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What is Mu-Law?

By Carol Francois
Updated May 16, 2024
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The mu-law is an algorithm used to compress the dynamic range for an audio signal transmitted over a telecommunication system. An algorithm is a series of calculations processed in a predefined order to within a computer or processor. The dynamic range of an audio signal is the ratio of the loudest undistorted sound over the background noise level picked up by the microphone.

Human speech has a dynamic range of 40 dB and human hearing has dynamic range of approximately 140 dB. When transmitting the human voice over a telecommunication system, the mu-law algorithm is applied to compress the data or audio signal. The purpose of this calculation is to improve the dynamic range ratio to ensure the sound of the human voice is carried with as little distortion as possible.

This formula is in use with both the older analog based systems as well as the newer digital systems. Under the analog system, the mu-law algorithm was used to reduce the impact of background noise or static. In the digital system, this formula compresses the digital signal to 8 bits, while keeping the same approximate level of noise.

The mathematical formula for mu-law is:

F(x) = sgn(x)*ln(1 + μ׀x׀))
ln(1 + μ)

where μ = 255 (8 bits) and for a given input x.

Mu-law is used for the digital telecommunication networks in North American and Japan to change the data to the supported 8 bit format. This is done after the signal or sound is received and processed by the digital computer system. In an analog system, this change is complete with the use of a non-linear gain amplifier. When working with an analog network, signal compression and adjustments should be completed internal to the analog system, where ever possible.

To convert an analog signal to digital, a converted with quantization levels set for an analog to digital conversion is required. Quantization refers to conversion of an analog signal into a digital one. The level must be of unequal space in accordance with the mu-law algorithm.

If the signal is already digital, then there is no need to convert anything and the mu-law formula can be applied directly. The 8 bit data file size is ideal for a digital file and matches the symbol size of most computers. This coincidence removed a large potential barrier to the digitization of the telecommunication system and was instrumental in the transition and acceptance of digital telecommunications.

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