Filter Banks

phonometry supports several filter types, each with its own transfer function characteristic. All banks place their −3 dB points on the ANSI S1.11 band edges, so band levels are comparable across architectures.

Fractional octave bands: the math

IEC 61260-1:2014 builds every band from the base-10 octave ratio $G = 10^{3/10} \approx 1.99526$ (so “one octave” is not exactly 2). For band fraction $1/b$, the mid frequencies and band edges follow (5.2-5.5):

$$f_m = 1000 \cdot G^{x/b} \quad (b\ \text{odd}), \qquad f_1 = f_m G^{-1/2b}, \quad f_2 = f_m G^{+1/2b}$$

so every 1/3-octave band spans $G^{1/3} \approx 1.2589 \approx 10^{1/10}$ — ten bands per decade, which is why the nominal frequencies (25, 31.5, 40 …) repeat scaled by 10. phonometry designs each band as an SOS cascade whose −3 dB points land exactly on $f_1$ and $f_2$ for every architecture — for Chebyshev II, Elliptic and Bessel that requires pre-warping the analytic band-edge mapping rather than trusting SciPy’s default parametrization.

Multirate decimation

A 25 Hz one-third-octave band at 48 kHz spans about 5.8 Hz — 0.024 % of Nyquist — coefficients so stiff they go numerically unstable. The bank avoids that by filtering low bands at a decimated rate:

octavefilter() / OctaveFilterBank parameters

Parameter	Type	Units	Range / default	Notes
x	1D or 2D array	digital units	non-empty	2D is [channels, samples]
fs	int	Hz	> 0
fraction	int	—	default 1; common 3; any b ≥ 1	Bands per octave = b
order	int	—	default 6	SOS order per band
limits	list [lo, hi]	Hz	default [12, 20000]	Analysis range
filter_type	str	—	'butter' (default), 'cheby1', 'cheby2', 'ellip', 'bessel'	See comparison above
ripple / attenuation	float	dB	required by cheby/ellip types	Passband ripple / stopband attenuation
show	bool	—	default False	Plot the bank response (needs matplotlib)
sigbands	bool	—	default False	Also return the per-band time signals
zero_phase	bool	—	default False	Forward-backward filtering (offline)
stateful / steady_ic (class)	bool	—	default False	Streaming state; see Block Processing

verify_filter_class(bank) checks the designed bank against the IEC 61260-1 Table 1 acceptance limits and reports the class (1, 2 or None if outside both) with per-band margins.

Filter Comparison and Zoom

We use Second-Order Sections (SOS) for all filters to ensure numerical stability. The following plot compares the architectures focusing on the -3 dB crossover point.

Type	Name	Usage Example	Best For
butter	Butterworth	octavefilter(x, fs, filter_type='butter')	General acoustic measurement.
cheby1	Chebyshev I	octavefilter(x, fs, filter_type='cheby1', ripple=0.1)	Sharper roll-off at the cost of ripple.
cheby2	Chebyshev II	octavefilter(x, fs, filter_type='cheby2', attenuation=60)	Flat passband with stopband zeros.
ellip	Elliptic	octavefilter(x, fs, filter_type='ellip', ripple=0.1, attenuation=60)	Maximum selectivity.
bessel	Bessel	octavefilter(x, fs, filter_type='bessel')	Preserving transient waveform shapes.

Gallery of Filter Bank Responses

Full spectral view of the filter banks for Octave (1/1) and 1/3-Octave fractions.

Architecture	1/1 Octave (Fraction=1)	1/3 Octave (Fraction=3)
Butterworth
Chebyshev I
Chebyshev II
Elliptic
Bessel

Filter Usage and Examples

1. Butterworth (butter)

The Butterworth filter is known for its maximally flat passband. It is the standard choice for acoustic measurements where no ripple is allowed within the frequency bands.

from phonometry import octavefilter
# Default standard measurement
spl, freq = octavefilter(x, fs, filter_type='butter')

2. Chebyshev I (cheby1)

Chebyshev Type I filters provide a steeper roll-off than Butterworth at the expense of ripples in the passband. Useful when high selectivity is needed near the cut-off frequencies.

# Selectivity with 0.1 dB passband ripple
spl, freq = octavefilter(x, fs, filter_type='cheby1', ripple=0.1)

3. Chebyshev II (cheby2)

Also known as Inverse Chebyshev, it has a flat passband and ripples in the stopband. It provides faster roll-off than Butterworth without affecting the signal in the passband. The stopband edges are placed automatically so that the −3 dB points land on the band edges (attenuation must be > 3.01 dB).

