Electronics Guide

Linearization

Many sensors exhibit nonlinear relationships between the physical quantity being measured and their electrical output. Linearization corrects these nonlinearities to produce outputs that are directly proportional to the measured quantity, simplifying subsequent processing and improving measurement accuracy across the full operating range.

Sources of Sensor Nonlinearity

Sensor nonlinearity arises from various physical mechanisms depending on the sensor type. Thermistors, for example, follow an exponential relationship between temperature and resistance described by the Steinhart-Hart equation. Strain gauges exhibit nonlinearity at high strain levels due to geometric effects. Photodiodes show logarithmic response characteristics over wide intensity ranges. Pressure sensors based on piezoresistive elements display nonlinearity from stress-dependent carrier mobility.

Even sensors designed for linear operation exhibit some degree of nonlinearity due to manufacturing tolerances, material imperfections, and second-order physical effects. Understanding the underlying physics helps engineers select appropriate linearization methods and anticipate residual errors.

Lookup Tables

Lookup tables provide a straightforward linearization approach, storing pre-calculated output values corresponding to specific input values. During operation, the system finds the two table entries bracketing the current input and interpolates between them. Linear interpolation suffices for tables with sufficient density, while higher-order interpolation schemes reduce table size requirements for highly curved characteristics.

Lookup table advantages include simplicity of implementation, deterministic execution time, and the ability to represent arbitrary nonlinearities without mathematical modeling. Disadvantages include memory consumption, especially for multi-dimensional corrections, and the need for dense tables when sensor characteristics have sharp features.

Polynomial Approximation

Polynomial approximation fits a mathematical function to the sensor characteristic, enabling continuous output calculation rather than discrete lookup. A polynomial of degree n can perfectly fit n+1 data points, and least-squares fitting produces best approximations for larger datasets. The polynomial form enables efficient calculation through nested multiplication (Horner's method).

The appropriate polynomial order depends on the sensor characteristic's complexity. Linear sensors need only offset and gain correction (first-order). Moderately nonlinear sensors often require second or third-order polynomials. Higher orders provide closer fits but increase computational cost and can introduce oscillatory artifacts between calibration points, a phenomenon known as Runge's phenomenon.

Piecewise Linear Approximation

Piecewise linear approximation divides the input range into segments, applying different linear equations within each segment. This approach combines the simplicity of linear calculations with the flexibility to approximate complex curves. Breakpoints between segments are chosen where the sensor characteristic changes significantly, concentrating correction effort where it matters most.

The implementation requires storing segment boundaries and corresponding slopes and offsets. During operation, the system determines which segment contains the current input and applies the appropriate linear equation. This method works particularly well for sensors with distinct regions of different behavior, such as saturation regions or transition zones.

Spline Interpolation

Spline interpolation connects calibration points with smooth polynomial segments while ensuring continuity of derivatives at segment boundaries. Cubic splines, the most common form, provide continuous first and second derivatives, eliminating the slope discontinuities of piecewise linear approaches. Natural splines add the constraint of zero second derivatives at the endpoints.

Spline interpolation excels when smooth output is important, as in control systems where derivative terms would amplify discontinuities from piecewise linear methods. The trade-off is increased computational complexity and the need to store polynomial coefficients for each segment.

Inverse Function Implementation

When the sensor's mathematical model is known, linearization may involve computing the inverse function directly. For example, linearizing a thermistor requires solving the Steinhart-Hart equation for temperature given resistance. Transcendental equations like this require iterative numerical methods such as Newton-Raphson iteration or can be approximated with rational functions.

Direct inverse calculation provides the most accurate linearization when the model precisely describes the sensor, but model accuracy depends on having correct parameters. Combined approaches use the theoretical model structure with empirically determined coefficients, benefiting from physical understanding while accommodating individual sensor variations.

Calibration Curves

Calibration curves establish the relationship between sensor output and the true value of the measured quantity, accounting for individual sensor variations, system gains, and offsets. Proper calibration is essential for achieving specified measurement accuracy and maintaining traceability to measurement standards.

