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This paper proposes a framework for designing robust precoders for a multi-input single-output (MISO) system that performs integrated sensing and communication (ISAC) across multiple cells and users. We use Cramer-Rao-Bound (CRB) to measure the sensing performance and derive its expressions for two multi-cell scenarios, namely coordinated beamforming (CBF) and coordinated multi-point (CoMP). In the CBF scheme, a BS shares channel state information (CSI) and estimates target parameters using monostatic sensing. In contrast, a BS in the CoMP scheme shares the CSI and data, allowing bistatic sensing through inter-cell reflection. We consider both block-level (BL) and symbol-level (SL) precoding schemes for both the multi-cell scenarios that are robust to channel state estimation errors. The formulated optimization problems to minimize the CRB in estimating the parameters of a target and maximize the minimum communication signal-to-interference-plus-noise-ratio (SINR) while satisfying a given total transmit power budget are non-convex. We tackle the non-convexity using a combination of semidefinite relaxation (SDR) and alternating optimization (AO) techniques. Simulations suggest that neglecting the inter-cell reflection and communication links degrades the performance of an ISAC system. The CoMP scenario employing SL precoding performs the best, whereas the BL precoding applied in the CBF scenario produces relatively high estimation error for a given minimum SINR value., Comment: 13 pages, 9 figures, subnitted to IEEE Trans. on Wireless Communications

Aiming at overcoming practical issues of successive interference cancellation (SIC), this paper proposes a dual-polarized rate-splitting multiple access (RSMA) technique for a downlink massive multiple-input multiple-output (MIMO) network. By modeling the effects of polarization interference, an in-depth theoretical analysis is carried out, in which we derive tight closed-form approximations for the outage probabilities and ergodic sum-rates. Simulation results validate the accuracy of the theoretical analysis and confirm the effectiveness of the proposed approach. For instance, under low to moderate cross-polar interference, our results show that the proposed dual-polarized MIMO-RSMA strategy outperforms the single-polarized MIMO-RSMA counterpart for all considered levels of residual SIC error., Comment: arXiv admin note: substantial text overlap with arXiv:2211.00851

Unmanned Aerial Vehicle (UAV) swarms are often required in off-grid scenarios, such as disaster-struck, war-torn or rural areas, where the UAVs have no access to the power grid and instead rely on renewable energy. Considering a main battery fed from two renewable sources, wind and solar, we scale such a system based on the financial budget, environmental characteristics, and seasonal variations. Interestingly, the source of energy is correlated with the energy expenditure of the UAVs, since strong winds cause UAV hovering to become increasingly energy-hungry. The aim is to maximize the cost efficiency of coverage at a particular location, which is a combinatorial optimization problem for dimensioning of the multivariate energy generation system under non-convex criteria. We have devised a customized algorithm by lowering the processing complexity and reducing the solution space through sampling. Evaluation is done with condensed real-world data on wind, solar energy, and traffic load per unit area, driven by vendor-provided prices. The implementation was tested in four locations, with varying wind or solar intensity. The best results were achieved in locations with mild wind presence and strong solar irradiation, while locations with strong winds and low solar intensity require higher Capital Expenditure (CAPEX) allocation., Comment: To be published in Transactions on Smart Grid

The polarization domain provides an extra degree of freedom (DoF) for improving the performance of multiple-input multiple-output (MIMO) systems. This paper takes advantage of this additional DoF to alleviate practical issues of successive interference cancellation (SIC) in rate-splitting multiple access (RSMA) schemes. Specifically, we propose three dual-polarized downlink transmission approaches for a massive MIMO-RSMA network under the effects of polarization interference and residual errors of imperfect SIC. The first approach implements polarization multiplexing for transmitting the users' data messages, which removes the need to execute SIC in the reception. The second approach transmits replicas of users' messages in the two polarizations, which enables users to exploit diversity through the polarization domain. The third approach, in its turn, employs the original SIC-based RSMA technique per polarization, and this allows the BS to transmit two independent superimposed data streams simultaneously. An in-depth theoretical analysis is carried out, in which we derive tight closed-form approximations for the outage probabilities of the three proposed approaches. Accurate approximations for the ergodic sum-rates of the two first schemes are also derived. Simulation results validate the theoretical analysis and confirm the effectiveness of the proposed schemes. For instance, under low to moderate cross-polar interference, the results show that, even under high levels of residual SIC error, our dual-polarized MIMO-RSMA strategies outperform the conventional single-polarized MIMO-RSMA counterpart. It is also shown that the performance of all RSMA schemes is impressively higher than that of single and dual-polarized massive MIMO systems employing non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) techniques.

