FIR Equiripple Design — Control Systems/Polynomials

FIR Equiripple Design

Control Systems / Polynomials

Designs a linear-phase finite impulse response filter of order N by the Parks-McClellan method and reports its N + 1 taps, with the centre of the transition band arriving on a port. It is MATLAB's firpm for two bands: the taps minimise the largest weighted error W(f)·|H(f) − D(f)| over both bands, so the error ripples with equal height across each band – the transition between the bands being a region it 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 firpm(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 answer is the fixed point of the Remez exchange on a dense frequency grid, and the grid moves with the band edges, so the whole 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 3 to 64 (firpm refuses an order below 3). 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 10 f_s – see Notes. Default 10.
• Passband Weight – W_p, the weight on the passband error. Strictly positive. Default 1.
• Stopband Weight – W_s, the same for the stopband. Only the ratio matters: the passband ripple comes out W_s/W_p times the stopband ripple. 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 of firpm – the frequency grid, the Remez exchange and the final inverse transform – as loops over local arrays, and it is the same program the block runs, printed once per language rather than transcribed ten times. Its cost per step grows with the order: the grid holds at most about 8·N points, and the exchange repeats until its set of extremal frequencies stops changing – at most 250 passes, as in MATLAB.

The three HDL targets are simulation-only, and deliberately: an iterative exchange over real-valued arrays does 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). firpm 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 Filter and Highpass Filter blocks design an FIR from ripple and attenuation targets rather than from firpm's weights, the Lowpass/Highpass FIR Filter Design blocks are window-method, 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 exchange starts afresh on every step.
• 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.
• The answer is firpm's on firpm's grid, to the bit. An equiripple design is optimal on a grid of 16 points per cosine, not on the continuous band, and a grid one rounding off moves every tap. The grid is built the way MATLAB's colon operator builds it – which is not a + k·d – so the two agree to rounding rather than merely closely.
• The output moves in small steps as f_c slides, because the number of grid points in each band is a whole number that changes with the band edges. Measured: sweeping f_c over 0.2 Hz in steps of 10⁻⁵ Hz at orders 13, 14 and 24, no step in any tap exceeds twice the median step (largest 1.3e−5), so the grid changes are smaller than the design's own smooth movement and far below its ripple (0.07 to 0.11 on the same designs).
• 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; firpm quietly increments the order and warns, and this block refuses, because its output size is Order + 1. Measured: firpm(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 exchange stops converging: MATLAB's firpm warns that it did not converge, and the two can differ by whole units. Inside the limit the block agrees with it to 1.9e−10 or better on every tap.
• Measured against R2026a, 1546 designs across the domain: worst disagreement with firpm on any tap 1.9e−10, at the domain boundary; away from it, through order 33, 1.1e−14.
• 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.