Symbol-Level Precoding in Downlink Systems

• Symbol-level precoding is a technique that uses both CSI and real-time symbol data to actively shape multiuser interference into constructive signals.
• It designs symbol-dependent transmit vectors to place noiseless received signals within favorable decision regions for PSK and QAM, thereby enhancing error margins.
• SLP frameworks extend to robust, learning-based, and hardware-constrained implementations, balancing performance gains with computational complexity.

Symbol-level precoding (SLP) is a downlink precoding paradigm in which the transmitter exploits not only channel state information (CSI) but also the instantaneous data symbols to design the transmit vector on a symbol-by-symbol basis, or for each symbol vector, so that multi-user interference (MUI) can be shaped as constructive interference rather than treated as purely harmful. In its canonical MU-MISO form, the precoder is a symbol-dependent mapping such as $\mathbf{x}_l=\mathcal{F}(\mathbf{H},\mathbf{s}_l)$, and the central design question is no longer only how to suppress interference statistically, but how to place noiseless received samples inside modulation-dependent decision regions with favorable error margins, often under power, SINR, rate, or symbol-error constraints (Bo et al., 2021, Liu et al., 2018).

1. Canonical definition and signal model

A standard SLP setting is the downlink multi-user MISO system with a base station having $N_t$ antennas and $K$ single-antenna users. At symbol time or symbol-vector index $l$, the transmitted vector is designed specifically for the current symbol realization, and user $k$ observes

$r_{k,l}=\mathbf{h}_k^H\mathbf{x}_l+n_{k,l},$

with $n_{k,l}$ modeled as AWGN. This symbol dependence is the defining distinction from conventional block-level or linear precoding, where a fixed matrix is designed from CSI only and then applied across many symbols without redesign at every symbol interval (Bo et al., 2021).

The literature repeatedly contrasts SLP with block-level precoding (BLP). In conventional BLP, interference is primarily suppressed or canceled; in SLP, it is shaped using instantaneous symbol knowledge so that it pushes the received point deeper into the correct decision region. This transmitter-side exploitation of interference is especially natural in downlink systems because the base station knows the users’ intended symbols before transmission. In that sense, SLP is not merely a different solver for the same beamforming problem; it changes the granularity of design from stream level to symbol level (Li et al., 2020).

The standard formulations span MU-MISO and related downlink settings, but the same principle has been adapted to wider classes of channels and waveform domains. The common element is that the optimization variable is symbol dependent, whether it is a narrowband transmit vector, a grouped symbol-level waveform, an IRS-assisted one-bit transmit signal, or an AFDM-domain waveform whose entries are designed directly from the intended symbol tuple (Xiao et al., 2022).

2. Constructive interference geometry and decision regions

For PSK signaling, SLP is most often expressed through geometric constructive-interference constraints. After rotating the desired symbol to the positive real axis, correct detection is enforced by requiring the noiseless received point to lie inside the corresponding wedge. A representative QoS metric is

$$d_{k,l}=\Big(\Re\{\mathbf{h}_k^H \mathbf{x}_{l}e^{-j\angle \mathbf{s}_l(k)}\}\tan\phi-|\Im\{\mathbf{h}_k^H \mathbf{x}_{l}e^{-j\angle \mathbf{s}_l(k)}\}|\Big)\cos \phi,$$

where $\phi=\pi/M$ for $M$-PSK. Larger $N_t$0 means the rotated noiseless point is farther from the nearest decision boundary, so max-min SLP designs often maximize the minimum such margin over users and symbol vectors (Bo et al., 2021).

An equivalent PSK-oriented description appears in terms of a safety margin. After rotating the noiseless received symbol $N_t$1 by the conjugate of the intended symbol $N_t$2, one defines

$N_t$3

with $N_t$4. Requiring $N_t$5 yields two linear inequalities in the real and imaginary parts of the effective received signal, which is why many PSK SLP problems reduce to quadratic programs with linear inequality constraints after real-valued reformulation (Liu et al., 2023).

For QAM, the geometry is more heterogeneous because inner, edge, and corner symbols have different decision-region structure. The SEP-constrained QAM literature expresses this through symbol-dependent inequalities on the residual mismatch

$N_t$6

with two-sided bounds for inner components and one-sided bounds for outer components. This produces tractable linear inequalities in the real and imaginary parts of $N_t$7, and it is precisely this decision-region asymmetry that enables constructive interference to be interpreted as admissible symbol perturbation rather than exact symbol matching (Liu et al., 2018).

A broader geometric generalization is the constructive interference region (CIR). One generic parameterization writes the target point as

$N_t$8

so the received noiseless symbol is allowed to move inside a symbol-dependent constructive region rather than being forced to coincide with the nominal constellation point. This CIR viewpoint supports both PSK and QAM formulations and underlies later weighted-MMSE and NNLS-based SLP designs (Wang et al., 2022).

