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A new approach to the problem of time-variant filtering is presented. This approach is based upon the generation of the mixed time-frequency representation (MTFR) of a signal, multiplication of that representation by a time-frequency function H(ω,t), and obtaining a filtered output by an inverse operation. The resultant filter is linear if the time-frequency representation used is the complex spectrogram. In contrast, the filter is nonlinear if the Wigner distribution function is used. Not every function of two variables is an allowed MTFR of a signal; some conditions must be satisfied. If the function produced by the product of the signal MTFR and the filter function is not an allowed MTFR, an approximation based on projection onto the space of allowed MTFR functions is investigated. This approximation yields a filtered function whose MTFR is as close as possible (in the least-squared sense) to the desired.