In this work, we propose a framework for energy-efficient trajectory design of an unmanned aerial vehicle (UAV)-based portable access point (PAP) deployed to serve a set of ground nodes (GNs). In addition to the PAP and GNs, the system consists of a set of intelligent reflecting surfaces (IRSs) mounted on man-made structures to increase the number of bits transmitted per Joule of energy consumed measured as the global energy efficiency (GEE). The GEE trajectory for the PAP is designed by considering the UAV propulsion energy consumption and the Peukert effect of the PAP battery, which represents an accurate battery discharge profile as a non-linear function of the UAV power consumption profile. The GEE trajectory design problem is solved in two phases: in the first, a path for the PAP and feasible positions for the IRS modules are found using a multi-tier circle packing method, and the required IRS phase shift values are calculated using an alternate optimization method that considers the interdependence between the amplitude and phase responses of an IRS element; in the second phase, the PAP flying velocity and user scheduling are calculated using a novel multilap trajectory design algorithm. Numerical evaluations show that: neglecting the Peukert effect overestimates the available flight time of the PAP; after a certain threshold, increasing the battery size reduces the available flight time of the PAP; the presence of IRS modules improves the GEE of the system compared to other baseline scenarios; the multi-lap trajectory saves more energy compared to a single-lap trajectory developed using a combination of sequential convex programming and Dinkelbach algorithm., Comment: 12 pages, 11 Figures, accepted as a Paper in IEEE Transactions on Vehicular Technology

In this work, we study the trade-off between the reliability and the investment cost of an unmanned aerial system (UAS) consisting of a set of unmanned aerial vehicles (UAVs) carrying radio access nodes, called portable access points (PAPs)), deployed to serve a set of ground nodes (GNs). Using the proposed algorithm, a given geographical region is equivalently represented as a set of circular regions, where each circle represents the coverage region of a PAP. Then, the steady-state availability of the UAS is analytically derived by modelling it as a continuous time birth-death Markov decision process (MDP). Numerical evaluations show that the investment cost to guarantee a given steady-state availability to a set of GNs can be reduced by considering the traffic demand and distribution of GNs., Comment: 5 Pages, 6 Figures, Conference: VTC2022-FAll (Workshops)

In this work, we optimize the 3D trajectory of an unmanned aerial vehicle (UAV)-based portable access point (PAP) that provides wireless services to a set of ground nodes (GNs). Moreover, as per the Peukert effect, we consider pragmatic non-linear battery discharge for the battery of the UAV. Thus, we formulate the problem in a novel manner that represents the maximization of a fairness-based energy efficiency metric and is named fair energy efficiency (FEE). The FEE metric defines a system that lays importance on both the per-user service fairness and the energy efficiency of the PAP. The formulated problem takes the form of a non-convex problem with non-tractable constraints. To obtain a solution, we represent the problem as a Markov Decision Process (MDP) with continuous state and action spaces. Considering the complexity of the solution space, we use the twin delayed deep deterministic policy gradient (TD3) actor-critic deep reinforcement learning (DRL) framework to learn a policy that maximizes the FEE of the system. We perform two types of RL training to exhibit the effectiveness of our approach: the first (offline) approach keeps the positions of the GNs the same throughout the training phase; the second approach generalizes the learned policy to any arrangement of GNs by changing the positions of GNs after each training episode. Numerical evaluations show that neglecting the Peukert effect overestimates the air-time of the PAP and can be addressed by optimally selecting the PAP's flying speed. Moreover, the user fairness, energy efficiency, and hence the FEE value of the system can be improved by efficiently moving the PAP above the GNs. As such, we notice massive FEE improvements over baseline scenarios of up to 88.31%, 272.34%, and 318.13% for suburban, urban, and dense urban environments, respectively., Comment: 12 PAGES, 9 FIGURES

