Centroid Estimation

The most fundamental centroid estimator computes the arithmetic mean of all received IQ samples assigned to a specific cluster. For a set of N complex samples z_i, the centroid μ̂ is calculated as:

• Formula: μ̂ = (1/N) Σ z_i
• Optimality: This is the maximum likelihood estimator when the noise is additive white Gaussian (AWGN), minimizing the sum of squared Euclidean distances.
• Variance: The accuracy of the estimate improves with the square root of the number of samples; doubling the precision requires quadrupling the observation window.

In a blind scenario where symbol membership is unknown, centroid estimation is performed iteratively using the K-Means algorithm:

• Initialization: Initial centroids are often seeded using k-means++ to avoid poor local minima, or by partitioning the IQ plane into angular sectors for PSK.
• Assignment Step: Each IQ sample is assigned to the nearest centroid based on Euclidean distance in the complex plane.
• Update Step: New centroids are recalculated as the mean of all samples assigned to that cluster.
• Convergence: The process repeats until centroid movement falls below a threshold, typically converging in fewer than 20 iterations for high-SNR signals.

A Gaussian Mixture Model (GMM) provides a probabilistic alternative to hard K-Means assignment, modeling the received constellation as a sum of K bivariate Gaussian distributions:

• Expectation Step (E-Step): Calculates the responsibility of each Gaussian component for every data point, representing the probability that a sample belongs to a specific constellation point.
• Maximization Step (M-Step): Updates the mean (centroid), covariance, and mixing weight of each component using the weighted statistics from the E-Step.
• Advantage: GMM captures the elliptical spread of clusters caused by IQ imbalance and provides uncertainty estimates, unlike the spherical assumption of K-Means.

Blind centroid estimation recovers the relative geometry of a constellation but introduces an unknown fixed phase rotation. Resolving this phase ambiguity is critical for demodulation:

• Differential Encoding: Transmitting the information in the difference between successive symbols rather than the absolute phase, making the rotation irrelevant.
• Unique Word Correlation: Cross-correlating the recovered constellation against a known synchronization sequence to estimate the rotational offset.
• Asymmetry Exploitation: For non-circularly symmetric constellations like QAM, the rotational angle can be estimated by aligning the recovered centroids with the axes of the IQ plane using principal component analysis or by minimizing the fourth-power phase offset.

A Carrier Frequency Offset (CFO) between the transmitter and receiver local oscillators causes the entire constellation to rotate continuously at a constant angular velocity. This rotation must be halted before centroid estimation can succeed:

• Impact: Without correction, IQ samples from a single constellation point smear into a ring or arc, making the arithmetic mean converge to the origin and destroying cluster separation.
• Mitigation: Coarse CFO is estimated using the M-th power nonlinearity for M-PSK signals, which strips the modulation and reveals a spectral line at M times the offset frequency.
• Fine Tracking: A decision-directed Costas loop or digital phase-locked loop (PLL) can track and derotate the signal after initial centroid estimates are available.

In non-Gaussian noise environments, such as those with impulsive man-made interference, the sample mean becomes unreliable due to its sensitivity to outliers. Robust alternatives include:

• Spatial Median: The point in the complex plane that minimizes the sum of Euclidean distances to all samples, computed iteratively via the Weiszfeld algorithm. It is significantly less influenced by extreme outliers.
• Trimmed Mean: Discarding a fixed percentage of the most distant samples from a preliminary centroid estimate before computing the final average.
• Huber M-Estimator: A hybrid approach that behaves like the mean for inliers (small residuals) and like the median for outliers (large residuals), providing a balance between efficiency and robustness.

An unsupervised machine learning algorithm that partitions received IQ samples into a predefined number of clusters by iteratively minimizing the within-cluster sum of squares. Centroid estimation is the core step in each iteration, where the arithmetic mean of all samples assigned to a cluster is calculated to update the cluster's center. This enables blind estimation of the transmitted constellation points without prior knowledge of the modulation format.

A probabilistic model representing the distribution of received IQ samples as a weighted sum of Gaussian components, each corresponding to a constellation point. Unlike hard clustering, GMMs provide soft assignments where each sample belongs to multiple clusters with a probability. The Expectation-Maximization (EM) algorithm iteratively refines centroid estimates by weighting samples by their posterior probabilities, making it robust to overlapping clusters in low-SNR environments.

An optimal detection strategy that classifies a received signal point by selecting the constellation symbol with the smallest Euclidean distance to the observation. This requires accurate centroid knowledge of the reference constellation. In blind systems, estimated centroids serve as the reference points for this decision process, directly linking centroid estimation quality to symbol error rate performance.

A quantitative metric measuring the Euclidean distance between the ideal reference constellation point and the actual received signal point. EVM quantifies the combined impact of all transmitter and channel impairments on modulation fidelity. Centroid estimation accuracy directly influences EVM calculations in blind measurement scenarios, where the reference is the estimated rather than the known ideal constellation.

A signal processing technique that recovers the original transmitted constellation from a distorted received signal without requiring a known training sequence. Algorithms like the Constant Modulus Algorithm (CMA) rely on statistical properties of the signal envelope rather than explicit centroid estimation. However, after equalization converges, centroid estimation is often applied to the recovered clusters to identify the modulation format and complete the blind receiver chain.

An inherent uncertainty in the absolute phase rotation of a recovered constellation caused by blind synchronization. Even with perfect centroid estimation, the entire IQ diagram may have a fixed rotational offset. Resolving this requires differential encoding or unique word detection to determine the absolute phase reference, ensuring that estimated centroids map to the correct transmitted symbols.