CTR1: Coordinated Transmit-Receive Strategy

• CTR1 is a zero-forcing beamforming strategy that restricts each user to its strongest stream per PRB, reducing the stream-selection complexity in MU-MIMO uplink.
• It integrates coordinated ZF processing with per-time-slot power management and practical modulation and coding schemes in an OFDMA framework.
• CTR1 achieves near-optimal throughput and fairness with low complexity, outperforming or matching BD and CTRF under diverse user densities.

Coordinated-Transmit-Receive-1 (CTR1) is a Zero-Forcing (ZF) beamforming strategy for the uplink of OFDMA-based MU-MIMO systems with multi-antenna users in which exactly one, specifically the strongest, data stream per scheduled user is enabled in each scheduled Physical Resource Block (PRB). In the formulation reported for single-cell MU-MIMO uplink with multi-channel resource allocation, CTR1 is studied alongside Block Diagonalization (BD), which enables all possible streams per scheduled user, and Coordinated-Transmit-Receive-Flexible (CTRF), which allows an arbitrary subset of a user’s streams in a PRB. Its defining role is to reduce the stream-selection space to the user level while retaining the best eigenmode per user, thereby coupling ZF receive processing, per-time-slot power management, practical Modulation and Coding Schemes (MCS), and proportional-fair scheduling within a tractable radio-resource-management (RRM) framework (Marques et al., 28 Sep 2025).

1. Definition and operating assumptions

CTR1 is defined by the restriction that only stream $1$, described as the strongest stream, is eligible for transmission for any scheduled user in any scheduled PRB. In the notation of the uplink RRM problem, the stream-selection variable $v^c_{u,s}\in\{0,1\}$ indicates whether stream $s$ of user $u$ is selected in PRB $c$, and the CTR1 constraint is

$$v^c_{u,s} = 0, \quad \forall u \in \mathcal{U}, \forall c \in \mathcal{C}, ~s \neq 1.$$

The total number of scheduled streams in each PRB is additionally limited by the number of BS antennas: $$\sum_{u \in \mathcal{U}} \sum_{s \in \mathcal{M}_U} v^c_{u,s} \leq M_B, \quad \forall c \in \mathcal{C}.$$ For CTR1, only $s=1$ can be active, and only that stream can receive non-zero power under the per-user power constraint across the time-slot: $$\sum_{c \in \mathcal{C}} \sum_{s \in \mathcal{M}_U} P^{c}_{u,s} \leq P_{U}, \quad P^{c}_{u,s} \geq 0.$$ The system model assumes a single-cell MU-MIMO uplink with multi-antenna users, OFDMA, and multi-channel resource allocation; each user has $M_U$ antennas, the base station has $v^c_{u,s}\in\{0,1\}$0 antennas, and scheduling and resource management are carried out over an entire time-slot because of power management. The evaluation reported for CTR1 assumes ZF beamforming at the BS with perfect channel state information (CSI), no signaling overhead, full buffers, and practical MCS for rate adaptation rather than the Shannon formula (Marques et al., 28 Sep 2025).

2. Mathematical role in uplink radio resource management

CTR1 is embedded in a weighted-sum-rate optimization over the full time-slot. The general uplink RRM objective is

$v^c_{u,s}\in\{0,1\}$1

subject to per-user sum-power limits, the per-PRB ZF stream cap, binary stream-selection variables, and the CTR1 stream restriction above. The rate of stream $v^c_{u,s}\in\{0,1\}$2 of user $v^c_{u,s}\in\{0,1\}$3 in PRB $v^c_{u,s}\in\{0,1\}$4 is represented through the MCS-based function

$v^c_{u,s}\in\{0,1\}$5

where $v^c_{u,s}\in\{0,1\}$6 is the effective channel gain for that stream under the selected stream set $v^c_{u,s}\in\{0,1\}$7.

Within this formulation, CTR1 converts the stream-allocation problem into a user-selection problem. For each user and PRB, only the strongest, also described as the principal eigenmode, is considered; in practice, this is the first singular vector in the user’s channel matrix for the given PRB. Because only one stream per user is present, intra-user interference is never an issue, and the ZF processing is applied across the scheduled users’ best streams. This suggests that CTR1 preserves the inter-user nulling structure of ZF while eliminating the intra-user stream-coupling choices that dominate the combinatorics of BD and CTRF (Marques et al., 28 Sep 2025).

3. Position relative to BD and CTRF

The CTR1, BD, and CTRF strategies differ primarily in how many streams per user per PRB they permit and, correspondingly, in scheduling granularity and ZF nulling scope.

