Fir Bandpass Bandstop Filter Design

FIR Highpass, Bandpass and Bandstop Filter Design (cross-cutting/numerics/fir-bandpass-bandstop-filter-design)

Use when you must design a linear-phase finite-impulse-response filter that passes or rejects frequencies above a cutoff (highpass) or inside or outside a band between two specified edges (bandpass, bandstop), by the windowed-sinc method: build the windowed ideal lowpass prototype at the cutoff (the half bandwidth for the band types), then form the requested tap set by spectral inversion (highpass: windowed unit sample minus the prototype), cosine frequency translation (bandpass: twice the prototype times the center cosine), or spectral inversion of the translated bandpass (bandstop). Every tap is computed from closed-form sine, cosine and window algebra, none looked up, no iteration and no search: deterministic, offline, pure Python stdlib (math only). It pairs with cross-cutting/numerics/digital-filter-design, which owns the Butterworth IIR lowpass and highpass members with a hard ftype fence, with cross-cutting/numerics/bandpass-bandstop-filter-design, which owns the Butterworth IIR bandpass and bandstop members by the z-domain frequency transformation (both its edges at exactly -3.0103 dB), and with cross-cutting/numerics/fir-filter-design, which designs FIR lowpass filters only. This leaf is the FIR highpass/bandpass/bandstop cell of the digital filter 2x2 grid; standalone lowpass FIR design with unity DC normalization stays with fir-filter-design (this leaf uses the prototype only as internal translation material and exposes no lowpass design function).

Domain quick reference

• Window weights w[n], n = 0..num_taps - 1, denominator num_taps - 1: rectangular all 1.0; hann 0.5 - 0.5cos(2pin/(num_taps - 1)); hamming 0.54 - 0.46cos(2pin/(num_taps - 1)); blackman 0.42 - 0.5cos(2pin/(num_taps - 1)) + 0.08cos(4pin/(num_taps - 1)). Every window is symmetric about the integer center M = (num_taps - 1)/2 and evaluates to exactly 1.0 at the center tap (cos(pi) = -1), so windowed taps stay symmetric (linear phase). num_taps = 1 is the degenerate case: the window is [1.0] and every design normalizes to the single tap [1.0].
• Ideal lowpass prototype, center M: h_lp[n] = sin(wc*(n - M))/(pi*(n - M)) for n != M, center limit h_lp[M] = wc/pi = 2*fc_proto/fs. Windowed prototype p[n] = w[n]*h_lp[n] (NOT DC-normalized; its DC gain is only approximately 1, which limits the spectral-inversion null depth).
• Prototype cutoff: for the highpass fc_proto = cutoff_hz; for the band types fc_proto = (high_cutoff_hz - low_cutoff_hz)/2, the half bandwidth, so the requested edges are the images of the prototype cutoff wc at w0 - wc and w0 + wc with center frequency w0 = 2picenter_frequency_hz/fs and center_frequency_hz = (low_cutoff_hz + high_cutoff_hz)/2 (the arithmetic mean of the requested edges, where the bandpass gain is normalized to exactly 1).
• Constructions (Oppenheim and Schafer Section 8.4 / Proakis and Manolakis Chapter 10 closed forms): highpass by spectral inversion b_raw[n] = w[n]delta[n - M] - p[n]; bandpass by cosine frequency translation b_raw[n] = 2p[n]cos(w0(n - M)); bandstop by spectral inversion of the translated bandpass b_raw[n] = w[n]delta[n - M] - 2p[n]cos(w0(n - M)). The constructions are per-sample additions and scalings, so raw bandstop plus raw bandpass equals the windowed unit sample w[n]*delta[n - M] elementwise (the complement identity the contract test recomputes).
• Unity passband gain normalization: b[n] = b_raw[n]/G with G the passband reference scale, |cosine sum of the raw taps at the reference|: Nyquist fs/2 for the highpass, the center frequency for the bandpass, DC for the bandstop. After normalization the passband gain is exactly 1 to float precision at the reference.
• Magnitude response of the symmetric tap set: H(f) = sum_n b[n]* cos(2pif/fs*(n - M)) (the linear phase term dropped); gain_at returns |H(f)| and magnitude_response_db returns 20*log10(|H(f)|). Cosine sums accumulate row by row in index order, deterministic and bitwise reproducible, no RNG.
• Band-edge geometry: the requested edges carry the windowed-sinc transition midpoint. The prototype response at its own cutoff is the gain-0.5 midpoint, so measured edge gains sit on the -6.020599913279624 dB (20log10(0.5)) band-edge geometry within window ripple and image leakage (worked Hamming anchors -5.995505518 to -6.073441862 dB). The -3.010299956639813 dB level (20log10(1/sqrt(2))) is the Butterworth prototype-3 dB-point convention of the IIR siblings; for these FIR filters it is crossed strictly INSIDE the nominal passband (bandpass at 215.960 and 784.004 Hz inside 200 to 800 Hz), at the bandstop notch shoulders (183.781 and 816.188 Hz), or above the highpass cutoff (103.994 Hz above 100 Hz), never at the requested edges.
• Group delay of the symmetric tap set: (num_taps - 1)/2 samples, constant for every frequency because the phase is exactly linear. All taps real and symmetric, all-zero transfer function: unconditionally stable, no feedback path. Direct-form convolution filter y[n] = sum_k b[k]*x[n - k] with the input zero outside its range and the output the same length as the input. Units are SI throughout: Hz for frequencies, samples per second for fs.

