Local Zero-Forcing in Wireless Systems

• Local Zero-Forcing (LZF) is a family of techniques that employ localized channel state information to cancel interference in distributed wireless systems.
• In cell-free Massive MIMO, LZF variants such as full, partial, and protective zero-forcing enable scalable per-AP precoding and combining without global CSI exchange.
• LZF methods, including subarray processing in XL-MIMO and graph-theoretic approaches, achieve reduced computational complexity while balancing interference suppression and signal enhancement.

Local Zero-Forcing (LZF) is a non-uniform term used across several research areas to denote zero-forcing operations carried out with localized information, localized subarrays, or localized user/pilot subspaces rather than by a single centralized global inverse. In wireless communications, the term most often refers to per-access-point or per-subarray precoding/combining based only on local CSI, as in cell-free Massive MIMO and XL-MIMO (Interdonato et al., 2019, Zhang et al., 2022, Cao et al., 27 Sep 2025). In other literatures, however, the same acronym can mean something different, including linear zero-forcing in interference alignment (Razaghi et al., 2011), while some papers do not define an LZF variant at all even though the underlying zero-forcing rule is intrinsically local (Brimkov et al., 2017). This suggests that LZF is best treated as a family of localized zero-forcing constructions rather than a single canonical algorithm.

1. Terminological scope and recurring meanings

The literature supports several distinct uses of “locality” in zero-forcing. In distributed multi-antenna systems, locality typically means that each AP or subarray computes its own precoder or combiner from locally available CSI, without global instantaneous CSI exchange. In graph-theoretic zero forcing, locality refers to the forcing rule itself, since whether a force is legal depends only on a vertex and its immediate neighborhood. In contrast, some papers use a similar acronym for concepts that are not local in this sense.

Context	Meaning of “local” or “LZF”	Representative source
Cell-free Massive MIMO downlink	Per-AP zero-forcing from local CSI; often extended to partial/protective variants	(Interdonato et al., 2019)
Cell-free Massive MIMO uplink	Per-AP local combining followed by LSFD at the CPU	(Zhang et al., 2022)
RIS-assisted XL-MIMO	Per-subarray zero-forcing with local channel matrices	(Cao et al., 27 Sep 2025)
Interference alignment	“LZF” means linear zero-forcing, not local zero-forcing	(Razaghi et al., 2011)
Graph zero forcing	No explicit LZF variant; the basic forcing rule is already local	(Brimkov et al., 2017)
MIMO-OTFS	“LZ” means low-complexity zero-forcing, not local zero-forcing	(Singh et al., 2020)

A useful working distinction is therefore between explicit LZF schemes, such as per-subarray or per-AP ZF, and adjacent constructions that share a locality rationale without using the same name. The latter include local partial zero-forcing, local regularized ZF, grouped subarray ZF approximations, and graph-theoretic zero forcing processes driven by local neighborhoods.

2. Cell-free Massive MIMO downlink: local full-pilot ZF, partial ZF, and protective ZF

A concrete wireless interpretation of LZF appears in the cell-free Massive MIMO downlink, where each AP performs precoding using only its own local CSI. The system in (Interdonato et al., 2019) is a TDD cell-free Massive MIMO downlink with $L$ distributed APs, $M$ antennas per AP, $K$ single-antenna UEs, independent Rayleigh fading, pilot contamination, and no instantaneous CSI exchange with the CPU or among APs. The paper works with the full-rank pilot-domain matrix

$$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$

and with local full-pilot zero-forcing (FZF) as the local-ZF benchmark: $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$ Its ZF property is local and pilot-domain specific: AP $l$ cancels interference toward all pilot directions other than the co-pilot one, but co-pilot UEs cannot be spatially separated.

The same paper then introduces local partial zero-forcing (PZF) and local protective partial zero-forcing (PPZF). Each AP partitions users into strong and weak sets, $S_l$ and $W_l$, with co-pilot users grouped together. PZF zero-forces only the pilot directions associated with $S_l$, using

$$\mathbf{w}^{PZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} \left(\mathbf{E}_{S_l}^{H}\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l\mathbf{E}_{S_l}\right)^{-1} \mathbf{e}_{j_{l,k}}} {\sqrt{\mathbb{E}\!\left\{ \left\| \bar{\mathbf{H}}_{l} \left(\mathbf{E}_{S_l}^{H}\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l\mathbf{E}_{S_l}\right)^{-1} \mathbf{e}_{j_{l,k}} \right\|^2 \right\}}}.$$

This only orthogonalizes the $M$0 selected channels in $M$1. PPZF strengthens the construction by projecting MRT for weak users onto the orthogonal complement of the strong-user pilot subspace, so that strong users are protected from interference generated by weak-user transmissions.

