In order to improve noise filtering for high-resolution X-ray spectroscopy using silicon drift detectors, optimum finite impulse response filters are calculated and tested experimentally. Common matched filter theory cannot be applied for this problem since the filters need to fulfill several requirements regarding their time domain transfer function, like, for example, the presence of a flat-top, zero-filter area. Therefore, the utilization of digital penalized least mean square method with silicon drift detectors is presented. Adaptations to the method are applied in order to suit the considered application targeting low dead-time at high count rates using modern silicon drift detectors with fast charge amplifiers. The workflow for filter calculation is presented: Signal data are acquired using a spectroscopy setup in order to obtain pulse shape and noise information. Desired constraints regarding the time-domain transfer function of the filters are written out, and limits for the precision of their fulfillment are derived. Optimization is carried out, and resulting filters are presented. In order to experimentally test the calculated filters, implementation in hardware is done. 55 Fe spectra are acquired and evaluated. Noise reduction is rated by calculation of residual electronic noise in the recorded 55 Fe spectra. Comparison to the noise reduction of trapezoid filters is done. Improvement up to (5.2 ± 0.3)% was found using the calculated optimum filter, proving the successful utilization of digital penalized least mean square method for signal processing with modern silicon drift detectors.