I/Q Constellation Ellipticity

Ellipticity arises from the combined effect of I/Q gain imbalance and quadrature skew. When the I and Q signal paths have mismatched amplitudes, the constellation scales unevenly along one axis. When the phase difference between the I and Q local oscillators deviates from the ideal 90 degrees, the axes become non-orthogonal. The interaction of these two impairments transforms a circular point cluster into an ellipse, with the major axis orientation determined by the relative severity of gain versus phase error.

Two parameters fully characterize the ellipse:

• Tilt Angle (θ): The angular orientation of the major axis relative to the ideal I-axis. A tilt of 0° or 90° indicates pure gain imbalance; a 45° tilt indicates dominant phase error.
• Axial Ratio (AR): The ratio of the major axis length to the minor axis length. An AR of 1.0 indicates a perfect circle (no ellipticity), while AR > 1.0 quantifies the severity of distortion.

These parameters are extracted via eigenvalue decomposition of the 2D covariance matrix of the constellation point cluster.

The I/Q impairment model is expressed as a 2×2 mixing matrix:

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[ I' ]   [ α   0   ] [ cos(φ/2)  sin(φ/2) ] [ I ]
[ Q' ] = [ 0   β   ] [ sin(φ/2)  cos(φ/2) ] [ Q ]

where α and β are the I and Q path gains, and φ is the quadrature phase error. The ellipticity parameters (tilt angle and axial ratio) are the observable geometric manifestations of this underlying matrix. Solving the inverse problem—recovering α, β, and φ from the ellipse geometry—is a standard calibration technique.

The specific ellipticity parameters (tilt angle and axial ratio) are highly repeatable for a given transmitter under stable conditions, yet vary measurably between devices due to manufacturing tolerances in analog components:

• DAC gain mismatches in the I and Q paths
• Mixer quadrature errors in the local oscillator distribution
• Baseband filter cutoff frequency variations

These parameters form a low-dimensional but highly discriminative feature vector for physical layer authentication, often used as inputs to one-class classifiers for device verification.

Ellipticity must be differentiated from other constellation deformations:

• DC Offset: Shifts the entire ellipse away from the origin; does not change its shape.
• Phase Noise: Causes angular smearing, thickening the ellipse boundary without changing its axial ratio.
• Nonlinear Compression: Warps the constellation into a non-elliptical shape (e.g., a rounded square for severe AM-AM distortion).
• Inter-Symbol Interference: Creates multiple displaced ellipses depending on adjacent symbol patterns.

Pure ellipticity is a linear, static distortion that preserves the relative positions of symbol centroids.

Practical estimation of ellipticity involves:

• Centroid Calculation: Compute the mean (I,Q) position for each symbol cluster over many captured bursts.
• Covariance Estimation: Build the 2×2 covariance matrix for the cluster's point distribution.
• Eigendecomposition: Extract eigenvalues (λ₁, λ₂) and eigenvectors (v₁, v₂).
  • Axial Ratio = √(λ₁/λ₂)
  • Tilt Angle = arctan(v₁_Q / v₁_I)
• Averaging: Average parameters across multiple symbol clusters to reduce estimation variance.

This process is computationally lightweight and suitable for real-time implementation on embedded SDR platforms.

The angular orientation of the major axis of an elliptical constellation point cluster, measured relative to the ideal I-axis. This tilt is a direct, sensitive measure of quadrature skew—the deviation of the I/Q phase difference from the ideal 90 degrees.

• A tilt angle of 0° or 90° indicates pure gain imbalance with no phase error
• A tilt of 45° indicates dominant phase imbalance with equal gain
• The tilt angle combined with the ellipse's eccentricity fully characterizes the I/Q imbalance matrix

The ratio of amplitude gain in the I signal path to that in the Q signal path. A value deviating from unity (1.0) indicates gain imbalance, which stretches a nominally circular constellation cluster into an ellipse aligned with the I or Q axis.

• Gain Ratio > 1: I-channel amplification exceeds Q-channel, elongating the cluster horizontally
• Gain Ratio < 1: Q-channel dominates, producing vertical elongation
• This ratio is one of two parameters—along with quadrature skew—that define the ellipticity of a constellation cluster

The deviation of the phase difference between the I and Q local oscillator signals from the ideal 90 degrees. Quadrature skew causes the I and Q axes to become non-orthogonal, shearing the constellation and rotating the ellipse's major axis away from the coordinate axes.

• Measured in degrees of phase error
• A skew of just 1-2 degrees can produce measurable ellipticity in high-order QAM constellations
• Combined with I/Q gain ratio, quadrature skew determines the eccentricity and tilt of the elliptical cluster

The comprehensive study of the shape, symmetry, and statistical structure of constellation point clusters. Morphological analysis extracts a multi-dimensional feature vector from each symbol's point cloud for emitter identification.

• Ellipticity is one key morphological descriptor, capturing the combined gain/phase imbalance
• Other descriptors include centroid offset, cluster kurtosis, and inter-symbol spacing variance
• Morphological features are robust against translation and rotation, making them suitable for channel-robust fingerprinting

Quantitative descriptors of the shape of a constellation point distribution, used as robust features for machine learning-based fingerprinting. The second-order moments (covariance matrix eigenvalues) directly quantify ellipticity.

• Variance: Spread of points along each axis; unequal variances indicate gain imbalance
• Covariance: Correlation between I and Q errors; non-zero covariance indicates quadrature skew
• Higher-order moments (skewness, kurtosis) capture non-Gaussian distortion patterns from amplifier non-linearity

The geometric deformation of an ideal constellation diagram into a non-uniform shape caused by the combined effects of I/Q gain and phase imbalances. While ellipticity describes the stretching of individual point clusters, warping describes the global distortion of the entire constellation lattice.

• A perfect square QPSK constellation becomes a parallelogram under quadrature skew
• A rectangular 16-QAM grid becomes a skewed rectangle with non-orthogonal axes
• Warping is the macroscopic manifestation of the microscopic ellipticity affecting each symbol cluster