This work investigates the performance of intelligent reflective surfaces (IRSs) assisted uplink non-orthogonal multiple access (NOMA) in energy-constrained networks. Specifically, we formulate and solve two optimization problems; the first aims at minimizing the sum of users' transmit power, while the second targets maximizing the system level energy efficiency (EE). The two problems are solved by jointly optimizing the users' transmit powers and the beamforming coefficients at IRS subject to the users' individual uplink rate and transmit power constraints. A novel and low complexity algorithm is developed to optimize the IRS beamforming coefficients by optimizing the objective function over a \textit{complex circle manifold} (CCM). To efficiently optimize the IRS phase shifts over the manifold, the optimization problem is reformulated into a feasibility expansion problem which is reduced to a max-min signal-plus-interference-ratio (SINR). Then, with the aid of a smoothing technique, the exact penalty method is applied to transform the problem from constrained to unconstrained. The proposed solution is compared against three semi-definite programming (SDP)-based benchmarks which are semi-definite relaxation (SDR), SDP-difference of convex (SDP-DC) and sequential rank-one constraint relaxation (SROCR). The results show that the manifold algorithm provides better performance than the SDP-based benchmarks, and at a much lower computational complexity for both the transmit power minimization and EE maximization problems. The results also reveal that IRS-NOMA is only superior to orthogonal multiple access (OMA) when the users' target achievable rate requirements are relatively high.

Rate-splitting multiple access (RSMA) has recently appeared as a powerful technique for improving the downlink performance of multiple-input multiple-output (MIMO) systems. By flexibly managing interference, RSMA can deliver high spectral and energy efficiency, as well as robustness to imperfect channel state information (CSI). In another development, an intelligent reflecting surface (IRS) has emerged as a method to control the wireless environment through software-configurable, near-passive, sub-wavelength reflecting elements. This article presents the potential of synergy between IRS and RSMA. Three important improvements achievable by IRS-RSMA schemes are identified, supported by insightful numerical examples, and mapped to beyond-5G use cases, along with future research directions., Comment: IEEE Communications Magazine, May 2022

In this paper, the appealing features of a dual-polarized intelligent reflecting surface (IRS) are exploited to improve the performance of dual-polarized massive multiple-input multiple-output (MIMO) with non-orthogonal multiple access (NOMA) under imperfect successive interference cancellation (SIC). By considering the downlink of a multi-cluster scenario, the IRSs assist the base station (BS) to multiplex subsets of users in the polarization domain. Our novel strategy alleviates the impact of imperfect SIC and enables users to exploit polarization diversity with near-zero inter-subset interference. Our results show that when the IRSs are large enough, the proposed scheme always outperforms conventional massive MIMO-NOMA and MIMO-OMA systems even if SIC error propagation is present. It is also confirmed that dual-polarized IRSs can make cross-polar transmissions beneficial to the users, allowing them to improve their performance through polarization diversity., Comment: arXiv admin note: substantial text overlap with arXiv:2012.03639

In this paper, we propose an energy-efficient optimal altitude for an aerial access point (AAP), which acts as a flying base station to serve a set of ground user equipment (UE). Since the ratio of total energy consumed by the aerial vehicle to the communication energy is very large, we include the aerial vehicle's energy consumption in the problem formulation. After considering the energy consumption model of the aerial vehicle, our objective is translated into a non-convex optimization problem of maximizing the global energy efficiency (GEE) of the aerial communication system, subject to altitude and minimum individual data rate constraints. At first, the non-convex fractional objective function is solved by using sequential convex programming (SCP) optimization technique. To compare the result of SCP with the global optimum of the problem, we reformulate the initial problem as a monotonic fractional optimization problem (MFP) and solve it using the polyblock outer approximation (PA) algorithm. Numerical results show that the candidate solution obtained from SCP is the same as the global optimum found using the monotonic fractional programming technique. Furthermore, the impact of the aerial vehicle's energy consumption on the optimal altitude determination is also studied., Comment: This version of the work has been accepted for publication in the IEEE 31st PIMRC 2020 - Workshop on UAV Communications for 5G and Beyond; 7 pages; 5 figures

In this letter, we propose an energy-efficient 3-dimensional placement of multiple aerial access points (AAPs), in the desired area, acting as flying base stations for uplink communication from a set of ground user equipment (UE). The globally optimal energy-efficient vertical position of AAPs is derived analytically by considering the inter-cell interference and AAP energy consumption. The horizontal position of AAPs which maximize the packing density of the AAP coverage area are determined using a novel regular polygon-based AAP placement algorithm. We also determine the maximum number of non-interfering AAPs that can be placed in the desired area. The effect of the AAP energy consumption on the optimal placement and the analytic findings are verified via numerical simulations., Comment: This version of the work has been accepted for publication in the IEEE COMMUNICATIONS LETTERS. 6 pages; 5 figures

