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Constructive Multiuser Interference in Symbol Level Precoding for the MISO Downlink Channel

Maha Alodeh, Symeon Chatzinotas, Björn Ottersten

arXiv:1408.4700v1cs.IT

TL;DR

The paper addresses harmful interference in simultaneous multiuser MISO downlink transmissions. It designs symbol-level precoders from channel and data information to exploit interference constructively, develops CIMRT and related multicast-linked optimization methods, and reports improved performance across power, fairness, and sum-rate objectives.

  • Problem

    Simultaneous multiuser downlink transmissions require precoding to manage interference, while conventional MRT is unsuitable for multiuser MISO downlinks.

  • Method

    The paper jointly uses channel state information and data information in symbol-level precoding, including CIMRT and multicast-linked designs for power, fairness, and sum-rate objectives.

  • Results

    The proposed constructive-interference schemes address transmit-power minimization, max-min SINR fairness, and fixed-power sum-rate maximization, with max-min SINR reported as outperforming conventional approaches.

  • Takeaways & Limitations

    Constructive interference can be incorporated into symbol-level precoding designs for multiple downlink objectives, with heuristic sum-rate performance varying across SNR regimes.

Abstract

from arXiv · show

This paper investigates the problem of interference among the simultaneous multiuser transmissions in the downlink of multiple antennas systems. Using symbol level precoding, a new approach towards the multiuser interference is discussed along this paper. The concept of exploiting the interference between the spatial multiuser transmissions by jointly utilizing the data information (DI) and channel state information (CSI), in order to design symbol-level precoders, is proposed. In this direction, the interference among the data streams is transformed under certain conditions to useful signal that can improve the signal to interference noise ratio (SINR) of the downlink transmissions. We propose a maximum ratio transmission (MRT) based algorithm that jointly exploits DI and CSI to glean the benefits from constructive multiuser interference. Subsequently, a relation between the constructive interference downlink transmission and physical layer multicasting is established. In this context, novel constructive interference precoding techniques that tackle the transmit power minimization (min power) with individual SINR constraints at each user's receivers is proposed. Furthermore, fairness through maximizing the weighted minimum SINR (max min SINR) of the users is addressed by finding the link between the min power and max min SINR problems. Moreover, heuristic precoding techniques are proposed to tackle the weighted sum rate problem. Finally, extensive numerical results show that the proposed schemes outperform other state of the art techniques.

I. INTRODUCTION

The paper distinguishes multicast, unicast, group-level, user-level, and symbol-level downlink precoding, focusing on exploiting rather than suppressing multiuser interference. It proposes constructive-interference designs using both channel and symbol information for power, fairness, and sum-rate objectives.

  • Transmission and precoding categories: Multicast sends a common data stream to multiple receivers, whereas unicast sends individual messages simultaneously over the downlink.Unicast motivates precoding because multiple simultaneous streams create multiuser interference.
  • Transmission and precoding categories: Symbol-level precoding designs each transmitted symbol using both channel state information and users’ data information.Unlike conventional schemes that fully decorrelate streams, it aims to constructively correlate interference.
  • Constructive interference: Constructive-interference schemes classify interference by its effect on reception, eliminating destructive components while retaining or rotating them into useful signal.The cited prior schemes target BPSK and QPSK scenarios and report gains over conventional precoding.
  • Constructive interference: Symbol-level precoding requires precoder recalculation every symbol period rather than every channel-coherence interval, increasing calculation, switching, and hardware demands.This follows from τc ≫ τs in slow-fading channels and the resulting update rates 1/τc versus 1/τs.
  • Paper contributions: The paper introduces M-PSK constructive-interference characterization and CIMRT, then relates constructive-interference precoding to physical-layer multicasting.CIMRT is presented as addressing weaknesses of CIZF.
  • Paper contributions: Its proposed designs target transmit-power minimization under SNR targets, minimum-SNR fairness under a power constraint, and sum-rate maximization within permissible power.The paper evaluates the proposed algorithms numerically after developing these problem formulations.

II. SYSTEM AND SIGNAL MODELS

The paper models a single-cell MISO downlink with M transmit antennas, K single-antenna users, M-PSK signaling, and CSI/DI available at the transmitter. It contrasts conventional beamforming, which suppresses non-intended inner products, with symbol-level precoding that can align interference constructively.

