Instantaneous Amplitude

Instantaneous amplitude A[n] is computed from the raw IQ sample pair using the Pythagorean theorem:

• Formula: A[n] = sqrt(I[n]² + Q[n]²)
• I[n]: The in-phase component at sample index n
• Q[n]: The quadrature component at sample index n
• Result: A real-valued, non-negative scalar representing the signal's momentary magnitude

This calculation converts the two-dimensional complex sample into a one-dimensional envelope value, discarding phase information while preserving amplitude dynamics.

Instantaneous amplitude directly reveals whether a modulation scheme maintains a constant envelope or exhibits amplitude variation:

• Constant Envelope: PSK, FSK, GMSK — A[n] remains theoretically flat, with variations indicating noise or channel distortion
• Variable Envelope: QAM, APSK, OFDM — A[n] fluctuates according to the symbol mapping, creating distinct amplitude levels

This property makes instantaneous amplitude a powerful discriminative feature for coarse modulation classification before finer phase or frequency analysis.

Statistical moments of the instantaneous amplitude sequence provide robust features for machine learning classifiers:

• Mean: Indicates average signal power
• Variance: Distinguishes constant-envelope from amplitude-modulated signals
• Kurtosis: Reveals the peakedness of the amplitude distribution, useful for separating QAM orders
• Rician K-factor: Derived from the envelope, estimates the ratio of line-of-sight to scattered power in fading channels

These statistics are often concatenated with phase and frequency features to form a comprehensive feature vector for automatic modulation classification.

Raw instantaneous amplitude values are scale-dependent on receiver gain and propagation distance, requiring normalization before neural network ingestion:

• Z-score normalization: Centers amplitude to zero mean and unit variance, removing gain dependency
• Min-max scaling: Maps amplitude to a fixed range like [0, 1] or [-1, 1]
• Per-segment normalization: Applied independently to each IQ segment to handle time-varying gain

Proper normalization ensures the classifier focuses on modulation structure rather than absolute signal power, critical for robust operation across varying receive conditions.

Instantaneous amplitude directly maps to the radial distance of constellation points from the origin:

• PSK constellations: All points lie on a single circle — constant instantaneous amplitude
• 16-QAM: Points occupy three concentric rings — three distinct amplitude levels
• APSK: Designed with multiple amplitude rings for spectral efficiency

Analyzing the histogram of instantaneous amplitude values reveals the number of amplitude levels, a key clue for identifying the modulation order and type without requiring full constellation recovery.

Instantaneous amplitude is vulnerable to several real-world distortions that must be accounted for in preprocessing:

• Fading: Multipath propagation causes rapid amplitude fluctuations, distorting the envelope
• Nonlinear amplification: Power amplifier saturation compresses amplitude peaks, reducing modulation distinctiveness
• Impulsive noise: Creates sharp amplitude spikes that corrupt statistical features

Robust classification pipelines often apply median filtering or outlier rejection to the amplitude sequence before feature extraction to mitigate these effects.

The angular component of the complex IQ sample at a given instant, calculated as the arctangent of Q/I. It represents the signal's momentary phase angle and is critical for identifying Phase-Shift Keying (PSK) modulations. Phase unwrapping algorithms are required to remove discontinuities when the angle crosses the ±π boundary, producing a continuous phase trajectory for feature extraction.

The time derivative of the instantaneous phase, representing the rate of change of the signal's frequency at a specific sample point. This is a key discriminative feature for Frequency Modulated (FSK) signals, where discrete frequency shifts correspond to transmitted symbols. Practical computation uses the phase difference between consecutive samples divided by the sampling interval to avoid numerical instability from direct differentiation.

Synonymous with instantaneous amplitude, the envelope traces the outer boundary of a modulated waveform. For constant-envelope modulations like GMSK or QPSK, the envelope is theoretically flat, while QAM signals exhibit amplitude variations that map directly to the constellation rings. Envelope statistics—mean, variance, and Rician K-factor—serve as robust features for hierarchical modulation classifiers.

The process of scaling the amplitude of an IQ sample stream to a standard range, typically using Z-score (zero mean, unit variance) or min-max scaling. This preprocessing step prevents numerical instability during neural network training and ensures the classifier focuses on modulation structure rather than absolute signal power. Without normalization, high-power signals would dominate gradient updates, biasing the model.

A preprocessing operation that shifts the complex baseband signal to exactly zero mean frequency by removing residual Carrier Frequency Offset (CFO). CFO causes the received IQ constellation to rotate continuously over time, corrupting instantaneous amplitude and phase measurements. Centering algorithms estimate the residual rotation rate and apply a counter-rotating phasor to stabilize the constellation before feature extraction.

The application of signal processing algorithms—such as wavelet thresholding, Savitzky-Golay filtering, or principal component analysis—to suppress additive noise in the IQ stream before classification. Denoising improves the effective Signal-to-Noise Ratio (SNR), making instantaneous amplitude transitions sharper and more distinguishable. This is particularly critical for low-SNR scenarios where amplitude-based features are obscured by thermal noise.