A dual-polarized intelligent reflecting surface (IRS) can contribute to a better multiplexing of interfering wireless users. In this paper, we use this feature to improve the performance of dual-polarized massive multiple-input multiple-output (MIMO) with non-orthogonal multiple access (NOMA) under imperfect successive interference cancellation (SIC). By considering the downlink of a multi-cluster scenario, the IRSs assist the base station (BS) to multiplex subsets of users in the polarization domain. Our novel strategy alleviates the impact of imperfect SIC and enables users to exploit polarization diversity with near-zero inter-subset interference. To this end, the IRSs are optimized to mitigate transmissions originated at the BS from the interfering polarization. The formulated optimization is transformed into quadratic constrained quadratic sub-problems, which makes it possible to obtain the optimal solution via interior-points methods. We also derive analytically a closed-form expression for the users' ergodic rates by considering large numbers of reflecting elements. This is followed by representative simulation examples and comprehensive discussions. The results show that when the IRSs are large enough, the proposed scheme always outperforms conventional massive MIMO-NOMA and MIMO-OMA systems even if SIC error propagation is present. It is also confirmed that dual-polarized IRSs can make cross-polar transmissions beneficial to the users, allowing them to improve their performance through diversity.

Massive multiple-input multiple-output (MIMO) and non-orthogonal multiple access (NOMA) are two key techniques for enabling massive connectivity in future wireless networks. A massive MIMO-NOMA system can deliver remarkable spectral improvements and low communication latency. Nevertheless, the uncontrollable stochastic behavior of the wireless channels can still degrade its performance. In this context, intelligent reflecting surface (IRS) has arisen as a promising technology for smartly overcoming the harmful effects of the wireless environment. The disruptive IRS concept of controlling the propagation channels via software can provide attractive performance gains to the communication networks, including higher data rates, improved user fairness, and, possibly, higher energy efficiency. In this article, in contrast to the existing literature, we demonstrate the main roles of IRSs in MIMO-NOMA systems. Specifically, we identify and perform a comprehensive discussion of the main performance gains that can be achieved in IRS-assisted massive MIMO-NOMA (IRS-NOMA) networks. We outline exciting futuristic use case scenarios for IRS-NOMA and expose the main related challenges and future research directions. Furthermore, throughout the article, we support our in-depth discussions with representative numerical results.

Wireless Energy Transfer (WET) is emerging as a potential green enabler for massive Internet of Things (IoT). Herein, we analyze Channel State Information (CSI)-free multi-antenna strategies for powering wirelessly a large set of single-antenna IoT devices. The CSI-free schemes are AA-SS (AA-IS), where all antennas transmit the same (independent) signal(s), and SA, where just one antenna transmits at a time such that all antennas are utilized during the coherence block. We characterize the distribution of the provided energy under correlated Rician fading for each scheme and find out that while AA-IS and SA cannot take advantage of the multiple antennas to improve the average provided energy, its dispersion can be significantly reduced. Meanwhile, AA-SS provides the greatest average energy, but also the greatest energy dispersion, and the gains depend critically on the mean phase shifts between the antenna elements. We find that consecutive antennas must be $\pi$ phase-shifted for optimum average energy performance under AA-SS. Our numerical results evidenced that correlation is beneficial under AA-SS, while a greater line of sight (LOS) and/or number of antennas is not always beneficial under such scheme. Meanwhile, both AA-IS and SA schemes benefit from small correlation, large LOS and/or large number of antennas., Comment: 16 pags, 10 figs, 1 table, submitted to IEEE JSAC 2020

Single-user multiple-input / multiple-output (SU-MIMO) communication systems have been successfully used over the years and have provided a significant increase on a wireless link's capacity by enabling the transmission of multiple data streams. Assuming channel knowledge at the transmitter, the maximization of the mutual information of a MIMO link is achieved by finding the optimal power allocation under a given sum-power constraint, which is in turn obtained by the water-filling (WF) algorithm. However, in spectrum sharing setups, such as Licensed Shared Access (LSA), where a primary link (PL) and a secondary link (SL) coexist, the power transmitted by the SL transmitter may induce harmful interference to the PL receiver. While such co-existing links have been considered extensively in various spectrum sharing setups, the mutual information of the SL under a constraint on the interference it may cause to the PL receiver has, quite astonishingly, not been evaluated so far. In this paper, we solve this problem, find its unique optimal solution and provide the power allocation policy and corresponding precoding solution that achieves the optimal capacity under the imposed constraint. The performance of the optimal solution and the penalty due to the interference constraint are evaluated over some indicative Rayleigh fading channel conditions and interference thresholds. We believe that the obtained results are of general nature and that they may apply, beyond spectrum sharing, to a variety of applications that admit a similar setup., Comment: Work very similar to this was reported in another related area. This is something that was not acknowledged at the time of the submission

