Integral Non-Linearity (INL)

INL is the running sum of DNL errors from the first to the last code. While DNL captures local step-size errors, INL reveals the global curvature of the transfer function. This cumulative nature means that even small, seemingly random DNL errors can accumulate into a large, distinctive INL bow or S-shape that is highly repeatable for a specific device. The exact shape of this deviation curve—whether it bows upward, downward, or oscillates—is a direct consequence of the front-end sample-and-hold amplifier's non-linearity and the reference ladder's voltage gradient errors, making it a robust, low-frequency signature.

A device's static non-linearity can be modeled as a memoryless polynomial:

• Linear term (a₁x): Ideal gain.
• Quadratic term (a₂x²): Causes even-order harmonic distortion and a DC offset shift.
• Cubic term (a₃x³): Causes odd-order harmonic distortion and gain compression/expansion. The coefficients (a₂, a₃, ...) are unique to each converter due to random dopant fluctuations and lithographic variations during fabrication. Fingerprinting systems can extract these polynomial coefficients as a compact, physics-based feature vector for device identification, rather than storing the entire INL curve.

The shape of the INL curve is heavily influenced by the converter topology:

• Flash ADCs: Exhibit a 'sawtooth' or random INL pattern due to random comparator offsets in the resistor ladder.
• Successive Approximation Register (SAR) ADCs: Often show a smooth, low-order bow due to capacitor array mismatch in the internal DAC.
• Pipelined ADCs: Display a periodic INL pattern repeating at the boundary of each pipeline stage, caused by inter-stage gain and sub-ADC reference errors.
• Sigma-Delta ADCs: Typically have very low INL due to noise shaping, but residual tones from idle tones and limit cycles can create a unique, low-amplitude signature.

INL is not perfectly static; it drifts predictably with Process-Voltage-Temperature (PVT) variations:

• Temperature: Changes the bandgap reference voltage and amplifier bias currents, causing a slow, global scaling of the INL curve. The temperature coefficient of INL is itself a distinguishing parameter.
• Supply Voltage: Variations modulate the headroom of analog stages, altering the saturation characteristics of transistors and shifting the INL profile, particularly near the rails. Robust fingerprinting systems model this drift using polynomial compensation or track it with an on-chip temperature sensor to maintain authentication accuracy over environmental changes.

Severe INL can manifest as missing codes or wide codes:

• Missing Code: When INL exceeds ±0.5 LSB, the transfer function becomes non-monotonic, and one or more digital output codes are never produced for any analog input. This creates a permanent, unclonable gap in the output histogram.
• Wide Code: Conversely, a code with an abnormally large width appears more frequently in the output, creating a statistical peak in the code density. These extreme non-idealities are highly deterministic and serve as strong, high-contrast features for device identification, especially when combined with a sinusoidal histogram test.

INL is practically extracted using the sinusoidal histogram method:

• A high-purity sine wave (with THD < -100 dBc) is applied to the ADC input.
• The probability density of a sine wave is known analytically, peaking at the rails.
• By comparing the actual code occurrence histogram to the ideal expected probability, the DNL per code is computed.
• INL is then derived as the cumulative sum of DNL. The resulting INL vector is a high-dimensional feature (e.g., 4096 points for a 12-bit ADC) that captures the complete static fingerprint of the device, suitable for input to a 1D-CNN or similarity metric like cosine distance.

A memoryless, amplitude-dependent distortion modeled by a polynomial transfer function. This is the core physical phenomenon that INL attempts to quantify.

• Creates harmonic distortion and intermodulation products.
• Forms a consistent, time-invariant component of the RF fingerprint.
• Can be expressed as a Taylor series: Vout = a0 + a1*Vin + a2*Vin^2 + a3*Vin^3 + ...

Amplitude distortion with a dependence on signal history or frequency. Unlike the static non-linearity measured by INL, dynamic effects introduce a complex, history-dependent signature.

• Encompasses slew-rate limiting and memory effects.
• Thermal time constants in the die cause slow, signal-dependent bias shifts.
• Much harder to clone or emulate than static INL, making it a robust fingerprinting feature.

The deviation of the actual slope of the transfer function from the ideal slope. This is a linear imperfection that scales the entire output.

• Contributes a systematic, identifiable bias to the signal.
• Often expressed as a percentage of full-scale range.
• When combined with offset error, forms the first-order (linear) component of the total INL budget.

A constant, static voltage difference between the ideal and actual transfer function. This introduces a fixed DC bias that shifts the entire quantization characteristic.

• A simple yet persistent component of a device's analog fingerprint.
• Caused by transistor threshold voltage mismatches in the input differential pair.
• Easily measured but difficult to eliminate entirely without trimming.

A severe manifestation of Differential Non-Linearity (DNL) where a converter completely skips one or more digital output codes.

• Creates a permanent, highly distinctive gap in the transfer function.
• Acts as a strong, unambiguous identifying feature for fingerprinting.
• Guaranteed to occur when DNL < -1 LSB at a specific code transition.