Calibration Fundamentals

Calibration compares sensor readings against known reference values, called standards, to determine correction parameters. Primary standards derive from fundamental physical constants or definitions; secondary standards are calibrated against primary standards. Working standards, calibrated against secondary standards, provide practical references for routine calibration. This hierarchy ensures traceability, meaning that all measurements can be related back to accepted definitions through documented comparisons.

Calibration uncertainty quantifies how well the calibration parameters are known, setting a fundamental limit on achievable measurement accuracy. Uncertainty analysis considers the reference standard's uncertainty, measurement repeatability, environmental effects during calibration, and the mathematical fitting process.

Single-Point Calibration

Single-point calibration adjusts only the offset, assuming the sensor's sensitivity matches its nominal specification. This simplest approach suits applications where sensors are well-matched to nominal values and only zero drift needs correction. Common examples include zeroing a pressure sensor at atmospheric pressure or nulling a scale with no load.

The limitation of single-point calibration is that gain errors remain uncorrected, causing measurement errors that grow with distance from the calibration point. This approach works best when measurements concentrate near the calibration point or when sensor gain accuracy exceeds requirements.

Two-Point Calibration

Two-point calibration determines both offset and gain by measuring at two known reference points. The points should span a significant portion of the measurement range, ideally near the endpoints, to minimize interpolation errors. Linear calibration assumes the sensor characteristic between points is linear; residual nonlinearity causes errors at intermediate values.

The two-point calibration equations calculate gain as the ratio of reference span to output span, and offset from the gain and either measurement pair. This approach corrects the dominant error sources for most sensors and represents the standard minimum calibration for industrial measurements.

Multi-Point Calibration

Multi-point calibration measures at three or more reference points, enabling correction for nonlinearity as well as offset and gain. More points provide better characterization of the sensor's actual behavior and reduce the impact of random measurement errors through averaging effects in the curve fit.

The reference points should be distributed across the measurement range, with additional points in regions of rapid characteristic change. End-point weighting ensures accurate calibration at range limits, while interior points capture nonlinearity. For polynomial fitting, the number of points should exceed the polynomial order by at least two to three times for statistical robustness.

Least Squares Fitting

Least squares fitting determines calibration parameters by minimizing the sum of squared differences between the calibration model and the measured data. This approach provides optimal parameter estimates when measurement errors are normally distributed with constant variance. The method extends naturally from linear to polynomial and more general nonlinear models.

For linear and polynomial models, least squares fitting has closed-form solutions involving matrix operations. Nonlinear models require iterative numerical methods that may converge to local rather than global minima. Proper initial parameter estimates and constraint handling ensure reliable convergence.

Calibration Verification

Calibration verification confirms that the calibration remains valid by measuring at check points and comparing against tolerance limits. Check points may differ from calibration points to test interpolation accuracy. Regular verification catches drift before measurements exceed accuracy requirements, triggering recalibration when necessary.

Verification frequency depends on sensor stability, environmental exposure, and accuracy requirements. Critical measurements may require verification before each use, while stable sensors in controlled environments may need only annual checks. Historical verification data guides adjustment of verification intervals based on actual drift behavior.

Calibration Certificate Documentation

Calibration certificates document the calibration process and results, providing the information necessary to assess measurement validity and maintain traceability. Essential elements include identification of the calibrated instrument and reference standards, environmental conditions during calibration, measured data and derived parameters, estimated uncertainties, and the calibration date and responsible personnel.

For regulated industries, calibration certificates must follow specific formats and include required statements. ISO/IEC 17025 accreditation provides international recognition of calibration laboratory competence, with accredited certificates carrying particular weight for quality system compliance.

Temperature Compensation

Temperature affects nearly all sensors, changing their sensitivity, offset, and sometimes their nonlinearity characteristics. Temperature compensation corrects for these effects, maintaining measurement accuracy across the operating temperature range. Effective compensation requires understanding the temperature dependencies and implementing appropriate correction algorithms.

