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In the envelope-constrained filtering problem, one attempts to optimize a filter's impulse response subject to the constraint that its response to a specified input lies within a prescribed envelope. This constrained optimization problem is reduced to an unconstrained dual problem with a nondifferentiable cost and then solved using a steepest ascent algorithm based on directional differentials. The resulting technique provides a versatile alternative to the least-squares methods that are commonly applied to such problems. Applications to a wide variety of signal processing problems in areas such as communication channel equalization, television transmission, radar and sonar detection, filter and antenna design, seismology, and ultrasonic imaging are discussed, and some numerical results are presented.