Audio bit depth

A PCM signal is a sequence of digital audio samples containing the data providing the necessary information to reconstruct the original analog signal. Each sample represents the amplitude of the signal at a specific point in time, and the samples are uniformly spaced in time. The amplitude is the only information explicitly stored in the sample, and it is typically stored as either an integer or a floating point number, encoded as a binary number with a fixed number of digits: the sample's bit depth, also referred to as word length or word size.

The resolution indicates the number of discrete values that can be represented over the range of analog values. The resolution of binary integers increases exponentially as the word length increases. Adding one bit doubles the resolution, adding two quadruples it, and so on. The number of possible values that an integer bit depth can represent can be calculated by using 2ⁿ, where n is the bit depth.[1] Thus, a 16-bit system has a resolution of 65,536 (2¹⁶) possible values.

Integer PCM audio data is typically stored as signed numbers in two's complement format.[2]

Today, most audio file formats and digital audio workstations (DAWs) support PCM formats with samples represented by floating point numbers.[3][4][5][6] Both the WAV file format and the AIFF file format support floating point representations.[7][8] Unlike integers, whose bit pattern is a single series of bits, a floating point number is instead composed of separate fields whose mathematical relation forms a number. The most common standard is IEEE 754 which is composed of three fields: a sign bit which represents whether the number is positive or negative, an exponent, and a mantissa which is raised by the exponent. The mantissa is expressed as a binary fraction in IEEE base-two floating point formats.[9]

The bit depth limits the signal-to-noise ratio (SNR) of the reconstructed signal to a maximum level determined by quantization error. The bit depth has no impact on the frequency response, which is constrained by the sample rate.

Quantization error introduced during analog-to-digital conversion (ADC) can be modeled as quantization noise. It is a rounding error between the analog input voltage to the ADC and the output digitized value. The noise is nonlinear and signal-dependent.

An 8-bit binary number (149 in decimal), with the LSB highlighted

In an ideal ADC, where the quantization error is uniformly distributed between ${\displaystyle \scriptstyle {\pm {\frac {1}{2}}}}$ least significant bit (LSB) and where the signal has a uniform distribution covering all quantization levels, the signal-to-quantization-noise ratio (SQNR) can be calculated from

${\displaystyle \mathrm {SQNR} =20\log _{10}({\sqrt {1.5}}\cdot 2^{\mathrm {b} })\approx 1.76+6.02{\mathrm {b} }\;\mathrm {dB} \,\!}$

where b is the number of quantization bits and the result is measured in decibels (dB).[10][11]

Therefore, 16-bit digital audio found on CDs has a theoretical maximum SNR of 98 dB, and professional 24-bit digital audio tops out as 146 dB. As of 2011, digital audio converter technology is limited to an SNR of about 123 dB[12][13][14] (effectively 21-bits) because of real-world limitations in integrated circuit design.[lower-alpha 2] Still, this approximately matches the performance of the human auditory system.[17][18] Multiple converters can be used to cover different ranges of the same signal, being combined to record a wider dynamic range in the long-term, while still being limited by the single converter's dynamic range in the short term, which is called dynamic range extension.[19][20]

The resolution of floating-point samples is less straightforward than integer samples because floating-point values are not evenly spaced. In floating-point representation, the space between any two adjacent values is in proportion to the value.

The trade-off between floating points and integers is that the space between large floating-point values is greater than the space between large integer values of the same bit depth. Rounding a large floating-point number results in a greater error than rounding a small floating-point number whereas rounding an integer number will always result in the same level of error. In other words, integers have a round-off that is uniform, always rounding the LSB to 0 or 1, and the floating point has uniform SNR, the quantization noise level is always of a certain proportion to the signal level.[21] A floating-point noise floor will rise as the signal rises and fall as the signal falls, resulting in audible variance if the bit depth is low enough.[22]

