Equivalent Rectangular Bandwidth¶
The equal-peak, equal-power-area rectangular width of a specified auditory or cochlear filter response.
Core Idea¶
For a specified auditory or cochlear filter response, equivalent rectangular bandwidth (ERB) is the width of an ideal rectangle that has the response's peak height and the same area under its power response. If \(P(f)\geq0\) is the power response over a declared frequency domain \(D\), with finite area and \(P_{\max}>0\), then
For a linear filter described by frequency response \(H(f)\), the relevant power weighting is \(P(f)=|H(f)|^2\). The integration convention, such as a positive-frequency interval or a two-sided domain, must be stated consistently; changing it changes the numerical area. The Acoustical Society of America defines the auditory term by equal maximum height and area. Shera, Guinan and Oxenham restate it as an equal-peak rectangle passing the same total power under white-noise drive.[1][2]
This is a bandwidth of a specified response at a condition, not one universal width of human hearing. Moore, Peters and Glasberg report distinct ERBs at four low signal frequencies and tested levels; Shera and colleagues use ERB in a comparison involving animal cochlear tuning and human behavioral estimates.[3][2]
Structural Signature¶
Signature: specified auditory or cochlear filter response + declared frequency domain + nonnegative power weighting + positive peak + equal-area rectangle → ERB width in hertz.[1][2]
- Response carrier. A frequency-selective auditory or cochlear filter is what the width describes. Its response may be inferred from masking or from physiological tuning evidence; the inference method is part of the interpretation, not a universal defining apparatus.[3][2]
- Power response and domain. The area uses a nonnegative weighting over an explicit frequency range. Squaring transfer magnitude is necessary when a linear transfer is interpreted as transmitted white-noise power. Integrating amplitude instead would answer a different question.[1][2]
- Peak reference. The hypothetical rectangle has the same peak power height as the represented filter. Without a fixed height, area alone allows rectangles of many widths.[1][2]
- Area-preserving width. The chosen rectangle's width is area divided by peak. Its edges need not reproduce the real filter's slopes or asymmetry. A threshold-crossing bandwidth does not perform this normalization.[1][2]
- Condition and inference scope. Center or characteristic frequency, level, organism or listener, and estimation model identify which response was summarized. They can change a reported ERB without changing the equal-area rule.[1][3][4][2]
What It Is Not¶
ERB is not a literal rectangular biological filter. The rectangle is a comparison object with the real or inferred filter's peak and integrated power-response area. It is also not the width between -3 dB or -10 dB crossings: such thresholds can yield a different width for the same response. Shera and colleagues distinguish the ERB-based \(Q_{\mathrm{ERB}}\) from the physiological \(Q_{10}\) convention.[1][2]
The normal-hearing \(\mathrm{ERB}_{N}\) formula in the ASA annotation is a fitted frequency relation under specified conditions, not the definition. Likewise, \(Q_{\mathrm{ERB}}(\mathrm{CF})=\mathrm{CF}/\mathrm{ERB}(\mathrm{CF})\) is a derived dimensionless sharpness ratio, and ERB-rate is a later frequency-coordinate construction. An electronic filter's equivalent noise bandwidth uses an analogous area/peak calculation but should not be silently presented as a measured auditory ERB.[1][2]
Scope of Application¶
The named identity applies when an auditory or cochlear filter response is summarized by the same-peak, same-area width. In human psychophysics, an auditory-filter shape can be inferred from notched-noise masking. In comparative cochlear work, animal neural threshold-frequency tuning data can be processed by published algorithms to estimate an effective ERB and report \(Q_{\mathrm{ERB}}\). These are different evidence routes to a related width, not interchangeable direct observations of one transfer function.[3][2]
Sound level, frequency and inference method matter. Moore and colleagues tested low signal frequencies at two masker levels; their reported lower-level mean widths belong to those conditions and participants. Shera and colleagues caution that apparent high-characteristic-frequency flattening in animal \(Q_{\mathrm{ERB}}\) could be a measurement artifact. Neither result establishes a level-invariant, species-independent auditory-filter width.[3][2]
Clarity¶
“Equivalent” refers to the power-response area at a fixed peak, not equal shape, equal -3 dB span or equal phase. For a linear filter, a white-noise thought experiment explains why \(|H(f)|^2\) matters: transmitted power scales with the squared magnitude of the transfer. The formula assumes a well-defined finite area and positive peak. Multiplying the entire response by one gain factor changes both area and peak together, leaving ERB unchanged.[1][2]
