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In this paper, we address adaptive acoustic echo cancellation in the presence of an unknown memoryless nonlinearity preceding the echo path. We approach the problem by considering a basis-generic expansion of the memoryless nonlinearity. By absorbing the coefficients of the nonlinear expansion into the unknown echo path, the cascade observation model is transformed into an equivalent multichannel structure, which we further augment with a multichannel first-order Markov model. For the resulting multichannel state-space model, we then derive a recursive Bayesian estimator that takes the form of an adaptive Kalman algorithm in the discrete Fourier transform (DFT) domain. We show that such a recursive estimator can be realized via a stable and structurally efficient multichannel state-space frequency-domain adaptive filter. We demonstrate that our algorithm, which stems from a contained framework, provides effective nonlinear echo cancellation in the presence of continuous double-talk, varying degree of nonlinear distortion, and changes in the echo path.