A bound on signal-to-quantizing noise ratios for digital encoding systems

When an analog signal is encoded into digital form and then decoded back into an analog signal, quantizing noise is always introduced. The amount of quantizing noise which contaminates the decoded analog signal is inextricably tied to the amount of redundancy present in the signal and in the digital bit stream. Reducing the quantizing noise and, therefore, increasing the fidelity of the resulting signal requires that the redundancy in the digital bit stream be reduced or eliminated. There is a point, however, beyond which the quantizing noise cannot be further reduced. This is discussed in quantitative terms by deriving an upper bound on the signal-to-quantizing noise power ratio possible for a given bit rate and signal ensemble. Ratios of signal-to-quantizing noise greater than this bound are not possible for digital encoding systems. This bound is compared with the operation of pulse code modulation, differential pulse code modulation, and delta modulation systems.