A neural threshold tuning curve is not itself a directly measured linear power-transfer function. Shera and colleagues say the animal \(Q_{\mathrm{ERB}}\) values in their Fig. 1 were computed from single-fiber threshold-frequency tuning curves by standard algorithms; they do not print each raw area integral. The resulting ERB is an inferred cochlear-filter bandwidth, not a claim that an experiment exposed a physical rectangle or measured a full \(H(f)\) at every frequency.[2]
Manages Complexity¶
ERB compresses a potentially asymmetric frequency response to one comparable width. The peak-and-area normalization makes a narrow steep response distinguishable from a broad one without relying on one arbitrary decibel crossing. It also lets investigators express selectivity by the inverse ratio \(Q_{\mathrm{ERB}}=\mathrm{CF}/\mathrm{ERB}\) when comparing characteristic frequencies.[1][2]
That compression discards slope, skirt asymmetry, fine structure and the details of the inference model. Glasberg and Moore warn that fitting only symmetric notches can distort an asymmetric auditory-filter estimate, and that some correction choices matter at extreme frequencies or in hearing impairment. An ERB value alone cannot reconstruct the original response or repair an invalid estimate.[4][3]
Abstract Reasoning¶
To analyze a candidate ERB, first identify the auditory or cochlear response carrier and its condition. State whether the response is empirical, fitted, or model-based, and declare the frequency domain. Next choose its power weighting and peak, integrate over that domain, and divide area by peak. If the source reports \(Q_{\mathrm{ERB}}\) rather than ERB, \(\mathrm{ERB}=\mathrm{CF}/Q_{\mathrm{ERB}}\) follows algebraically at the same characteristic frequency and inference conditions.[1][2]
Check what was actually measured or calculated. A threshold-frequency curve may be input to an algorithm estimating filter width; an ERB-spaced collection of modeled filters does not, just by its spacing, prove the equal-area width of each realized channel. The calculation, its surrogate data, and its biological interpretation must remain separate.[2]
Knowledge Transfer¶
The human and animal cases transfer the same peak-and-area width rule across unlike evidence: behavioral masking-derived filter shapes on one side and single auditory-nerve-fiber tuning-derived estimates on the other. Each can yield a width in hertz, while its organism, frequency, level and estimation assumptions remain attached. This is literal within auditory or cochlear filtering, not a claim that the physiological curves and behavioral fits measure exactly the same object without modeling assumptions.[3][2]
The area/peak mathematics can be used for an engineering filter's noise-equivalent bandwidth. That mathematical analogy does not by itself transfer the named auditory ERB identity to every electronic filter. Transfer of the computation and transfer of the domain-specific label are different claims.[1][2]
Examples¶
Human low-frequency masking study. Moore, Peters and Glasberg used symmetric and asymmetric notched-noise masking at signal frequencies 100, 200, 400 and 800 Hz and two overall noise levels, 77 and 87 dB SPL. They report lower-level mean ERBs of 36, 47, 87 and 147 Hz, respectively. Mapped roles: response carrier → the auditory-filter shapes inferred for listeners; power response/domain → the frequency weighting around each tested signal frequency in the fitted response; peak reference → the inferred shape's maximum; equal-area width → the four reported lower-level ERB means; condition → the particular frequencies, masker levels, subjects and masking model. Their indexed abstract supports these design and result values, not individual raw filter curves.[3][1]
Cat and guinea-pig neural tuning comparison. Shera, Guinan and Oxenham's Fig. 1 reports animal \(Q_{\mathrm{ERB}}=\mathrm{CF}/\mathrm{ERB}(\mathrm{CF})\) values computed from single auditory-nerve-fiber threshold-frequency tuning curves by standard algorithms. Mapped roles: response carrier → inferred cochlear frequency selectivity in each species; power-response/domain → the effective filter response underlying the algorithm's width estimate, not the raw threshold curve read as \(H(f)\); peak reference → the hypothetical rectangle's same-peak condition in the paper's ERB definition; equal-area width → the ERB denominator implied by each computed \(Q_{\mathrm{ERB}}\); condition → species, characteristic frequency and neural-tuning inference. The article compares these animal curves with a separate human behavioral masking-fit curve. It does not publish each integration step, and the apparent high-frequency flattening may be an artifact.[2]
