Instantaneous Phase

The instantaneous phase φ[n] is computed as the four-quadrant arctangent of the quadrature component over the in-phase component:

• Formula: φ[n] = atan2(Q[n], I[n])
• Range: Wrapped to the interval (-π, π] radians
• Discontinuities: Phase jumps of 2π occur when the angle crosses the ±π boundary
• Unwrapping required: A phase unwrapping algorithm must be applied to reconstruct the continuous phase trajectory for frequency calculation

The atan2 function is essential because it preserves quadrant information that would be lost using a simple arctan(Q/I) division.

Raw instantaneous phase exhibits sawtooth-like discontinuities due to angular wrapping at the ±π boundary. This artifact must be addressed before the phase can be used for feature extraction:

• Wrapped phase: φ_wrapped ∈ (-π, π]
• Unwrapped phase: φ_unwrapped = φ_wrapped + 2πk, where k is an integer chosen to minimize the difference between consecutive samples
• Phase jumps: A 2π discontinuity occurs when the true phase crosses the branch cut
• Unwrapping algorithms: Standard implementations track cumulative phase shifts and add integer multiples of 2π to maintain continuity

For modulation classification, the wrapped phase is often sufficient for constellation-based methods, while unwrapped phase is necessary for instantaneous frequency calculation.

Instantaneous phase is a primary discriminative feature for identifying phase-modulated signals and distinguishing them from frequency or amplitude modulation schemes:

• PSK signals: Exhibit discrete phase states (e.g., BPSK: 0 and π; QPSK: ±π/4, ±3π/4)
• QAM signals: Show combined phase and amplitude variations forming a grid pattern
• FM signals: Display continuous, smooth phase variation without discrete clustering
• AM signals: Show minimal phase variation concentrated around a constant carrier phase

Neural network classifiers often use phase histograms or phase transition statistics as input features derived from the instantaneous phase sequence.

Instantaneous phase estimation is highly sensitive to additive noise, particularly at low signal amplitudes where the phase angle becomes ill-defined:

• Low SNR regime: Phase estimates become uniformly distributed, losing all modulation information
• Amplitude-dependent reliability: Phase accuracy degrades as the instantaneous amplitude approaches the noise floor
• Threshold effect: Below a certain SNR threshold, phase-based features become unreliable for classification
• Mitigation strategies: Weighting phase estimates by instantaneous amplitude or using phase-locked loops for carrier recovery improves robustness

This sensitivity is why classifiers often combine phase with amplitude and frequency features for reliable operation across varying channel conditions.

The differential phase—the difference between consecutive instantaneous phase samples—reveals the symbol transition patterns that characterize different modulation formats:

• Differential PSK (DPSK): Information is encoded in phase differences rather than absolute phase
• Phase transition histogram: A histogram of Δφ[n] = φ[n] - φ[n-1] shows distinct peaks corresponding to allowed symbol transitions
• Symbol rate estimation: The rate of phase changes correlates with the baud rate of the signal
• π/4-DQPSK detection: Characteristic ±π/4 and ±3π/4 transitions distinguish this format from standard QPSK

Differential phase features are inherently robust to slow carrier phase drift, making them valuable for non-coherent classification scenarios.

The instantaneous phase and instantaneous frequency are mathematically coupled through differentiation, forming complementary feature pairs for signal analysis:

• Definition: f_inst[n] = (1/2π) × dφ/dt ≈ (1/2π) × (φ[n] - φ[n-1]) / T_s
• FSK discrimination: Frequency-shift keyed signals show discrete frequency levels in the derivative of phase
• Linear phase = constant frequency: A linearly increasing unwrapped phase indicates a constant frequency offset (CFO)
• Joint features: Combining phase and frequency features in a dual-input neural network improves classification accuracy for hybrid modulation schemes

Computing instantaneous frequency from unwrapped phase avoids the noise amplification inherent in direct frequency discrimination methods.

The absolute magnitude of the complex IQ sample at a given instant, calculated as the square root of I² + Q². This represents the signal envelope and is a critical feature for distinguishing constant-envelope modulations like Frequency Modulation (FM) from amplitude-varying schemes like Quadrature Amplitude Modulation (QAM).

• Formula: A[n] = sqrt(I[n]² + Q[n]²)
• Use Case: Envelope detection for On-Off Keying (OOK) classification
• Robustness: Insensitive to phase rotations, making it a stable feature under Carrier Frequency Offset (CFO)

The time derivative of the instantaneous phase, representing the rate of change of the signal's frequency at a specific sample point. This derived feature is essential for discriminating Frequency Shift Keying (FSK) variants and identifying frequency-hopped signals.

• Calculation: f[n] = (1 / 2π) * dφ[n] / dt
• Practical Implementation: Phase differencing with unwrapping to avoid 2π discontinuities
• Key Insight: Constant instantaneous frequency indicates an unmodulated carrier, while discrete jumps reveal FSK modulation orders

A discrete time-domain measurement representing the instantaneous state of a modulated signal, composed of an In-Phase (I) and Quadrature (Q) component. The instantaneous phase is derived directly from these two orthogonal components via the arctangent function.

• Representation: Complex number s[n] = I[n] + jQ[n]
• Relationship: The I component is the projection onto the cosine axis; Q is the projection onto the sine axis
• Context: Without the IQ sample pair, instantaneous phase computation is impossible in a coherent receiver

A signal processing operation that corrects the discontinuities in the instantaneous phase calculation caused by the arctangent function's limited range of [-π, π]. Without unwrapping, the phase jumps by 2π radians at each boundary crossing, corrupting derived features like instantaneous frequency.

• Algorithm: Detects phase jumps greater than a tolerance (typically π) and adds or subtracts 2π to restore continuity
• Critical For: Computing instantaneous frequency and phase-based cumulants
• Failure Mode: Unwrapped phase errors cascade into incorrect modulation identification

The residual frequency difference between the transmitter and receiver local oscillators, causing the received IQ constellation to rotate continuously over time. This directly manifests as a linear ramp in the instantaneous phase, obscuring the modulation's true phase trajectory.

• Effect on Phase: φ_observed[n] = φ_modulation[n] + 2π * Δf * n * T_s
• Mitigation: I/Q centering algorithms estimate and remove the linear phase trend before classification
• Classifier Impact: Uncompensated CFO makes phase-based features useless for modulation recognition

A preprocessing operation that shifts the complex baseband signal to exactly zero mean frequency by removing residual Carrier Frequency Offset (CFO). This centers the constellation in the complex plane and stabilizes the instantaneous phase, making it a reliable feature for downstream neural network classifiers.

• Method: Estimates the frequency offset via the phase slope and applies a counter-rotating complex exponential
• Result: A stationary constellation where the instantaneous phase reflects only the modulation content
• Pipeline Position: Must be applied before phase computation or feature extraction