Temperature Effects on Sensors

Temperature influences sensors through multiple mechanisms. Resistive sensors experience resistance changes in both sensing elements and associated circuitry. Semiconductor devices show strong temperature dependence in carrier concentrations and mobilities. Mechanical sensors face thermal expansion affecting dimensions and stresses. Piezoelectric sensors experience temperature-dependent charge sensitivity and dielectric properties.

Temperature effects typically divide into offset drift and sensitivity drift. Offset drift shifts the output at zero input, often specified in units per degree. Sensitivity drift changes the slope of the input-output characteristic, typically specified as a percentage per degree. Some sensors also show temperature-dependent nonlinearity, requiring more complex compensation schemes.

Hardware Compensation Techniques

Bridge circuits inherently provide some temperature compensation when all arms experience the same temperature. Matched resistors in the bridge drift together, maintaining balance at zero input. This approach works well when temperature gradients across the bridge are small, as in integrated pressure sensors with all resistors on a single die.

Thermistor compensation networks add temperature-dependent elements to counteract sensor drift. A negative temperature coefficient thermistor can compensate for decreasing sensor sensitivity with temperature by increasing circuit gain. These networks require careful design to match the sensor's temperature characteristic across the operating range.

Software Compensation Algorithms

Software compensation uses a separate temperature measurement to correct sensor readings algorithmically. The simplest approach applies linear correction factors for offset and sensitivity drift. More sophisticated algorithms use polynomial functions of temperature, possibly combined with lookup tables for complex behaviors.

The correction equation typically takes the form: corrected = (raw - offset(T)) / sensitivity(T), where offset(T) and sensitivity(T) are functions of temperature T. These functions may be polynomials, lookup tables, or combinations thereof, determined during calibration at multiple temperatures.

Calibration for Temperature Compensation

Temperature compensation requires calibration at multiple temperatures across the operating range, measuring the sensor characteristic at each temperature. A minimum of three temperatures (low, ambient, high) captures linear temperature effects; more temperatures reveal nonlinear dependencies. The calibration points should span the expected temperature extremes plus margin for uncertainty.

For each calibration temperature, full sensor characterization establishes the offset, sensitivity, and nonlinearity at that temperature. Fitting these parameters versus temperature yields the compensation functions. Cross-validation at intermediate temperatures verifies compensation accuracy.

Dynamic Temperature Effects

Rapid temperature changes can cause transient measurement errors beyond steady-state temperature effects. Thermal gradients within the sensor create differential expansion stresses. Temperature sensor lag causes compensation to use stale temperature data. Thermal time constants of different sensor components may differ, causing temporary mismatches.

Mitigating dynamic effects requires understanding the thermal behavior of the complete sensor assembly. Thermal modeling identifies potential gradient sources. Proper thermal coupling between temperature sensor and main sensor reduces lag effects. Filtering temperature measurements appropriately balances responsiveness against noise.

Integrated Temperature Sensors

Many modern sensors include integrated temperature sensing elements for compensation purposes. MEMS inertial sensors typically include on-die temperature sensors. Smart pressure sensors provide temperature outputs alongside pressure readings. These integrated sensors experience the same thermal environment as the primary sensing element, minimizing gradient and lag errors.

Integrated temperature sensors may require their own calibration for absolute accuracy, though relative accuracy across the primary sensor's temperature range is usually sufficient for compensation purposes. The sensor data sheet specifies the temperature measurement interface and accuracy.

Digital Filtering

Digital filtering removes noise and unwanted signal components from sensor data while preserving the information of interest. Unlike analog filters that operate on continuous signals, digital filters process discrete samples and can implement filter characteristics that would be difficult or impossible to achieve with analog components.

Moving Average Filters

The moving average filter, the simplest digital filter, outputs the average of the most recent N samples. This filter smooths noise by averaging out random variations while passing slowly changing signals. The frequency response shows a sinc function shape with zeros at multiples of the sample rate divided by N.