  • System model: The system comprises one multi-antenna base station serving K single-antenna users over quasi-static block-fading channels with M-PSK modulation.Each user channel is represented by h_j, and receiver noise is modeled as an i.i.d. complex Gaussian variable.
  • Signal model: The received signal is formed from the transmitted vector, user-specific precoding vectors, data symbols, and additive receiver noise.The compact formulation stacks users’ signals, channels, precoders, data symbols, and power allocation into matrix expressions.
  • Conventional precoding: Conventional beamforming seeks maximal inner products with intended channels and minimal inner products with non-intended channels.This includes pseudo-inverse precoding and formulations based on power minimization with SINR constraints or SINR-margin maximization under a power constraint.
  • Symbol-level precoding: Symbol-level precoding uses data symbols and channel information to steer non-intended contributions into directions that constructively add at each receiver.Interfering signals can move received points deeper into their correct detection regions and redesign cross-stream terms to correlate constructively with the desired term.
  • Constructive-interference design: Different constructive-interference precoders are proposed by redesigning cross-user terms in the received-signal and SINR formulations.The proposed designs target constructive correlation between a_j_i and the intended coefficient a_j_j.

A. Power constraints for user based and symbol based precodings

User-based precoding enforces an average power constraint over the channel coherence interval, whereas symbol-based precoding must satisfy power constraints for every transmitted symbol vector. The section defines constructive interference geometrically for M-PSK and relates channel correlation to its potential benefit and limits.

  • Power constraints: User-based precoding enforces the power constraint over the coherence interval because its precoder remains fixed while transmitted symbols vary.The constraint is expressed through the expected squared norm of the transmitted signal over τ_c.
  • Power constraints: Symbol-level precoding requires the transmit-power constraint to hold separately for each symbol vector and symbol period.The instantaneous constraint is expressed through the squared norm of x, formed from the precoding vectors and current data symbols.
  • Constructive interference: In conventional transmission, interference can move received symbols outside their detection regions, while symbol-level design can push them deeper inside the correct regions.Interference is therefore classified as constructive or destructive according to whether it facilitates or deteriorates correct symbol detection.
  • Constructive-interference definition: For M-PSK, constructive interference requires the received interfering signal to lie within the target symbol’s correct detection region, with sign compatibility selecting the valid quadrant.The detection region is specified by an angular interval around the target symbol, and the transmitter is assumed to know both DI and CSI.
  • Constructive-interference properties: Constructive interference is mutual, and fully correlated signals can contribute to received power rather than behaving solely as interference.The paper contrasts this benefit with conventional precoding and notes that CSI and DI yield higher performance than conventional techniques in the stated setting.
  • Channel geometry: When user channels become co-linear, channel inversion fails and interference cannot be mitigated; physically orthogonal channels instead eliminate multiuser interference.The co-linear case is identified as the worst case, while the orthogonal case is the optimal case for conventional interference behavior.

V. CONSTRUCTIVE INTERFERENCE PRECODING FOR MISO DOWNLINK CHANNELS

The paper develops constructive-interference precoding for MISO downlinks using symbol-level DI and CSI, first relating a correlation-rotation design to multicast transmission and then proposing an MRT-based alternative. The latter uses unitary-plane rotations to make simultaneous transmissions constructive while preserving other beamforming directions.

  • Constructive-interference precoding: The transmitter is assumed capable of designing precoding at symbol level using both channel state information and data information.The design also combines precoding with power allocation.
  • CIZF: The initial design combines zero forcing with a correlation rotation that aims to make transmitted signals constructively received at each user.Its formulation has a multicast interpretation because each user can be viewed as receiving a common symbol with a user-dependent rotation.
  • CIZF: The correlation-rotation zero-forcing design fails for co-linear users, and the paper attributes this contradiction to the zero-forcing step.The authors therefore propose a new precoding technique to address this case.
  • CIMRT: The proposed CIMRT approach uses Givens rotations in selected beamforming-vector planes to grant constructive interference while preserving unitarity.A rotation changes the selected pair while leaving the remaining beamforming directions fixed, and rotations of different planes are decoupled.
  • CIMRT: The rotation parameters are obtained from nonlinear equations, with candidate roots evaluated to select an optimal solution and adjust associated values.The algorithm includes constructing the relevant matrices, selecting a beamforming plane, and solving for the rotation parameters.

VI. CONSTRUCTIVE INTERFERENCE FOR POWER MINIMIZATION

Constructive-interference power minimization designs the transmit vector directly from channel and data information, rather than optimizing a conventional precoding matrix independently of the symbols. The formulation enforces constructive reception of each user's data symbol under individual SNR targets.

  • Power minimization constraints: The power-minimization formulation constrains each user to receive its corresponding data symbol constructively at target SNR ζj.The constraints C1 guarantee correct symbol detection, while ζ contains the users' SNR targets.
  • Power minimization constraints: The reformulation with variables ŵk = wkdk shows that exact symbols need not be physically transmitted separately if users detect their intended symbols correctly.The data symbols can instead be incorporated into the precoding design.
  • Relation to conventional precoding: Under conventional linear precoding, the transmit-power-minimizing solution can use a unit-rank precoding matrix W.The corresponding transmit vector is represented through a common vector structure.
  • Direct symbol-level design: Conventional precoding optimizes W from CSI and then carries data symbols, whereas constructive-interference precoding jointly uses CSI and DI to design x.The conventional transmit vector has a linear dependence on DI and CSI; the constructive scheme optimizes x directly.
  • Direct symbol-level design: Constructive-interference precoding directly optimizes the transmit vector x using both channel-state and data information.This skips the intermediate optimization of W used in conventional linear beamforming.

