Extended Total Squared Correlation (ETSC)

• ETSC is a metric that extends classical total squared correlation by incorporating weighted interference power factors for both intra-cell and inter-cell scenarios.
• It establishes a closed-form lower bound that generalizes the Welch bound, guiding optimal pilot sequence design in overloaded multi-cell settings.
• The ETSC-MM algorithm employs a majorization–minimization framework for iterative sequence optimization, effectively mitigating pilot contamination in dense networks.

The Extended Total Squared Correlation (ETSC) is a metric and framework designed to quantify and minimize interference in multi-cell wireless communication systems where non-orthogonal sequence sets are necessary. ETSC generalizes the classical concept of total squared correlation by weighting inner products among user sequences according to an interference power factor matrix, thus capturing both intra-cell and inter-cell interference. It establishes theoretical lower bounds that unify and extend prior results such as the Welch bound, and underpins novel algorithmic approaches to optimal or nearly optimal pilot sequence construction.

1. Definition and Motivation

The ETSC is introduced to address the challenges of interference in multi-cell wireless networks, such as pilot contamination in cellular systems or overloaded code-division multiple-access (CDMA), where the number of users per cell exceeds the available pilot or signature sequence length (user overload regime).

Let $\mathcal{S}$ denote the full set of pilot (or signature) sequences, partitioned into $J$ cells, each with $K$ sequences of length $\tau$ (with $K \geq \tau$). The key issues are:

• Non-Orthogonality: Orthogonal sequence allocation is often infeasible in overloaded settings.
• Interference Modeling: Both intra-cell (within a cell) and inter-cell (between cells) interference must be considered.

To capture these, the ETSC is defined as the squared Frobenius norm of an extended Gram matrix $G_B$ whose elements are scaled by interference power factors $\beta_{i,j}$ reflecting channel strength disparities among cells. The metric thus extends traditional total squared correlation (TSC) to more realistic multi-cell environments.

2. Mathematical Formulation and Generalized Lower Bound

For the multi-cell setting with $J$ cells, each cell $j$ has a pilot matrix $S_j$ ($\tau \times K$), and the interference power factor matrix $B = [\beta_{i,j}]$ encodes the relative importance of inter- and intra-cell correlations.

The extended Gram matrix is constructed as:

$$G_B = \left[\sqrt{\beta_{i,j}}\, S_{i}^H S_{j} \right]_{0 \leq i,j < J}$$

where $S_{i}^H S_{j}$ is the cross-correlation matrix between the sequences of cells $i$ and $j$.

The ETSC metric is then:

$$\mathrm{ETSC}(\mathcal{S}, B) = \| G_B \|_{F}^2 = \sum_{i=0}^{J-1} \sum_{j=0}^{J-1} \beta_{i,j} \| S_i^H S_j \|_F^2$$

The principal theoretical result is a closed-form lower bound on the ETSC for unimodular sequences when $K \geq \tau$ and $B$ is positive definite:

$$\mathrm{ETSC}(\mathcal{S}, B) \geq \frac{K^2}{\tau} \sum_{i=0}^{J-1} \sum_{j=0}^{J-1} \beta_{i,j}$$

Special cases include:

• Classical Welch bound: $B$ is all-ones ($\beta_{i,j} = 1$), recovering $\frac{N^2}{\tau}$ with $N$ total users.
• Extended Welch bound: Particular parameterizations for two-cell systems with possible $K \leq \tau$.

This bound describes the fundamental minimum achievable ETSC for any sequence set in overloaded multi-cell systems.

3. Optimality Conditions and the Role of the Power Factor Matrix

Achieving the ETSC lower bound requires strong optimality conditions:

• Each cell’s sequence set must be a Welch-bound-equality (WBE) set, fulfilling the optimality constraints for squared correlations known from classical sequence design.
• The interference power factor matrix $B$ must be positive definite. Under this condition, the quadratic form $\sum_{i,j} \beta_{i,j}$ is well-posed, and the lower bound applies tightly.

