Pulse Shaping Analysis

The impulse response of a transmitter's pulse shaping filter—typically a root-raised cosine (RRC)—exhibits subtle deviations from the theoretical ideal. These deviations arise from component tolerance mismatches in the analog filter implementation, including resistor and capacitor value variations of 1-5%. The resulting time-domain waveform contains unique ringing artifacts and sidelobe asymmetries that serve as a persistent hardware fingerprint. Key measurement points include:

• Excess bandwidth roll-off: The alpha factor deviates from specification by 0.01-0.05
• Sidelobe symmetry: Asymmetric sidelobe amplitudes indicate I/Q path imbalance
• Truncation artifacts: Finite impulse response (FIR) filter length limitations create unique endpoint distortions

The clock recovery circuit in each transmitter introduces a unique pattern of symbol timing jitter that modulates the pulse shaping filter's output. This jitter, caused by phase-locked loop (PLL) noise and oscillator phase noise, creates a device-specific stochastic warping of the symbol transitions. The statistical distribution of timing errors—including its standard deviation and higher-order moments—forms a distinctive signature. Critical parameters include:

• RMS jitter: Typically 1-10 picoseconds for high-quality oscillators
• Jitter power spectral density: The frequency-domain distribution reveals PLL loop bandwidth characteristics
• Cycle-to-cycle correlation: Adjacent symbol timing errors exhibit device-specific autocorrelation patterns

The Gibbs phenomenon and filter implementation constraints produce characteristic overshoot and ringing at symbol transitions. Each transmitter exhibits a unique pattern of pre-shoot and post-shoot amplitude due to:

• Filter order truncation: Practical FIR filters with 40-80 taps create distinct ringing envelopes
• Amplifier slew rate limitations: The power amplifier's finite response speed shapes the rising and falling edges
• Impedance mismatches: Reflections from antenna connectors create secondary ringing patterns The amplitude, decay rate, and frequency of these ringing artifacts constitute a measurable device fingerprint that persists across different modulation schemes.

The passband ripple and group delay distortion of the pulse shaping filter are never perfectly flat in physical implementations. Manufacturing variances in analog filter components create device-specific patterns of:

• Amplitude ripple: Periodic variations of 0.1-0.5 dB across the passband caused by filter component tolerances
• Group delay variation: Frequency-dependent phase distortion that smears symbol timing, typically 1-10 nanoseconds peak-to-peak
• Ripple periodicity: The spectral spacing of ripple peaks correlates with physical filter topology and component values These frequency-domain signatures are highly stable over time and temperature, making them reliable long-term identifiers.

The residual inter-symbol interference (ISI) pattern created by imperfect pulse shaping is a rich source of device-specific information. Even when the filter meets the Nyquist zero-ISI criterion on paper, hardware imperfections introduce:

• Deterministic ISI: A repeatable pattern of symbol interference dependent on the transmitted data sequence
• Asymmetric eye diagram: The eye opening exhibits unique closure patterns in amplitude and timing
• Data-dependent jitter: The timing of zero crossings shifts based on surrounding symbol values in a device-specific manner By analyzing the ISI pattern across known training sequences, the unique filter response can be characterized and used for authentication.

The out-of-band emission profile of a transmitter is directly shaped by its pulse shaping filter's stopband attenuation characteristics. Each device exhibits a unique spectral mask due to:

• Stopband attenuation depth: Variations of 5-15 dB in the suppression of adjacent channel power
• Spectral regrowth shoulders: Non-linear amplifier effects create asymmetric spectral spreading unique to each PA-filter combination
• Transition band slope: The rate of power roll-off at the channel edge varies with filter component precision These spectral containment characteristics are measurable using standard spectrum analyzers and remain consistent across different operating frequencies, providing a robust cross-band fingerprint.

The foundational principle governing pulse shaping to eliminate intersymbol interference (ISI). A raised-cosine filter satisfies the Nyquist ISI criterion, where the impulse response crosses zero at multiples of the symbol period. The roll-off factor (α) controls excess bandwidth—lower α conserves spectrum but increases sensitivity to timing jitter. Deviations from the ideal Nyquist response, caused by analog component tolerances, create the unique spectral signatures exploited for fingerprinting.

Manufacturing variances in resistors, capacitors, and active components cause the actual filter response to deviate from the design specification. Key measurable impairments include:

• Passband ripple: Amplitude variation within the filter's passband
• Group delay distortion: Non-constant phase slope causing frequency-dependent time shifts
• Stopband attenuation asymmetry: Uneven rejection of out-of-band signals These analog imperfections are stable over time and unique to each physical transmitter chain.

The scatter plot of in-phase (I) vs. quadrature (Q) signal samples provides a direct visual representation of pulse shaping fidelity. An ideal root-raised-cosine filter pair produces tight, well-separated symbol clusters. Hardware impairments manifest as:

• Cluster warping from non-linear filter phase response
• Symbol trajectory overshoot from filter ringing
• I/Q imbalance causing elliptical cluster shapes The statistical distribution of these deviations forms a robust device fingerprint.

An eye diagram overlays multiple symbol periods to visualize signal integrity. For fingerprinting, quantitative metrics extracted from the eye include:

• Eye height and width: Reduced by filter bandwidth limitations and jitter
• Zero-crossing jitter: Timing uncertainty at decision thresholds
• Rise/fall time asymmetry: Caused by unmatched filter component slew rates
• Overshoot percentage: Filter ringing exceeding steady-state amplitude Each transmitter exhibits a characteristic eye pattern due to its unique baseband filter response.

Pulse shape fingerprints must remain distinguishable despite multipath fading and Doppler shift introduced by the wireless channel. Techniques for robustness include:

• Domain-adversarial neural networks that force the feature extractor to ignore channel variations
• Contrastive learning that clusters same-device signals across diverse channel conditions
• Cyclostationary processing that isolates the transmitter's symbol-rate periodicity from channel-induced distortions These methods ensure the extracted filter-response signature remains stable in dynamic environments.

A two-dimensional function S_x^α(f) measuring correlation between spectral components separated by a cycle frequency (α). Pulse shaping introduces cyclostationarity at multiples of the symbol rate. The spectral correlation function reveals:

• Filter roll-off shape encoded in the cycle frequency profile
• Excess bandwidth visible as correlation width around the symbol rate
• Transmitter-unique spectral redundancy patterns from filter imperfections This representation is inherently robust to stationary noise and interference.