Gaussian Fit — Control Systems/Curve Fitting

Gaussian Fit

Control Systems / Curve Fitting

Fits a sum of N Gaussians to the last W samples and emits its amplitudes:

ŷ(x) = Σ_i=1..n a_i·exp( −((x − b_i) ÷ c_i)² )

This is MATLAB's gauss<n> model, and the output carries its amplitudes in that model's own order – a₁, a₂, …, a_n. The newest sample sits at x = 0 and the sample i steps back at x = −i·h, the same abscissa Polynomial Fit and Fourier Fit use. It is the useful half of peak fitting: known peaks, unknown strengths.

Ports

• u – the sampled signal being fitted. Scalar: one channel and its own window – see Notes.
• a – the amplitude column, [n, 1], in the order above. Its height follows Number of Terms alone.

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 amplitudes cannot be determined by fewer than n points. Bounded above because the dot products are unrolled at export.
• Number of Terms – n, a whole number from 1 to 8, matching MATLAB's gauss1…gauss8. Each term adds one Gaussian and one output row.
• Centres – [b₁ … b_n], an n element vector giving where each bump sits on the fitted axis, in the same units as Abscissa Step. The window covers −(W−1)·h to 0, so a centre far outside that range contributes a nearly flat column and conditions the fit badly.
• Widths – [c₁ … c_n], an n element vector of strictly positive numbers. MATLAB's c is the 1/e half width, not a standard deviation: c = σ·√2.
• Abscissa Step – h, the spacing between consecutive samples on the fitted axis. A single positive number. Set it to the sampling period and give the centres and widths in seconds, or leave it at 1 and give them in samples.
• 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 exponential is evaluated in any generated core – every one of them 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 amplitude, with the products accumulated at full width and shifted back once per row.

Simulink bridge

None (Support::None). Fitting a sum of Gaussians to a running window is a Curve Fitting Toolbox function (fit with a gauss<n> model), 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 centres and widths are parameters here and MATLAB solves for them. That is the one real departure from fit(x, y, 'gauss2'), and it is what makes the block possible: with b and c unknown the fit is nonlinear and needs an iterative solver every sample; with both given it is an ordinary linear least squares whose design matrix is fixed by the configuration. If the peak positions are what you are looking for, this is the wrong block – find them first and set them here.
• ⚠ Its output is NOT a polynomial, and Evaluate Fit cannot read it. Evaluate Fit, Fit Derivative and Fit Integral all speak coefficients in descending powers; these are amplitudes against a basis of exponentials. Wiring this block into any of the three produces a number rather than an error, so the mistake is silent.
• One solve, then arithmetic. The small normal-equation system is solved once when the configuration loads; every sample afterwards is n fixed dot products.
• ⚠ Not every setting is a fit, and the block refuses rather than guessing. Two terms given the same centre and width are the same column, and the fit has no unique answer; two bumps that overlap almost that far have one, but the window can barely tell them apart and the normal matrix is ill conditioned long before it is singular. The singular case is detected when the configuration loads and reported with a reason – rather than emitting the arbitrary answer a pseudo-inverse would have produced. Keep the centres at least a width apart and neither case arises.
• Verified against MATLAB. At W = 13, n = 2, h = 0.4, centres [−1.1 −3.3] and widths [0.9 1.7], R2026a's own least-squares solve of the same design matrix answers a = [0.3030382460638365, 0.21965420016626469] on a fixed 13-sample window; this block reproduces both entries to 1.0e−15 relative, with cond(A'A) = 2.23. Solving the normal equations squares the condition number where a QR solve 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; it costs nothing at a conditioning like that one.
• MATLAB's width is not a standard deviation. exp(-((x-b)/c)^2) has no factor of two in the exponent, so c = σ·√2 and the bump's full width at half maximum is 2·c·√(ln 2).
• The window is zero-prefilled, and the zeros count. The first W−1 amplitude vectors of a run are a startup transient, matching Moving Median, Detrend, Savitzky-Golay Filter, Polynomial Fit and Fourier 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.