FIR Window Design — Control Systems/Polynomials

FIR Window Design

Control Systems / Polynomials

Designs a finite impulse response filter of order N by the window method and reports its N + 1 taps, with the cutoff arriving on a port. It is MATLAB's fir1: the ideal brick-wall impulse response, truncated to N + 1 samples, multiplied by a window and scaled so the gain is 1 in the passband.

With W_n = 2f_c/f_s and t_k = k − N/2 the tap's distance from the centre, every tap is one sine of the cutoff: b_k = w_k·sin(πW_nt_k) / (πt_k).

It designs; it does not filter. Unlike the five IIR designers in this family it has one output, because an FIR filter's denominator is the scalar 1: feed b to Discrete / Transfer Fcn Direct Form II Time Varying's Num port and a Constant of 1 to its Den. The two need not be the same length.

Ports

• fc – the cutoff frequency in Hz. Scalar. Clamped into (0, f_s/2) – 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. Default 12.
• Filter Type – which ideal response is truncated:
  • Lowpass – passes below the cutoff, scaled to unity at zero frequency. Default.
  • Highpass – passes above it, scaled to unity at Nyquist. Requires an even order – see Notes.
• Window – which taper multiplies the truncated response. All four are the symmetric forms, as MATLAB's are:
  • Hamming – 0.54 − 0.46·cos(2πk/N). Default, and fir1's.
  • Hann – 0.5 − 0.5·cos(2πk/N); the end taps are zero.
  • Blackman – 0.42 − 0.5·cos(2πk/N) + 0.08·cos(4πk/N); the lowest sidelobes and the widest transition.
  • Rectangular – no taper at all, which is the plain truncated sinc and the worst sidelobes.
• Sample Rate (Hz) – f_s, the rate the filter is designed FOR, and the rate the cutoff is 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. The order, the type, the window and the rate are structural and are baked into the generated body, so nothing is exposed as a tunable parameter on the generated core; every tap's window value and its two constants are inlined. No generated body contains a branch: whether a tap sits at the centre is decided by the order, so the centre tap is emitted as its own expression rather than tested for.

The three HDL targets are simulation-only, and deliberately: a sine and a reciprocal do not belong in a Q16.16 datapath. Both are evaluated in real and the values convert at the port boundary, which is what the Trigonometry family does for the same reason. The seven software targets are exact.

Simulink bridge

None (Support::None). fir1 is a MATLAB function, and the Simulink blocks that carry a windowed-sinc design – Lowpass FIR Filter Design and its Highpass, Bandpass and Bandstop siblings, in the DSP System Toolbox – do the design internally and expose no part of it: each takes a signal in and gives a filtered signal out, so none has taps a diagram could read. 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.
• 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. fir1 quietly increments the order and warns; this block refuses, because its output size is Order + 1 and a silently longer vector is worse than a stated refusal. Measured: fir1(9, Wn, 'high') returns eleven taps, not ten.
• Nothing divides by the cutoff. Writing W_n·sinc(W_nt) as sin(πW_nt)/(πt) cancels W_n out of the divisor, so a cutoff of zero is an ordinary value and not a special case.
• The two scalings are different rules, not one. A lowpass is divided by Σb and a highpass by |Σ(−1)^kb_k| – unity at zero frequency and unity at Nyquist respectively, which is what fir1 does.
• The clamp is a live branch, not a safety net. f_c is clamped into (0, f_s/2) by (|x−lo| − |x−hi| + lo + hi)/2, branchless and identical in this block and in all ten emitted bodies, so they agree bit for bit.
• The windows are the SYMMETRIC forms, as MATLAB's hamming, hann and blackman are by default. The periodic forms differ in the divisor (N rather than N − 1 points) and would put the design a visible distance from fir1.
• Measured against R2026a, at f_s = 4 Hz over orders 8, 9, 16, 21 and 32, cutoffs 0.7 Hz and 1.1 Hz, all four windows and both types: the largest disagreement with fir1 on any tap is 3.3e−16 – measured through a harness carrying the shipped C++, not the prototype it was written from.
• 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.