In order to realize the blind separation of multi-Gaussian source and related source signals, based on the fast approximation joint diagonalization(FAJD) algorithm, the time-varying autoregressive theory in the field of fault diagnosis is successfully applied to the blind separation of correlated source signals and blind separation of multi-Gaussian source signals. Firstly, the time-varying autoregressive(TVAR) model is used to model the source signal. Then, the whitened pre-processing makes the modeled source signal have a structure that can be combined diagonally, and the time-varying parameters are approximated by the weighted sum of the basis functions. For the form of the weighted sum of the known basis functions, it becomes a time-invariant parameter, and then the model coefficient matrix group is solved using recursive least squares method, and it is used as the target matrix group of fast approximation and diagonalization. Finally, the separation of mixed signals is achieved using the FAJD algorithm. Matlab simulation experiments verify that the proposed algorithm is effective for the separation of correlated source signals and multi-Gaussian source signals. Due to the excellent characteristics of the TVAR model in the algorithm, this algorithm is very suitable for blind separation of mixed communication signals.