Moving averages are trivial to implement, requiring only addition, subtraction, and division. A circular buffer stores the N most recent samples. Each new sample adds to the sum while the oldest sample subtracts, maintaining a running total that divides by N for the output. Integer arithmetic implementations avoid division by choosing N as a power of two.

Exponential Moving Average

The exponential moving average (EMA), also called single-pole IIR filter or exponential smoothing, weights recent samples more heavily than older ones with exponentially decaying weights. The recursive formula is: output = alpha * input + (1 - alpha) * previous_output, where alpha controls the smoothing amount.

EMA advantages include minimal memory requirements (storing only the previous output), computational simplicity, and adjustable smoothing through a single parameter. The equivalent time constant relates to alpha and sample rate, enabling direct specification of the desired response time. Unlike moving averages, EMA has no finite impulse response; all past samples influence the output, though older samples contribute negligibly.

Finite Impulse Response Filters

Finite impulse response (FIR) filters compute outputs as weighted sums of current and past input samples, with the weights called filter coefficients or taps. The number of taps determines the filter order and the sharpness of the frequency response. FIR filters are inherently stable and can provide linear phase response, preserving waveform shapes.

FIR filter design involves selecting coefficients to achieve desired frequency response characteristics. Window methods apply a smoothing window to the ideal impulse response. Parks-McClellan and similar algorithms optimize coefficients for minimax error criteria. Many design tools and coefficient tables are available for standard filter specifications.

Infinite Impulse Response Filters

Infinite impulse response (IIR) filters use feedback, incorporating past output values as well as past and current inputs. This feedback enables much sharper frequency responses with fewer coefficients than equivalent FIR filters, but at the cost of potential instability and nonlinear phase response.

IIR designs often derive from classical analog filter prototypes such as Butterworth, Chebyshev, and elliptic filters, transformed to digital equivalents using bilinear transformation or impulse invariance. These designs inherit the well-understood characteristics of their analog counterparts while adding the flexibility and precision of digital implementation.

Median Filters

Median filters output the middle value of a sorted window of samples, effectively rejecting outliers while preserving edges. Unlike linear filters, median filters are nonlinear, providing fundamentally different noise rejection characteristics. They excel at removing impulsive noise such as electrical spikes while maintaining sharp transitions in the underlying signal.

The primary disadvantage of median filters is computational cost; sorting operations require more processing than linear filter multiply-accumulate operations. Efficient implementations for small window sizes use sorting networks or maintain partially sorted lists updated incrementally as the window slides.

Filter Parameter Selection

Selecting appropriate filter parameters requires balancing noise rejection against signal bandwidth and response time. Heavier filtering (lower cutoff frequency, more averaging) provides better noise reduction but slows response to actual signal changes. The application requirements determine acceptable trade-offs between smoothness and responsiveness.

Characterizing the noise spectrum helps target filter design. If noise concentrates at specific frequencies (such as power line interference), narrow notch filters remove it without affecting other frequencies. Broadband noise benefits from low-pass filtering with cutoff above the signal bandwidth. Impulsive noise suggests median or other nonlinear filters.

Real-Time Implementation Considerations

Real-time filter implementation must complete calculations within the sample period to avoid missing data or accumulating latency. Fixed-point arithmetic provides deterministic timing on processors without floating-point hardware. Pipelining and parallel processing enable high-order filters at fast sample rates.

Filter initialization affects the startup transient when the filter begins operating or after a discontinuity. Options include initializing internal states to zero (causing a slow settling from zero), to the first sample value, or using a shorter filter during startup. The choice depends on how the system handles initial measurements.

Sensor Fusion

Sensor fusion combines data from multiple sensors to achieve better estimates than any single sensor provides alone. By integrating complementary sensor characteristics, fusion systems overcome individual sensor limitations, providing improved accuracy, extended operating range, or enhanced reliability. Modern navigation, robotics, and motion tracking systems rely heavily on sensor fusion techniques.

