In telecommunication and signal processing companding (occasionally called compansion) is a method of mitigating the detrimental effects of a channel with limited dynamic range. The name is a portmanteau of compressing and expanding. The use of companding allows signals with a large dynamic range to be transmitted over facilities that have a smaller dynamic range capability. Companding is employed in telephony and other audio applications such as professional wireless microphones and analog recording.
How it works 
While the dynamic range compression used in audio recording and the like depends on a variable-gain amplifier, and so is a locally linear process (linear for short regions, but not globally), companding is non-linear and takes place in the same way at all points in time. The dynamic range of a signal is compressed before transmission and is expanded to the original value at the receiver.
The electronic circuit that does this is called a compandor and works by compressing or expanding the dynamic range of an analog electronic signal such as sound recorded by a microphone. One variety is a triplet of amplifiers: a logarithmic amplifier, followed by a variable-gain linear amplifier and an exponential amplifier. Such a triplet has the property that its output voltage is proportional to the input voltage raised to an adjustable power.
Companded quantization is the combination of three functional building blocks – namely, a (continuous-domain) signal dynamic range compressor, a limited-range uniform quantizer, and a (continuous-domain) signal dynamic range expander that inverts the compressor function. This type of quantization is frequently used in telephony systems.
In practice, companders are designed to operate according to relatively simple dynamic range compressor functions that are designed to be suitable for implementation using simple analog electronic circuits. The two most popular compander functions used for telecommunications are the A-law and μ-law functions.
Companding is used in digital telephony systems, compressing before input to an analog-to-digital converter, and then expanding after a digital-to-analog converter. This is equivalent to using a non-linear ADC as in a T-carrier telephone system that implements A-law or μ-law companding. This method is also used in digital file formats for better signal-to-noise ratio (SNR) at lower bit rates. For example, a linearly encoded 16-bit PCM signal can be converted to an 8-bit WAV or AU file while maintaining a decent SNR by compressing before the transition to 8-bit and expanding after a conversion back to 16-bit. This is effectively a form of lossy audio data compression.
Professional wireless microphones since the dynamic range of the microphone audio signal itself is larger than the dynamic range provided by radio transmission. Companding also reduces the noise and crosstalk levels at the receiver.
In the transmission of pictures by electric currents, the method which consists in sending currents varied in a non-linear relation to the light values of the successive elements of the picture to be transmitted, and at the receiving end exposing corresponding elements of a sensitive surface to light varied in inverse non-linear relation to the received current.—A. B. Clark patent
In 1970, H. Kaneko developed the uniform description of segment (piecewise linear) companding laws that had by then been adopted in digital telephony.
In the 1980s (and 90's), many of the music equipment manufacturers (Roland, Yamaha, Korg) used companding for data compression in their digital synthesizers. This dates back to the late 80's when memory chips would often come as one the most costly parts in the instrument. Manufacturers usually expressed the amount of memory as it is in the compressed form. i.e. 24MB waveform ROM in Korg Trinity is actually 48MB of data. Still the fact remains that the unit had a 24MB physical ROM. In the example of Roland SR-JV expansion boards, they usually advertised them as 8MB boards which contain '16MB-equivalent content'. Careless copying of the info and omitting the part that stated "equivalent" can often lead to confusion.
- W. R. Bennett, "Spectra of Quantized Signals", Bell System Technical Journal, Vol. 27, pp. 446–472, July 1948.
- Robert M. Gray and David L. Neuhoff, "Quantization", IEEE Transactions on Information Theory, Vol. IT-44, No. 6, pp. 2325–2383, Oct. 1998. doi:10.1109/18.720541
- A description of companding in wireless microphones
- US patent, A. B. Clark, "Electrical picture-transmitting system", issued 1928-11-13, assigned to AT&T
- Randall K. Nichols and Panos C. Lekkas (2002). Wireless Security: Models, Threats, and Solutions. McGraw-Hill Professional. ISBN 0-07-138038-8.
- B. Smith, "Instantaneous Companding of Quantized Signals," Bell System Technical Journal, Vol. 36, May 1957, pp. 653–709.
- H. Kaneko, "A Unified Formulation of Segment Companding Laws and Synthesis of Codecs and Digital Compandors," Bell System Technical Journal, Vol. 49, September 1970, pp. 1555–1558.