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The digital filter with complex coefficients finds applications in the digital processing of analytic signals and complex envelopes. A theory is developed for designing such filters based on low-pass analog prototypes and digital design techniques for real filters. An example of a filter designed according to this theory is presented. The relative advantages of real and analytic signal processing are discussed. It is shown that the filtering required by either processing technique requires essentially the same amount of signal operations, which is reasonable in view of the fact that the same amount of information is processed in both classes of filter.