The central design tradeoff is explicit. MRT uses all $M$2 spatial dimensions for signal enhancement. Local FZF spends $M$3 dimensions to null all locally resolvable pilot directions, leaving array gain $M$4. Local PZF spends only $M$5, leaving $M$6. The paper states that PZF and PPZF can substantially outperform maximum ratio transmission and zero-forcing, and that their performance is comparable to regularized zero-forcing; it also emphasizes that the schemes are fully distributed, require no additional front-hauling overhead, and are suitable for APs with very few antennas (Interdonato et al., 2019).

3. Cell-free Massive MIMO uplink: local ZF-type combining and LSFD

The uplink counterpart develops a closely related family of local ZF-inspired combiners. In (Zhang et al., 2022), the system is a cell-free Massive MIMO uplink with $M$7 APs, $M$8 antennas per AP, $M$9 single-antenna UEs, independent Rayleigh fading, MMSE channel estimation, pilot contamination, and large-scale fading decoding (LSFD). The local pilot-space estimate matrix is

$K$0

and each AP forms a local soft estimate

$K$1

after which the CPU combines these via

$K$2

The full-pilot ZF combiner is

$K$3

and satisfies

$K$4

Thus, as in the downlink, local ZF is pilot-subspace zero-forcing rather than user-wise separation: users sharing a pilot remain inseparable.

The paper then studies PFZF, PWPFZF, and LRZF. PFZF is the main local partial-ZF method: strong UEs are combined with partial full-pilot ZF over a reduced pilot subspace, while weak UEs use MR. PWPFZF modifies the weak-user treatment by replacing MR with projected MR through the orthogonal projector

$K$5

LRZF is the local regularized ZF combiner, trading hard nulling for regularized suppression.

A major analytical result is that the paper derives closed-form expressions of the uplink SE for FZF, PFZF, and PWPFZF with LSFD, explicitly incorporating channel estimation errors and pilot contamination. Its numerical conclusions are that LRZF provides the highest SE, while PWPFZF is preferable when the number of pilot sequences is large and the number of antennas per AP is small; PWPFZF also improves the performance of weak UEs and realizes uniformly good service for all UEs in a scalable fashion when combined with fractional power control (Zhang et al., 2022).

4. XL-MIMO and RIS-assisted XL-MIMO: subarray-local ZF

A direct and explicit definition of Local Zero-Forcing appears in RIS-assisted XL-MIMO. In (Cao et al., 27 Sep 2025), an XL-MIMO downlink array with $K$6 antennas is divided into $K$7 subarrays of size

$K$8

and serves near-field users (NFUEs) directly and far-field users (FFUEs) through a RIS. The key statement is that each subarray $K$9 independently applies precoding to nullify intra-group interference within its own subarray, but cannot eliminate intra-group interference caused by other subarrays. This is locality in both a geometric and an algebraic sense.

For NFUEs, the per-subarray LZF precoder is

$$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$0

with local channel matrix

$$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$1

For FFUEs, the corresponding local ZF precoder is

$$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$2

with

$$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$3

These are local pseudo-inverses defined on subarray channel matrices rather than on a full-array channel matrix.

The paper contrasts LZF with centralized ZF (CZF), which uses one global ZF inversion across the full $$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$4-antenna array. LZF instead uses $$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$5 independent local inversions and allows per-subarray per-user power control. It also uses visibility regions (VRs): users are assigned only a subset of subarrays that contribute meaningfully to received power, and LZF uses the same VRs as CZF for simplicity. The stated complexity reduction for the precoding stage is from $$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$6 to $$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$7, and the simulations report that CZF achieves the best performance, while LZF offers comparable results with lower complexity; when prioritizing NFUEs or FFUEs, LZF achieves strong performance for the prioritized group (Cao et al., 27 Sep 2025).

A related XL-MIMO direction does not explicitly use the term LZF but follows the same local-processing rationale. In (Ribeiro et al., 2021), mean-angle based zero-forcing (MZF) partitions the URA into smaller vertical subarrays, groups users by elevation, performs intra-group ZF in the azimuth domain, performs inter-group ZF in the elevation domain, and assembles the beamformer as

$$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$8

The complexity drops from full-array ZF scaling $$\bar{\mathbf{H}}_l = \mathbf{Y}_l \boldsymbol{\Phi} \in \mathbb{C}^{M\times \tau_p},$$9 to $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$0. This suggests a closely related notion of local ZF in XL-MIMO: grouped subarray-based ZF approximations can substantially reduce complexity, but their performance depends on the validity of local plane-wave and angular-grouping assumptions (Ribeiro et al., 2021).

