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The analysis of the noise temporal correlation effects on spectral properties and relaxation phenomena in hysteretic systems driven by arbitrary colored noise processes is presented. The calculations are performed under the framework of Monte Carlo techniques for simulating the stochastic processes and Preisach formalism for modeling magnetic materials. It is found that the correlation time of the input signal significantly influences the spectral and relaxation properties of the hysteretic system at moderate and low noise strength. Both the power spectral density at low frequencies and the viscosity coefficient increase with the correlation time of the input. The high-frequency spectrum of the output is less affected by the correlation time and decays as 1/f2 in the case of the Ornstein-Uhlenbeck (OU) noise.