Equivalent Rectangular Bandwidth¶
The equal-peak, equal-power-area rectangular width of a specified auditory or cochlear filter response.
Core Idea¶
Equivalent rectangular bandwidth (ERB) is the width, in hertz, of an ideal rectangle with the same peak height and integrated power-response area as a specified auditory or cochlear filter response. For a nonnegative response \(P(f)\) over a declared frequency domain \(D\), with finite area and peak \(P_{\max}>0\), \(\mathrm{ERB}_{D}=\int_D P(f)\,df/P_{\max}\). A linear filter transfer \(H(f)\) contributes power weighting \(P(f)=|H(f)|^2\), not its unsquared amplitude. The same domain convention must be used for area and peak. The rectangle is an equal-area comparison, not a claim that a biological filter has vertical sides.[ref-b1f2c0bfa99f][ref-ca44dc8dcc15]
The width belongs to a response at stated conditions. A normal-hearing center-frequency fit such as the ASA annotation's \(\mathrm{ERB}_N\), and the ratio \(Q_{\mathrm{ERB}}=\mathrm{CF}/\mathrm{ERB}\), are conditional uses of the width rather than its definition.[ref-b1f2c0bfa99f][ref-ca44dc8dcc15]
Scope of Application¶
The metric applies to inferred human auditory filters and to cochlear-filter estimates from animal neural tuning, with separate methods and limits. Moore, Peters and Glasberg estimated human low-frequency filter shapes from notched-noise masking at 100, 200, 400 and 800 Hz under two masker levels. Shera, Guinan and Oxenham report cat and guinea-pig \(Q_{\mathrm{ERB}}\) values computed from earlier single-fiber threshold-frequency tuning curves by standard algorithms, and compare them with a separately derived human masking-fit curve. A threshold curve is evidence for an inferred filter bandwidth, not itself a directly measured linear power-transfer function.[ref-0c8ba19b820f][ref-ca44dc8dcc15]
Frequency, sound level, organism, subject and inference model remain attached to a width. The Moore values below are sample means for the lower tested masker level. Shera and colleagues caution that high-characteristic-frequency flattening in animal \(Q_{\mathrm{ERB}}\) may be a measurement artifact. None of these sources establishes one level-invariant ERB shared by all ears or species.[ref-0c8ba19b820f][ref-ca44dc8dcc15]
Clarity¶
“Equivalent” means same peak and same power-response area. It does not mean identical shape, equal phase, or equal -3 dB or -10 dB span. If the peak is not fixed, many rectangles can share an area; if a decibel-threshold crossing replaces area matching, a different bandwidth is computed. Shera and colleagues distinguish the ERB-based sharpness ratio from \(Q_{10}\), which uses a -10 dB bandwidth.[ref-b1f2c0bfa99f][ref-ca44dc8dcc15]
For animal neural data, Shera's Fig. 1 reports computed \(Q_{\mathrm{ERB}}=\mathrm{CF}/\mathrm{ERB}(\mathrm{CF})\), so a width at a given CF is algebraically \(\mathrm{CF}/Q_{\mathrm{ERB}}\). The paper defines the same-white-noise-power rectangle but does not publish each raw area integral. Do not present its threshold-tuning curves as directly measured \(H(f)\) responses.[^ref-ca44dc8dcc15]
Manages Complexity¶
A single ERB makes filter selectivity comparable without choosing an arbitrary decibel crossing. Dividing area by peak removes a common gain scaling; the derived \(Q_{\mathrm{ERB}}\) ratio additionally relates that width to characteristic frequency. This compresses a frequency response into a scalar, while preserving enough information for a defined width comparison.[ref-b1f2c0bfa99f][ref-ca44dc8dcc15]
The compression loses slopes, asymmetry and estimation details. Glasberg and Moore warn that only symmetrical notches can give seriously misleading shapes for asymmetric filters and that corrections at extreme frequencies or in hearing impairment can matter. A reported ERB cannot by itself reconstruct the shape or certify the inference method.[^ref-49d3c9823087]
Abstract Reasoning¶
To test a candidate ERB, specify the auditory or cochlear filter being summarized and the response conditions. Declare the frequency domain, choose a nonnegative power weighting, determine its peak, integrate the weighting and divide by that peak. If the available publication instead gives \(Q_{\mathrm{ERB}}\), recover ERB only at the same CF and under the same inference conditions. Separate the definition's equal-area step from what an experiment measured and what a model inferred.[ref-b1f2c0bfa99f][ref-ca44dc8dcc15]
This width presupposes a frequency-selective filter response: without one there is no peak or area to summarize. The live Filter (signal processing) identity can exist without an ERB computation, and a scalar ERB is not itself a signal-to-signal filter. The recorded edge is therefore strict composition/presupposes rather than subsumption. Empirical Measurement may supply an estimate, and mathematical Measure supplies integration, but neither whole identity replaces that nearer filter prerequisite.[ref-b1f2c0bfa99f][ref-ca44dc8dcc15]
Knowledge Transfer¶
