Encoding law

In digital communications, an encoding law (or quantization law) is a (typically non-uniform) allocation of quantization levels across the possible analog signal levels in an analog-to-digital converter (ADC) system. These laws are especially important in voice encoding, where they help optimize signal-to-noise ratio and bandwidth usage.[1]

Encoding laws can be viewed as a simple form of instantaneous companding, a process that compresses the dynamic range of a signal before quantization and expands it after reconstruction.[2] This technique improves the performance of systems where the signal amplitude distribution is not uniform.

Common Encoding Laws

The two most widely used encoding laws are:

  • μ-law (mu-law): Used mainly in North America and Japan. It offers better dynamic range at low amplitudes by applying a logarithmic compression function. Its mathematical formula is defined as:
 : F(x) = sign(x) · ln(1 + μ|x|) / ln(1 + μ), where 0 ≤ |x| ≤ 1 and μ = 255.[3]
  • A-law: Preferred in Europe and most of the rest of the world. It applies a piecewise linear-logarithmic compression function with μ ≈ 87.6.

Both encoding laws are defined in the ITU-T standard G.711, which specifies pulse code modulation (PCM) at a bit rate of 64 kbit/s for digital telephony systems.[4]

Applications

Encoding laws are essential in:

These laws allow for a more efficient use of limited bandwidth in systems where human speech is the primary signal, as most voice energy is concentrated at low amplitudes.

Technical Rationale

Human hearing is more sensitive to changes in low-level signals than high-level signals. Encoding laws exploit this non-linearity by assigning more quantization levels to lower amplitude signals, thus improving perceptual audio quality.[6]

This is a form of **non-uniform quantization**, which provides higher resolution where it matters most. Without such companding laws, a linear quantizer would require more bits to achieve the same subjective quality.

See also

References

  1. ^ Proakis, John G.; Manolakis, Dimitris G. (2007). Digital Signal Processing: Principles, Algorithms, and Applications. Pearson. p. 1056. ISBN 9780131873742.
  2. ^ Oppenheim, Alan V.; Schafer, Ronald W. (2009). Discrete-Time Signal Processing (3rd ed.). Pearson. ISBN 9780131988422.
  3. ^ Jayant, N. S.; Noll, Peter (1984). Digital Coding of Waveforms: Principles and Applications to Speech and Video. Prentice Hall. p. 180. ISBN 9780132119139.
  4. ^ "Pulse code modulation (PCM) of voice frequencies – ITU-T Recommendation G.711". International Telecommunication Union. Retrieved 2025-06-23.
  5. ^ "Companding in Digital Audio Systems". Analog Devices. Retrieved 2025-06-23.
  6. ^ Fischer, Robert (1991). "Precoding and Signal Shaping for Digital Transmission". IEEE Communications Magazine. 29 (12): 26–34. doi:10.1109/35.103855.
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