Sanjay Karmakar

Multigroup Decodable STBCs From Clifford Algebras

A space-time block code (STBC) in <i>K</i> symbols (variables) is called a <i>g</i>-group decodable STBC if its maximum-likelihood (ML) decoding metric can be written as a sum of <i>g</i> terms, for some positive integer <i>g</i> greater than one, such that each term is a function of a subset of the <i>K</i> variables and each variable appears in only one term. In this paper, we provide a general structure of the weight matrices of multigroup decodable codes using Clifford algebras. Without assuming that the number of variables in each group is the same, a method of explicitly constructing the weight matrices of full-diversity, delay-optimal multigroup decodable codes is presented for arbitrary number of antennas. For the special case of <i>2</i> ^a number of transmit antennas, we construct two subclass of codes: 1) a class of <i>2</i> ^a -group decodable codes with rate <i>[(a</i>)/(2⁽ <i>a</i>-1))], which is, equivalently, a class of single-symbol decodable codes, and 2) a class of <i>(2a</i>-2)-group decodable codes with rate <i>[((a</i>-1))/(2⁽ <i>a</i>-2))], i.e., a class of double-symbol decodable codes.

The Capacity Region of the MIMO Interference Channel and Its Reciprocity to Within a Constant Gap

The capacity region of the two-user multi-input multioutput (MIMO) Gaussian interference channel (IC) is characterized to within a constant gap that is independent of the channel matrices for the general case of the MIMO IC with an arbitrary number of antennas at each node. An achievable rate region and an outer bound to the capacity region of a class of interference channels were obtained in previous work by Telatar and Tse as unions over all possible input distributions. In contrast to that previous work on the MIMO IC, a simple and an explicit achievable coding scheme are obtained here and shown to have the constantgap-to-capacity property and in which the sub-rates of the common and private messages of each user are explicitly specified for each achievable rate pair. The constant-gap-to-capacity results are thus proved in this work by first establishing explicit upper and lower bounds to the capacity region. A reciprocity result is also proved which is that the capacity of the reciprocal MIMO IC is within a constant gap of the capacity region of the forward MIMO IC.

Minimum-Decoding-Complexity, Maximum-rate Space-Time Block Codes from Clifford Algebras

It is well known that Alamouti code and, in general, space-time block codes (STBCs) from complex orthogonal designs (CODs) are single-symbol decodable/symbol-by-symbol decodable (SSD) and are obtainable from unitary matrix representations of Clifford algebras. However, SSD codes are obtainable from designs that are not CODs. Recently, two such classes of SSD codes have been studied: (i) coordinate interleaved orthogonal designs (CIODs) and (ii) minimum-decoding-complexity (MDC) STBCs from quasi-ODs (QODs). In this paper, we obtain SSD codes with unitary weight matrices (but not CODs) from matrix representations of Clifford algebras. Moreover, we derive an upper bound on the rate of SSD codes with unitary weight matrices and show that our codes meet this bound. Also, we present conditions on the signal sets which ensure full-diversity and give expressions for the coding gain

Multigroup-Decodable STBCs from Clifford Algebras

A space-time block code (STBC) in K symbols (variables) is called g-group decodable STBC if its maximum-likelihood decoding metric can be written as a sum of g terms such that each term is a function of a subset of the K variables and each variable appears in only one term. In this paper we provide a general structure of the weight matrices of multi-group decodable codes using Clifford algebras. Without assuming that the number of variables in each group to be the same, a method of explicitly constructing the weight matrices of full-diversity, delay-optimal g-group decodable codes is presented for arbitrary number of antennas. For the special case of Nt = 2a we construct two subclass of codes: (i) A class of 2a-group decodable codes with rate a/(2(a-1)), which is, equivalently, a class of single-symbol decodable codes, (ii) A class of (2a-2)-group decodable with rate (a-1)/(2(a-2)), i.e., a class of double-symbol decodable codes. Simulation results show that the DSD codes of this paper perform better than previously known quasi-orthogonal designs

The Generalized Degrees of Freedom Region of the MIMO Interference Channel and Its Achievability

The generalized degrees of freedom (GDoF) region of the MIMO Gaussian interference channel (IC) is obtained for the general case of an arbitrary number of antennas at each node and where the signal-to-noise ratios (SNRs) and interference-to-noise ratios vary with arbitrary exponents to a nominal SNR. The GDoF-optimal coding scheme involves message splitting and partial interference decoding and consists of linear Gaussian superposition coding of the private and common submessages that can be seen as jointly performing signal-space and signal-level interference alignment. The admissible degree of freedom (DoF)-splits between the private and common messages are also specified. A study of the GDoF region reveals various insights through the joint dependence of optimal interference management techniques at high SNR on the SNR exponents and the numbers of antennas at the four terminals. For instance, it reveals that, unlike in the scalar IC, treating interference as noise is not always GDoF-optimal even in the very weak interference regime. Moreover, while the DoF-optimal strategy that relies just on transmit/receive zero-forcing beamforming and time sharing is not GDoF optimal (and thus has an unbounded gap to capacity), the precise characterization of the very strong interference regime-where single-user DoF performance can be achieved simultaneously for both users-depends on the relative numbers of antennas at the four terminals and thus deviates from what it is in the single-input single-output case. For asymmetric numbers of antennas at the four nodes, the shape of the symmetric GDoF curve can be a “distorted W” curve to the extent that for certain multiple-input multiple-output ICs it is a “V” curve.

