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We describe a systematic method for designing highly accurate and efficient infinite impulse response (IIR) and finite impulse response (FIR) filters given their specifications. In our approach, we first meet the specifications by constructing an IIR filter with, possibly, a large number of poles. We then construct, for any given accuracy, an optimal IIR version of such filter (with a minimal number of poles). Finally, also for any given accuracy, we convert the IIR filter to an efficient FIR filter cascade (either serial or parallel). Since in this FIR approximation the non-causal part of the IIR filter only introduces an additional delay (as a function of the desired accuracy), our IIR construction does not have to enforce causality. Thus, we obtain a simple method for constructing linear phase filters if the specifications so require. All of these procedures are accomplished via robust, fast algorithms. We provide several illustrative examples of our method.