Filtering and state estimation

• Implement and analyze a moving-average or exponential low-pass filter.
• Separate measurement, estimate, residual, and true-but-unknown state in a diagnostic record.

• probability-uncertainty
• signals-systems

Sensors do not reveal truth directly. A camera-derived position jitters, an ADC includes quantization and electrical noise, and a distance sensor may occasionally return an outlier. Filtering combines measurements over time to make a more useful estimate. The cost is usually delay or reduced ability to follow rapid change.

An N-sample moving average is ȳ_k = (1/N)Σx_{k-i}. For independent zero-mean noise with standard deviation σ, its standard deviation falls ideally to σ/√N, but the window delays changing signals. An exponential filter uses ŷ_k = αx_k + (1-α)ŷ_{k-1}, with 0 < α ≤ 1. Smaller α smooths more and responds more slowly. A state estimator goes further by combining a process model and measurement uncertainty; even a simple “predict, compare residual, correct” loop makes its assumptions explicit.

The residual r_k = measurement - predicted measurement is diagnostically valuable. Random residuals near zero may support the model; persistent bias suggests calibration error or an unmodelled force; abrupt large residuals may signal dropout or impact. Do not silently discard inconvenient readings. Record raw data, estimate, filter parameters, timestamp, saturation, and rejection decisions. For a camera mount, the build decision could be a modest filter that reduces pixel jitter while preserving enough bandwidth to track intentional pan motion.

A common misconception is that a smoother plot is more accurate. Heavy filtering can produce a clean but late estimate and hide a loose connector. Another is treating a filter's output as a physical measurement; it is a computed estimate based on assumptions. Averaging correlated drift does not remove the drift as σ/√N predicts. Choose filter bandwidth relative to system dynamics, examine residuals, and test known changes. If control stability worsens after filtering, added delay may outweigh reduced noise.

Problem

A distance sensor reports 100, 104, 98, and 102 mm. Compute the four-sample mean and the ideal standard-deviation reduction factor for independent noise.

Steps

1. Sum readings: 100 + 104 + 98 + 102 = 404 mm.
2. Mean is 404 mm ÷ 4 = 101 mm.
3. For independent zero-mean noise, standard deviation of the mean is σ/√4 = σ/2.

Result

The estimate is 101 mm, with an ideal random-noise standard deviation half that of one reading.

Sanity check

The mean lies within the observed 98–104 mm range; the uncertainty claim does not cover bias or correlated drift.

A larger moving-average window makes a stationary trace smoother but causes a robot to react late to obstacles. What explains both effects?

A. The window reduces high-frequency variation but adds delay B. The sensor's physical range increases C. Averaging removes every bias D. The sample clock becomes continuous

Your answer: ______________________________

Reasoning:

Goal

Compare raw, filtered, and residual signals during stationary and moving tests.

Equipment

• Low-voltage distance or light sensor
• Microcontroller
• Serial logger
• Movable target
• Ruler

Procedure

1. Log raw readings from a fixed target and predict how windows N = 1, 4, and 16 will affect scatter and response time.
2. Move the target by a measured amount at a marked time; compute each moving average without deleting raw samples.
3. Compare actual scatter, time to cross 90% of the change, and residuals for each window.
4. Deliberate fault: briefly cover or disconnect the sensor within its safe interface; diagnose the outlier/dropout from raw data and residual rather than masking it with a long average.

What to observe

• Expected: longer windows reduce stationary scatter but delay the measured step.
• Measure: actual raw range, filtered crossing time, residuals, and dropout code; identify assumptions behind any uncertainty estimate.

Transfer challenge

Choose and justify a filter for a desk robot that must ignore millimetre jitter yet detect a hand approaching within 0.3 s.

Cautions

• Use only sensor-safe low-voltage wiring and disconnect power before changing connections.

Learner: ____________________ Date: ____________________

Evidence artifact or file: ____________________________________________

Notes: