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ΣΔ modulation is used in many different ways because it allows a high-precision signal to be generated from binary quantized data. An example of such an application is half-tone printing in which each dot can be either on or off and the image to be rendered is gray scale. The authors examine ΣΔ modulation as an optimization algorithm and show that, although the first order ΣΔ is optimum, there are usually significantly superior techniques. They present some new list decoding techniques applicable to filters which are infinite state machines.