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The effect of charge recombination on the noise associated with the signal current at the anode of a semiconductor drift detector is studied for both time and amplitude measurements. The analysis is performed by fully taking into account the diffusion of electrons in the semiconductor, the electrostatic induction process that gives rise to the signal, and the correct boundary condition at the anode for the electron density. Both the time variance and the amplitude noise-to-signal ratio are calculated as a function of the filter width for different values of the time constant of the recombination process. A comparison with previous results based on more simplified treatments is also given.