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The aim of the present work is the development of the hardware, firmware and software architecture of a data acquisition (DAQ) system suitable for the readout of high-speed high-resolution silicon drift detectors and of controlled-drift detectors for spectroscopic imaging of X-rays. The novelty of the developed DAQ system is to be sensitive to multiple pulses present in the sampled waveform without loss in measurement resolution with respect to a single pulse algorithm. This feature is particularly important at high event rates as it would improve the detected rate and also avoid errors in the pulse measurements. A multi-maximum search algorithm has been developed and implemented in order to demonstrate the feasibility of this approach. The focus of the paper is on the description of the general architecture and on the full test of the DAQ chain in experimental conditions. The resolution of the measured amplitude and time-of-arrival has been evaluated in the case of two pulses per waveform, including the case of partially overlapping events. In the case of Poissonian occurrence of pulses we compared the main performance parameters of the developed multiple-pulse DAQ (case of n = 2 pulses) with respect to a single-pulse DAQ which confirmed the expected increase of the event throughput in good agreement with the theoretical expectations and showed no degradation of the measurement resolution.