Sum Of Sines Fit — Control Systems/Curve Fitting

Sum of Sines Fit

Control Systems / Curve Fitting

Fits a sum of N sinusoids at given frequencies to the last W samples and emits each term's quadrature pair. The model is MATLAB's sin<n>,

ŷ(x) = Σ_i=1..n a_i·sin( b_i·x + c_i )

and with the frequencies b_i given, each term expands into the equivalent pair

a_i·sin(b_ix + c_i) = A_i·sin(b_ix) + B_i·cos(b_ix)

which is linear. The output carries A₁, B₁, …, A_n, B_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.

Ports

• u – the sampled signal being fitted. Scalar: one channel and its own window – see Notes.
• c – the coefficient column, [2n, 1], in the order above: the sine amplitude then the cosine amplitude of each term. 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 2n: 2n coefficients cannot be determined by fewer than 2n 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 sin1…sin8. Each term adds a sine and a cosine, so the output grows by two.
• Frequencies (rad per unit) – [b₁ … b_n], an n element vector of strictly positive angular frequencies, in radians per unit of x. They need not be related to one another – that is what separates this block from Fourier Fit. ⚠ Only the products b_i·h reach the arithmetic, and each must stay clear of a multiple of π – see Notes.
• Abscissa Step – h, the spacing between consecutive samples on the fitted axis. A single positive number. Setting it to the sampling period and giving the frequencies in radians per second is the same fit as leaving it at 1 and giving them in radians per sample.
• 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 2n×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 four settings, and changing any of them changes how many multiplies and how many outputs the core contains. Re-export after changing them. No sine or cosine is evaluated in any generated core – they were 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. Emitting the amplitude and phase instead would have cost that – see Notes.

Simulink bridge

None (Support::None). Fitting a sum of sinusoids to a running window is a Curve Fitting Toolbox function (fit with a sin<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 output is the quadrature pair, not MATLAB's amplitude and phase. The two carry the same information: a_i = √(A_i² + B_i²) and c_i = atan2(B_i, A_i), one Trigonometry / Atan2 and one Sqrt away. The pair is what this block emits because the arctangent has a branch cut: the phase of a fit to a moving window walks around the circle, and every crossing of ±π is a jump of 2π that two implementations only agree about when they agree to the last bit. The quadrature coefficients are continuous everywhere, and they are what keeps the three HDL targets synthesizable.
• ⚠ The frequencies are parameters here and MATLAB solves for them. With b unknown the fit is nonlinear and needs an iterative solver every sample. If the frequencies are what you are looking for, this is the wrong block – reach for Machine Learning / Feature Engineering / FFT Magnitude or Transforms / Goertzel first, then set the result here.
• ⚠ This is not Fourier Fit with different numbers. Fourier Fit fits harmonics of one fundamental and carries a constant term a₀; this fits n independent frequencies and carries none. A signal made of two unrelated tones has no fundamental to be a harmonic of, which is why MATLAB ships fourier<n> and sin<n> as separate models. Use Fourier Fit when the content really is harmonic: it determines one frequency parameter instead of n, and it fits the mean.
• There is no constant term. Every basis function averages to nearly zero over a long window, so a signal with an offset is fitted through that offset rather than having it removed. Put a Signal Smoothing / Detrend ahead of the block if the offset is not wanted.
• One solve, then arithmetic. The small normal-equation system is solved once when the configuration loads; every sample afterwards is 2n fixed dot products.
• ⚠ Not every setting is a fit, and the block refuses rather than guessing. The basis degenerates on a sample grid: a product b_i·h at an exact multiple of π makes that term's sine column vanish identically, and two frequencies that alias to the same value on the grid cannot be told apart. Either makes the normal matrix singular, which is detected when the configuration loads and reported with a reason – rather than emitting the arbitrary answer a pseudo-inverse would have produced. Keep every b_i·h under π, and the frequencies apart by more than one bin of the window, and neither case arises.
• Verified against MATLAB. At W = 13, h = 0.3 and frequencies [0.9 2.3], R2026a's own least-squares solve of the same design matrix answers [−0.20722798579525406, −0.40790884283512951, −0.63871062749727303, −0.12000205236902012] on a fixed 13-sample window; this block reproduces all four entries to 3.1e−16 relative, with cond(A'A) = 1.47. 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.
• 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 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.