FIR Least Squares Design — Control Systems/Polynomials

FIR Least-Squares Design

Control Systems / Polynomials

Designs a linear-phase finite impulse response filter of order N by weighted least squares and reports its N + 1 taps, with the centre of the transition band arriving on a port. It is MATLAB's firls for two bands: the taps minimise W_p·∫_pass|H − 1|² + W_s·∫_stop|H|², the transition between the bands being a region the error ignores.

With f_c on the port and t_w the transition width, the bands are [0, f_c − t_w/2] and [f_c + t_w/2, f_s/2], which is firls(N, [0 fc−tw/2 fc+tw/2 fs/2]/(fs/2), [1 1 0 0], [Wp Ws]) for a lowpass and the same with [0 0 1 1] and the weights swapped for a highpass. The minimiser is one linear solve whose matrix moves with the band edges, so the design is redone on every step.

It designs; it does not filter. Feed b to Discrete / Transfer Fcn Direct Form II Time Varying's Num port and a Constant of 1 to its Den, as for FIR Window Design.

Ports

• fc – the centre of the transition band, in Hz. Scalar. Clamped into [t_w, f_s/2 − t_w] – see Notes.
• b – the filter's taps, in the order a delay line consumes them, as a column of N + 1 entries. The response is symmetric, so the vector reads the same forwards and backwards.

Parameters

• Order – N, a whole number from 1 to 64. The output is N + 1 long and the group delay is N/2 samples. An even order is a type I filter and an odd one a type II, whose response is zero at Nyquist. Default 12.
• Filter Type – which band is the passband:
  • Lowpass – desired response 1 below the transition and 0 above it. Default.
  • Highpass – 0 below and 1 above. Requires an even order – see Notes.
• Transition Width (Hz) – t_w, the width of the don't-care region centred on f_c. Strictly positive and below f_s/4, and Order × t_w must not exceed 6 f_s – see Notes. Default 10.
• Passband Weight – W_p, how much an error in the passband costs. Strictly positive. Default 1.
• Stopband Weight – W_s, the same for the stopband; raising it trades passband ripple for attenuation. Strictly positive. Default 1.
• Sample Rate (Hz) – f_s, the rate the filter is designed FOR and the rate the band edges are measured against. Strictly positive. This is not the block's own rate. Default 100.
• 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. Every setting is structural and is baked into the generated body, so nothing is exposed as a tunable parameter on the generated core. The body is the whole design – the band integrals, the Cholesky factorisation of the least-squares system and the two triangular solves – as loops over local arrays, and it is the same program the block runs, printed once per language rather than transcribed ten times.

The three HDL targets are simulation-only, and deliberately: a sine, a square root and a linear solve do not belong in a Q16.16 datapath. The design runs in real (VHDL inside a procedure) and the taps convert at the port boundary. The seven software targets are exact.

Simulink bridge

None (Support::None). firls is a MATLAB function and no Simulink library block carries it. Every design block in the DSP System Toolbox's filter-design library is a filter – signal in, filtered signal out – so none exposes taps a diagram could read: the Lowpass/Highpass FIR Filter Design blocks are window-method, the Lowpass Filter and Highpass Filter blocks design from ripple and attenuation targets, and Digital Filter Design is an interactive designer with no parameter a model could set. There is therefore no library path this block could name. The bridge reports it rather than dropping it silently, and it has no parity testbench; code export verification still covers it across all ten languages. No configuration of it crosses either, including "Sampling Time (s)", which has no counterpart to be written to.

Notes

• Algebraic, with no state: the output depends only on the current input.
• The port carries the CENTRE of the transition, not a band edge, so a lowpass and a highpass on the same wire split the spectrum in the same place. It is clamped into [t_w, f_s/2 − t_w] by (|x−lo| − |x−hi| + lo + hi)/2, branchless and identical in the block and in all ten emitted bodies, so each band always keeps at least half a transition width.
• An odd order is refused for a highpass, and MATLAB does something else. A symmetric FIR of odd order has a zero at Nyquist and so cannot pass the top of the band; firls quietly increments the order and warns, and this block refuses, because its output size is Order + 1. Measured: firls(9, ...) with a highpass response returns eleven taps, not ten.
• The order-times-width limit is measured, and it is where MATLAB stops being an answer. A transition wide for its order makes a design whose error is at the floor of double precision, and there the least-squares matrix is numerically singular: its condition number grows like e^0.6·N·D (D the width as a fraction of Nyquist) and MATLAB's own solve warns that the matrix is singular. Inside the limit the block agrees with firls to 9.9e−9 or better on every tap; past it the two can differ by whole units.
• Measured against R2026a, 1362 designs across the domain: worst disagreement with firls on any tap 9.9e−9, reached at the domain boundary where the system's condition number is largest – and numpy's LAPACK solve of the same matrix lands as far from MATLAB, so the spread is the problem's and not the block's. Away from the boundary, at order 16 or less, it is 5e−15.
• No state space: one scalar in and one vector out, so there is no A/B/C/D to merge and model reduction correctly declines it.