Chinmay S. Vaze

The Degree-of-Freedom Regions of MIMO Broadcast, Interference, and Cognitive Radio Channels With No CSIT

The degree-of-freedom (DoF) regions are characterized for the multiple-input multiple-output (MIMO) broadcast channel (BC), interference channels (ICs), including X and multihop ICs, and the cognitive radio channel (CRC), when there is no channel state information at the transmitter(s) (CSIT) and for fading distributions in which transmit directions are statistically indistinguishable. For the K-user MIMO BC, the exact DoF region is obtained, which shows that time division is DoF-region optimal. For the two-user MIMO IC and CRC, inner and outer bounds are obtained that coincide for a vast majority of the relative numbers of antennas at the four terminals. Finally, the DoF of the K-user MIMO IC, the CRC, and X networks are obtained for certain classes of these networks. The results herein are derived for fading distributions and additive noises that are more general than those considered in other simultaneous related works. The DoF with and without CSIT are compared and conditions under which a lack of CSIT does, or does not, result in the loss of DoF are identified, thereby 1) providing robust no-CSIT schemes that have the same DoF as their previously found CSIT counterparts and 2) identifying situations where CSI feedback to transmitters would provide gains that are significant enough that even the DoF could be improved.

The Degrees of Freedom Region and Interference Alignment for the MIMO Interference Channel With Delayed CSIT

The degrees of freedom (DoF) region of the two-user multiple-input multiple-output (MIMO) interference channel (IC) is studied under the assumptions of fast fading and delayed channel state information (CSI) at the transmitters (CSIT). Under our fast fading assumption, the channel matrices vary independently across time, so that delayed CSIT is equivalent to outdated CSIT. In particular, the DoF region under delayed CSIT is established for the general MIMO IC with an arbitrary numbers of antennas at each of the four terminals. Toward this end, a set of outer bounds to the DoF region of the general MIMO IC is derived. These bounds are then shown to be tight by developing DoF-region-optimal interference alignment schemes. A comparison of the DoF region of the MIMO IC under the delayed CSIT assumption with those under the two extremes of instantaneous CSIT and no CSIT assumptions is made. This comparison reveals that there are nonempty classes of MIMO ICs, defined by certain relationships between the numbers of antennas at the four terminals, that correspond to each of the following four scenarios: the no CSIT DoF region is strictly contained by, or is equal to, the delayed CSIT DoF region, which in turn is strictly contained by, or is equal to, the instantaneous CSIT DoF region. It is notable that within the class of MIMO ICs for which the delayed CSIT DoF region is strictly larger than the no CSIT DoF region, there is a subclass for which the interference alignment scheme which uses just delayed CSIT achieves the entire DoF region previously known to be achievable only with instantaneous CSIT.

The degrees of freedom region of the two-user MIMO broadcast channel with delayed CSIT

The degrees of freedom (DoF) region of the K-user MIMO (multiple-input multiple-output) Gaussian broadcast channel (BC) is studied under i.i.d. fading when there is delayed channel state information at the transmitter (CSIT). The general case of the MIMO BC is considered where each terminal has an arbitrary number of antennas. The delayed CSIT assumption is that the transmitter has perfect knowledge of ‘stale’ channel states (i.e., with some delay) but no knowledge of current CSI. An outer-bound to the DoF region is derived. This bound is shown to be tight in the 2-user case via an interference alignment scheme that optimally accounts for multiple and possibly distinct number of antennas at the two receivers.

The Degrees-of-Freedom Region of the MIMO Interference Channel With Shannon Feedback

The two-user multiple-input multiple-output (MIMO) fast-fading interference channel (IC) with an arbitrary number of antennas at each of the four terminals is studied under the settings of Shannon feedback and output feedback, wherein channel matrices and outputs, or just the channel outputs, respectively, are available to the transmitters with a delay. While for most numbers of antennas at the four terminals, the degree-of-freedom (DoF) regions with Shannon feedback are equal to that with just delayed channel state information (CSIT), it is shown that for the rest of the MIMO ICs, the DoF region with Shannon feedback is strictly larger than the DoF region with just delayed CSIT. To realize these DoF gains with Shannon feedback, a new interference alignment scheme is obtained wherein transmitter cooperation made possible by output feedback (in addition to delayed CSIT) is employed to effect a more efficient form of interference alignment than is feasible with previously known schemes that use just delayed CSIT. The DoF region for output-only feedback is also obtained for all but a class of MIMO ICs that satisfy one of two inequalities involving the numbers of antennas. Moreover, the DoF region for Shannon feedback is shown to be applicable to two limited Shannon feedback settings where the transmitters have knowledge only of certain channel matrices and certain outputs with delay.

