ETSC-MM: Optimizing Pilot Sequences

• ETSC-MM is a majorization–minimization algorithm that designs non-orthogonal pilot sequences to reduce multi-user interference in overloaded multi-cell systems.
• It iteratively minimizes the extended total squared correlation metric using surrogate quadratic functions, accommodating both unimodular and non-unimodular constraints.
• The algorithm approaches theoretical lower interference bounds, generalizing previous methods and enabling improved performance in massive MIMO and dense cellular deployments.

The ETSC-MM algorithm is a majorization–minimization (MM) approach developed to design non-orthogonal pilot sequences for multi-cell interference networks, focusing on minimizing overall multi-user interference when the number of users exceeds the sequence length. By optimizing the extended total squared correlation (ETSC) metric, which accounts for both intra-cell and inter-cell interference via a general interference power factor matrix, the ETSC-MM framework produces sequence sets that achieve (or approach) fundamental theoretical lower bounds on total interference, even under challenging system parameters that defeat previous two-cell or orthogonality-based constructions (Gu et al., 19 Oct 2025). The method flexibly accommodates both unimodular (constant-envelope) and non-unimodular constraints, enabling practical implementation across massive MIMO, CDMA, and dense cellular deployments.

1. Problem Background and Motivation

Classical pilot sequence designs in wireless systems are centered on orthogonality to eliminate pilot contamination. However, when the user load $K$ per cell surpasses the pilot sequence length $\tau$ (i.e., $K \geq \tau$), mutually orthogonal sequences are impossible, leading to inescapable interference. Moreover, the disparity in channel power between home-cell and neighboring-cell users, characterized by the interference power factor matrix $\mathbf{B}$, necessitates refined sequence design criteria beyond the standard total squared correlation (TSC). The ETSC metric generalizes TSC to account for these inter-cell disparities by defining

$$\mathrm{ETSC}(\mathcal{S}, \mathbf{B}) = \| \mathbf{G}_{(\mathbf{B})} \|_F^2,$$

where $\mathbf{G}_{(\mathbf{B})}$ is the extended Gram matrix whose $(i,j)$ block is $\sqrt{\beta_{i,j}}\,\mathbf{S}_i^H \mathbf{S}_j$. The goal is to design $\{\mathbf{S}_j\}$ to minimize $\mathrm{ETSC}(\mathcal{S}, \mathbf{B})$ under per-sequence norm (or unimodular) constraints, thereby reducing pilot contamination in multi-cell environments.

2. Theoretical Foundation: Extended Welch Bound and Optimality

The ETSC-MM algorithm is underpinned by a generalization of the Welch bound, termed the “new extended Welch bound,” which quantifies the theoretical minimum achievable ETSC for a given system: $$\mathrm{ETSC}(\mathcal{S},\mathbf{B}) \geq \frac{K^2}{\tau} \sum_{i=0}^{J-1} \sum_{j=0}^{J-1} \beta_{i,j}$$ provided $\mathbf{B}$ is positive definite. Equality holds if and only if each cell’s sequence set $\mathbf{S}_j$ forms a Welch-bound-equality (WBE) sequence set, which simultaneously minimizes mutual correlations within cells and across cells weighted by interference factors. This extends prior two-cell, $K \leq \tau$ results, making the approach applicable to arbitrary cell/user configurations.

3. Majorization–Minimization (MM) Optimization Scheme

Direct minimization of the ETSC metric leads to a quartic objective in the sequence elements, complicating classical optimization. The ETSC-MM algorithm solves this by employing an MM framework:

• The current sequence stack $x^{(l)}$ is used to construct a surrogate quadratic function $u(x, x^{(l)})$ that majorizes the original ETSC objective at $x^{(l)}$.
• This surrogate is obtained by reformulating the ETSC in terms of the vectorized variable, applying Lemma 1 (matrix upper bounding), and using Kronecker product algebra.
• The update at each iteration solves

$$x^{(l+1)} = \arg\min_{x \in \mathcal{X}} u(x, x^{(l)}),$$

where $\mathcal{X}$ is the constraint set (e.g., per-sequence unit norm or per-symbol unimodularity).

For unimodular constraints,

$$x_n[t] = \exp(\mathrm{i} \cdot \phi(y[n\tau + t])),$$

while for general unit norm constraints,

$x_n = y_n / \|y_n\|,$

where $y$ is computed from the current surrogate (eqs. (37)–(38) in the paper).

This leads to monotonic descent in ETSC, with each update being computationally efficient and parallelizable. The iterative MM steps are continued until convergence.

4. Comparison to Prior Methods and Advantages

The ETSC-MM algorithm generalizes and improves upon prior art:

• Unlike [Wang et al., 2020], which is restricted to two-cell scenarios and $K \leq \tau$, ETSC-MM applies to arbitrary cell ($J$) counts, user numbers $K$, and sequence lengths $\tau$.
• It can produce both unimodular (beneficial for low PAPR hardware) and non-unimodular sequences by appropriate update rules.
• The method does not require a priori knowledge of equality-achieving conditions for the extended Welch bound; instead, it adaptively approaches the lowest permitted ETSC given $\mathbf{B}$ and system size.
• Existing solutions often generate only non-unimodular sequences or require restrictive parameter relationships.

ETSC-MM represents a robust, general-purpose solution that subsumes many previously distinct design cases within a single iterative algorithmic template.

5. Applications in Wireless Communications

Minimizing the ETSC is central to reducing pilot contamination in contemporary wireless systems, especially in:

• Massive MIMO deployments where pilot overhead is a bottleneck,
• CDMA and overloaded signature sequence design,
• Coordinated multi-point (CoMP) and cell-edge scenarios with high frequency reuse,
• Systems with requirements for constant envelope (unimodular) pilots to maximize transmission efficiency.

By producing nearly ETSC-optimal sequence sets for multi-cell deployments with arbitrary user-to-sequence ratios and interference matrices, ETSC-MM enables improved channel estimation quality and thus communication reliability and spectral efficiency.

6. Implementation and Practical Considerations

Algorithm 1 in the source describes the practical implementation, including initialization (e.g., random or WBE-based), memory structure, and projection steps. Each MM update is computationally lightweight, and the framework supports acceleration via SQUAREM or similar schemes. Convergence is monotonic and guaranteed due to the boundedness of the ETSC metric below zero and the majorization construction. For unimodular sequences, the constant envelope update is efficient and hardware-friendly.

The overall computational cost is dominated by linear algebra associated with surrogate construction and vector projections; these are feasible for typical $K$ and $\tau$ up to a few hundred.

7. Future Directions and Open Issues

The ETSC-MM framework establishes a foundation for adaptive sequence design under dynamic interference scenarios. Possible future research avenues include:

• Extending the MM framework to scenarios with time-varying or non-stationary $\mathbf{B}$,
• Adapting the approach for joint pilot and data sequence optimization,
• Applying MM-based design to other performance metrics (e.g., total mean-squared error, SINR-based objectives),
• Incorporating additional hardware constraints (e.g., finite alphabet, quantized amplitudes).

The flexibility of the MM approach and the generality of the ETSC metric suggest the framework may be adaptable to a wider array of interference management problems in heterogeneous and dynamic network environments, provided the surrogate functions and update rules can be appropriately designed to majorize complex, coupled system-level metrics.