Central Zero-Forcing (CZF)

• Central Zero-Forcing (CZF) is a centralized beamforming method that computes a global zero-forcing precoder from full channel state information to eliminate inter-user interference.
• It is applied in cooperative C-RAN, massive MIMO, and RIS-assisted systems, relying on ample spatial degrees of freedom and accurate CSI feedback.
• The technique introduces challenges such as CSI quantization errors and fronthaul data load, prompting schemes like PAQ to balance feedback accuracy and beamforming gain.

Searching arXiv for recent and foundational papers on Central Zero-Forcing and closely related formulations. Central Zero-Forcing (CZF) is a centralized linear precoding architecture for multiuser downlink transmission in which a single processing entity computes a zero-forcing beamformer from global or full channel state information and coordinates multiple transmit dimensions so that inter-user interference is nulled in the estimated channel domain. In cooperative C-RAN, this entity is the joint precoding matrix computation unit (JPCU); in massive and XL-MIMO, it is the base-station-side global precoder; and in RIS-assisted XL-MIMO, it is the full-array inversion applied to the selected visibility-region channel matrix (Arad et al., 2020).

1. Centralized zero-forcing principle

In the communications literature represented here, CZF denotes zero-forcing performed from a central controller with access to global CSI, rather than per-transmitter or per-subarray local CSI only. The defining property is that beamforming vectors are designed jointly across all active transmit dimensions. In the C-RAN formulation, a cluster of $M$ RRHs, each with $N_t$ antennas, jointly serves $Q$ single-antenna mobile stations under single-user decoding, and the JPCU reconstructs each global channel vector

$$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$

The standard CZF precoder is then the ZF beamformer designed at the JPCU from the quantized global CSI (Arad et al., 2020).

The null-space construction is the usual one. For user $q$, the beamformer is formed from the null space of the estimated channels of all other users,

$$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$

where $Z_q$ has columns forming an orthonormal basis for the null space of $\{\hat h_j\}_{j\in\mathcal{Q}_{-q}}$. Consequently, $q_q$ is orthogonal to the estimated interfering channels, not necessarily to the true channels, because the central unit only knows quantized CSI. Under perfect CSI, the corresponding ideal precoder $q_q^\star$ is obtained by replacing $N_t$0 with $N_t$1 (Arad et al., 2020).

Feasibility follows the standard broadcast-channel ZF condition. The composite transmit dimension must be large enough to null all multiuser interference, so CZF is meaningful only when the cluster has enough spatial degrees of freedom to support the simultaneously served users; in the fully cooperative view this requires $N_t$2, and more precisely the null space used for user $N_t$3 has dimension $N_t$4 (Arad et al., 2020).

2. Cooperative C-RAN formulation

The canonical C-RAN CZF model is a cooperative downlink in which local channel estimates are pushed upward to a centralized processor over rate-limited links. Each RRH $N_t$5 estimates its local channels $N_t$6, quantizes them, and sends the indices to the JPCU, which reconstructs the global channel seen by each user. The received signal at user $N_t$7 is

$N_t$8

and after stacking the per-RRH transmit vectors and applying linear precoding, the system is equivalent to a MISO broadcast channel with desired beam $N_t$9, residual multiuser interference, and noise (Arad et al., 2020).

The paper first analyzes a simplified equal-power allocation under a short-term per-RRH power constraint $Q$0, with

$Q$1

With $Q$2, the imperfect-CSI SINR is written as

$Q$3

and the ergodic achievable rate is

$Q$4

The perfect-CSI benchmark is

$Q$5

with rate loss

$Q$6

This formulation makes explicit that CZF in C-RAN is not only a beamforming problem but also a fronthaul-CSI architecture: the central processor is only as effective as the CSI that can be transported to it (Arad et al., 2020).

3. Imperfect CSI, feedback scaling, and rate-loss analysis

The core analytical result in the C-RAN treatment is an upper bound on the rate loss induced by incomplete CSI. The model assumes that large-scale coefficients $Q$7 are known perfectly, while each directional component $Q$8 is quantized independently using RVQ, with estimated channel

$Q$9

Under this model, the paper proves

$$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$0

where the two terms separately bound desired-signal mismatch and residual interference effects (Arad et al., 2020).

The asymptotic implication of the bound is the most important systems-level statement: the gap shrinks like

$$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$1

so the CSI feedback budget must increase with SNR and user count to prevent the network from becoming interference-limited. The paper states explicitly that, as in classic finite-feedback ZF for the broadcast channel, keeping the degrees of freedom requires the bit budget per channel to grow roughly linearly with SNR in dB and with $$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$2 (Arad et al., 2020).

This identifies the central weakness of CZF in fronthaul-limited architectures. Zero-forcing itself is not the principal obstacle; the bottleneck is the cost of making global CSI available at the central processor with sufficient fidelity. A plausible implication is that centralized interference nulling and CSI transport must be treated as a single design problem rather than as separable PHY and fronthaul layers.

4. Precode-and-Quantize as a CZF-compatible reduction of CSI overhead

The same paper proposes a CZF-compatible CSI-sharing scheme called precode and quantize (PAQ). Instead of quantizing full $$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$3-dimensional local channels, each RRH first applies a local front-end precoding matrix $$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$4 to create lower-dimensional effective channels

$$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$5

The overall effective channel is then

$$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$6

Each RRH chooses which users to null locally through a selection policy $$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$7, and $$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$8 is the projection matrix into the null space of the discarded users’ channels (Arad et al., 2020).