As network deployments become denser, interference arises as a dominant performance degradation factor. To confront with this trend, Long Term Evolution (LTE) incorporated features aiming at enabling cooperation among different base stations, a technique termed as Coordinated Multi Point (CoMP). Recent field trial results and theoretical studies of the performance of CoMP schemes revealed, however, that their gains are not as high as initially expected, despite their large coordination overhead. In this paper, we review recent advanced Coordinated Beamforming (CB) schemes, a special family of CoMP that reduces the coordination overhead through a joint choice of transmit and receive linear filters. We focus on assessing their resilience to uncoordinated interference and Channel State Information (CSI) imperfections, which both severely limit the performance gains of all CoMP schemes. We present a simple yet encompassing system model that aims at incorporating different parameters of interest in the relative interference power and CSI errors, and then utilize it for the performance evaluation of the state-of-the-art in CB schemes. It is shown that blindly applying CB in all system scenarios can indeed be counter-productive., Comment: 6 pages, 4 figures, IEEE WCNC 2017. arXiv admin note: text overlap with arXiv:1604.04592

Modern cellular networks in traditional frequency bands are notoriously interference-limited especially in urban areas, where base stations are deployed in close proximity to one another. The latest releases of Long Term Evolution (LTE) incorporate features for coordinating downlink transmissions as an efficient means of managing interference. Recent field trial results and theoretical studies of the performance of joint transmission (JT) coordinated multi-point (CoMP) schemes revealed, however, that their gains are not as high as initially expected, despite the large coordination overhead. These schemes are known to be very sensitive to defects in synchronization or information exchange between coordinating bases stations as well as uncoordinated interference. In this article, we review recent advanced coordinated beamforming (CB) schemes as alternatives, requiring less overhead than JT CoMP while achieving good performance in realistic conditions. By stipulating that, in certain LTE scenarios of increasing interest, uncoordinated interference constitutes a major factor in the performance of CoMP techniques at large, we hereby assess the resilience of the state-of-the-art CB to uncoordinated interference. We also describe how these techniques can leverage the latest specifications of current cellular networks, and how they may perform when we consider standardized feedback and coordination. This allows us to identify some key roadblocks and research directions to address as LTE evolves towards the future of mobile communications., Comment: 16 pages, 6 figures, accepted to IEEE Communications Magazine

Recently, several advanced multi-antenna radio communications technologies have emerged to meet the increased capacity demands in wireless multi-user networks. Despite their great potential, the extent of these techniques' practical applicability still remains questionable, since they have to face either backhaul limitations or cost and hardware constraints. In this paper, we propose a new system solution which includes network architecture, antenna technology and radio transmission protocol to reduce drastically the hardware complexity and cost as well as the channel state information / user data feedback requirements of multi-user multi-antenna wireless networks. We focus on the forward link of an interference channel in a cloud radio access network setup wherein an arbitrary number of remote radio heads are each equipped with a single radio frequency module parasitic antenna array and wish to send data to their respective single-antenna user terminals, while co-existing in time and frequency. Base stations select cooperatively the optimal combination of pre-determined beams prior transmission. Our proposed approach is able to achieve the aforementioned goals, while offering significant downlink sum-rate gains due to the available spatial degrees of freedom.

