Derivative

Shippingstateful

derivative

Numerical differentiation with state persistence across blocks

Signature

Inputs

signalSignalrequired

Outputs

resultSignal

Parameters

Key	Type	Default	Notes
`method`	enum	central	one of: central, 5point, second

Description

Derivative estimates the time derivative $\overset{x}{˙} (t) = d x / d t$ of an incoming signal by finite differences. It consumes a single signal input and emits a result of the same length, sample-aligned with the input. The finite-difference scheme is selected by the method parameter: central (second-order two-sided), 5point (fourth-order five-point stencil), or second (the second derivative $\overset{x}{¨}$ ).

The node is stateful. Its DerivState carries the last sample value and its timestamp across processing blocks, so streaming DAQ data split into consecutive buffers is differentiated continuously without a discontinuity or a lost sample at every block boundary. The spacing $h$ is taken from the signal's time base, so non-uniform or per-block sample rates are handled through the stored timestamps rather than an assumed constant step.

Units propagate: differentiating a quantity of unit $U$ yields unit $U / s$ (e.g. m becomes m/s, and second method gives m/s^2). Because differentiation is a linear operation on samples, uncertainty (sigma) propagates through the stencil coefficients; note that finite differencing amplifies high-frequency noise, so output sigma is typically larger than input sigma. Consider pre-smoothing noisy inputs.

Mathematics

\frac{d x}{d t}_{i} \approx \frac{x _{i + 1} - x _{i - 1}}{2 h}

\frac{d x}{d t}_{i} \approx \frac{- x _{i + 2} + 8 x _{i + 1} - 8 x _{i - 1} + x _{i - 2}}{12 h}

\frac{d ^{2} x}{d t ^{2}}_{i} \approx \frac{x _{i + 1} - 2 x _{i} + x _{i - 1}}{h ^{2}}

Examples

Velocity from a position track

Feed a position signal in metres into Derivative with method = central. The output carries unit m/s. For a track sampled at 1 kHz ( $h = 0.001 s$ ), a linear ramp of $2 m$ over $1 s$ yields a flat output at $2 m/s$ , and the stored DerivState makes the estimate seamless across buffer boundaries in a live acquisition.

Acceleration in one node

Set method = second to obtain $\overset{x}{¨}$ directly from a position signal instead of chaining two Derivative nodes. Input unit m produces output unit m/s^2. This uses the three-point second-difference stencil $\frac{x _{i + 1} - 2 x _{i} + x _{i - 1}}{h ^{2}}$ , avoiding the extra noise amplification of two successive first derivatives.

Applications

Deriving velocity and acceleration from position or displacement sensors in motion control and robotics.
Computing rate-of-change alarms in process monitoring (e.g. pressure or temperature slope detection).
Edge and slope detection as a preprocessing stage before peak detection or thresholding.
Estimating current from measured charge, or EMF from flux, in electromagnetics and instrumentation pipelines.

Neat

The stored `last_value`/`last_ts` state means splitting a stream into arbitrary block sizes gives bit-identical results to processing it as one buffer — the block boundary is invisible.

The `5point` stencil is fourth-order accurate ($O(h^4)$ error) versus second-order for `central`, so on smooth signals it is markedly more accurate at the same sample rate, but it is also more sensitive to noise.

Known issues

Finite differencing amplifies high-frequency noise; broadband sensor noise can dominate the output. Smooth or filter noisy signals before differentiating.

One-sided stencils are used at the very first block start (no prior state) and at signal boundaries, which are lower-order and slightly less accurate than the interior central/5-point estimates.