Dirty Paper Coding for fading channels with partial transmitter side information

The problem of dirty paper coding (DPC) over the fading dirty paper channel (FDPC) Y=H(X+S)+Z, a more general version of Costas channel, is studied for the case in which there is partial and perfect knowledge of the fading process H at the transmitter (CSIT) and the receiver (CSIR), respectively. A key step in this problem is to determine the optimal inflation factor (under Costas choice of auxiliary random variable) when there is only partial CSIT. Towards this end, two iterative numerical algorithms are proposed. Both of these algorithms are seen to yield a good choice for the inflation factor. Finally, the high-SNR (signal-to-noise ratio) behavior of the achievable rate over the FDPC is dealt with. It is proved that FDPC (with t transmit and r receive antennas) achieves the largest possible scaling factor of min(t,r) log SNR even with no CSIT. Furthermore, in the high SNR regime, the optimality of Costas choice of auxiliary random variable is established even when there is partial (or no) CSIT in the special case of FDPC with t les r. Using the high-SNR scaling-law result of the FDPC (mentioned before), it is shown that a DPC-based multi-user transmission strategy, unlike other beamforming-based multi-user strategies, can achieve a single-user sum-rate scaling factor over the multiple-input multiple-output Gaussian Broadcast Channel with partial (or no) CSIT.

A New Outer Bound via Interference Localization and the Degrees of Freedom Regions of MIMO Interference Networks With No CSIT

The two-user multiple-input multiple-output (MIMO) interference and cognitive radio channels are considered, where the <i>i</i>th transmitter and the <i>i</i>th receiver have <i>M</i>_i and <i>N</i>_i antennas, respectively. In particular, the degrees of freedom (DoF) regions of these channels are studied under the assumption of having no channel state information at the transmitters. Making certain assumptions about the distributions of channel matrices of increasing generality respectively, Huang, Zhu and Guo, and the authors of this paper characterized the DoF region of the MIMO interference channel for all values of the four-tuple (<i>M</i>₁, <i>M</i>₂, <i>N</i>₁, <i>N</i>₂), except when min(<i>M</i>₁, <i>N</i>₁) >; <i>N</i>₂ >; <i>M</i>₂ or min(<i>M</i>₂, <i>N</i>₂) >; <i>N</i>₁ >; <i>M</i>₁. More recently, for isotropic fading distributions, Zhu and Guo have solved this latter case by providing a tight outer bound to the DoF region. Here, a simpler and more widely applicable proof of that outer bound is given based on the idea of interference localization. Using the same idea, under Rayleigh fading, the DoF region is then also established for the MIMO cognitive radio channel (when the second transmitter is cognitive) with min(<i>M</i>₁+<i>M</i>₂, <i>N</i>₁) >; <i>N</i>₂ >; <i>M</i>₂-the only class for which the inner and outer bounds previously reported by the authors were not tight-thereby completing the DoF region characterization of this channel under Rayleigh fading for all values of (<i>M</i>₁, <i>M</i>₂, <i>N</i>₁, <i>N</i>₂).

The degrees of freedom of the 2×2×2 interference network with delayed CSIT and with limited Shannon feedback

The 2×2×2 interference network, which is a layered two-hop, two-user interference channel, consists of two transmitters, two relays, and two receivers with both the first hop network between the transmitters and the relays and the second hop network between the relays and the receivers being i.i.d. Rayleigh fading Gaussian interference channels. Two feedback models are studied. In the first one, called the delayed channel state information at the transmitters (delayed CSIT) model, the transmitters are assumed to know the first and second hop channel coefficients with a finite delay but the relays have no side information whatsoever. In the second feedback model, referred to as the limited Shannon feedback model, one of the two relays knows the first and second hop channel coefficients with a finite delay and the received signal of one of the receivers with a finite delay and the other relay and the transmitters have no side information whatsoever. It is shown in this paper that under both these settings, the 2×2×2 interference network has 4/3 degrees of freedom. The result is obtained by developing a broadcast-channel-type upper-bound and two new achievability schemes termed retro-cooperative interference alignment. Generalizations of the results are provided for networks with multiple but fully-connected layers, multiple relays per layer, and M-antenna source and destination terminals with the total number of antennas at the relays being no less than 2M.