Workflow

1. Fix the design inputs and choose the prototype (the design-input fixing step): the sample rate fs, the requested edges (a single cutoff for the highpass; low and high edges for the band types, 0 < low < high < fs/2), an odd num_taps >= 1, and the window (rectangular, hann, hamming, blackman). The prototype lowpass cutoff is the cutoff itself for the highpass and the half bandwidth (high - low)/2 for the band types. Nonphysical inputs raise ValueError; even tap counts are rejected because the center index M must stay integer for symmetric linear-phase taps.
2. Build the windowed ideal lowpass prototype (the windowed prototype build step): window_coefficients(window, num_taps) returns w[n] and ideal_lowpass_taps(cutoff_hz, sample_rate_hz, num_taps) returns h_lp[n] = sin(wc*(n - M))/(pi*(n - M)) with the center limit 2*fc/fs; the module helper _windowed_prototype multiplies them into p[n] = w[n]*h_lp[n].
3. Translate or invert to the raw band taps (the raw-construction translate-invert step): the helper _translate_invert builds the raw (unnormalized) tap set from the defining relation of the requested type, spectral inversion w[n]delta[n - M] - p[n] for the highpass, cosine frequency translation 2p[n]cos(w0(n - M)) for the bandpass, and spectral inversion of the translated bandpass for the bandstop. The three public design functions call it: design_highpass(cutoff_hz, sample_rate_hz, num_taps, window), design_bandpass(low_cutoff_hz, high_cutoff_hz, sample_rate_hz, num_taps, window) and design_bandstop(...) with the same signature.
4. Normalize to unity passband gain (the unity passband gain normalization step): divide the raw taps by the passband reference cosine sum G (Nyquist for the highpass, the center frequency for the bandpass, DC for the bandstop), so the returned dicts carry nyquist_gain, center_gain and dc_gain equal to 1.0 within 1e-12. The highpass is then also near-unity away from DC, the bandstop near-unity at DC and Nyquist away from the notch, because the notch complement is not perfect.
5. Verify the band edges (the band-edge verification step): band_edge_checks(coefficients, ftype, sample_rate_hz, low_cutoff_hz, high_cutoff_hz=None) probes the magnitude response at the requested edges and returns {low_edge_db, high_edge_db, target_db, low_edge_ok, high_edge_ok, passband_ok, extra, verdict} with target_db the module constant -6.020599913279624; each measured edge must sit within 0.1 dB of the midpoint target and the passband reference gain within 0.1 dB of 0 dB for a PASS verdict.
6. Probe, read the group delay and filter (the probe-filter step): gain_at and magnitude_response_db evaluate the real cosine sum at any probe in [0, fs/2] (0 Hz and fs/2 are valid probes), group_delay_samples returns the constant (num_taps - 1)/2, and filter_signal applies the direct-form convolution to a sampled record, output the same length as the input with the first (num_taps - 1) samples carrying the transient. The deterministic checks are in the contract test scripts/test_fir_bandpass_bandstop_filter_design.py.