After this local dimensionality reduction, the JPCU designs a reduced-space CZF precoder,

$$h_q = [h_{q,1}^\dagger,\ldots,h_{q,M}^\dagger]^\dagger \in \mathbb{C}^{MN_t\times 1}.$$9

and the transmitted beamformer is

$q$0

Because the effective channels live in $q$1 dimensions rather than $q$2, the same per-RRH feedback budget is distributed over fewer active users, so each channel gets $q$3 bits instead of $q$4, and the quantization error scales like

$q$5

The scheme also reduces fronthaul data load, since each RRH handles only $q$6 effective streams instead of all $q$7 (Arad et al., 2020).

The analysis shows a three-term rate-loss bound in which the quantization-related terms have the same structure as in standard CZF after the substitutions $q$8 and $q$9, but there is an additional irreducible array-gain penalty

$$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$0

which does not vanish with $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$1. The conclusion is therefore not that PAQ dominates CZF uniformly, but that it trades some array gain for more accurate CSI feedback and lower fronthaul load. When the feedback budget is limited, PAQ may outperform conventional CZF because it suppresses quantization-induced interference more efficiently, even though it sacrifices some beamforming gain (Arad et al., 2020).

5. Generalizations in massive MIMO and XL-MIMO

In massive MIMO literature, CZF typically refers to a centralized precoder computed at the base station using global CSI, and the Reduced Channel Zero-Forcing (RCZF) class can be read as a generalization of this idea. RCZF replaces full-channel nulling by nulling on a reduced channel $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$2, with conditions $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$3 for $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$4 and $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$5. Ordinary ZF is recovered by choosing $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$6. The paper proves that interference cancellation is possible only if the precoding is RCZF, and that if precoding is RCZF, then there exists a detection matrix $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$7 that cancels interference; with RCZF precoding, MMSE-IRC and QR-MLD provide complete interference suppression asymptotically (Mineev et al., 2022).

In XL-MIMO, classical ZF functions as the centralized benchmark that later approximations try to emulate at lower complexity. The low-complexity XL-MIMO paper states that classical ZF is a full centralized array-wide computation and proposes Mean-Angle Based Zero-Forcing (MZF) and Tensor Zero-Forcing (TZF) as approximations. The reported computational scaling is $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$8 for classical ZF, $$q_q = \frac{(\hat h_q^\dagger Z_q)^\dagger}{\|\hat h_q^\dagger Z_q\|},$$9 for MZF, and $Z_q$0 for TZF, reflecting the cost of replacing a global inversion by structured reduced-dimension problems (Ribeiro et al., 2021).

An explicit contemporary use of the term appears in RIS-assisted XL-MIMO, where three schemes are compared: central zero-forcing (CZF), local zero-forcing (LZF), and maximum ratio transmission (MRT). There, CZF uses the whole selected VR-based channel matrix across the array, rather than independent subarray-local inversions. The simulation results state that, when equal priority is given to near-field and far-field users, the proposed design improves the sum of the weighted minimum spectral efficiency by $Z_q$1, $Z_q$2, and $Z_q$3 with CZF, LZF, and MRT, respectively, compared to equal power allocation and random phase shifts; CZF achieves the best performance, while LZF offers comparable results with lower complexity (Cao et al., 27 Sep 2025).

6. Related terminology, contrasts, and recurring ambiguities

Several nearby terms are distinct from CZF. Constructive interference zero forcing (CIZF) is not centralized zero forcing; it is a symbol-by-symbol linear precoding method that cancels only interference terms that do not add to the intended signal power. In the BPSK formulation, a cross-user term is constructive when

$Z_q$4

and the precoder retains such terms instead of nulling them all. This makes CIZF a modification of the zero-forcing principle rather than a synonym for CZF (Christopoulos et al., 2013).

A separate and common ambiguity is purely acronymic. In mathematical logic, “CZF” usually denotes Constructive Zermelo–Fraenkel set theory, not central zero-forcing. Lubarsky proves that CZF + full Separation is equiconsistent with Second Order Arithmetic (Lubarsky, 2015), and Haykazyan studies $Z_q$5 with a Reinhardt set and shows that it interprets $Z_q$6 with a cofinal elementary embedding $Z_q$7 (Jeon, 2021). This collision of notation is terminological rather than conceptual.

Zero forcing in graph theory is likewise unrelated to communication-theoretic CZF. There, zero forcing is an iterative coloring process, and the zero forcing polynomial

$Z_q$8

counts zero forcing sets by size rather than beamforming configurations (Boyer et al., 2018). The shared phrase “zero forcing” refers broadly to an elimination mechanism, but the graph-theoretic and linear-precoding usages belong to different research programs.

Across these contexts, the communications meaning of Central Zero-Forcing remains stable: a centrally computed, globally coordinated ZF precoder whose effectiveness depends on the available spatial degrees of freedom and on how accurately global CSI can be acquired and transported. The strongest recurring theme is therefore architectural rather than purely algebraic: CZF is valuable when centralized CSI is sufficiently accurate, and it becomes costly when the system is limited by feedback, fronthaul, or full-array inversion complexity (Arad et al., 2020).