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Technology / Technical guide
Numerical Derivatives: Signal Slope & Noise Amplification
Differentiation estimates how quickly a quantity changes. It also changes the contribution of measurement errors: nearby readings are subtracted, then divided by a time interval. A sharper-looking derivative is therefore not automatically a better account of the underlying change. Noise, correlation and approximation error all participate.
Slope uses a difference and a time scale.
The uncertainty of a derivative cannot be inferred from the uncertainty of one reading without describing how successive errors are related.
01 / Difference
A discrete slope is an interval estimate.
For readings y[n] separated by Ts seconds, a forward difference is d̂[n] = (y[n+1] − y[n])/Ts.
For readings y[n] separated by Ts seconds, a forward difference is d̂[n] = (y[n+1] − y[n])/Ts. If y is measured in metres, the result is in m/s. The reading interval is part of the expression, rather than an optional display conversion.
A smooth underlying signal makes this quotient approach a derivative as the interval becomes small. With finite spacing, it represents an average slope over the interval. Assigning that value to the beginning, middle or end changes its timing interpretation.
02 / Error model
Subtraction combines two errors with their covariance.
Write each reading as its true value plus an error ε.
Write each reading as its true value plus an error ε. The derivative error is (ε[n+1] − ε[n])/Ts. With equal error variance σ² and neighbouring covariance c, its variance is (2σ² − 2c)/Ts². Those assumptions must accompany the result.
For independent neighbouring errors, c is zero and the standard deviation becomes √2 σ/Ts. Positive correlation reduces the difference error relative to that independent model; negative correlation increases it. Consequently, a single-channel noise amplitude alone does not determine derivative uncertainty.
03 / Toy calculation
A shorter interval can increase slope noise.
Suppose, only for illustration, that independent reading errors have standard deviation 0.1 unit at either spacing.
Suppose, only for illustration, that independent reading errors have standard deviation 0.1 unit at either spacing. With Ts = 0.2 s, the slope-error standard deviation is about 0.707 unit/s. With Ts = 0.1 s, it becomes about 1.414 unit/s.
This comparison holds per-reading uncertainty unchanged. It does not establish how a real sensor’s uncertainty changes with sampling, averaging or bandwidth. Altering the acquisition model can change both σ and the correlation, so the arithmetic is not a universal sampling rule.
04 / Frequency weighting
Differentiation gives greater weight to rapid variation.
MathWorks explains why a difference-based derivative makes noisy signal content more prominent.
MathWorks explains why a difference-based derivative makes noisy signal content more prominent. For the chosen discrete difference, the magnitude response is 2|sin(Ω/2)|/Ts, where Ω is angular frequency in radians per sample. Near zero frequency it approximates continuous differentiation.
The bounded discrete response should not be confused with an unlimited continuous-frequency extension. Frequencies already indistinguishable after sampling cannot be recovered by differentiation. A derivative operates on the available representation, including its missing or misrepresented information.
05 / Competing errors
Noise and finite-spacing approximation are different terms.
A curved signal does not have a constant slope across the interval.
A curved signal does not have a constant slope across the interval. Finite spacing can therefore introduce approximation error even when the readings contain no noise. Reducing spacing can reduce this error for a sufficiently smooth underlying function.
Meanwhile, independent reading noise is divided by that smaller spacing. These effects can move in opposite directions. A constant common offset cancels in a difference, but a drifting offset need not. Removing one kind of error does not establish that the derivative is unbiased.
06 / Scope
A derivative quantity is not a controller setting.
Higher derivatives involve further differences and stronger dependence on timing and smoothness.
Higher derivatives involve further differences and stronger dependence on timing and smoothness. They require their own error model. A numerical slope also differs from the derivative term inside a complete feedback architecture, which has additional relationships and constraints.
No MARSTON differentiation method, noise level or derivative-control behaviour is specified. This page contains no live-data transformation, filter tuning or sensor adjustment; it explains the mathematical consequences of a declared difference quotient.
Related technical guides: Estimator Bias, Variance & Mean-Squared Error · Least-Squares Fitting: Residuals, Weights & Input Errors · Vibration Displacement, Velocity & Acceleration.
General background: MathWorks: signal differentiation and measurement noise. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Interval Ts | Seconds between represented readings | Use actual model timing rather than an assumed display rate |
| Slope units | Quantity unit divided by seconds | A change of physical dimension accompanies differentiation |
| Reading variance σ² | Squared input units | Defined for the stated error process |
| Neighbour covariance c | Shared variation between errors | Independence cannot be assumed from separate timestamps |
| Approximation error | Difference between interval slope and derivative | Depends on smoothness and interval placement |
Engineering questions.
Does higher sampling always improve a derivative?
No. The result depends on spacing, per-reading uncertainty and correlation.
Why does a fixed offset cancel?
The same additive offset appears with opposite signs in the difference.
Does a drifting offset also cancel?
Generally not; its change contributes to the estimated slope.
Can differentiation recover aliased information?
No. It cannot reverse information loss already present in sampled readings.
Is the derivative error always √2 σ/Ts?
Only for the stated equal-variance, independent neighbouring errors.
Is a PID derivative setting recommended?
No. This page concerns finite-difference mathematics, not field control changes.
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