MIMO Transmitter Crosstalk

• MIMO transmitter crosstalk is interference between transmit paths arising from hardware feedback, joint signaling, or electromagnetic coupling.
• Effective mitigation techniques, such as SVD-based precoding and Bussgang decomposition, enable enhanced spectral efficiency and reduced distortion.
• Advanced schemes like LC-PW and FF-PW provide significant improvements in adjacent channel power ratios and data rates across both xDSL and mMIMO systems.

Multiple-input multiple-output (MIMO) transmitters with crosstalk constitute a key area of research and practical engineering in both wireline and wireless systems. In such transmitters, the presence of crosstalk—mutual coupling or interference between different transmit paths—can originate from intentional joint signaling (as in xDSL vectoring), hardware-induced feedback (e.g., backward crosstalk in PA arrays), or electromagnetic coupling in antenna subarrays. Accurate characterization, mitigation, and exploitation of crosstalk are fundamental to maximizing spectral efficiency, minimizing distortion, and enabling massive MIMO scalability.

1. Crosstalk Phenomena in MIMO Transmitters

Crosstalk in MIMO transmitters manifests through diverse mechanisms depending on the physical medium and circuit architecture. In xDSL wireline systems, crosstalk splits into in-domain (“self-crosstalk”)—arising from capacitive and inductive coupling inside a cable binder—and out-of-domain (“external” or “alien” crosstalk), which reflects interference from non-coordinated adjacent cables (e.g., ADSL2+ disturbing VDSL2). In mmWave and massive MIMO wireless transmitters, crosstalk arises through electromagnetic coupling at power amplifier (PA) inputs/outputs, mutual coupling in densely packed antenna arrays, and imperfect isolation in multi-chain RF hardware. Backward crosstalk, in particular, is caused by signal leakage from the output of one PA into the input of another, or via impedance mismatches in the transmitter network (Händel et al., 2019).

The distinction between self and external crosstalk is vital for signal processing. Self-crosstalk, often deterministic and confined within a managed system, is amenable to joint precoding or detection. External crosstalk typically requires more advanced statistical modeling and is harder to coordinate or cancel.

2. System Modeling under Crosstalk

2.1 xDSL: Tone-by-Tone MIMO Model

Discrete multi-tone (DMT) xDSL systems are structured as a set of orthogonal tones, each described by:

$$\mathbf{y}(f) = \mathbf{H}(f) \mathbf{x}(f) + \mathbf{n}(f)$$

where $\mathbf{x}(f)$ is the $N$-dimensional transmit vector, $\mathbf{H}(f)$ encodes both the desired and self-crosstalk gains, and $\mathbf{n}(f)$ is colored Gaussian noise aggregating AWGN and external crosstalk. External noise exhibits nontrivial spatial covariance across lines (Nir et al., 2010).

2.2 Wireless Massive MIMO with Nonlinear Crosstalk

In mmWave mMIMO transmitter arrays, crosstalk at the PA level is modeled as linear feedback:

$\mathbf{u} = \Gamma \mathbf{x} + K \mathbf{r}$

$\Gamma$ is the PA gain matrix, $K$ is the crosstalk-coupling matrix (zero diagonal, complex off-diagonals), and $\mathbf{r}$ is the nonlinear PA output. Each PA is generally modeled as a memoryless nonlinear function (cubic polynomial, dual input in advanced models) incorporating both the intended input and crosstalk terms. The aggregate model yields a closed-loop "dirty" transmitter whose outputs deviate from ideal linear amplification (Händel et al., 2019, Prasad et al., 2023).

3. Crosstalk Mitigation Techniques

3.1 Linear Decoupling and Whitening

In xDSL, self-crosstalk can be cancelled using linear detectors such as the zero-forcing (ZF) equalizer. When facing spatially colored external noise, whitening is imperatively applied: the noise covariance $\mathbf{R}_n(f)$ is factorized via Cholesky or square-root decomposition as $\mathbf{x}(f)$0, and the received signal is filtered by $\mathbf{x}(f)$1, yielding a whitened channel:

$\mathbf{x}(f)$2

$\mathbf{x}(f)$3

This transformation facilitates optimal joint precoding and detection under general MIMO crosstalk and arbitrary colored noise (Nir et al., 2010).

3.2 SVD-Based Precoding/Decoding

On each tone, the whitened MIMO channel is diagonalized via SVD:

$\mathbf{x}(f)$4

with $\mathbf{x}(f)$5 diagonalizing the system into parallel eigenmodes. The transmitter applies the precoder $\mathbf{x}(f)$6 and the receiver applies $\mathbf{x}(f)$7, realizing $\mathbf{x}(f)$8 non-interfering SISO subchannels with spatially whitened noise (Nir et al., 2010).

3.3 Digital Predistortion and Post-Weighting in mMIMO

Beam-oriented digital predistortion (BO-DPD) is designed to linearize the output in the desired beam direction but leaves residual nonlinear distortion and crosstalk in other spatial directions. The introduction of a post-weighting (PW) block—after BO-DPD—enables suppression of out-of-beam nonlinear radiation by adaptively mixing nonlinear basis functions across all PAs in a subarray.