Worked example

Primary geometry (band types): fs = 4000 Hz, band edges low = 200 Hz and high = 800 Hz, num_taps = 101, Hamming window; the prototype lowpass cutoff is the half bandwidth fc_proto = 300 Hz (wc = 0.15pi) and the translation center w0 = 0.25pi, center_frequency_hz = 500 Hz. Highpass geometry: fs = 1000 Hz, cutoff 100 Hz (wc = 0.2*pi), num_taps = 101, Hamming. All values below are REAL outputs of scripts/fir_bandpass_bandstop_filter_design_logic.py (pure stdlib math, deterministic; the contract test exits 0 with all checks passed):

• Highpass design_highpass(100, 1000, 101, "hamming"): group delay 50.0 samples, nyquist_gain 1.000000000000000, symmetry error 1.4e-17. Center tap b[50] = 0.800131546113 (the windowed unit sample minus the prototype center, scaled by the Nyquist normalization), neighbors b[49] = b[51] = -0.186958764181, b[48] = b[52] = -0.150841106810, outer taps b[1] = 0.000308982598, b[2] = 0.000527514464, b[0] = 0.000000000000 (float floor of the sine at an integer multiple of pi). Edge gain at the requested 100 Hz cutoff: -6.013188638 dB, 0.007411 dB from the -6.020599913 dB midpoint target; DC gain -56.644897249 dB (the spectral-inversion null floor); 50 Hz probe -55.337139253 dB; passband +0.009876223 dB at 150 Hz, -0.000850773 dB at 300 Hz, -0.000006122 dB at 480 Hz. The -3.010299956640 dB level is crossed at 103.994 Hz, above the requested cutoff.
• Bandpass design_bandpass(200, 800, 4000, 101, "hamming"): center_frequency_hz 500.000000000 = (200 + 800)/2, group delay 50.0, center_gain 1.000000000000000, symmetry error 2.8e-17. Center tap b[50] = 0.299984336175, b[49] = b[51] = 0.204171360832, b[48] = b[52] = 0.000000000000 (the w0M = 12.5pi structural zeros), b[1] = -0.000662242631, b[2] = -0.000651902497, b[3] = -0.000132065332, b[0] = 0.000000000000. Edge gains: 200 Hz -5.995505518 dB and 800 Hz -6.018418812 dB, within 0.025094 and 0.002181 dB of the midpoint target; center +0.000000000 dB; stopband probes -50.065791442 dB at DC, -62.087519915 dB at 100 Hz, -57.420533724 dB at 1000 Hz, -61.325771718 dB at 1200 Hz, -65.756900774 dB at 1600 Hz, -66.996568467 dB at Nyquist; passband probe -0.002015073 dB at 400 Hz. band_edge_checks verdict PASS. The -3.010299956640 dB level is crossed at 215.960 and 784.004 Hz, strictly inside the requested passband.
• Bandstop design_bandstop(200, 800, 4000, 101, "hamming"): dc_gain 0.999999999999999, group delay 50.0, symmetry error 2.8e-17. Center tap b[50] = 0.697809868401 (one minus the bandpass center tap to scale), b[49] = b[51] = -0.203543185312, b[48] = b[52] = -0.000000000000, b[1] = 0.000660205105, b[2] = 0.000649896784, b[3] = 0.000131659006. Edge gains: 200 Hz -6.073441862 dB and 800 Hz -6.050454002 dB, within 0.052842 and 0.029854 dB of the midpoint target; DC gain -0.000000000 dB and Nyquist gain -0.031101109 dB (unity passband, small notch-complement droop); passband probes -0.020390713 dB at 100 Hz and -0.034678594 dB at 1200 Hz; in-notch probe -74.933184863 dB at 400 Hz; the notch center null at 500 Hz measures -85.671233701 dB, the float floor of the imperfect complement. band_edge_checks verdict PASS. The -3.010299956640 dB shoulders sit at 183.781 and 816.188 Hz, strictly outside the notch band.
• Window trade at the bandpass geometry (101 taps, 200 to 800 Hz at 4000 Hz), edge gains and stopband probes at 900 and 1200 Hz: rectangular low edge -5.701966 dB, high edge -5.994135 dB, 900 Hz -46.380 dB, 1200 Hz -38.919 dB; hann low edge -6.021533 dB, high edge -6.020536 dB, 900 Hz -58.142 dB, 1200 Hz -85.570 dB; hamming low edge -5.995506 dB, high edge -6.018419 dB, 900 Hz -62.438 dB, 1200 Hz -61.326 dB; blackman low edge -6.020932 dB, high edge -6.020591 dB, 900 Hz -60.091 dB, 1200 Hz -94.916 dB. The midpoint geometry holds across all four windows (the rectangular lower edge carries the largest image-leakage bias); the stopband floor follows the window with Blackman deepest and rectangular shallowest at the 1200 Hz probe.
• Deterministic signal tests: an impulse through the bandpass reproduces the coefficient vector exactly (max error 0.0); a constant 5.0 input settles to a bandstop tail mean of 5.000000000 (unity DC), a bandpass tail mean of -1.569208e-02 (the DC stopband, -50.065791442 dB times 5.0) and a highpass tail mean of 7.357413e-03 (the DC null floor); alternating +5/-5 settles through the bandpass to a tail RMS of 4.468601e-04 relative to 5.0 (the Nyquist stopband) and through the bandstop to 0.996425755 relative (the near-unity Nyquist gain); two identical design calls are bitwise identical.
• Recorded low-cutoff pitfall (corpus query-2 geometry): the windowed-sinc geometry needs the tap count large enough that the window main lobe resolves the cutoff, fcnum_taps/fs roughly 2 or more. At fc = 0.5 Hz, fs = 100 Hz, num_taps = 51 (fcnum_taps/fs = 0.255) the highpass is degenerate (the -6.0206 dB crossing sits below 0.001 Hz, not at the requested 0.5 Hz cutoff; the design cannot realize the cutoff). At num_taps = 501 (fc*num_taps/fs = 2.505) the geometry is restored: -50.768 dB at 0.1 Hz and the crossing at 0.5004 Hz. No contract anchor asserts the degenerate 51-tap numbers; the contract pins the worked geometries above.