Most processing operations on digital audio involve the re-quantization of samples and thus introduce additional rounding errors analogous to the original quantization error introduced during analog-to-digital conversion. To prevent rounding errors larger than the implicit error during ADC, calculations during processing must be performed at higher precisions than the input samples.[23]

Digital signal processing (DSP) operations can be performed in either fixed point or floating-point precision. In either case, the precision of each operation is determined by the precision of the hardware operations used to perform each step of the processing and not the resolution of the input data. For example, on x86 processors, floating-point operations are performed with single or double precision and fixed-point operations at 16-, 32- or 64-bit resolution. Consequently, all processing performed on Intel-based hardware will be performed with these constraints regardless of the source format.[lower-alpha 3]

Fixed point digital signal processors often supports specific word lengths to support specific signal resolutions. For example, the Motorola 56000 DSP chip uses 24-bit multipliers and 56-bit accumulators to perform multiply-accumulate operations on two 24-bit samples without overflow or truncation.[24] On devices that do not support large accumulators, fixed point results may be truncated, reducing precision. Errors compound through multiple stages of DSP at a rate that depends on the operations being performed. For uncorrelated processing steps on audio data without a DC offset, errors are assumed to be random with zero means. Under this assumption, the standard deviation of the distribution represents the error signal and quantization error scales with the square root of the number of operations.[25] High levels of precision are necessary for algorithms that involve repeated processing, such as convolution.[23] High levels of precision are also necessary in recursive algorithms, such as infinite impulse response (IIR) filters.[26] In the particular case of IIR filters, rounding error can degrade frequency response and cause instability.[23]

Headroom and noise floor at audio process stages for the purpose of comparison with dither level

The noise introduced by quantization error, including rounding errors and loss of precision introduced during audio processing, can be mitigated by adding a small amount of random noise, called dither, to the signal before quantizing. Dithering eliminates non-linear quantization error behavior, giving very low distortion, but at the expense of a slightly raised noise floor. Recommended dither for 16-bit digital audio measured using ITU-R 468 noise weighting is about 66 dB below alignment level, or 84 dB below digital full scale, which is comparable to the microphone and room noise level, and hence of little consequence in 16-bit audio.

24-bit and 32-bit audio does not require dithering, as the noise level of the digital converter is always louder than the required level of any dither that might be applied. 24-bit audio could theoretically encode 144 dB of dynamic range, and 32-bit audio can achieve 192 dB, but this is almost impossible to achieve in the real world, as even the best sensors and microphones rarely exceed 130 dB.[27]

Dither can also be used to increase the effective dynamic range. The perceived dynamic range of 16-bit audio can be 120 dB or more with noise-shaped dither, taking advantage of the frequency response of the human ear.[28][29]

Dynamic range is the difference between the largest and smallest signal a system can record or reproduce. Without dither, the dynamic range correlates to the quantization noise floor. For example, 16-bit integer resolution allows for a dynamic range of about 96 dB. With the proper application of dither, digital systems can reproduce signals with levels lower than their resolution would normally allow, extending the effective dynamic range beyond the limit imposed by the resolution.[30] The use of techniques such as oversampling and noise shaping can further extend the dynamic range of sampled audio by moving quantization error out of the frequency band of interest.

If the signal's maximum level is lower than that allowed by the bit depth, the recording has headroom. Using higher bit depths during studio recording can make headroom available while maintaining the same dynamic range. This reduces the risk of clipping without increasing quantization errors at low volumes.

Bit depth is a fundamental property of digital audio implementations. Depending on application requirements and equipment capabilities, different bit depths are used for different applications.

Bit depth affects bit rate and file size. Bits are the basic unit of data used in computing and digital communications. Bit rate refers to the amount of data, specifically bits, transmitted or received per second. In MP3 and other lossy compressed audio formats, bit rate describes the amount of information used to encode an audio signal. It is usually measured in kb/s.[51]

• Audio system measurements
• Color depth, the corresponding concept for digital images
• Effective number of bits