Transmitter (TX) cooperation at various levels has been shown to increase the sum throughput of multiuser multiple-input multiple-output (MIMO) systems. In this paper we consider a K-user MIMO system where TXs have only global channel state knowledge. It has been theoretically shown that interference alignment (IA) achieves the K/2 degrees of freedom of this K-user MIMO interference channel. However, results on IA and all proposed transceiver techniques for this channel up to date, assume conventional antenna arrays at the transceivers with multiple radio-frequency (RF) chains, each connected to a different antenna element. To reduce the consequent hardware burden and power dissipation imposed by such arrays, we propose in this paper the utilization of compact single-RF electronically steerable parasitic (passive) array radiators (ESPARs) at the cooperating TXs. A signal model capable of capturing the characteristics of the considered antenna arrays is first described and then a general precoding design methodology for the tunable parasitic loads at the TXs' ESPARs is introduced. Specific precoding techniques and an indicative ESPAR design are presented for a 3-user 2x2 MIMO system with one ESPAR TX, and the obtained performance evaluation results show that the gains of TX cooperation are still feasible., Comment: 6 pages, 3 figures, IEEE GLOBECOM 2014

This paper presents the current state-of-the-art of massive antenna array architectures with significant front-end hardware savings, as an enabler for future small and powerful cell nodes that will be able to carry massive MIMO technology. Radio frequency (RF) hardware architectures with a single power amplifier are reviewed, compared, and found superior to conventional MIMO implementations in terms of cost, dissipated heat, and physical size. This progress on the RF-side allows to merge the two competing cellular concepts of virtual and massive MIMO into a hybrid approach of remote radio heads with massive MIMO arrays., Comment: 10 pages, 4 figures

It is already well-known that interference alignment (IA) achieves the sum capacity of the K-user interference channel at the high interference regime. On the other hand, it is intuitively clear that when the interference levels are very low, a sum-rate scaling of K (as opposed to K/2 for IA) should be accessed at high signal-to-noise ratio values by simple ("myopic") single-link multiple-input multiple-output (MIMO) techniques such as waterfilling. Recent results have indicated that in certain low-to-moderate interference cases, treating interference as noise may in fact be preferable. In this paper, we present a distributed iterative algorithm for K-user MIMO interference networks which attempts to adjust itself to the interference regime at hand, in the above sense, as well as to the channel conditions. The proposed algorithm combines the system-wide mean squared error minimization with the waterfilling solution to adjust to the interference levels and channel conditions and maximize accordingly each user's transmission rate. Sum-rate computer simulations for the proposed algorithm over Ricean fading channels show that, in the interference-limited regime, the proposed algorithm reconfigures itself in order to achieve the IA scaling whereas, in the low-to-moderate interference regime, it leads itself towards interference-myopic MIMO transmissions., Comment: 5 pages, 4 figures, accepted for IEEE ICC 2013

Interference Alignment (IA) is the process of designing signals in such a way that they cast overlapping shadows at their unintended receivers, while remaining distinguishable at the intended ones. Our goal in this paper is to come up with an algorithm for IA that runs at the transmitters only (and is transparent to the receivers), that doesn't require channel reciprocity, and that alleviates the need to alternate between the forward and reverse network as is the case in Distributed IA (Gomadam, Cadambe, Jafar 08'), thereby inducing significant overhead in certain environments where the channel changes frequently. Most importantly, our effort is focused on ensuring that this one-sided approach does not degrade the performance of the system w.r.t. Distributed IA (since it cannot improve it). As a first step, we model the interference in each receiver's desired signal as a function of the transmitters' beamforming vectors. We then propose a simple steepest descent (SD) algorithm and use it to minimize the interference in each receiver's desired signal space. We mathematically establish equivalences between our approach and the Distributed IA algorithm (Gomadam, Cadambe, Jafar 08') and show that our algorithm also converges to an alignment solution (when the solution is feasible)., Comment: Submitted to the IEEE GLOBECOM 2011 conference, under the Signal Processing for Communications Track: 5 pages, double columns, 5 figures

We construct a class of linear space-time block codes for any number of transmit antennas that have controllable ML decoding complexity with a maximum rate of 1 symbol per channel use. The decoding complexity for $M$ transmit antennas can be varied from ML decoding of $2^{\lceil \log_2M \rceil -1}$ symbols together to single symbol ML decoding. For ML decoding of $2^{\lceil \log_2M \rceil - n}$ ($n=1,2,...$) symbols together, a diversity of $\min(M,2^{\lceil \log_2M \rceil-n+1})$ can be achieved. Numerical results show that the performance of the constructed code when $2^{\lceil \log_2M \rceil-1}$ symbols are decoded together is quite close to the performance of ideal rate-1 orthogonal codes (that are non-existent for more than 2 transmit antennas)., Comment: 5 pages, to appear in ISIT, June 2007, Nice, France