Approximating a bandlimited function using very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order

Digital signal processing has revolutionized the storage and transmission of audio and video signals as well as still images, in consumer electronics and in more scientific settings (such as medical imaging). The main advantage of digital signal processing is its robustness: although all the operations have to be implemented with, of necessity, not quite ideal hardware, the a priori knowledge that all correct outcomes must lie in a very restricted set of well-separated numbers makes it possible to recover them by rounding off appropriately. Bursty errors can compromise this scenario (as is the case in many communication channels, as well as in memory storage devices), making the “perfect” data unrecoverable by rounding off. In this case, knowledge of the type of expected contamination can be used to protect the data, prior to transmission or storage, by encoding them with error correcting codes; this is done entirely in the digital domain. These advantages have contributed to the present widespread use of digital signal processing. Many signals, however, are not digital but analog in nature; audio signals, for instance, correspond to functions f(t), modeling rapid pressure oscillations, which depend on the “continuous” time t (i.e. t ranges over or an interval in , and not over a discrete set), and the range of f typically also fills an interval in .F or this reason, the first step in any digital processing of such signals must consist in a conversion of the analog signal to the digital world, usually abbreviated as A/D conversion. For different types of signals, different A/D schemes are used; in this paper, we restrict our attention to a particular class of A/D conversion schemes adapted to audio signals. Note that at the end of the chain, after the signal has been processed, stored, retrieved, transmitted, ..., all in digital form, it needs to be reconverted to an analog signal that can be understood by a human hearing system; we thus need a D/A conversion there.