3. Optimization criteria and structural interpretations

A large part of the SLP literature concerns the choice of objective. One classical family maximizes a symbol-wise margin subject to a power constraint, or minimizes power subject to constructive-region or SEP guarantees. In the PSK case, the benchmark max-min fairness problem can be stated as maximizing the minimum $N_t$9 subject to $K$0, and the corresponding convexity is one reason this class became a standard reference point for later low-complexity and learning-based approximations (Bo et al., 2021).

A more structural line of work shows that SLP is not, in general, an arbitrary nonlinear mapping. Under energy minimization for QAM, the optimal SLP solution can be written exactly as

$K$1

so the effective design variable is a symbol perturbation $K$2 constrained by decision-region inequalities. In this sense, SLP is exactly symbol-perturbed zero forcing under the corresponding energy-minimization criterion (Liu et al., 2018). A later treatment extends the same viewpoint and shows that SLP can be represented as perturbed ZF and, with modulo detection, as a vector-perturbation (VP) scheme augmented by additional continuous perturbations and, for some objectives, nullspace terms (Liu et al., 2021).

The objective need not be geometric margin alone. For high-order QAM, average symbol-error-rate minimization has been formulated directly using exact Q-function expressions that depend jointly on the normalized transmit signal $K$3 and a symbol-level rescaling factor $K$4. This yields a non-convex but explicitly noise-aware SLP problem,

$K$5

and it shifts the design interpretation from region feasibility to probabilistic error minimization (Wang et al., 2023).

A closely related direction replaces hard constructive-region targeting by weighted MMSE inside the CIR. In CI-WMMSE, the desired point is $K$6 with $K$7, and the objective minimizes the expected weighted MSE between the scaled received vector and this constructive target. After eliminating the transmit vector in closed form, the remaining optimization becomes a nonnegative least-squares problem. The same framework contains WMMSE/MMSE and CI-ZF/ZF as special cases, which makes explicit that SLP can interpolate between interference-exploitation and classical estimation-oriented precoding viewpoints (Wang et al., 2022).

4. Practicality, computational burden, and learning-based implementations

The main practical obstacle in SLP is the symbol-by-symbol optimization burden. Even when each individual problem is convex or efficiently solvable, a traditional design may require one distinct precoder for every symbol vector. For $K$8-PSK and $K$9 users, that scaling is $l$0, which becomes prohibitive as the number of users grows (Xiao et al., 2022).

One route to practicality is architectural simplification. Grouped SLP (G-SLP) divides users into $l$1 groups and designs symbol-level precoders per group while exploiting intra-group constructive interference and suppressing inter-group interference. This reduces the number of required precoders from $l$2 to $l$3, where $l$4 is the size of group $l$5. The resulting trade-off is explicit: fewer groups preserve more of the full SLP gain, while more groups produce much lower computational complexity (Xiao et al., 2022).

A second practical issue arises in multi-level constellations. In conventional QAM-oriented SLP, the receiver often needs a symbol-dependent rescaling factor for correct demodulation. This creates a symbol-level feed-forward overhead that can offset the gains of SLP. A block-level rescaling method addresses this by first solving the conventional per-symbol SLP problem to obtain constructive scaling values $l$6, and then performing in-block power allocation so that

$l$7

which implies a common rescaling factor over the block. The optimal allocation has the exact closed form

$l$8

and the paper states that this does not increase BS-side complexity (Li et al., 2020).

A third line of work uses machine learning. An early example is the Efficient Precoding Neural Network (EPNN), which learns a CSI-to-precoder map in an unsupervised-like manner through a loss tied directly to the constructive-interference QoS metric rather than to optimizer-generated labels. For $l$9 and $k$0, the reported online execution time is about 1.42\% of CVX on CPU and 0.15\% on GPU, while the SER degradation relative to convex SLP is reported as less than 2 dB (Bo et al., 2021). A related deep-unfolded design, SLP-SDNet, unfolds a proximal interior-point method for strict PSK SLP and is trained without target labels using a Lagrangian-based loss; it matches optimization-based strict SLP below $k$1 dB SINR and requires about 8% more transmit power above that level, while giving about a $k$2 decrease in average execution time per symbol (Mohammad et al., 2021).

Learning-based SLP has also moved beyond fixed modulation and fixed receiver rules. The AMPD-NN framework jointly learns per-user modulation orders, the symbol-level transmit precoder, and the receiver detection rule under a sum-rate requirement and a transmit power constraint, with the objective of reducing average SER. Its two modules are MOP-NN for modulation-order prediction and SLPD-NN for symbol-level precoding and detection. This departs from conventional SLP formulations in which $k$3 and $k$4 are fixed in advance, and it suggests that the communication chain itself can be learned jointly rather than only the precoder (Liu et al., 2022).

When deployment memory is critical, unfolded robust SLP networks have also been quantized. The robust relaxed-phase designs RSLP-BDNet and RSLP-TDNet report model compression of approximately 21.33\times and 13\times, respectively, relative to full-precision RSLP-DNet, while maintaining about 89%–95% of optimization-based robust SLP performance (Mohammad et al., 2021).