Complementary Sensor Characteristics

Effective sensor fusion exploits complementary sensor properties. Accelerometers provide accurate short-term motion sensing but drift when integrated for position. GPS provides absolute position but with limited update rate and accuracy. Combining them yields smooth, drift-free position tracking unavailable from either alone.

Gyroscopes measure angular rate accurately for short periods but accumulate integration drift. Magnetometers provide absolute heading reference but suffer from local magnetic disturbances. Fusing gyroscope and magnetometer data produces stable heading estimates that reject both drift and momentary disturbances.

Complementary Filters

Complementary filters, the simplest fusion approach, combine sensor outputs through parallel high-pass and low-pass filters whose responses sum to unity. High-frequency content comes from one sensor, low-frequency content from another, with the crossover frequency chosen to exploit each sensor's strengths.

For attitude estimation, a complementary filter might take high-frequency angular rate from a gyroscope (integrated to angle) and low-frequency absolute reference from accelerometer/magnetometer measurements. The filter continuously blends these sources, providing responsive angle estimates free from both gyroscope drift and accelerometer noise.

Weighted Averaging

When multiple sensors measure the same quantity, weighted averaging combines their outputs with weights reflecting their relative accuracies. Optimal weights are inversely proportional to variance; more accurate sensors receive higher weights. This approach assumes sensor errors are independent and unbiased, conditions that should be verified.

Dynamic weighting adjusts sensor contributions based on operating conditions. A GPS/INS system might weight GPS heavily when satellite geometry is good but reduce its influence during poor visibility. Fault detection algorithms can set weights to zero for failed sensors, providing graceful degradation.

Redundancy and Fault Tolerance

Multiple sensors measuring the same quantity provide redundancy for fault detection and tolerance. Comparing sensor outputs reveals disagreements indicating potential failures. Voting schemes select the majority value when sensors disagree. Analytical redundancy uses physical relationships to cross-check measurements without duplicate sensors.

Fault-tolerant fusion requires at least enough sensors to detect failures (two for detection, three for isolation, four for continued operation after single failures). Aerospace and safety-critical systems specify minimum redundancy levels based on failure probability requirements.

Multi-Rate Fusion

Sensors often operate at different sample rates, requiring multi-rate fusion techniques. GPS may update once per second while inertial sensors run at hundreds of hertz. The fusion algorithm must accommodate these rate differences, typically by predicting fast-sensor states between slow-sensor updates.

Between slow sensor updates, fast sensor data provides state propagation. When slow sensor measurements arrive, they correct accumulated drift. The correction applies instantaneously or distributes over time to avoid output discontinuities. Proper handling of timing, including sensor latency and processing delays, is essential for accurate fusion.

Sensor Fusion Architectures

Loosely coupled fusion processes each sensor independently, then combines processed outputs. This approach simplifies design and allows sensor modules to operate independently, but may lose information available at raw data levels. GPS/INS systems often use loosely coupled fusion, combining GPS position with INS-computed position.

Tightly coupled fusion processes raw measurements from all sensors together, enabling optimal use of available information. The fusion algorithm directly uses GPS pseudoranges rather than computed positions, for example, extracting more information and providing better performance in challenging conditions. The trade-off is increased complexity and interdependence between sensor processing.

Kalman Filtering

The Kalman filter provides an optimal framework for sensor fusion and state estimation, recursively combining predictions from a system model with sensor measurements while tracking uncertainty. Named after Rudolf Kalman, who published the algorithm in 1960, this technique forms the core of navigation systems, tracking algorithms, and countless other applications requiring estimates of system states from noisy measurements.

State Space Representation

Kalman filtering operates on state space system representations. The state vector contains all quantities needed to predict future system behavior: positions, velocities, biases, and other relevant parameters. State equations describe how states evolve over time, while measurement equations relate states to sensor outputs.

The discrete-time linear state equations have the form: x(k+1) = F*x(k) + B*u(k) + w(k), where x is the state vector, F is the state transition matrix, u is a known input, B maps inputs to states, and w represents process noise. The measurement equation is: z(k) = H*x(k) + v(k), where z is the measurement vector, H is the observation matrix, and v is measurement noise.