Verification

• design_highpass(100, 1000, 101, "hamming") returns the seven documented keys with the tap anchors above and nyquist_gain 1.0 within 1e-12; design_bandpass and design_bandstop return their nine documented keys with center_frequency_hz 500.0 and unity center_gain / dc_gain within 1e-12.
• The measured edge gains sit within 0.1 dB of the -6.020599913279624 dB midpoint target (real Hamming anchors -5.995505518 to -6.073441862 dB); band_edge_checks returns verdict PASS for all three worked designs, and the bandpass and bandstop passband references measure 0.000000000 dB (unity by construction).
• The -3.010299956639813 dB Butterworth level is NOT at the requested edges: contract crossing measurements lie strictly between the bandpass edges (215.960 and 784.004 Hz inside 200 to 800 Hz), strictly outside the bandstop notch band (183.781 and 816.188 Hz) and above the highpass cutoff (103.994 Hz); the requested edge gains differ from -3.010299956640 dB by more than 1 dB.
• Raw-construction identities: rebuilding the window, the prototype, the raw tap sets and the passband scale G with math only gives max |b[n]*G - raw[n]| below 1e-9 for every worked design, and the bandstop complement identity b_bs_raw[n] + b_bp_raw[n] = w[n]*delta[n - M] holds elementwise below 1e-9.
• Symmetry and group delay: max |b[n] - b[num_taps - 1 - n]| below 1e-12 for every worked design (real anchors 1.4e-17 highpass, 2.8e-17 bandpass and bandstop) and group_delay_samples(101) = 50.0.
• Deterministic signal identities: the multi-tone records through filter_signal reproduce the designed |H| at each tone by quadrature projection over the settled integer-cycle tail within 1e-9 relative; the bandpass impulse round trip is exact; constant and alternating inputs settle to the designed DC and Nyquist behavior.
• Nonphysical inputs raise ValueError: fs <= 0, cutoff_hz <= 0 or >= fs/2, low_cutoff_hz <= 0 or >= high_cutoff_hz, high_cutoff_hz >= fs/2, even or zero num_taps, unknown window names (bartlett), probes outside [0, fs/2], empty or non-finite sample and coefficient lists, unknown ftype values, and high_cutoff_hz not None with ftype "highpass".
• Run the contract test offline under both interpreters, exit 0: python3 scripts/test_fir_bandpass_bandstop_filter_design.py and the pyenv 3.13 hook interpreter. All numeric asserts are tolerance-based (no exact float equality on computed sums), so the suite holds identically on both.

