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The optimization problem for the design of multistage decimators and interpolators is considered. The corresponding objective function for sample rate increase or decrease is based upon the number of multiplies and adds per second. The structure of the multidimensional gradient equations for the decimation or interpolation ratios is investigated. A drastic simplification of the minimization process is demonstrated. Even for an arbitrary number of stages K, the solution of the K-1-dimensional problem can be reduced to one dimension, giving an enormous saving in computation. The highly applicable cases of K=3 and K=4 are treated explicitly.