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This paper considers the synthesis of optimum sampled data multipole filters with n inputs and m outputs. The signal portion of each input is assumed to consist of a stationary random component and a polynomial with unknown coefficients but known maximum order. Each signal is corrupted by stationary random noise. The filter under investigation is linear, time-invariant, and has finite memory. Each input to the filter consists of a sequence of impulses with a constant period T. Each impulse is assumed to have an area equal to the value of the signal plus noise at the sampling instant. The synthesis procedure to be developed is to specify the weighting functions of the filter such that the system error, which is defined as the difference between the actual and ideal outputs, has zero ensemble mean and the system ensemble mean square error is minimum. The weighting functions thus obtained will have, in general, abrupt jumps at the sampling instants but they are continuous within the sampling intervals. The synthesis procedure is extended to the case shown in Appendix A where each of the nonrandom signals can be expressed as an arbitrary linear combination of a set of known time functions. Further generalization is possible to the synthesis of time-varying filter with sampled nonstationary random inputs as given in Appendix B.