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In this paper, we introduce a Markov-switching generalized autoregressive conditional heteroscedasticity (GARCH) model for nonstationary processes with time-varying volatility structure in the short-time Fourier transform (STFT) domain. The expansion coefficients in the STFT domain are modeled as a multivariate complex GARCH process with Markov-switching regimes. The GARCH formulation parameterizes the correlation between sequential conditional variances while the Markov chain allows the process to switch between regimes of different GARCH formulations. We obtain a necessary and sufficient condition for the asymptotic wide-sense stationarity of the model, and develop a recursive algorithm for signal restoration in a noisy environment. The conditional variance is estimated by iterating propagation and update steps with regime conditional probabilities, while the model parameters are evaluated a priori from a training data set. Experimental results demonstrate the performance of the proposed algorithm.