Strategy	Streams per user per PRB	Noted properties
CTR1	1 (strongest)	Low complexity; user-level scheduling; only inter-user ZF nulling
BD	All ($v^c_{u,s}\in\{0,1\}$8)	High complexity; user-level scheduling; both intra- and inter-user nulling
CTRF	Flexible ($v^c_{u,s}\in\{0,1\}$9)	Highest complexity; user × stream-level scheduling; both intra- and inter-user nulling

CTR1 is therefore not merely a low-rank approximation of BD. Its defining simplification is that the search space is restricted to deciding whether the strongest stream of a user is scheduled in a PRB. The reported heuristic correspondingly searches over users rather than over streams. By contrast, BD enables all possible streams per scheduled user, and CTRF allows an arbitrary subset of a user’s streams in a PRB. The paper characterizes CTR1 as simpler, more robust to power management, and often close to CTRF in throughput when multiuser diversity is strong, especially for moderate-to-large user populations (Marques et al., 28 Sep 2025).

4. Stream enabling, scheduling, and power management

CTR1 uses per-user strongest-stream enabling. For each user $s$0 and PRB $s$1, the channel matrix is decomposed, eigenmodes are ordered by strength, and only the first eigenmode is eligible for selection. The proposed heuristic, described as an efficient heuristic based on greedy-up searches for stream-sets that provides feasible solutions, operates over the entire time-slot and considers fairness, practical MCS, and all RRM processes. In the CTR1 case, the heuristic selects which users’ strongest streams to schedule in which PRBs, subject to the power and ZF constraints (Marques et al., 28 Sep 2025).

The RRM framework includes stream selection, power management, beamforming, MCS selection, and proportional fairness weights updated in each time-slot. Two power-management schemes are reported. The first is two-step power management (TPM), a more complex method with per-user joint power allocation over all scheduled PRBs and streams using water-filling over a continuous approximation, followed by MCS-aware greedy adjustment. The second is equal power management (EPM), in which a user’s total power budget is split equally across all enabled streams. For CTR1, where only one stream per scheduled user is allowed, the power allocation is described as straightforward, and the strategy is reported as robust to suboptimal power allocation. The paper attributes this robustness to the fact that each user sends only on its best channel eigenmode, so equal splitting does not waste power on weak intra-user streams (Marques et al., 28 Sep 2025).

5. Reported performance regimes and fairness properties

The reported performance of CTR1 is scenario-dependent. In Rural Macro (RMa) scenarios, BD can outperform CTR1 when the number of users is small, because enabling multiple streams per user is then beneficial. For large user counts, however, CTR1’s performance is reported to be very close to CTRF’s, with geometric mean rates typically within 10–15% of CTRF’s, and it often surpasses BD at high user counts. In Urban Macro (UMa) scenarios, CTR1 is reported to match CTRF across the board, even for small user numbers, and the abstract concludes that CTR1 emerges as an alternative to CTRF due to similar performance (Marques et al., 28 Sep 2025).

The same study reports that system parameters can substantially impact the performance of the ZF strategies, and that BD performance is more impaired by a simpler power-management scheme than CTR1 and CTRF. Under EPM versus TPM, CTR1 and CTRF experience only a $s$2 drop in geometric mean throughput, whereas BD can lose up to 35% in performance. The paper also emphasizes proportional fairness: the optimization weights users by past achieved rates, and because CTR1 schedules only the principal stream of each user, balancing fairness among users is described as easier when the number of users is moderate to large. A plausible implication is that CTR1 derives much of its operating advantage not from aggressive per-user multiplexing, but from exploiting multiuser diversity under realistic MCS-constrained RRM (Marques et al., 28 Sep 2025).

6. Terminological context and relation to earlier coordinated Tx-Rx research

The term “coordinated transmit-receive” predates the specific CTR1 designation. In an earlier downlink MU-MIMO setting, coordinated Tx-Rx beamforming referred to joint transmit and receive processing in which each user applies a pre-receiver filter, an effective channel $s$3 is formed, and block diagonalization is applied on the stacked effective channels to null inter-user interference after receive filtering. That work also proposed adaptive stream selection based on QR-decomposition and the minimization of the noise-amplification metric

$s$4

with a fairness constraint ensuring that each user gets at least one stream, and reported a BER improvement of about $s$5 dB at a target BER of $s$6 relative to a conventional coordinated Tx-Rx beamforming algorithm (An et al., 2010).

A common source of confusion is therefore the shared acronym family. CTR1 is not the adaptive coordinated Tx-Rx scheme of the downlink paper. The 2010 scheme adapts the number of streams per user according to instantaneous channel conditions, whereas CTR1 fixes the uplink allocation rule to exactly one strongest stream per scheduled user per PRB (An et al., 2010). Related coordination concepts also appear in networked integrated sensing and communications, where multiple BSs jointly optimize coordinated transmit beamforming for information and sensing signals and jointly process echoes for target detection; in that setting the optimization is formulated as a non-convex QCQP and solved via semi-definite relaxation with provable rank recovery (Cheng et al., 2023). This suggests that CTR1 should be understood as a specific uplink ZF stream-enabling strategy within a broader coordinated transmit/receive lineage, rather than as a generic label for all joint transmit-receive processing.