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Contents

- 1. Information
- 2. Expressed
- 3. Floating-point
- 4. Resolutions
- 5. Instability
- 6. Quantization
- 7. Oversampling
- 8. Oversampling
- 9. Respectively
- 10. A-weighted
- 11. Quantization

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The number of bits of information recorded for each digital audio sample An analog signal (in red) encoded to 4- bit PCM digital samples (in blue); the bit depth is four, so each sample's amplitude is one of 16 possible values.

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. Thus, a 16- bit system has a resolution of 65,536 (2 16) possible values.

Many audio file formats and digital audio workstations (Days) now support PCM formats with samples represented by floating point numbers. Both the WAS file format and the RIFF file format support floating point representations.

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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. It is a rounding error between the analog input voltage to the ADC and the output digitized value.

An 8- bit binary number (149 in decimal), with the LSB highlighted In an ideal ADC, where the quantization error is uniformly distributed between ±12{\display style \script style {\pm {\franc {1}{2}}}} the least significant bit (LSB) and where the signal has a uniform distribution covering all quantization levels, the signal-to-quantization-noise ratio (SNR) can be calculated from Where Q is the number of quantization bits and the result is measured in decibels (dB).

Therefore, 16- bit digital audio found on CDs has a theoretical maximum SNR of 96 dB and professional 24- bit digital audio tops out as 144 dB. As of 2011 , digital audio converter technology is limited to an SNR of about 123 dB (effectively 21-bits) because of real-world limitations in integrated circuit design.

Still, this approximately matches the performance of the human auditory system. 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.

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This greatly increases the SNR compared to an integer system because the accuracy of a high-level signal will be the same as the accuracy of an identical signal at a lower level. The trade-off between floating point 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 round-off that is uniform, always rounding the LSB to 0 or 1, and floating point has SNR that is uniform, the quantization noise level is always of a certain proportion to the signal level.

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. Most processing operations on digital audio involve the re-quantization of samples and thus introduce additional rounding error analogous to the original quantization error introduced during analog-to-digital conversion.

To prevent rounding error larger than the implicit error during ADC, calculations during processing must be performed at higher precision than the input samples. 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. Fixed point digital signal processors often support specific word lengths in order to support specific signal resolutions.

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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 mean. 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.

High levels of precision are necessary for algorithms that involve repeated processing, such as convolution. High levels of precision are also necessary in recursive algorithms, such as infinite impulse response (AIR) filters.

In the particular case of AIR filters, rounding error can degrade frequency response and cause instability. Headroom and noise floor at audio process stages for the purpose of comparison with dither level 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 prior to 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 microphone and room noise level, and hence of little consequence in 16- bit audio.

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24- 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, but based on manufacturer's data sheets no ADC sexist that can provide higher than ~125 dB.

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.

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.

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.

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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.

Oversampling is an alternative method to increase the dynamic range of PCM audio without changing the number of bits per sample. Because quantization error is assumed to be uniformly distributed with frequency, much of the quantization error is shifted to ultrasonic frequencies and can be removed by the digital to analog converter during playback.

For an increase equivalent to n additional bits of resolution, a signal must be over sampled by Over sampled PCM, therefore, exchanges fewer bits per sample for more samples in order to obtain the same resolution.

Each sample at reconstruction would be unique in that for each of the original sample points sixteen are inserted, all having been calculated by a digital reconstruction filter. The mechanism of increased effective bit depth is as previously discussed, that is, quantization noise power has not been reduced, but the noise spectrum has been spread over 16× the audio bandwidth.

This caused confusion in the marketplace and even in professional circles, because 14- bit PCM allows for 84 dB SNR, 12 dB less than 16- bit PCM. Philips had implemented 4× oversampling with first order noise shaping which theoretically realized the full 96 dB dynamic range of the CD format.

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In practice the Philips CD100 was rated at 90 dB SNR in the audio band of 20Hz-20kHz, the same as Sony's CDP-101. Oversampling a signal results in equal quantization noise per unit of bandwidth at all frequencies and a dynamic range that improves with only the square root of the oversampling ratio.

Noise shaping is a technique that adds additional noise at higher frequencies which cancels out some error at lower frequencies, resulting in a larger increase in dynamic range when oversampling. For nth-order noise shaping, the dynamic range of an over sampled signal is improved by an additional 6 n dB relative to oversampling without noise shaping.

For example, for a 20 kHz analog audio sampled at 4× oversampling with second-order noise shaping, the dynamic range is increased by 30 dB. A 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. Graceland '11 (version 6) DAY by Apple Inc. 16- bit default with 24- bit real instrument recording Audacity Open source audio editor 16- and 24- bit PCM and 32- bit floating point FL Studio DAY by Image-Line 16- and 24- bit int and 32- bit floating point (controlled by OS) ^ DVD-Audio also supports optional Meridian Lossless Packing, a lossless compression scheme.

^ Blu-ray supports a variety of non-LPCM formats but all conform to some combination of 16, 20 or 24 bits per sample. ^ ITU-T specifies the A-law and law commanding algorithms, compressing down from 13 and 14 bits respectively.

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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. ^ For example, in MP3, quantization is performed on the frequency domain representation of the signal, not on the time domain samples relevant to bit depth.

Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications, Second Edition, online book. Pro Tools 10 Advanced Music Production Techniques, pg.

“Taking the Mystery out of the Infamous Formula, “SNR = 6.02 N + 1.76dB,” and Why You Should Care” (PDF). Dynamic Range (–60dB input, A-weighted): 124dB typical Dynamic Range (–60dB input, 20 kHz Bandwidth): 122dB typical ^ “WM8741 : High Performance Stereo DAC”.

128dB SNR (‘A’-weighted mono @ 48 kHz) 123dB SNR (non-weighted stereo @ 48 kHz) ^ “The great audio myth: why you don't need that 32- bit DAC”. Retrieved 2 December 2016. All the '32 bit capable' DAC chips existent today have actual resolution less than 24 bites.

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The practical dynamic range could be said to be from the threshold of hearing to the threshold of pain ^ US6317065B1, “Multiple A to D converters for enhanced dynamic range”, issued 1999-07-01 ^ Christodoulou, Lakes; Lane, John; Kasparov, Takes (1 March 2010). “Dynamic range extension using multiple A/D converters”.

“Determining Appropriate Precision for Signals in Fixed-Point AIR Filters”. ^ Choosing a high-performance audio ADC, retrieved 7 May 2019 ^ Montgomery, Chris (25 March 2012).

With use of shaped dither, which moves quantization noise energy into frequencies where it's harder to hear, the effective dynamic range of 16-bit audio reaches 120dB in practice, more than fifteen times deeper than the 96dB claim. 120dB is greater than the difference between a mosquito somewhere in the same room and a jackhammer a foot away.... or the difference between a deserted 'soundproof' room and a sound loud enough to cause hearing damage in seconds.

One of the great discoveries in PCM was that, by adding a small random noise (that we call dither) the truncation effect can disappear. Even more important was the realization that there is a right sort of random noise to add, and that when the right dither is used, the resolution of the digital system becomes infinite.

^ “Sweetwater Knowledge Base, Master link: What is a “Red Book” CD?” ^ “White paper Blu-ray Disc Format, 2. B Audio Visual Application Format Specifications for BD-ROM Version 2.4” (PDF).

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^ “G.711 : Pulse code modulation (PCM) of voice frequencies” (PDF). ^ “DIGITAL SOUND SIGNALS: tests to compare the performance of five commanding systems for high-quality sound signals” (PDF).