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New cyclic multiuser detection (MUD) algorithms are presented in this paper which are based on blind adaptive MUD algorithms but have been modified such that they operate directly on the frequency shift (FRESH) architecture. The operation of both stochastic gradient descent (SGD) and recursive least squares (RSL) algorithms are examined. Suboptimal FRESH architectures are also investigated. The relative performance of the adaptive algorithms on the FRESH architecture is compared with an implementation on the filter bank architecture. This paper also investigates the relative sensitivity of the adaptive algorithms to timing jitter on these competing architectures.