High-Rate, Multisymbol-Decodable STBCs From Clifford Algebras

It is well known that space-time block codes (STBCs) obtained from orthogonal designs (ODs) are single-symbol decodable (SSD) and from quasi-orthogonal designs (QODs) are double-symbol decodable (DSD). However, there are SSD codes that are not obtainable from ODs and DSD codes that are not obtainable from QODs. In this paper, a method of constructing g-symbol decodable ( g-SD) STBCs using representations of Clifford algebras are presented which when specialized to g=1,2 gives SSD and DSD codes, respectively. For the number of transmit antennas 2a the rate (in complex symbols per channel use) of the g -SD codes presented in this paper is [(a+1-g)/(2a-g)]. The maximum rate of the DSD STBCs from QODs reported in the literature is [(a)/(2a-1)] which is smaller than the rate [(a-1)/(2a-2)] of the DSD codes of this paper, for 2a transmit antennas. In particular, the reported DSD codes for 8 and 16 transmit antennas offer rates 1 and 3/4 , respectively, whereas the known STBCs from QODs offer only 3/4 and 1/2, respectively. The construction of this paper is applicable for any number of transmit antennas. The diversity sum and diversity product of the new DSD codes are studied. It is shown that the diversity sum is larger than that of all known QODs and hence the new codes perform better than the comparable QODs at low signal-to-noise ratios (SNRs) for identical spectral efficiency. Simulation results for DSD codes at various spectral efficiencies are provided.

The Diversity-Multiplexing Tradeoff of the Dynamic Decode-and-Forward Protocol on a MIMO Half-Duplex Relay Channel

The diversity-multiplexing tradeoff of the dynamic decode-and-forward protocol is characterized for the half-duplex three-terminal (m, k, n)-relay channel where the source, relay and the destination terminals have m, k and n antennas, respectively. The tradeoff curve is obtained as a solution to a simple, two-variable, convex optimization problem which is explicitly solved in closed-form for certain special classes of relay channels, namely, the (1, k, 1) relay channel, the (n, 1, n) relay channel and the (2, k, 2) relay channel. Moreover, the tradeoff curves for a certain class of relay channels, such as the (m, k, n >; k) channels, are found to be identical to those for the decode-and-forward protocol for the full duplex channel while for other classes of channels they are marginally lower at high multiplexing gains. Our results also show that for some classes of relay channels and at low multiplexing gains the diversity orders of the dynamic decode-and-forward protocol are greater than those of the static compress-and-forward protocol which in turn is known to be tradeoff optimal over all static half duplex protocols. In general, the dynamic decode-and-forward protocol has a performance that is comparable to that of the static compress-and-forward protocol which, unlike the dynamic decode-and-forward protocol, requires global channel state information at the relay node. Its performance is also close to that of the decode-and-forward protocol over the full-duplex relay channel thereby indicating that the half-duplex constraint can be compensated for by the dynamic operation of the relay wherein the relay switches from the receive to the transmit mode based on the source-relay channel quality.

Diversity-multiplexing tradeoff of the dynamic decode and forward protocol on a MIMO half-duplex relay channel

We compute the diversity-multiplexing tradeoff (DMT) curve for a three node multi-input-multi-output (MIMO) half-duplex (HD) relay network, operating in the dynamic decode-and-forward (DDF) [1] mode. We consider the case where the source and the destination have n antennas each and the single relay node has m antennas. Denoting such a channel as a (n, m)-relay channel, we provide an analytical characterization of the DMT curve for certain simple configurations such as (n, 1), (1, m) and (2, 2). We employ a numerical method to compute the DMT for more general channel configurations. Interestingly, for low multiplexing gains the achievable diversity orders of the HD-DDF protocol coincides with the diversity orders achieved by the full-duplex decode-and-forward (FDDF) protocol analyzed in [2]. In fact, the HD-DDF and FDDF protocols achieve the same diversity orders for all integer multiplexing gains. Thus, the half duplex constraint does not significantly affect the achievable DMT for the DF protocol when the source and the destination have the same number of antennas.

The diversity-multiplexing tradeoff of the MIMO Z interference channel

The fundamental diversity-multiplexing tradeoff (DMT) of the quasi-static fading, MIMO Z interference channel (ZIC), with M1 and M2 antennas at the transmitters and N1 and N2 antennas at the corresponding receivers, respectively, is derived. Channel state information at the transmitters (CSIT) and a short-term average power constraint is assumed. The achievability of the DMT is proved by showing that a simple Gaussian superposition coding scheme can achieve a rate region which is within a constant (independent of signal-to-noise ratio (SNR)) number of bits from an upper bound to the capacity region of the ZIC. We also characterize an achievable DMT of the ZIC with No-CSIT and show that in a small region of multiplexing gains (MG), the full CSIT DMT of the ZIC can be achieved with no CSIT at all. The size of this MG region depends on the system parameters such as the number of antennas at the four nodes (referred to hereafter as “antenna configuration”), SNRs and interference-to-noise ratio (INR) of the direct and cross links. Interestingly, for some antenna configurations this MG region covers the entire MG region of the ZIC. Thus, under these circumstances, the optimal DMT of the MIMO ZIC with F-CSIT is same as that of a corresponding ZIC with No-CSIT and availability of CSIT can not further improve the DMT. Finally, we identify a class of ZICs with M1 = M2 = M ≤ N1 over 2, N1 ≤ N2 and SNR ≤ INR where the achievable DMT with No-CSIT coincides with the optimal DMT with F-CSIT.

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.