The equal-peak/equal-area rule transfers from a human masking-derived filter shape to an animal neural-tuning-derived estimate, while each source's evidence route stays visible. A reported \(Q_{\mathrm{ERB}}\) is an operative use of the denominator ERB, not merely a choice to space model-filter center frequencies by an ERB scale. The two settings differ in organism and method yet retain response carrier, power-area criterion, peak, width and condition roles.[ref-0c8ba19b820f][ref-ca44dc8dcc15]
The analogous area/peak mathematics can define noise-equivalent bandwidth for an engineering filter. That analogy does not automatically make every electronic-filter bandwidth an auditory ERB. The named identity here remains tied to auditory or cochlear filtering.[ref-b1f2c0bfa99f][ref-ca44dc8dcc15]
Example¶
Human notched-noise masking. At the lower tested noise level, Moore and colleagues report mean ERBs of 36, 47, 87 and 147 Hz for signal frequencies 100, 200, 400 and 800 Hz, respectively. Mapped roles: carrier → listeners' inferred auditory filters; power response → fitted frequency-weighting shape around each signal frequency; peak → maximum of the inferred shape; equal-area width → each reported ERB mean; condition → the tested frequencies, masker level, participants and masking model. Their indexed abstract supports the design and values but not individual raw curves.[ref-0c8ba19b820f][ref-b1f2c0bfa99f]
Cat and guinea-pig neural tuning. Shera and colleagues' Fig. 1 plots animal \(Q_{\mathrm{ERB}}\) derived from single auditory-nerve-fiber threshold-frequency tuning curves by standard algorithms. Mapped roles: carrier → inferred cochlear frequency selectivity; power-response area → the effective filter width produced by the tuning-curve algorithm, not raw thresholds read as \(H(f)\); peak → the same-peak reference in the paper's ERB definition; width → the ERB denominator of each computed \(Q_{\mathrm{ERB}}\); condition → animal species, CF and inference method. The paper does not print each integration step, and the animal curves are not interchangeable with its human behavioral fit.[^ref-ca44dc8dcc15]
Relationships to Other Abstractions¶
Current abstraction Equivalent Rectangular Bandwidth Domain-specific
Parents (1) — more general patterns this builds on
-
Equivalent Rectangular Bandwidth presupposes Filter (Signal Processing) Domain-specific
An auditory ERB requires a frequency-selective auditory or cochlear filter response to summarize.
Hierarchy path (1) — routes to 1 parentless root
- Equivalent Rectangular Bandwidth → Filter (Signal Processing) → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Equivalent Rectangular Bandwidth sits in a sparse region of the domain-specific corpus (92nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Audio Recording & Acoustic Phenomena (10 abstractions)
Nearest neighbors
- Adaptive feedback cancellation — 0.80
- Log Gabor filter — 0.79
- Reverberation — 0.79
- Binaural recording — 0.79
- Microtonality — 0.78
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
A -3 dB or -10 dB bandwidth uses threshold crossings, and \(Q_{10}\) should not be read as \(Q_{\mathrm{ERB}}\). \(\mathrm{ERB}_N\) is a conditional normal-hearing fit, not a rule defining every filter. An ERB-rate coordinate or ERB-spaced filterbank is downstream use, not proof that each channel's response-area width was computed. Finally, human psychophysical ERB values and animal neural-tuning-derived estimates have different inference limits even though they share the named width rule.[ref-b1f2c0bfa99f][ref-0c8ba19b820f][^ref-ca44dc8dcc15]
References¶
[^ref-b1f2c0bfa99f]: Acoustical Society of America, Equivalent rectangular bandwidth, ASA/ANSI acoustical terminology entry 6.31, definition and Annotation 1. The annotation's normal-hearing formula specifies a cochlear input level of 51 dB; the entry does not make that fitted relation the definition.
[^ref-0c8ba19b820f]: Brian C. J. Moore, Robert W. Peters and Brian R. Glasberg, Auditory filter shapes at low center frequencies, Journal of the Acoustical Society of America 88 (1990), 132–140, DOI 10.1121/1.399960; original indexed abstract, methods and results. The accessible abstract supports the four frequencies, masker levels and lower-level mean ERBs; no individual curve is reproduced here.
[^ref-49d3c9823087]: Brian R. Glasberg and Brian C. J. Moore, Derivation of auditory filter shapes from notched-noise data, Hearing Research 47 (1990), 103–138, DOI 10.1016/0378-5955(90)90170-T; original indexed abstract, method and correction cautions.
[^ref-ca44dc8dcc15]: Christopher A. Shera, John J. Guinan Jr. and Andrew J. Oxenham, Revised estimates of human cochlear tuning from otoacoustic and behavioral measurements, Proceedings of the National Academy of Sciences 99 (2002), 3318–3323, DOI 10.1073/pnas.032675099, printed p.3319 / PDF p.2, “Comparing Cochlear Tuning Across Species” and Fig. 1 caption. The paper defines equal-white-noise-power ERB and reports computed animal \(Q_{\mathrm{ERB}}\) from earlier single-fiber threshold tuning curves; it does not print each underlying area integral.