Total Harmonic Distortion (THD)

THD is the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency. It is typically expressed as a percentage or in dB. For a signal with a fundamental amplitude A1 and harmonic amplitudes A2, A3, ..., An, THD is calculated as the square root of the sum of the squares of the harmonic amplitudes divided by the fundamental amplitude. This single figure of merit captures the aggregate effect of static non-linearity in amplifiers and data converters.

When a signal passes through a non-linear device, such as a power amplifier, spectral regrowth occurs. This generates energy at integer multiples of the input frequency. The exact amplitude and phase of each harmonic (2nd, 3rd, etc.) are determined by the unique polynomial transfer function of the specific analog components. This pattern is highly sensitive to Process-Voltage-Temperature (PVT) variation, making it a reliable, device-specific signature.

THD is a single-tone measurement, but it directly predicts a device's behavior under more complex, multi-tone excitation. The same non-linear transfer function that creates harmonics also creates Intermodulation Distortion (IMD) products when multiple signals are present. A device's Third-Order Intercept Point (IP3) is a key figure of merit derived from this behavior, and it correlates strongly with the amplitude of the third harmonic in a THD measurement.

In a transmitter chain, both the DAC and subsequent analog stages contribute to THD. DAC non-idealities like Integral Non-Linearity (INL) and Differential Non-Linearity (DNL) create a static, memoryless distortion. This is compounded by the dynamic non-linearity of amplifiers, which may exhibit memory effects due to thermal time constants. The composite THD fingerprint is a convolution of these cascaded imperfections.

THD is measured by applying a high-purity, single-tone sine wave to the device under test and analyzing the output spectrum. Key steps include:

• Windowing: Applying a low-leakage window function (e.g., Blackman-Harris) to the time-domain data.
• FFT: Computing the frequency spectrum.
• Harmonic Identification: Locating the fundamental and its integer multiples.
• Power Summation: Calculating the ratio of harmonic power to fundamental power. The specific amplitudes of the 2nd through 5th harmonics are often used as a feature vector for a fingerprinting model.

A key advantage of THD-based fingerprints is their resilience to linear channel distortions like multipath fading. While the channel can alter the absolute amplitude and phase of the fundamental and its harmonics, the relative amplitude ratios between harmonics are primarily a function of the transmitter's non-linearity. Machine learning models can be trained on these ratios to achieve channel-robust feature learning, ensuring reliable identification even in dynamic environments.

SFDR measures the ratio of the fundamental signal's RMS amplitude to the highest spurious component in the output spectrum. While THD captures the total power of all harmonics, SFDR identifies the single loudest artifact, which is often a harmonic or an interleaving spur. A device's specific SFDR profile, including the frequency and amplitude of the worst spur, is a critical, easily identifiable feature for emitter identification.

IMD occurs when two or more signals pass through a non-linear system, generating new frequencies at the sums and differences of the originals and their harmonics. Unlike THD, which is measured with a single tone, IMD reveals the complex polynomial transfer function of the transmitter chain. The specific pattern of third-order products (2f1-f2, 2f2-f1) is a highly unique, device-specific spectral regrowth pattern.

INL is a static measure of a data converter's deviation from an ideal straight-line transfer function. This low-frequency, process-dependent curvature is the root cause of the harmonic distortion quantified by THD. The unique shape of the INL curve—whether it's S-shaped, bow-shaped, or irregular—directly determines the amplitude and phase of the generated harmonics, forming a time-invariant component of the fingerprint.

Memory effect describes a non-linearity where the current output depends on past signal values, not just the instantaneous input. This is caused by thermal time constants, bias network impedances, and charge trapping. While static non-linearity produces consistent harmonics, memory effects cause the amplitude and phase of those harmonics to vary with signal envelope history, creating a rich, history-dependent signature that is significantly harder to clone.

SINAD is a composite figure of merit that combines the power of all noise and distortion components, including the harmonics measured by THD. It represents the total degradation of the signal. For fingerprinting, SINAD provides a single-value summary of a device's aggregate analog imperfections, while the individual breakdown of SINAD into its noise and distortion constituents provides the detailed feature vector.

Phase noise represents rapid, random fluctuations in a signal's phase, originating from oscillator and clock instabilities. While THD quantifies amplitude distortion, phase noise creates a unique spectral skirt around the carrier. The combined profile of harmonic amplitude distortion and phase noise sidebands forms a complete, two-dimensional spectral signature that is extremely effective for deep learning-based device classification.