If $B$ loses positive definiteness, the theoretical argument for the bound fails and direct analytic constructions become nonviable, necessitating more general algorithmic approaches.

4. ETSC-MM Algorithm: Majorization–Minimization Optimization

For cases where analytic sequence construction is infeasible, such as when $B$ is not positive definite or $K<\tau$, the ETSC-MM algorithm is proposed.

Algorithmic features:

• Majorization–Minimization (MM) Framework: The quartic ETSC objective is computationally intractable in general. The MM approach replaces this with a surrogate (majorizer) that is easier to minimize.
• Quadratic Lower Surrogate: By vectorizing $G_B$ and expressing ETSC as a quadratic function in the sequence vectors, the update reduces to standard forms. For non-unimodular sequences, the update is $x_n \leftarrow y_n / \|y_n\|$. For unimodular sequences, each element is set by phase: $x_n[t] = \exp\{i \cdot \arg(y[t])\}$.
• Monotonic Descent and Fast Convergence: Each iteration provably decreases ETSC. Acceleration techniques such as SQUAREM can further speed convergence.

This facilitates efficient sequence construction for arbitrary $B$, $K$, and $\tau$, enabling practical application to a wide class of network configurations.

5. Applications in Wireless Communication Systems

The ETSC framework and ETSC-MM algorithm have several key applications:

• Pilot Contamination Mitigation: In dense multi-cell cellular systems (including 5G), pilot contamination is a major source of channel estimation error. Sequences designed to minimize ETSC simultaneously reduce mean squared error (MSE) in channel estimation.
• Overloaded CDMA: In classical CDMA with $K > \tau$, minimizing ETSC improves both user separation and symbol detection performance.
• Unimodular Sequence Design: ETSC-MS can optimize sequences with constant modulus, yielding a $0$ dB peak-to-average-power ratio (PAPR), beneficial for reducing power amplifier requirements.

Because the ETSC lower bound generalizes the Welch and extended Welch bounds, this approach is adaptable to a wider variety of network scenarios, including those with more than two cells or in the heavily overloaded regime.

6. Practical Implications and Theoretical Significance

• Unified Design Criterion: ETSC provides a comprehensive metric that encompasses intra-cell and inter-cell interference within a single analytic and algorithmic framework.
• Sequence Set Construction: The framework guides analytic and algorithmic design, depending on $B$ and system parameters, for sequence sets achieving near-optimal levels of total squared correlation.
• Extension of Classical Results: ETSC’s generalization of the Welch bound—and the rigorous treatment of weighted (heterogeneous) interference—represents a significant formal advance for both theoretical and applied signal design.
• System-Level Outcomes: Networks employing ETSC-minimizing sequences can achieve lower interference floors, enabling improved throughput, better reliability, and more efficient hardware utilization.

7. Summary Table: ETSC Construction and Application

Aspect	Description	Implication
ETSC Definition	Weighted squared Frobenius norm of extended Gram matrix	Captures total network interference
Lower Bound (Closed-form)	$\frac{K^2}{\tau} \sum_{i,j} \beta_{i,j}$ for unimodular, $B \succ 0$	Generalizes Welch and extended Welch bounds
Optimal Sequence Condition	Each cell must use WBE sequences if $B$ is positive definite	Achieves theoretical minimum interference
ETSC-MM Algorithm	MM-based iterative minimization (supports unimodular and non-unimodular)	Practical sequence generation for arbitrary $B$, $K$, $\tau$
Wireless Applications	Multi-cell pilot/ signature design, overloaded CDMA, PAPR reduction	Directly improves MSE, spectral efficiency, robustness

ETSC establishes the state-of-the-art framework for rigorous and practical non-orthogonal sequence design in modern multi-cell wireless systems, providing performance guarantees and efficient construction methods applicable across a broad class of communication architectures (Gu et al., 19 Oct 2025).