Dynamic Time Warping — Control Systems/Correlation And Convolution

Dynamic Time Warping

Control Systems / Correlation And Convolution

The warping distance between the last W samples of two streams: the smallest total distance achievable by stretching and compressing one onto the other.

D(i,j) = c(i,j) + min( D(i−1,j), D(i,j−1), D(i−1,j−1) ), and the answer is D(W,W)

This is MATLAB's dtw(x, y). Unlike a correlation it does not ask by how much is one shifted; it asks how alike are these two shapes if time itself may be stretched, which is what lets it match two signals that run at different and changing speeds.

Ports

• x – the first signal. Scalar, with its own window.
• y – the second signal. Scalar, with its own window of the same length. The two are compared, so they should carry the same kind of quantity.
• dist – the warping distance, scalar and never negative. It is a total along the path, not a mean, so it grows with the window length.

Parameters

• Window Length – W, how many samples of each stream are compared. A whole number from 3 to 24. Bounded above because the table is unrolled at export: the emitted code grows as W² even though the storage does not.
• Distance – the metric a single pair of samples is scored with:
  • absolute – |x_i − y_j|, MATLAB's default.
  • squared – (x_i − y_j)², which penalises one large mismatch far more than several small ones and so prefers a path that keeps every pair close.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text.

The window length and the metric are structural – together they decide how many cells the core contains – so re-export after changing either. One row of the table is kept, not all of it: a cell needs the one above, the one to its left and the one diagonally above, so a single row updated in place plus one carried diagonal supplies all three, and a core holds W+4 numbers rather than W².

The three HDL targets are genuine synthesizable Q16.16: no division, no transcendental, no table lookup – subtractions, a sign test, additions and comparisons. ⚠ The thing to watch there is range rather than precision: the distance is a total over roughly W terms, so a long window against large inputs can reach the format's ceiling of about ±32768 and saturate. The squared metric reaches it sooner.

Simulink bridge

None (Support::None). dtw is a Signal Processing Toolbox function and that toolbox ships no Simulink library, so there is no path a diagram could name. The bridge reports this block rather than dropping it silently, and it therefore has no parity testbench; code export verification still covers it across all ten languages.

Notes

• Stateful, and discrete by nature (setDiscreteOnlyBlock(true)): both windows advance once per sample.
• ⚠ It is not Cross Correlation, which it sits beside. Cross Correlation slides one signal past the other rigidly and reports how well they line up at each lag. This block lets the time axis stretch locally, so it matches signals whose speeds differ and change – and in exchange it cannot report a lag, because different parts of the window are shifted by different amounts. Reach for that block to find a delay, this one to score a similarity.
• ⚠ MATLAB's maxsamp band is not offered, and the reason is the datapath rather than the effort: constraining |i−j| makes the cells outside the band unreachable, which the algorithm expresses as +∞. A fixed-point datapath has no such value, and any finite stand-in is a number the accumulated cost can legitimately exceed – at which point the band stops being a constraint and silently becomes a shortcut. What this block computes is dtw(x, y) with no third argument.
• ⚠ dtw(x, x) = 0, and this block's own sample page is a picture of that. A signal warps onto itself at no cost, so two identical inputs give a flat zero however interesting each of them is – and the documentation harness that produces the sample below drives every input port with the same stimulus, so that is exactly what it recorded. The page is showing the identity rather than a broken block. To see the distance do anything, the two inputs have to differ.
• The distance is a total, not a mean. Two windows twice as long score roughly twice as far apart even when they are equally alike, so a threshold chosen at one window length does not carry to another.
• Both windows are zero-prefilled, so the first W−1 distances of a run compare partly-empty windows.
• Scalar only, on both inputs. One channel each; wire one block per pair.
• No state space. A minimum over paths is not a linear map of the inputs, so model reduction correctly declines to merge it.