Full scale

In electronics and signal processing, full scale or full code represents the maximum amplitude a system can represent.

In digital systems, a signal is said to be at digital full scale when its magnitude has reached the maximum representable value. Once a signal has reached digital full scale, all headroom has been utilized, and any further increase in amplitude will result in an error known as clipping. The amplitude of a digital signal can be represented in percent, full scale, or decibels, full scale (dBFS).

In analog systems, full scale may be defined by the maximum voltage available, or the maximum deflection (full scale deflection or FSD) or indication of an analog instrument such as a moving coil meter or galvanometer.

Binary represenation

Since binary integer representation range is asymmetrical, full scale is defined using the maximum positive value that can be represented.^[1]^[2] For example, 16-bit PCM audio is centered on the value 0, and can contain values from −32,768 to +32,767. A signal is at full-scale if it reaches from −32,767 to +32,767. (This means that −32,768, the lowest possible value, slightly exceeds full-scale.)

Signal processing in digital audio workstations often uses floating-point arithmetic, which can include values past full-scale, to avoid clipping in intermediate processing stages. In a floating-point representation, a full-scale signal is typically defined to reach from −1.0 to +1.0.^[3]^[4]

Processing

The signal passes through an anti-aliasing, resampling, or reconstruction filter, which may increase peak amplitude slightly due to ringing.

It is possible for the analog signal represented by the digital data to exceed digital full scale even if the digital data does not, and vice versa. In the analog domain there is no peak/clipping problem unless the d/a analog circuitry was badly designed. In the digital domain there are no peaks created by these conversions.

If a proper (no clipping/saturation) analog signal is converted to digital via A/D with sufficient samples, and then reconverted to analog via D/A, the Nyquist theorem guarantees that there will be no problem in the analog domain due to "peak" issues because the restored analog signal will be an exact copy of the original analog signal. (However, if the signal is normalized in the digital domain, it may contain "intersample peaks" which exceed full scale after analog reconstruction.)^[5]

References

^ "AES Standard » AES17-2015: AES standard method for digital audio engineering - Measurement of digital audio equipment". www.aes.org. Retrieved 2016-04-29. amplitude of a 997-Hz sine wave whose positive peak value reaches the positive digital full scale, leaving the negative maximum code unused. NOTE In 2's-complement representation, the negative peak is 1 LSB away from the negative maximum code.{{cite web}}: CS1 maint: url-status (link)
^ "IEC 61606-3:2008 Audio and audiovisual equipment - Digital audio parts - Basic measurement methods of audio characteristics - Part 3: Professional use". International Electrotechnical Commission. 2008. Retrieved 2018-07-27. amplitude of a 997 Hz sinusoid whose peak positive sample just reaches positive digital full-scale (in 2's-complement a binary value of 0111…1111 to make up the word length) and whose peak negative sample just reaches a value one away from negative digital full-scale (1000…0001 to make up the word length) leaving the maximum negative code (1000…0000) unused{{cite web}}: CS1 maint: url-status (link)
^ "Wave File Specifications". McGill University Telecommunications & Signal Processing Laboratory. Retrieved 2020-05-03. For float data, full scale is 1.{{cite web}}: CS1 maint: url-status (link)
^ "Adobe Audition User Guide" (PDF). 32-bit 0.24 normalized float (type 3 – 32-bit) is the standard floating point format for type 3 .wav files. Values are normalized to the range +/-1.0{{cite web}}: CS1 maint: url-status (link)
^ Nielsen, Søren H.; Lund, Thomas. "0dBFS+ Levels in Digital Mastering" (PDF). TC Electronic A/S. Denmark.

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[1] "AES Standard » AES17-2015: AES standard method for digital audio engineering - Measurement of digital audio equipment". www.aes.org. Retrieved 2016-04-29. amplitude of a 997-Hz sine wave whose positive peak value reaches the positive digital full scale, leaving the negative maximum code unused. NOTE In 2's-complement representation, the negative peak is 1 LSB away from the negative maximum code.{{cite web}}: CS1 maint: url-status (link)

[2] "IEC 61606-3:2008 Audio and audiovisual equipment - Digital audio parts - Basic measurement methods of audio characteristics - Part 3: Professional use". International Electrotechnical Commission. 2008. Retrieved 2018-07-27. amplitude of a 997 Hz sinusoid whose peak positive sample just reaches positive digital full-scale (in 2's-complement a binary value of 0111…1111 to make up the word length) and whose peak negative sample just reaches a value one away from negative digital full-scale (1000…0001 to make up the word length) leaving the maximum negative code (1000…0000) unused{{cite web}}: CS1 maint: url-status (link)

[3] "Wave File Specifications". McGill University Telecommunications & Signal Processing Laboratory. Retrieved 2020-05-03. For float data, full scale is 1.{{cite web}}: CS1 maint: url-status (link)

[4] "Adobe Audition User Guide" (PDF). 32-bit 0.24 normalized float (type 3 – 32-bit) is the standard floating point format for type 3 .wav files. Values are normalized to the range +/-1.0{{cite web}}: CS1 maint: url-status (link)

[5] Nielsen, Søren H.; Lund, Thomas. "0dBFS+ Levels in Digital Mastering" (PDF). TC Electronic A/S. Denmark.

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