Prediction Step

The prediction step propagates the state estimate and its uncertainty forward in time using the system model. The predicted state equals the state transition matrix times the previous state plus known inputs. The predicted covariance equals the transition matrix times the previous covariance times the transition matrix transposed, plus process noise covariance.

Process noise covariance represents uncertainty in the model itself: unmodeled disturbances, approximation errors, and unknown inputs. Larger process noise covariance causes the filter to weight measurements more heavily relative to predictions. Tuning process noise balances model confidence against measurement trust.

Update Step

The update step incorporates new measurements to correct the predicted state. The innovation, or measurement residual, equals the actual measurement minus the measurement predicted from the current state estimate. The Kalman gain determines how much the innovation adjusts the state, optimally balancing prediction and measurement uncertainties.

The Kalman gain computation involves the predicted covariance, observation matrix, and measurement noise covariance. Low measurement noise relative to prediction uncertainty yields high gain, weighting measurements strongly. High measurement noise yields low gain, trusting predictions more than measurements.

Covariance Estimation

The Kalman filter maintains a covariance matrix representing uncertainty in the state estimate. This covariance propagates through predictions (generally increasing uncertainty) and updates (generally decreasing uncertainty). The covariance provides confidence bounds for estimates and drives the Kalman gain calculation.

Proper covariance values require accurate noise characterization. Measurement noise covariance comes from sensor specifications or experimental determination. Process noise covariance is often tuned empirically, adjusting until filter performance meets expectations. Underestimating process noise causes overconfidence in the model; overestimating causes excessive sensitivity to measurement noise.

Extended Kalman Filter

The extended Kalman filter (EKF) adapts the linear Kalman filter for nonlinear systems by linearizing about the current state estimate. The state transition and observation functions are nonlinear, but their Jacobian matrices, evaluated at each time step, approximate the linear matrices used in the standard algorithm.

EKF limitations arise from linearization errors. Large nonlinearities or poor initial estimates can cause divergence. The linearization assumes small deviations from the operating point, violated during rapid dynamics or when estimates are significantly wrong. Despite these limitations, EKF works well for many practical applications and remains widely used.

Unscented Kalman Filter

The unscented Kalman filter (UKF) handles nonlinearity differently, using carefully chosen sample points (sigma points) to capture the statistical distribution of the state. These points propagate through the nonlinear functions exactly, then recombine to estimate the transformed distribution. This approach captures nonlinear effects more accurately than linearization.

UKF performance generally equals or exceeds EKF for nonlinear systems, sometimes dramatically so for highly nonlinear cases. Computational cost is somewhat higher due to multiple sigma point evaluations, but implementation complexity is similar. UKF is increasingly preferred for new designs where EKF limitations are concerns.

Implementation Considerations

Numerical stability requires careful implementation, particularly for covariance matrices that must remain positive definite. Square root filters factorize the covariance, operating on factors rather than the covariance directly to maintain positive definiteness. Joseph form covariance updates provide better numerical properties than simpler formulas.

Initialization establishes the starting state estimate and covariance. Poor initialization can cause slow convergence or filter divergence. When initial state is unknown, large initial covariance lets measurements dominate until the estimate converges. Some applications use special initialization procedures to establish estimates before entering normal operation.

Practical Tuning Guidelines

Tuning a Kalman filter involves selecting process and measurement noise covariances that yield desired performance. Start with physically motivated values: measurement noise from sensor specifications, process noise from expected disturbance magnitudes. Adjust based on observed performance, typically in simulation before field testing.

Common tuning symptoms guide adjustments. Slow response to actual changes suggests insufficient process noise (filter trusts model too much). Excessive noise in estimates suggests insufficient measurement noise weighting (filter trusts sensors too much). Innovation sequences should be white with magnitude consistent with their predicted covariance; violations indicate model mismatch or incorrect noise parameters.

Advanced Processing Techniques

Beyond the fundamental methods, advanced processing techniques address specific challenges in sensor data processing, including outlier rejection, adaptive algorithms, and methods for handling specific sensor types or operating conditions.

