Equivalent Rectangular Bandwidth (ERB): Auditory Filter Width Metric

The equivalent rectangular bandwidth (ERB) describes the bandwidth of a hypothetical rectangular filter that passes the same total power as a human auditory filter with unit gain at its center frequency. Psychoacousticians use ERB to quantify the frequency selectivity of the cochlea, enabling accurate models of masking, loudness, and speech intelligibility. Signal-processing engineers adopt ERB-based filter banks to mimic human perception in compression, noise reduction, and hearing-protection devices.

Definition and Formulas

Glasberg and Moore (1990) proposed an analytic ERB formula for normal-hearing listeners: ERB(f) = 24.7 × (4.37 × f/1000 + 1) where frequency f is in hertz and ERB is expressed in hertz. The formula reflects experimental masking data across a wide frequency range (100 Hz to 10 kHz). An alternative, the ERB-number (ERB_N) scale, integrates 1/ERB(f) over frequency to create a perceptually uniform frequency axis analogous to the Bark scale.

ERB relates to the equivalent noise bandwidth used in electronics, but the psychoacoustic ERB incorporates the ear’s nonlinear behavior and level dependence. In practice, researchers report ERB in hertz, often alongside center frequency and sound pressure level.

Historical Development

Early critical-band theories by Fletcher and Zwicker divided the spectrum into Bark bands based on masking data. As measurement precision improved, Glasberg and Moore refined the concept with ERB, offering a more accurate representation of auditory filters for moderate sound levels. Their work synthesized notched-noise masking experiments, otoacoustic emission data, and physiological evidence to derive the ERB formula widely used today.

The ERB-rate scale, measured in “ERBs,” now appears in standards such as ISO 226 (equal-loudness contours) and IEC 61260 (filter bank design). Hearing-aid algorithms, perceptual codecs, and psychoacoustic models routinely rely on ERB spacing to allocate filter bands efficiently.

Conceptual Foundations

Auditory Filter Shape

Real auditory filters resemble rounded exponentials rather than rectangles. The ERB captures their effective width while retaining the simplicity of rectangular filters for analytical work. In gammatone filter banks, designers set bandwidths equal to ERB(f) to emulate neural responses measured in the cochlear nerve.

Level Dependence

At high sound levels, cochlear nonlinearities broaden auditory filters. Researchers report level-dependent ERB by adjusting the constants in the Glasberg–Moore formula. When interpreting data, always note the SPL at which ERB was measured, aligning with sound-pressure level conventions.

Relation to Other Psychoacoustic Scales

One ERB corresponds roughly to one Bark at low frequencies but diverges above 5 kHz. The ERB-rate scale offers a convenient alternative to the Bark scale for filter-bank design, especially when modeling speech intelligibility or audio codecs. Researchers often convert between ERB and Bark to compare results across studies, referencing both the Bark and ERB frameworks.

Applications and Importance

Speech coders such as MPEG and Dolby AC-3 allocate bits according to ERB-spaced filter banks, ensuring perceptually uniform quantization noise. Hearing aids and cochlear implants implement ERB-inspired channel spacing to balance spectral resolution with computational efficiency. Psychoacoustic research uses ERB to design masking experiments, evaluate hearing protection, and assess audio product quality.

Occupational hygienists incorporate ERB-weighted analyses when predicting temporary threshold shifts from broadband noise. Pairing ERB insights with the noise exposure calculator helps set safe listening durations tailored to the spectral content of the environment.

Working with ERB Data

When reporting ERB, specify the measurement method (notched-noise masking, otoacoustic emissions, auditory brainstem response) and listener profile (normal-hearing, hearing-impaired). Provide center frequencies, sound levels, and any level-dependent corrections. Convert ERB-rate indices back to hertz using the inverse Glasberg–Moore equations to maintain compatibility with instrumentation readouts.

In applied acoustics, combine ERB-based filter design with regular breaks scheduled via the screen-break frequency reminder to mitigate listener fatigue. Designers of auditory displays and alarms can prototype filter spacing using the LC resonant frequency tool to map ERB center frequencies onto resonant circuits or digital oscillators.