B. The Relation Between Constructive Interference Precoding and Constrained Constellation Multicast

The constructive-interference power-minimization problem is related to constrained constellation multicast through a common transmit structure spanning users' channel subspaces. Uniformly scaling all SNR targets scales the output vector without changing its normalized direction.

  • Equivalence to constrained multicast: The optimal CIPM precoder is given through an equivalent channel construction, linking constructive-interference precoding to constrained constellation multicast.Theorem 1 expresses the optimal precoder using xe(d, A(d, dj)H).
  • Optimization structure: The minimum-power solution occurs when users meet their target SNR thresholds with equality, because the constructive alignment constraints determine the received phases.The original inequality constraints are therefore reformulated as equalities.
  • Equivalence to constrained multicast: CI power minimization and constrained constellation multicast must span the subspaces of every user's channel.This is stated as a corollary of the relation between the two formulations.
  • Equivalence to constrained multicast: The constructive-interference formulation uses a common precoder for all users, resembling multicast while retaining constraints for each user's distinct detected symbol.Unlike multicast's common-message setting, C1 guarantees that each user detects its corresponding data symbol.
  • Scaling property: Uniformly scaling all SNR targets by n scales the output vector by √n while leaving its normalized direction unchanged.The received symbol phases remain unchanged under this uniform scaling.

C. Constructive Interference Power minimization bounds

The section characterizes theoretical minimum-power benchmarks for constructive-interference transmission. One genie-aided bound assumes all multiuser transmissions are constructively aligned, while a tighter multicast-related bound imposes unit rank.

  • Genie-aided bound: The genie-aided bound can be found by solving the corresponding optimization using linear programming.It represents a theoretical upper bound for symbol-basis constructive-interference exploitation.
  • Genie-aided bound: The genie-aided minimum transmit power is obtained when all multiuser transmissions interfere constructively with respect to every user's desired symbol.This ideal condition is characterized by phase alignment across the transmissions.
  • Multicast-related bounds: Dropping the phase-alignment constraint C1 yields a theoretical upper bound associated with complete correlation among the communicated information.The comparison uses a multicast formulation with the same symbol for all users.
  • Multicast-related bounds: The bound from Eq. (39) is tighter because it imposes a unit-rank approximation on the multicast transmit covariance.This constraint enables comparison with unit-rank constructive-interference power minimization.

VII. WEIGHTED MAX MIN SINR ALGORITHM FOR CONSTRUCTIVE INTERFERENCE PRECODING (CIMM)

The constructive-interference max-min formulation targets weighted fairness by maximizing the worst user's SINR while preserving correct symbol detection. Its solution is a scaled minimum-power solution, with the scale found through optimization or bisection.

  • Fairness objective: Constructive-interference max-min SINR precoding improves fairness by maximizing the users' weighted worst-case SNR.The formulation includes additional constraints ensuring that the data are detected correctly at the receivers.
  • Relation to minimum power: The optimal max-min output vector is a scaled version of the minimum-power solution, linking the two optimization problems.The optimal scaling parameter is denoted t* in the formulation.
  • Relation to minimum power: The optimal value t* can be found by solving the minimum-power problem, so max-min SINR reduces to finding the appropriate scale.The resulting method uses this relationship to construct the max-min solution.
  • Fairness objective: Compared with the conventional max-min formulation, constructive interference introduces an additional 3K constraints that limit system performance.These constraints arise from requiring correct data detection in addition to the fairness objective.
  • Solution procedure: A bisection method can solve the constructive-interference max-min problem using the parameterized optimization formulation.The proposed procedure follows the same general solution style as prior bisection-based max-min methods.

VIII. WEIGHTED SUM RATE MAXIMIZATION ALGORITHMS FOR CONSTRUCTIVE INTERFERENCE PRECODING (CISR)

This section formulates weighted sum-rate maximization for constructive-interference precoding and develops heuristic algorithms under practical modulation constraints. It accounts for symbol-level interference exploitation while recognizing that the general optimization is difficult.

