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We present the design guidelines and the experimental characterization of a multichannel acquisition system that measures the amplitude and the time-of-arrival of the signal pulses delivered by position-sensing silicon drift detectors (SDDs). The readout system has been equally developed for multichannel SDDs and for controlled drift detectors (CDDs) intended for spectroscopic imaging of X-rays or charged particles. The analog section includes a very large scale integration (VLSI) front-end preamplifier and bias current generator for the on-chip JFET follower while the digital back-end is realized with 12 bit 100 MS/s 8-channel analog-to-digital converter (ADC) versa modular eurocard (VME) boards. Amplitude and time are measured by digitally processing each unipolar shaped pulse also in presence of a superposed background waveform. The VME modularity allows the expansion of the readout system up to 128 channels per VME crate. The overall linearity error is better than 0.05%, and the mean noise over all channels, expressed in terms of equivalent noise charge, is about 4 electrons r.m.s. The measured time resolution is 0.6 ns r.m.s. at a signal charge of 5000 electrons, corresponding to a position resolution of 2-3 μm r.m.s. along the drift direction. The developed readout system has been used for X-ray imaging tests with CDDs at Sincrotrone Trieste.