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An exact alternative approach to the diagrammatic technique for treating scattering from rough surfaces is developed. The magnetic field integral equation for the current induced on the rough perfectly conducting surface is multiplied by a Fourier kernel involving all orders of surface height derivatives and their associated transform variables. Averages of this weighted equation are converted to convolations in the transform domain. The result of this operation is a singular integral equation of the first kind of infinite dimensions (because of the infinite number of height derivatives) for the stochastic Fourier transform of the current. A procedure is developed for estimating the effects of ignoring one or more surface height derivatives in terms of the range of validity of the resulting approximate solution. Special limiting cases of very gently undulating surfaces and uniformly rough surfaces are examined. New and illuminating results are obtained for the latter case.