Related leaves

• cross-cutting/numerics/digital-filter-design: the immediate sibling; owns the Butterworth IIR lowpass and highpass members with a hard ftype fence (its contract test raises ValueError for any non-LP/HP type), single-cutoff prewarp anchors at the -3.0103 dB prototype point.
• cross-cutting/numerics/bandpass-bandstop-filter-design: the Butterworth IIR bandpass and bandstop cell by the z-domain frequency transformation, with both edges at exactly -3.010299956640 dB; no windowed-sinc taps.
• cross-cutting/numerics/fir-filter-design: linear-phase FIR lowpass taps by the windowed-sinc method with unity DC normalization; no band types, no spectral inversion, no frequency translation.
• cross-cutting/numerics/fast-fourier-transform: spectral analysis of a signal, not band filtering of it.
• cross-cutting/numerics/power-spectral-density: Welch averaged periodogram estimation of recorded time data; no filter coefficients.
• cross-cutting/numerics/finite-difference-derivatives: differentiating the filtered signal, not designing the band filter.
• cross-cutting/numerics/cross-correlation-analysis: delay estimation between two filtered traces, not band filtering.

Pitfalls

• Reading the requested edges at -3.0103 dB: the -3.010299956640 dB level is the Butterworth prototype-3 dB-point convention of the IIR siblings. A windowed-sinc FIR tap set has no such point at its requested edges: the prototype response at its own cutoff is the gain-0.5 midpoint of the transition, so the band edges sit on the -6.020599913279624 dB midpoint within window ripple and image leakage, and the -3.0103 dB level is reached strictly inside the passband (bandpass) or at the notch shoulders (bandstop). This leaf verifies its edges against the FIR midpoint target and says so in its description; the -3.0103 dB phrasing is the physically forced deviation from the IIR anchor convention, documented with measured crossing evidence in the Worked example.
• Designing below the resolution geometry: the windowed-sinc method needs fcnum_taps/fs roughly 2 or more for the window main lobe to resolve the cutoff. At fcnum_taps/fs around 0.25 the design is degenerate (measured gains near -2.7 dB at the reported rejection point) and no edge anchor holds. Use a larger tap count (501 taps restored the 0.5 Hz at 100 Hz geometry to -50.768 dB rejection with the crossing at 0.5004 Hz).
• Trusting the far-field window ordering: the classical first-sidelobe ordering does not extend to far stopband probes: Hann out-attenuates Hamming at the worked 1200 Hz probe (-85.570 dB vs -61.326 dB) because the Hamming far sidelobe envelope decays more slowly; report probes as measured.
• Reading structural zeros as general claims: at the worked 101-tap geometry w0M = 12.5pi, so bandpass and bandstop taps at offsets n - M = +-2 mod 4 are zero to float floor and the outer taps read 0.000000000000; these are geometry-specific, not general properties of the construction.
• Writing exact float equality on computed sums: the module and its contract test use tolerance-based asserts only, since accumulated rounding differs between interpreters (the pre-push hook resolves python3 to pyenv 3.13.12 while foreground shells use /usr/bin/python3 3.9.6).

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_fir_bandpass_bandstop_filter_design.py

It covers the full workflow: the design-input guard rails and ValueError set of every public function, the window anchor values and symmetry, the ideal-lowpass center limit, the worked highpass/bandpass/bandstop tap anchors at both geometries, the unity passband gains, the -6.020599913279624 dB midpoint edge geometry and PASS verdicts of band_edge_checks, the measured -3.010299956640 dB crossing locations, the raw-construction recovery and bandstop complement identities, the stopband and notch null floors, the four-window trade, the group delay identity, bitwise determinism with math as the only import, the direct-form impulse round trip, the constant and alternating settle behavior, and the three-tone quadrature gain identities for the highpass, bandpass and bandstop. 35 tests pass in under a second offline under python3 (3.9.6) and the pyenv 3.13 hook interpreter.

Compliance

• Standards referenced, not reproduced: naca-tr-824 (NACA Report 824, summary of low-speed airfoil data) is cited reference-only per standards-map.yaml as the numerics-convention anchor for the library; the windowed-sinc FIR design relations above are classical digital filter methodology (Oppenheim and Schafer, Discrete-Time Signal Processing, Section 8.4, FIR design by windowing, with the highpass prototype from spectral inversion of the lowpass response and the bandpass/bandstop prototypes from cosine modulation or frequency translation; identical closed forms in Proakis and Manolakis, Chapter 10, and Hamming, Digital Filters), paraphrase-only.
• compliance: STANDARDS-REF, gated: false.