Two main schemes exist:

• Fully-featured post-weighting (FF-PW): Applies independent weights to each nonlinear basis for each PA, maximizing flexibility.
• Low-complexity post-weighting (LC-PW): Groups weights according to a geometric rule, reducing hardware and computational complexity while maintaining most of the performance gains (Prasad et al., 2023).

Explicit crosstalk compensation signals are included in the DPD model to further reduce the impact of mutual coupling.

3.4 Bussgang Decomposition and Backward Crosstalk Mitigation

For feedback-induced crosstalk and PA nonlinearities, the Bussgang decomposition allows representing the overall MIMO transmitter output as:

$\mathbf{x}(f)$9

where $N$0 is a deterministic gain-attenuation matrix and $N$1 is uncorrelated distortion. This facilitates closed-form expressions for error analysis and guides the optimization of input power back-off to minimize mean-square error (NMSE) under joint crosstalk and nonlinearity (Händel et al., 2019).

4. Power Allocation and Optimization under Practical Constraints

In wireline vectoring, sum capacity can only be optimized subject to practical constraints, including per-modem power budgets and spectral mask limits:

$N$2

$N$3

Lagrangian dual optimization is employed, producing a generalized water-filling solution where the power allocated to each spatial eigenmode is:

$N$4

where $N$5, $N$6 are dual variables (per-modem and mask constraints) and $N$7 are entries of the unitary precoder (Nir et al., 2010). This dual-loop algorithm converges rapidly with modest computational load for realistic $N$8 and $N$9.

In feedback-crosstalk MIMO transmitters, optimal power back-off (PBO) is derived by minimizing the worst-case NMSE across transmitter branches, resulting in closed-form candidate roots due to the cubic form of the NMSE polynomial in the input power (Händel et al., 2019).

5. Performance Impact and Trade-offs

Extensive simulation in xDSL vectored systems demonstrates that full (Tx+Rx) SVD-based vectoring schemes, which optimize jointly at both ends, gain 20–40% in downstream data rate compared to one-sided ZF-like schemes. The advantage increases with more participant lines and becomes pronounced in high-frequency external crosstalk scenarios (e.g., VDSL2 interference) (Nir et al., 2010).

In mmWave mMIMO transmitters, FF-PW achieves significant suppression of nonlinear out-of-beam radiations—over 20 dB better than prior schemes—with measured adjacent channel power ratios (ACPR) of 63.5 dB. LC-PW achieves approximately 52.2 dB, with a substantially reduced hardware burden (20–50% of the coefficients required by FF-PW), while BO-DPD alone with crosstalk preprocessing achieves as little as 33 dB ACPR (Prasad et al., 2023).

In 2×2 MIMO transmitters, even −50 dB backward crosstalk measurably alters gain and inter-branch distortion correlation. Bussgang-based analysis shows a nonzero optimal PBO; extremely low or high input power both degrade NMSE due to noise dominance or PA nonlinearity saturation. Customized "dirty" precoders that leverage knowledge of the hardware transfer function outperform naive maximum-ratio transmission, but simple sub-optimal variants generally approach optimal performance under mild crosstalk (Händel et al., 2019).

Scheme	Avg. ACPR (dB)	Degree of Hardware Complexity
FF-PW	63.5	Very High
LC-PW	52.2	Moderate
BO-DPD + CTP	33	Low

6. Extensions and Generalizations

The presented analysis and architectures generalize directly to $\mathbf{H}(f)$0 MIMO transmitters and receivers. Bussgang-based decompositions, NMSE back-off optimizations, and generalized SVD-based Tx/Rx design extend naturally to higher dimensional systems, with the primary limitation being computational complexity of matrix factorizations and the hardware scalability of real-time implementations. In mMIMO arrays, reductions in RF-chain count and adaptive grouping (as in LC-PW) are critical for feasibility at scale (Prasad et al., 2023, Händel et al., 2019).

In xDSL, observed rate loss due to granular per-modem and spectral mask constraints is negligible compared to single total-power approaches, supporting their adoption in full-scale deployments (Nir et al., 2010).

7. Key Insights and Physical Implications

Crosstalk in MIMO transmitters is not simply a source of signal degradation but an axis for architectural optimization. System-level mitigation exploits joint transmitter and receiver coordination, advanced nonlinear modeling, and practical constraints-driven power allocation. Explicit recognition and modeling of crosstalk—either as colored spatial noise (xDSL), feedback nonideality (RF circuits), or mutual antenna coupling (massive MIMO)—enables algorithms that approach near-capacity performance under realistic channel and hardware limits.

A plausible implication, supported by the referenced works, is that the combination of (i) empirical crosstalk estimation, (ii) SVD-based or Bussgang-aware Tx/Rx signal processing, and (iii) complexity-optimized implementations (e.g., LC-PW, dual-loop water-filling) collectively constitute the current best practice for scalable, high-fidelity MIMO transmission in the presence of crosstalk (Nir et al., 2010, Prasad et al., 2023, Händel et al., 2019).