5. Acronym collisions and adjacent constructions in communications

The acronym “LZF” is not stable across communication-theoretic subfields. In the interference-alignment paper (Razaghi et al., 2011), LZF means linear zero-forcing, not local zero-forcing. There the receiver projects onto the orthogonal complement of the aligned interference subspace; for user 1,

$$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$1

and the detection rule is

$$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$2

The paper states that LZF and lattice decoding achieve the same degrees of freedom, but that LZF performs very poorly at finite SNR when channel amplitudes vary, especially under Rayleigh fading.

A different source of ambiguity appears in MIMO-OTFS, where “LZ” denotes low-complexity zero-forcing rather than local zero-forcing. The receiver in (Singh et al., 2020) is a globally exact ZF detector implemented efficiently through the doubly-circulant structure of the OTFS channel matrix, 2D DFT diagonalization, block-wise inversion, and the Schur complement. The paper explicitly states that the proposed LZ receiver provides exactly the same solution as the conventional ZF receiver and reduces complexity from $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$3 to essentially $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$4 under the regime $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$5.

Locality also appears indirectly in lattice-reduction analyses of ZF decoding. The paper on the success probability of the standard ZF decoder proves that SQRD and V-BLAST, which act only by column permutations, have no effect on $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$6; in dimension $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$7, LLL reduction can improve $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$8, with larger $$\mathbf{w}^{FZF}_{l,k} = \frac{\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}} {\sqrt{\mathbb{E}\!\left\{\left\|\bar{\mathbf{H}}_{l} (\bar{\mathbf{H}}_l^{H}\bar{\mathbf{H}}_l)^{-1} \mathbf{e}_{i_k}\right\|^2\right\}}}.$$9 yielding no worse and potentially better $l$0; but for $l$1, LLL can decrease $l$2 (Wen et al., 2018). Since the paper explicitly remarks that its conclusions are highly relevant to localized, blockwise, or reduced-complexity ZF schemes, a plausible implication is that local ZF preprocessing should not assume that “more reduced” always means “more reliable,” especially once local blocks interact.

6. Graph-theoretic locality and broader structural lessons

In graph theory, the literature does not formalize a named variant called Local Zero-Forcing, because the standard zero-forcing rule is already local. In (Brimkov et al., 2017), the rule is: at each integer-valued timestep, a colored vertex $l$3 with a single uncolored neighbor $l$4 forces that neighbor to become colored. The closure $l$5, zero forcing sets, connected zero forcing, restrained zero forcing, bounded-timestep forcing, and forts are all defined from this neighborhood-local mechanism. The paper’s computational approaches reinforce the same point: its infection model uses edge- and neighborhood-level constraints, and its fort formulations encode local obstruction patterns. The closure algorithm runs in $l$6 time because only the closed neighborhood of a newly colored vertex can change forcing status.

The digraph paper (Ferrero et al., 2017) makes the same locality even sharper. In its loopless form, a white vertex is forced when it is the only white out-neighbor of a blue vertex. The paper defines critical and strongly critical sets through the local condition $l$7, and proves exact formulas for iterated line digraphs. If $l$8 satisfies $l$9 and $S_l$0, then

$S_l$1

and if $S_l$2 is $S_l$3-regular with $S_l$4, then

$S_l$5

These results are not about wireless LZF, but they show that localized forcing rules can admit exact combinatorial characterizations when the neighborhood geometry is sufficiently structured.

Taken together, these works suggest three general lessons. First, “locality” in zero-forcing usually means a restriction on the information set or operator domain: local CSI at an AP, a pilot-domain subspace, a subarray, or a graph neighborhood. Second, locality almost always buys scalability or reduced complexity, but does not remove global couplings: co-pilot users remain inseparable in cell-free systems, residual cross-subarray interference remains in subarray-local ZF, and local lattice improvements do not guarantee global ZF improvement beyond $S_l$6 structure. Third, the most successful LZF constructions are not pure local nulling schemes but localized tradeoff schemes—partial ZF, protective ZF, regularized ZF, or grouped subarray approximations—that allocate only part of the available degrees of freedom to interference suppression while retaining signal gain (Interdonato et al., 2019, Zhang et al., 2022, Cao et al., 27 Sep 2025, Wen et al., 2018).