  • Relation to prior work: The proposed weighted sum-rate formulation extends prior user-level and multicast approaches by exploiting interference among multiuser transmissions.Earlier work includes MRT rotation for interference reduction, multicast high-SNR designs, and low-complexity iterative or heuristic methods.
  • Problem formulation: The weighted sum-rate problem is reformulated to account for constructive interference among simultaneous multiuser data streams.The formulation uses a unit-rank precoding assumption to enable tractable heuristic solutions, although the optimal solution need not be unit rank.
  • Practical modulation constraints: Symbol-level modulation cannot adapt independently for every symbol because the resulting signaling overhead is impractical.Instead, each user’s modulation remains fixed during the coherence time and is selected at its beginning.
  • Practical modulation constraints: The modulation order for each user is selected over the transmission frame using an optimal multicast formulation.This determines the highest modulation order that users can support during the frame.
  • Practical modulation constraints: The transmitter can assume the highest-order PSK expected by users because higher-order PSK symbols can be decoded as lower-order PSK when the target modulation is known.The receiver’s demapping and coding depend on the assigned modulation and coding scheme.

B. Genie aided sum rate upper bound

The genie-aided sum-rate upper-bound analysis addresses a difficult phase- and threshold-constrained optimization using approximations and two heuristic algorithms. The proposed methods use phase alignment or user selection to obtain tractable designs.

  • Genie-aided upper bound: The maximum sum rate is obtained by solving an optimization with both phase-alignment and threshold constraints.Because these constraint types differ, the optimal solution cannot be found straightforwardly.
  • Low-SNR approximation: A low-SINR approximation simplifies the sum-rate objective in the noise-limited regime σ^2 → ∞.Under this approximation, the formulation resembles generic power-minimization precoding and the user weights vanish.
  • Heuristic algorithms: Two heuristic algorithms are proposed because the original sum-rate maximization problem is difficult to solve directly.They are the phase-alignment algorithm CISR-PA and the greedy sum-rate algorithm CISR-G.
  • Heuristic algorithms: CISR-PA uses an approximate optimization with phase-alignment constraints and channel-matrix eigenvectors to construct the precoding direction.The resulting direction is based on eigenvectors of H^H H and is used to formulate received SINR.
  • Heuristic algorithms: CISR-G combines unconstrained multicast optimization with constrained constellation power minimization to select compatible user subsets.It uses projections and projection angles relative to the maximum-eigenvalue multicast direction before evaluating power minimization.

IX. ALGORITHMS COMPLEXITY

This section compares the computational complexity and simulated performance of the proposed algorithms. Results show regime-dependent trade-offs among energy efficiency, power consumption, and sum rate, while multicast remains an upper bound.

  • Complexity: CIPM has lower complexity than CIMRT because it requires solving 2K linear equations simultaneously.CIMRT additionally requires an SVD and solving K^2−K over 2 sets of nonlinear equations.
  • Complexity: CIMM requires solving 2K linear equations at each bisection iteration, with bisection complexity log2(max r_k).CISR-PA instead requires numerical solution of a convex optimization and eigenvalue decomposition.
  • Complexity: CISR-G requires convex optimization, a maximum-eigenvalue eigenvector, user-combination search, and 2K simultaneous linear equations.Its user-subset search is performed to identify the most suitable users to serve during coherence time.
  • Numerical results: Optimal multicast achieves the highest energy efficiency, while the gap to the genie-aided bound decreases as target rate increases.CIZF performs worse than the depicted techniques, whereas CIPM is more energy-efficient than CIZF because channel inversion wastes energy.
  • Numerical results: CIMRT performs close to CIPM at high targets and outperforms CIZF at the cost of higher complexity.For power consumption, the gaps between CIPM, optimal multicast, and the genie-aided upper bound remain fixed across target rates in the stated scenario.
  • Numerical results: CISR-G outperforms CISR-PA at low SNR, whereas CISR-PA outperforms CISR-G at high SNR as the performance gap increases with SNR.Low-SNR operation favors preselecting compatible users, while high-SNR operation favors serving all users.
  • Numerical results: CIMM underperforms CISR-G and CISR-PA at low SNR, but at high SNR it outperforms CISR-G and remains below CISR-PA.Multicast is an upper bound and cannot deliver different messages to different users.

XI. CONCLUSIONS

The paper exploits CSI and DI to align multiuser symbols constructively, connects constructive-interference precoding with multicast, and develops designs for power, fairness, and sum-rate objectives. Simulations show that the preferred heuristic depends on SNR.

  • Conclusions: The proposed symbol-level precoding jointly uses CSI and DI to constructively correlate transmitted symbols for M-PSK modulation.This enables interference exploitation among multiuser transmissions.
  • Conclusions: The paper establishes a connection between constructive-interference precoding and multicast precoding.This connection supports several constructive-interference designs with different optimization objectives.
  • Conclusions: The proposed designs address transmit-power minimization with SINR thresholds, fairness through max-min SINR, and sum-rate maximization with fixed transmit power.The max-min SINR problem is related to the power-minimization problem, while heuristic solutions are proposed for sum-rate maximization.
  • Conclusions: CISR-G performs well at low SNR, while CISR-PA performs well at high SNR.The preferred sum-rate heuristic therefore changes with the operating SNR regime.
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