Outlier Detection and Rejection

Outliers, measurements grossly inconsistent with expected values, can severely degrade estimation performance. Detection methods compare measurements against predictions, flagging those exceeding statistical thresholds. The Kalman filter innovation covariance provides a natural threshold: innovations exceeding several standard deviations are suspect.

Rejection strategies range from simple exclusion to soft weighting. Hard rejection discards outliers entirely, risking information loss if the threshold is too aggressive. Soft rejection downweights suspicious measurements rather than excluding them completely. Robust estimation methods like M-estimators automatically reduce outlier influence without explicit detection.

Adaptive Filtering

Adaptive filters adjust their parameters based on observed data, accommodating changing conditions without manual tuning. Adaptive Kalman filters estimate noise covariances online from innovation sequences. When measurements become noisier, the filter automatically reduces their weight; when the system model becomes less accurate, process noise estimates increase.

Multiple model approaches run parallel filters with different parameter sets, combining outputs based on how well each model explains the observations. This technique handles mode switches, such as maneuvering versus non-maneuvering target tracking, where a single parameter set cannot cover all conditions.

Nonlinear Estimation Beyond EKF and UKF

Particle filters represent probability distributions with collections of samples (particles), avoiding the Gaussian assumption of Kalman-based methods. Each particle represents a possible state, weighted by measurement likelihood. This approach handles arbitrary distributions and severe nonlinearities but requires many particles for high-dimensional states.

Gaussian sum filters approximate non-Gaussian distributions as sums of Gaussian components, each processed by a separate Kalman filter. The components capture multi-modal distributions that single Gaussian estimates cannot represent. Component management (splitting, merging, pruning) keeps computational cost manageable.

Fixed-Lag Smoothing

Standard Kalman filtering is causal, using only past and current measurements. Smoothing incorporates future measurements to improve estimates, accepting latency for accuracy. Fixed-lag smoothing delays outputs by a specified interval, refining estimates with measurements received during the lag period.

Smoothing provides significantly better estimates when latency is acceptable. Navigation systems may use smoothed estimates for map generation while providing filtered estimates for real-time control. Post-processing applications (trajectory reconstruction, scientific analysis) typically use full smoothing over entire data records.

Integrity Monitoring

Safety-critical applications require integrity monitoring to detect when estimates may be unreliable. Receiver autonomous integrity monitoring (RAIM) in GPS detects satellite failures by comparing redundant measurements. Protection levels bound the position error with specified probability, enabling go/no-go decisions based on required accuracy.

Extending integrity concepts to fused sensor systems remains an active research area. The challenge lies in characterizing error bounds when combining multiple imperfect sensors with imperfect models. Fault detection, isolation, and recovery (FDIR) procedures specify responses to detected anomalies.

Summary

Sensor data processing transforms raw sensor outputs into accurate, reliable measurements through a combination of correction, filtering, and fusion techniques. Linearization compensates for inherent sensor nonlinearities using lookup tables, polynomials, or splines. Calibration curves establish the relationship between sensor outputs and true values, with multi-point calibration enabling nonlinearity correction. Temperature compensation addresses the ubiquitous temperature sensitivity of real sensors through hardware networks or software algorithms.

Digital filtering removes noise while preserving signal information, with filter type and parameters chosen based on noise characteristics and application requirements. Sensor fusion combines multiple sensors to overcome individual limitations, exploiting complementary characteristics for improved accuracy and reliability. The Kalman filter provides an optimal framework for fusing measurements with system models, recursively estimating states while tracking uncertainty.

These techniques form an integrated processing chain in practical systems. Raw sensor data passes through linearization and temperature compensation, yielding corrected measurements. Digital filters reduce noise. Fusion algorithms combine multiple corrected, filtered measurements with system models to produce final estimates. Understanding each technique and how they interact enables engineers to design sensor processing systems that extract maximum information from available sensors while maintaining robust operation across varying conditions.