# Flat passband with 60 dB stopband attenuation
spl, freq = octavefilter(x, fs, filter_type='cheby2', attenuation=60)

4. Elliptic (ellip)

Elliptic (Cauer) filters have the shortest transition width (steepest roll-off) for a given order. They feature ripples in both the passband and stopband.

# Maximum selectivity for extreme band isolation
spl, freq = octavefilter(x, fs, filter_type='ellip', ripple=0.1, attenuation=60)

5. Bessel (bessel)

Bessel filters are optimized for linear phase response and minimal group delay. They preserve the shape of filtered waveforms (transients) better than any other type, but have the slowest roll-off.

# Best for pulse analysis and transient preservation
spl, freq = octavefilter(x, fs, filter_type='bessel')

6. Linkwitz-Riley (linkwitz_riley)

Specifically designed for audio crossovers. Linkwitz-Riley filters (typically 4th order, but any even order is supported) allow splitting a signal into bands that, when summed, result in a perfectly flat magnitude response and zero phase difference between bands at the crossover.

from phonometry import linkwitz_riley
# Split signal into Low and High bands at 1000 Hz
low, high = linkwitz_riley(signal, fs, freq=1000, order=4)
# Reconstruction: low + high == signal (flat response)

Verifying the IEC 61260-1 class

verify_filter_class checks every band of a bank against the acceptance limits of IEC 61260-1:2014 (Table 1, with the fractional-octave breakpoint mapping and log-frequency interpolation from the standard) and reports the performance class per band with its margin in dB:

from phonometry import OctaveFilterBank, verify_filter_class

bank = OctaveFilterBank(fs=48000, fraction=3, order=6)
result = verify_filter_class(bank)
print(result["overall_class"])          # 1, 2 or None
print(result["bands"][0])               # {'freq': ..., 'class': 1, 'margin_class1_db': ...}

The order-6 Butterworth response (blue) threads between the forbidden regions: it must attenuate at least the red mask outside the band and no more than the purple mask inside it.

With default parameters (order 6), Butterworth meets class 1. Chebyshev II lands in class 2 — capped exactly by its attenuation=60 versus the 70 dB far-stopband requirement (raise attenuation to reach class 1). Chebyshev I, Elliptic and Bessel do not meet class limits at order 6: passband ripple (cheby1/ellip) and slow roll-off (bessel) violate the mask.

Signal Decomposition and Stability

By setting sigbands=True, you can retrieve the time-domain components of each band. This allows for advanced analysis or comparing how different architectures (e.g., Butterworth vs Chebyshev) affect the signal phase and transient response.

import numpy as np
from phonometry import octavefilter

# 1. Generate a signal (Sum of 250Hz and 1000Hz)
fs = 48000
t = np.linspace(0, 0.5, int(fs * 0.5), endpoint=False)
y = np.sin(2 * np.pi * 250 * t) + np.sin(2 * np.pi * 1000 * t)

# 2. Compare architectures (Butterworth vs Chebyshev II)
spl_b, freq, xb_butter = octavefilter(y, fs=fs, fraction=1, sigbands=True, filter_type='butter')
spl_c2, _, xb_cheby2 = octavefilter(y, fs=fs, fraction=1, sigbands=True, filter_type='cheby2')

# 'xb_butter' and 'xb_cheby2' contain the time-domain signals per band

The plot compares the Butterworth (solid blue) and Chebyshev II (dashed red) responses. The bottom plot shows the Impulse Response, highlighting the differences in stability and transient decay.

Note

Why do the signals look shifted in time? Digital IIR filters (like Butterworth or Chebyshev) have non-linear phase responses, which results in frequency-dependent Group Delay. In the 250 Hz band, you can see that the Chebyshev II filter has a different propagation delay compared to the Butterworth filter. This is a normal physical property of these architectures: more aggressive frequency roll-offs usually come at the cost of higher group delay and phase distortion.

Group delay, quantified

The group delay $\tau_g(\omega) = -\frac{d\phi(\omega)}{d\omega}$ of the 1 kHz octave band shows the trade-off directly: Bessel stays nearly flat across the passband (transient shapes survive), while Chebyshev I and Elliptic pay for their steep roll-off with strong delay peaks at the band edges.

Zero-phase filtering

For offline analysis you can eliminate group delay entirely: zero_phase=True filters each band forward-backward (scipy.signal.sosfiltfilt), keeping band signals time-aligned with the input. The effective attenuation doubles, and the option is incompatible with stateful (block) processing.

from phonometry import OctaveFilterBank

bank = OctaveFilterBank(fs=48000, fraction=3)
spl, freq, xb = bank.filter(y, sigbands=True, zero_phase=True)

Causal filtering delays the burst by the filter’s group delay; zero-phase filtering keeps it aligned with the input.