Spline Fit — Control Systems/Curve Fitting

Spline Fit

Control Systems / Curve Fitting

Fits a B-spline of order k to the last W samples and emits its B-form coefficients:

ŷ(x) = Σ_j=1..n c_j·B_j,k(x | t)

where t is the knot sequence and n = len(t) − k. This is MATLAB's spap2, and in its second Knot Placement mode it is spapi. The knots are k-fold at both ends of the window's own span (MATLAB's augknt), so the basis sums to one across it. The newest sample sits at x = 0 and the sample i steps back at x = −i·h, the same abscissa Polynomial Fit, Fourier Fit and Gaussian Fit use.

Ports

• u – the sampled signal being fitted. Scalar: one channel and its own window – see Notes.
• c – the coefficient column, [n, 1], in the basis's own order. Its height follows Knot Placement: (number of interior knots) + k in the explicit mode, and W in the averaged one.

Parameters

• Window Length – W, how many samples the fit sees. A whole number from 3 to 101, and it must be at least n: n coefficients cannot be determined by fewer than n points. Bounded above because the dot products are unrolled at export.
• Spline Order – k, a whole number from 2 to 6. Order is degree + 1, so 4 – the default, and MATLAB's – is the cubic spline; 2 is the broken line.
• Knot Placement – which of MATLAB's two functions this is:
  • Explicit interior knots – you give them, n is (number of interior knots) + k, and with n < W this is a least-squares fit. It is spap2(knots, k, x, y), and it is the default.
  • Averaged (interpolating) – n = W and the interior knots are derived, each the average of k−1 consecutive sample positions. The collocation matrix is then square and the curve passes through every sample of the window. It is spapi, and the averaging is MATLAB's aptknt. Interior Knots is ignored here – see Notes for why it must be.
• Interior Knots – a strictly increasing vector of positions inside the window's span (−(W−1)·h, 0), in the same units as Abscissa Step. Read only in Explicit interior knots mode, and it must name at least one knot. A spline with no interior knot is a single polynomial piece of degree k−1, which is exactly what Polynomial Fit already computes – reach for that block instead of leaving this one empty.
• Abscissa Step – h, the spacing between consecutive samples on the fitted axis. A single positive number.
• 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 n×W coefficient bank is structural and is inlined into the arithmetic at export time rather than exposed as a tunable parameter: it follows from all five settings, and changing any of them changes how many multiplies and how many outputs the core contains. Re-export after changing them. No Cox-de Boor recursion runs in any generated core – the basis was evaluated once, at export time, to build the bank.

The three HDL targets are genuine synthesizable Q16.16: a shift register and one fixed multiply-accumulate per coefficient, with the products accumulated at full width and shifted back once per row.

Simulink bridge

None (Support::None). Fitting a B-spline to a running window is a Curve Fitting Toolbox function (spap2, spapi), and that toolbox ships no Simulink library at all, 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)): the window advances once per sample.
• ⚠ The averaged mode derives its knots, and that is not a convenience. Putting the interior knots on the sample positions is the obvious way to reach n = W and it is the wrong one. Measured at W = 13, k = 4, h = 0.4: averaged knots give cond(A'A) = 15.3 and a largest bank entry of 2.35; knots on the sites give cond(A'A) = 3.6e10 and a bank entry of 1.0e5, which saturates a Q16.16 datapath outright and makes the three HDL targets meaningless. That is why the mode computes them.
• ⚠ Its output is NOT a polynomial, and Evaluate Fit cannot read it. Evaluate Fit, Fit Derivative and Fit Integral speak coefficients in descending powers; these are B-form coefficients, which mean nothing without the knot sequence. Wiring this block into any of the three produces a number rather than an error, so the mistake is silent.
• ⚠ It is not Cubic Spline either, and the difference is which way the data flows. Curve Fitting / Cubic Spline takes a curve as configuration and evaluates it at its input; this block takes a signal and fits a curve to it. Reach for Cubic Spline when the curve is known and you want its value, and for this block when the curve is what you are after.
• One solve, then arithmetic. The normal-equation system is solved once when the configuration loads; every sample afterwards is n fixed dot products.
• ⚠ Not every knot sequence is a fit, and the block refuses rather than guessing. Interior knots that repeat, that fall outside the window's span, or that leave a stretch of k consecutive knots with no sample between them make a basis column vanish or coincide with its neighbour, and the fit has no unique answer. That is detected when the configuration loads and reported with a reason – rather than emitting the arbitrary answer a pseudo-inverse would have produced.
• Verified against MATLAB, at the basis and at the answer. The Cox-de Boor recursion was diffed entry by entry against R2026a's spcol at W = 13, k = 4, h = 0.4, interior knots [−3.5 −2.2 −1.1]: all 91 entries of the 13×7 collocation matrix agree exactly – worst difference 0.0, not merely small. The fitted coefficients then reproduce spap2's own answer to 2.3e−15 relative at cond(A'A) = 21.4. Solving the normal equations squares the condition number where MATLAB's QR does not, which is the price of collapsing the whole fit into one inlined bank of numbers instead of putting a factorization in ten generated cores.
• The window is zero-prefilled, and the zeros count. The first W−1 coefficient vectors of a run are a startup transient, matching Moving Median, Detrend, Savitzky-Golay Filter, Polynomial Fit, Fourier Fit and Gaussian Fit.
• Scalar only. One channel and its own history; wire one block per channel.
• No state space. Linear in the input, but through a fixed filter bank over W past samples rather than an A/B/C/D pair, so model reduction correctly declines to merge it.