Dirty Paper Coding for the MIMO cognitive radio channel with imperfect CSIT

A Dirty Paper Coding (DPC) based transmission scheme for the Gaussian multiple-input multiple-output (MIMO) cognitive radio channel (CRC) is studied when there is imperfect and perfect channel knowledge at the transmitters (CSIT) and the receivers, respectively. In particular, the problem of optimizing the sum-rate of the MIMO CRC over the transmit covariance matrices is dealt with. Such an optimization, under the DPC-based transmission strategy, needs to be performed jointly with an optimization over the inflation factor. To this end, first the problem of determination of inflation factor over the MIMO channel Y = H1X +H2S +Z with imperfect CSIT is investigated. For this problem, two iterative algorithms, which generalize the corresponding algorithms proposed for the channel Y = H(X +S)+Z, are developed. Later, the necessary conditions for maximizing the sum-rate of the MIMO CRC over the transmit covariances for a given choice of inflation factor are derived. Using these necessary conditions and the algorithms for the determination of the inflation factor, an iterative, numerical algorithm for the joint optimization is proposed. Some interesting observations are made from the numerical results obtained from the algorithm. Furthermore, the high-SNR sum-rate scaling factor achievable over the CRC with imperfect CSIT is obtained.

Beamforming and aligned interference neutralization achieve the degrees of freedom region of the 2×2×2 MIMO interference network

The layered two-hop, two-unicast multi-input, multi-output (MIMO) interference network consists of two transmitters, two relays, and two receivers with the first and the second hop networks between transmitters and relays, and between relays and receivers, respectively, both being Gaussian MIMO interference channels. The degrees of freedom (DoF) region is established in the general case in which there are an arbitrary numbers of antennas at each of the six terminals. It is shown that the DoF region coincides with the min-cut outer bound for both time-varying or fixed-channel coefficients. The min-cut bound is shown to be achievable via a concatenated communication scheme that consists of linear vector space joint beamforming (that includes zero forcing and signal alignment) at all terminals as the inner precoding scheme and point-to-point coding and aligned interference neutralization as the outer precoding scheme.We study the layered two-hop, two-unicast multiinput, multi-output (MIMO) interference network, which consists of two transmitters, two relays, and two receivers with the first and the second hop networks between transmitters and relays, and between relays and receivers, respectively, both being Gaussian MIMO interference channels. The DoF region is established in the general case, where there are arbitrary numbers of antennas at all terminals. It is shown that the DoF region coincides with the min-cut outer-bound, and is achievable via a scheme involving beamforming and aligned interference neutralization.

On the generalized degrees of freedom region of the MIMO interference channel with no CSIT

The generalized degrees of freedom region (GDoF) of the multiple-input multiple-output (MIMO) interference channel (IC) is studied under the “no CSIT” assumption under which there is perfect channel state information (CSI) at the receivers and no CSI at the transmitters (CSIT). In the very weak interference regime, where the ratio of channel gains (in dB) of the interfering and direct links, α, is ≤ 0.5, the GDoF regions are characterized for the two classes of the MIMO ICs defined by (a) M<inf>1</inf> = N<inf>1</inf> > M<inf>2</inf> ≥ N<inf>2</inf> and (b) M<inf>1</inf> = N<inf>1</inf> > N<inf>2</inf> > M<inf>2</inf> (where M<inf>i</inf> is the number of antennas at transmitter i and N<inf>i</inf> is the number of antennas at receiver i, i ∈ {1, 2}). In particular, inner-bounds are obtained by developing CSI-independent coding schemes using which it is shown that for each of the two classes a significant portion of the perfect-CSIT GDoF region can be achieved even without CSIT. Furthermore, tight outer-bounds to the no-CSIT GDoF regions are obtained that simultaneously account for the interference encountered by both the receivers. These bounds are thus fundamentally different from those derived in earlier works which deal with the case of α = 1, i.e., the degrees of freedom (DoF) regions. Interestingly, it is found that the loss of DoFs due to lack of CSIT is much less pronounced for the α ≤ 1 over 2 than it is for α = 1.