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 basically inverts the compressor function. This type of quantization is frequently used in older speech telephony systems. The compander function of the compressor is key to the performance of such a quantization system. In principle, the compressor function can be designed to exactly map the boundaries of the optimal intervals of any desired scalar quantizer function to the equal-size intervals used by the uniform quantizer and similarly the expander function can exactly map the uniform quantizer reconstruction values to any arbitrary reconstruction values. Thus, with arbitrary compressor and expander functions, any possible non-uniform scalar quantizer can be equivalently implemented as a companded quantizer. 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.
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