# Audibility of noise in audio systems

Calculation of noise perception, detection threshold & dynamic range perceived by the human ear from SNR.

*April 5, 2023*

*Signal-to-noise ratio* (SNR) is the most widely used measurement for dynamic range. SNR is defined as the RMS of the peak sinusoidal signal divided by the RMS of the total noise. For audio measurements the measurement of noise RMS is band limited at 20 kHz and the noise is usually A-weighted.

However, the ear "integrates" the noise in small (approx. 1/9 octave) segments. In addition to the segmented integration sound pressure is weighted with the outer-ear and middle ear transfer function (inverse of ATH curve). Total RMS overestimates the noise relative to the ear, therefore SNR underestimates dynamic range relative to the ear.

Related article: Noise perception, detection threshold & dynamic range.

The purpose of this software is to predict the audibility of noise in audio systems with "flat" noise floor (amplifiers, DACs, some analog and digital formats). Shaped noise is not part of the simulation. Detailed instructions can be found further down the page.

### dBFS scale

dB (0-20 kHz)

### SPL scale

dB (0-20 kHz)

dBSPL

## Help

▶ Critical band, dBFS

▶ Signal-to-Noise Ratio

▶ Noise audibility threshold on dBFS scale (relative decibel scale)

▶ Calculation in the SPL domain

▶ Further readings

### Critical band, dBFS

**Critical band & critical bandwidth**

The frequency selectivity of our hearing system can be approximated by subdividing the intensity of the sound into parts that fall into critical bands. Level of a pure tone judged as loud as a band-pass noise with the same level up to a certain bandwidth. This bandwidth is called critical bandwidth.

**ERB vs. old (Zwicker) critical band:**

ERB scale is finer therefore the calculated noise levels are slightly lower with ERB bands than with the "older" critical bands. Fortunately, the difference between the two scales is very small in the most sensitive region of hearing (between 1 kHz and 10 kHz). Using the "old" (Zwicker) critical bands the noise power is one decibel higher at 1 kHz, two decibels higher at 4 kHz and three decibels higher at 10 kHz.

**Decibels relative to full-scale (dBFS)**

Decibels relative to full-scale (dBFS) is a unit of measurement for amplitude levels in digital systems. The level of 0 dBFS is assigned to the RMS value of the full scale sine wave. All level values smaller than the maximum are negative.

dBFS in Wikipedia ^{➚}

### Signal-to-Noise Ratio (SNR)

Since noise voltage and noise RMS is a function of bandwidth, a bandwidth has to be selected for the measurement. The most common bandwidth is 22 kHz with a 2 kHz margin for the bandwidth-limiting filter.

In this calculation the Signal-to-Noise Ratio is not weighted and bandwidth for the noise RSM is 20kHz.

**Converting A-weighted SNR to non-weighted SNR**

This conversion is valid only for flat spectrum noise. If the SNR is measured with an A-weighting filter plus a band limiting filter at ~22kHz, then:

SNR ≈ SNR' - 2.5 [dB]

Where SNR' is the A-weighted SNR in *dBA*. So a DAC with A-weighted SNR of *96 dBA* has a non-weighted SNR of *93.5 dB*.

**Digital system with TPDF dither**

Sampling frequency is 44.1 kHz.

SNR ≈ bits * 6 - 2.5 [dB],

where *bits* is the number of bits

**Reel-to-reel tape**

Non-weighted SNRs:

- best: 75 dB
- average: 70 dB

### Noise audibility threshold on dBFS scale (relative decibel scale)

Calculation:

- SNR converted to noise RMS and noise spectral density (NSD, power/Hz).
- Audible frequency range is divided into ERB bands (ERB = Equivalent Rectangular Bandwidth).
- Noise level (noise RMS, noise power) is calculated within each band. This gives the noise RMS per band curve. This is also a single tone curve with the same loudness as the noise in that ERB band.
- Values are weighted with the inverse Absolute Threshold of Hearing curve (this can be a loudness curve measured at low level SPL). The weighted curve is the noise audibility curve (noise audibility threshold).

The "noise-free" subjective dynamic range can be expressed with the absolute value of the maximum weighted noise RMS. Or more simply: the "noise-free" subjective dynamic range is defined by the top of the noise audibility curve. (since 0 dBFS is the RMS of the largest sine wave, the absolute value of noise RMS is the SNR in that band).

⇱ Back### Calculation in the SPL domain

(Note: SPL = Sound Pressure Level)

Calculation:

- SNR converted to noise RMS and noise spectral density (NSD, power/Hz).
- Audible frequency range is divided into ERB bands (ERB = Equivalent Rectangular Bandwidth)
- Noise RMS (or noise power) is calculated within each band (noise level per band)
- Noise RMS is converted from dBFS to SPL by adding the SPL of the full-scale sine (peak SPL). This noise SPL per band curve can be compared with the ATH curve.

When the gain set so that the noise SPL curve touches the ATH curve, the peak SPL can be used to express the "noise-free" subjective dynamic range.

⇱ Back### Further readings

**Noise audibility in audio systems:**

‘‘Minimally Audible Noise Shaping‘‘, S. P. Lipshitz, J. Vanderkooy, and R. A. Wannamaker, 1991

‘‘Noise: Methods for Estimating Detectability and Threshold‘‘, R. Stuart, 1994

**Critical bands, loudness of band-pass noise, threshold of complex tones:**

‘Psychoacoustics - Facts And Models‘‘, Zwicker, Fastl, 2007 (chapter 6: Critical Bands and Excitation; chapter 8: Loudness)

‘‘An Introduction to the Psychology of Hearing‘‘, Brian C.J. Moore, 2013 (chapter 3: Frequency Selectivity, Masking and the Critical Band; chapter 4: The Perception of Loudness)

More articles in this topic:

Noise perception, detection threshold & dynamic range

Listening test: Quantization noise & bit-depth