Structural Tensions¶
The cited works reveal an estimation caution, not a demonstrated intrinsic tradeoff of the ERB identity. A single width helps compare tuning, but it cannot retain the filter's slopes or certify that the inferred response was sound. Symmetric-notch assumptions can bias an asymmetric estimate; neural threshold tuning requires an algorithm before it is interpreted as an ERB. These are checks on measurement and inference, not evidence that every ERB computation must choose between two opposing goals.[4][2]
Structural–Framed Character¶
The equal-peak, equal-area computation is structural: given a response, a frequency convention and a peak, the width follows by integration and division. The frame enters when investigators decide how a human masking experiment or animal neural tuning curve represents a cochlear filter and what sound level or species makes the comparison meaningful. The width's arithmetic does not itself judge whether one organism has better hearing.[1][3][2]
No institution's preference or normative ranking defines the formula; ASA standardizes the term, while experiments supply conditional estimates. The vocabulary of an equal-area rectangle travels into electronic-filter noise bandwidth, but importing the auditory ERB label there without an auditory carrier is analogy rather than another admitted instance. The live Filter (signal processing) node supplies the response prerequisite. Its character: a structural scalar measure with domain-specific auditory inference conditions and no universal evaluative outcome.[1][2]
Structural Core vs. Domain Accent¶
The core is the response peak, power-response area and width of the equal-area rectangle over a declared domain. Remove the response or peak and the width cannot be computed; replace the area equality with a decibel threshold and the metric changes. Notched-noise task details, the four Moore signal frequencies, a particular animal species, a tuning-curve algorithm and a fitted \(\mathrm{ERB}_{N}\) relation are accents or conditional evidence, not all-instance clauses.[1][3][2]
An integral is a general mathematical ingredient, and Measurement can describe the empirical route, but neither is the nearest object-level typed parent. A frequency-selective auditory/cochlear filter response is required in every admitted instance; Filter (signal processing) can exist without an ERB computation, making the recorded strict presupposes edge appropriate. The named ERB remains auditory-domain specific. A broader cross-domain equal-area-bandwidth Prime would need separately established instances and identity conditions, not merely an analogous formula.[1][2]
Instantiates / Related Primes¶
This entry presupposes Filter (Signal Processing).
- Filter (signal processing) — strict prerequisite. The width presupposes a response-bearing auditory or cochlear filter. A filter is a signal-to-signal selective system, whereas ERB is one derived scalar; the relation is composition/presupposes, not subsumption.[1][2]
- Measurement — related empirical route. Masking and neural tuning studies contain measurements and inferential models, but the ERB formula is not the entire instrument-to-value and uncertainty chain.[3][2]
- Measure — related mathematical ingredient. Integration supplies area; an additive set-size rule alone does not define this auditory-filter width.
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.Both the human masking inference and the animal auditory-nerve tuning inference assign an ERB to a frequency-selective auditory or cochlear filter response. Remove that response and no peak or power-response area remains from which the width could be derived. The live Filter (signal processing) identity can exist without an ERB calculation; ERB adds the equal-peak/equal-area scalar rule. The width is not itself an input-to-output filter, so subsumption is false.
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 identifies threshold crossings; \(Q_{10}\) and \(Q_{\mathrm{ERB}}\) therefore need not coincide. An ERB-rate coordinate or the conditional normal-hearing \(\mathrm{ERB}_{N}\) fit is a use of ERB, not the defining equal-area operation. A modeled filterbank may be spaced by ERB values without documenting each channel's actual equal-area width. Human masking-derived widths and animal neural-tuning-derived widths should retain their separate inference limits.[1][3][2]
References¶
[1] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s
[2] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28
[3] 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. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l
[4] 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. registry ↩a ↩b ↩c