5. Robust, hardware-constrained, and application-specific extensions

Robust SLP has been developed for settings in which CSI is imperfect or shared only approximately. In overlay cognitive radio, the CBS exploits shared PU data and CSI to shape its transmission so that both the cognitive users and the primary users satisfy symbol-wise safety-margin constraints. Under perfect CSI, the resulting problem is a quadratic program with linear inequalities; under imperfect CSI, both norm-bounded and AQNM-based stochastic robustifications still reduce to quadratic objectives with linear inequality constraints. A central point is that cross-network interference is not treated as something that must always be canceled; it can be constructive in both directions (Liu et al., 2023).

Hardware constraints lead to different SLP formulations. In IRS-assisted MU-MISO downlink with one-bit DACs at the BS, the per-symbol transmit vector must lie in a four-point one-bit alphabet, and the joint problem over the one-bit SLP vectors and IRS phases becomes a mixed integer nonlinear program. The proposed solution alternates a dual mirror-descent plus maximum block improvement method for the one-bit SLP subproblem with accelerated projected gradient for the IRS phase update. The resulting design improves BER over one-bit quantized linear precoding and over no-IRS baselines, while explicitly optimizing under the one-bit constraint instead of quantizing a continuous solution afterward (Wang et al., 2020).

Integrated sensing and communications (ISAC) has enlarged the scope of SLP from communications-only beamforming to joint waveform design. In wideband FTN ISAC, the optimization variable is a whole block $k$5, because faster-than-Nyquist signaling induces temporal interference in addition to spatial multiuser interference. The communication side imposes blockwise constructive-interference constraints, while the sensing side minimizes the MMSE for target-response-matrix estimation. The resulting problem is non-convex and is handled by minorization or successive convex approximation, with binary penalty search used to solve the convex QCQP subproblems (Liao et al., 2023).

A different extension appears in AFDM. There, the transmitter uses uplink-estimated AF-domain CSI and channel reciprocity to design an AFDM-domain symbol-level waveform so that the received symbols already lie in the desired PSK constructive regions. The downlink SLP problem is first formulated as an SOCP and then reduced, through Lagrangian and KKT analysis, to a simplex-constrained quadratic program in the dual variable $k$6,

$k$7

with the primal transmit vector recovered as

$k$8

The explicit goal is to offload receiver-side equalization and channel-estimation burden to the base station, enabling direct symbol detection at the user (Tang et al., 17 Aug 2025).

6. Limitations, misconceptions, and research directions

A common misconception is that SLP is simply a vague nonlinear alternative to linear precoding. Several results contradict that interpretation. Under energy minimization, SLP is exactly symbol-perturbed ZF; with modulo detection, extended SLP becomes closely connected to vector perturbation; and for PPAP-type objectives, nullspace perturbations become important while for total-power minimization they can be useless (Liu et al., 2018, Liu et al., 2021). This suggests that SLP is better viewed as a family of structured symbol-domain optimization problems than as an unrestricted nonlinear mapping.

A second misconception is that SLP is only a high-SNR constructive-geometry tool. The average-SER minimization work for high-order QAM explicitly incorporates the noise variance through exact Q-function expressions and a rescaling factor, and CI-WMMSE incorporates noise distribution information through an expected weighted-MSE objective. Both were introduced precisely to address the limitation that purely geometric CI-region methods tend to show their strongest gains mainly in the high-SNR regime (Wang et al., 2023, Wang et al., 2022).

A third issue is that SLP gains are not uniform across all modulation orders and all objectives. Under total transmit power minimization for QAM, there is a formal result that plain ZF becomes near-optimal as the QAM order grows very large, which implies that the relative advantage of SLP is strongest for lower-order or moderate-order QAM in that criterion (Liu et al., 2021). This does not negate SLP; it narrows the conditions under which its extra symbol-level degrees of freedom are most valuable.

The literature also makes clear that many SLP designs remain dependent on assumptions that are favorable but restrictive: perfect CSI at the BS, symbol-level optimization latency that is acceptable in real time, fixed system dimensions, and training distributions that match deployment in learning-based implementations. Practical remedies such as grouped SLP, block-level rescaling, deep unfolding, adaptive modulation and learned detection, and application-specific reformulations in CR, IRS, ISAC, and AFDM all address parts of this gap, but they do not eliminate it (Xiao et al., 2022, Li et al., 2020, Liu et al., 2022).

Taken together, these developments portray SLP as a broad design framework rather than a single optimization template. Its unifying premise is consistent across formulations: the transmitter should exploit instantaneous symbol knowledge and modulation geometry to shape interference constructively. What varies is the objective—margin, power, SEP, weighted MSE, sensing MMSE, throughput, or robustness—and with it the mathematical machinery, which now ranges from convex programs, NNLS, MM, and dual QP reductions to deep unfolding and end-to-end learned transmitter-receiver designs.