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The problem to be considered in this paper is that of designing radar signals and receivers that are optimum for detecting a point target masked by a background of clutter returns and thermal noise. The problem of choosing an optimum signal when no constraints are placed on the type of signals allowed is discussed briefly, but the remainder of the paper is restricted to signals and receiver impulse responses that are uniformly spaced, phase and amplitude-tapered pulse trains. Expressions for the signal-to-interference ratio obtained when a signal is used with its matched filter (pm/) and with the optimum filter or clutter filter are then derived together with an explicit expression for the clutter filter. An iterative technique for maximizing is devised. This scheme has the useful property that it generates a sequence of signals whose 's form a monotone, nondecreasing sequence. This is followed by an application of the calculus of variations to derive the Euler equations for the stationary points of and . The form of the Euler equations suggests iterative techniques for their solution; in fact, the technique suggested for the solution of the Euler equation associated with is essentially the iterative technique that was described above.