2. Sensor characteristics and measurement principles

Every fusion pipeline is only as trustworthy as the individual measurements feeding it. Before algorithms can combine readings from multiple sensors, an engineer must understand what each sensor's output actually represents and how far it can be trusted. This chapter examines the core characteristics that define measurement quality — accuracy, precision, resolution, drift, noise, and calibration — and shows how they behave in real industrial environments.

Accuracy vs. Precision: Two Independent Properties

These two terms are frequently confused, yet they describe fundamentally different error mechanisms. Accuracy describes how close a measured value is to the true value. Precision (also called repeatability) describes how closely repeated measurements agree with each other, regardless of whether they are correct.

A pressure transmitter may consistently report 4.98 bar when the true line pressure is 5.00 bar — that is high precision but imperfect accuracy, a stable offset error. Another transmitter may scatter readings between 4.85 and 5.15 bar around the correct mean — that is good accuracy on average but poor precision, dominated by random noise. In fusion systems, precision directly determines how much a sensor's contribution should be weighted: a noisy sensor with correct mean still degrades a weighted average unless its variance is accounted for.

Accuracy is typically specified in one of three forms:

• Absolute error: ±0.5 °C — a fixed deviation regardless of reading.
• Percent of reading (% of reading): ±0.1% of the measured value — error scales with magnitude.
• Percent of full scale (% FS): ±0.25% FS — error is constant across the range, so relative error grows at low readings.

The distinction matters. A 0–10 bar transmitter rated ±0.25% FS carries ±0.025 bar uncertainty everywhere, which is 2.5% of a 1 bar reading but only 0.25% of a 10 bar reading. When a fusion algorithm combines a high-range and a low-range sensor, the low-range device usually dominates at low pressures precisely because its % FS error translates into a smaller absolute error there.

Resolution and Noise Floor

Resolution is the smallest change a sensor can detect or its output can represent. A 12-bit ADC over a 0–10 V span gives a resolution of 10 V / 4096 ≈ 2.44 mV per count. Resolution is a quantization limit, not an accuracy statement: a sensor can have fine resolution yet poor accuracy if it is miscalibrated.

Noise is the random fluctuation superimposed on the signal. It is usually characterized by its standard deviation σ or by peak-to-peak amplitude. A useful rule of thumb: if the noise is Gaussian, roughly 99.7% of samples fall within ±3σ. If a load cell's noise σ is 0.02 kg, the practical detection limit for a stable reading is around 0.06 kg — you cannot resolve a 0.01 kg change no matter how many digits the display shows.

Resolution and noise interact. Increasing ADC resolution beyond the sensor's noise floor produces more digits but no more information. A common industrial mistake is to specify a 24-bit ADC for a sensor whose own noise is equivalent to 14 bits; the extra bits only digitize noise.

Drift: The Slow Enemy

Drift is a gradual change in output over time that is unrelated to the measured variable. It is the dominant long-term error source in industrial settings and the primary reason periodic calibration is mandatory.

• Zero drift (offset drift): the output at zero input creeps upward or downward. A temperature transmitter reading 0.3 °C when the RTD is in an ice bath has zero drift.
• Span drift (sensitivity drift): the slope of the transfer function changes, so the error grows with the reading.
• Thermal drift: both zero and span shift with ambient temperature, specified as ppm/°C or %/°C.

Drift is not noise — it is systematic and slowly varying, which means it cannot be averaged away. It must be corrected by recalibration or compensated by a model. In fusion architectures, drift is often handled by cross-referencing sensors with different drift mechanisms: a piezoelectric accelerometer and a capacitive accelerometer drift differently, so their disagreement over time can be used to estimate and correct each one.

Calibration in Industrial Practice

Calibration is the process of comparing a sensor's output against a reference of known accuracy and adjusting or documenting the relationship. Two distinct activities are often conflated:

• Calibration: measuring the deviation and recording it (as-found data).
• Adjustment: physically or digitally altering the sensor to reduce the deviation (as-left data).

A standard industrial calibration follows a defined sequence. For a 4–20 mA pressure transmitter with a 0–10 bar range:

1. Isolate and vent the process; connect a certified pressure source and a precision mA meter.
2. Apply five points — 0%, 25%, 50%, 75%, 100% of span — in ascending order, then descending, to capture hysteresis.
3. Record as-found output at each point. At 5.00 bar (50%), the ideal output is 12.00 mA.
4. Compute error: if the meter reads 12.08 mA, the error is +0.08 mA, or +0.5% of the 16 mA span.
5. If error exceeds the tolerance (commonly ±0.25% or ±0.5% of span), perform zero and span adjustment and repeat the five-point run to obtain as-left data.
6. Document both data sets with the reference standard's traceable certificate number.

The as-found record is the most valuable part of the procedure. It reveals drift rate, which lets maintenance planners extend or shorten calibration intervals based on evidence rather than a fixed calendar. If a transmitter drifts 0.1% per year and tolerance is 0.5%, a two-year interval is defensible; if it drifts 0.4% per year, it is not.

How These Characteristics Propagate into Fusion

When multiple sensors are combined, their individual error characteristics determine the fusion result. A weighted average assigns each sensor a weight inversely proportional to its variance:

w_i = (1 / σ_i²) / Σ(1 / σ_j²)

Example: three temperature sensors
  T1: σ = 0.10 °C  → 1/σ² = 100
  T2: σ = 0.20 °C  → 1/σ² = 25
  T3: σ = 0.50 °C  → 1/σ² = 4
  Σ = 129
  w1 = 0.775, w2 = 0.194, w3 = 0.031

If readings are 100.2, 100.5, 99.8 °C:
  fused = 0.775(100.2) + 0.194(100.5) + 0.031(99.8)
        = 77.655 + 19.497 + 3.094 = 100.25 °C

This example shows why precision (σ) must be known before fusion. A sensor with a large offset but small σ will still pull the estimate toward its biased value. That is why calibration — removing systematic error — must precede fusion, while noise characterization determines the weights.

The figure below summarizes how these characteristics relate to one another and where each is addressed in the measurement chain.

Practical Takeaways for Fusion Design

Three engineering rules follow from these characteristics. First, calibrate before you fuse: systematic errors (offset, span) cannot be averaged out and will bias the fused result. Second, characterize noise to set weights: the inverse-variance weighting shown above is only as good as the σ values you supply, so measure them under realistic process conditions, not on a bench. Third, track drift over time: as-found calibration records are the data source for drift models and for deciding when a sensor should be replaced rather than recalibrated.

When these characteristics are understood and documented, the fusion algorithm has the information it needs to combine sensors intelligently. When they are ignored, even a sophisticated algorithm will faithfully propagate bad data into confident but wrong estimates.