Missing Codes

A missing code occurs when the Differential Non-Linearity (DNL) at a specific code transition equals or exceeds -1 LSB. In this condition, the step width for that code collapses to zero or becomes negative, meaning no analog input voltage will ever produce that digital output. This is a direct, measurable consequence of severe static non-linearity in the converter's architecture, often caused by mismatched unit elements in the internal DAC of a successive-approximation or pipelined ADC. The specific location of the missing code in the transfer function is a deterministic result of the manufacturing process variations.

Unlike transient noise or environmental drift, a missing code is a static, hard-wired defect in the silicon. It is physically impossible to stimulate the converter to produce the missing digital word, making this gap a permanent feature of the device's identity. This characteristic is highly valuable for supply chain hardware authentication because the specific pattern of missing codes is randomly distributed across a batch of chips due to Process-Voltage-Temperature (PVT) variation. A cloned or counterfeit device will have a different, or no, missing code pattern, making it instantly distinguishable.

A missing code introduces a severe, deterministic distortion in the digitized waveform. In the frequency domain, this manifests as a spike in the quantization noise floor and the generation of high-level, signal-dependent harmonic and intermodulation distortion products. This directly degrades key dynamic performance metrics:

• SINAD (Signal-to-Noise and Distortion Ratio): Sharply reduced due to the added distortion power.
• SFDR (Spurious-Free Dynamic Range): Dominated by the spurs created by the abrupt transfer function discontinuity.
• ENOB (Effective Number of Bits): Significantly lower than the stated resolution, as the missing code effectively removes usable quantization levels.

A missing code (DNL = -1 LSB) must be distinguished from a wide code (DNL > 0). A wide code means a specific digital output appears for a wider-than-ideal range of analog inputs, but the code is still present. A missing code is a complete absence. In practice, a severe wide code is often adjacent to a missing code because the total charge or current from the mismatched unit elements is redistributed. The neighboring code 'steals' the input range from the missing one, creating a paired anomaly that is an even stronger, more complex identifying feature.

The likelihood and pattern of missing codes are heavily dependent on the converter architecture:

• High-Speed Flash ADCs: Prone to 'sparkle codes' and missing codes due to comparator offsets and bubble errors in the thermometer-to-binary decoder.
• SAR ADCs: Missing codes are directly linked to the mismatch of binary-weighted capacitors in the internal Charge-Redistribution DAC. A major transition, like the MSB switch, is a common failure point.
• Time-Interleaved ADCs: Missing codes can result from a dead or severely offset sub-ADC in the interleaved array, creating a periodic pattern of missing codes that is a uniquely exploitable interleaving mismatch signature.

For RF fingerprinting, a missing code in the transmitter's DAC or the receiver's ADC is a gold-standard feature. When a DAC has a missing code, it cannot generate a specific amplitude level, creating a permanent 'notch' in the IQ constellation distortion pattern and generating unique intermodulation distortion (IMD) products. This feature is:

• Channel-Robust: The gap in the transfer function is independent of multipath or fading.
• High-Contrast: It provides a binary, yes/no feature for a classifier, simplifying deep learning signal identification models.
• Drift-Resistant: Unlike analog biases, a missing code does not drift with temperature; it is a structural defect.

The direct precursor to missing codes. DNL measures the deviation of an actual step width from the ideal 1 Least Significant Bit (LSB). A DNL of -1 LSB indicates a step width of zero, meaning the code is skipped entirely.

• Critical Threshold: Missing codes occur when DNL ≤ -1 LSB.
• Guaranteed Monotonicity: A converter with |DNL| < 1 LSB is guaranteed to have no missing codes.
• Fingerprint Utility: Even DNL values between -0.5 and -0.99 LSB create highly compressed codes that are statistically rare and identifiable.

The cumulative sum of DNL errors, representing the maximum deviation of the actual transfer function from an ideal straight line. While DNL causes a local code gap, INL describes the overall bow or warp of the converter's response.

• Signature Shape: INL curves (S-bow, parabolic) are highly process-dependent and serve as a macroscopic device fingerprint.
• Relationship: A missing code manifests as a vertical step discontinuity in the INL plot.
• End-Point vs. Best-Fit: The method used to define the ideal line changes the INL profile, a key consideration for fingerprint normalization.

The inherent sawtooth-shaped error between the analog input and its quantized digital output, bounded by ±0.5 LSB for an ideal converter. Missing codes catastrophically violate this bound.

• Ideal Model: Treated as uniformly distributed white noise uncorrelated with the input.
• Missing Code Impact: Creates a dead zone where quantization error exceeds ±1.5 LSB, introducing gross distortion and deterministic spurs.
• Dithering Interaction: Intentional noise injection can sometimes bridge narrow dead zones but cannot restore a fully missing code.

A dynamic performance metric that collapses all noise and distortion into a single resolution figure. A missing code directly degrades ENOB by introducing large, signal-dependent errors.

• Calculation: Derived from SINAD: ENOB = (SINAD_dB - 1.76) / 6.02.
• Missing Code Penalty: A single missing code can drop ENOB by 0.5–1.5 bits, depending on signal amplitude and frequency.
• Fingerprint Stability: ENOB degradation from missing codes is permanent and temperature-stable, making it a robust long-term identifier.

In high-speed time-interleaved ADCs, multiple sub-converters sample in sequence. Gain, offset, and timing mismatches between these sub-ADCs create periodic, deterministic spurs that can mimic or mask missing codes.

• Interleaving Spurs: Appear at f_s/M ± f_in, where M is the number of interleaved channels.
• Confusion Risk: A sub-ADC with internal missing codes produces a repeating pattern every M samples, which can be misdiagnosed as a simple gain mismatch.
• Exploitation: This periodic signature is a multiplier effect, making interleaved converters exceptionally easy to fingerprint.

Missing codes are a static, memoryless non-linearity—the output gap depends only on the instantaneous input voltage. Dynamic non-linearity, by contrast, depends on signal history.

• Static (Missing Codes): Fixed dead zones in the DC transfer curve. Easily mapped with a slow ramp test.
• Dynamic (Slewing, Memory): Distortion that varies with frequency and past signal values, caused by amplifier slewing or thermal time constants.
• Combined Fingerprint: A device exhibits both types; the static missing code map provides the invariant anchor, while dynamic effects add a rich, context-dependent layer.