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This article describes a significantly improved sum-of-sinusoids-based model for the accurate simulation of time-correlated Rayleigh and Rician fading channels. The proposed model utilises random walk processes instead of random variables for some of the sinusoid parameters to more accurately reproduce the behaviour of wireless radio propagation. Every fading block generated using our model has accurate statistical properties on its own and hence, unlike previously proposed models, there is no need for time-consuming ensemble-averaging over multiple blocks. Using numerical simulation it is shown that the important statistical properties of the generated fading samples have excellent agreement with the theoretical reference functions. A fixed-point hardware implementation of the corresponding Rayleigh and Rician fading channel simulator on a field-programmable gate array (FPGA) is presented. By efficiently scheduling the operations, the reconfigurable fading channel simulator is compact enough that it can be efficiently used to simulate multipath scenarios and multiple-antenna systems (e.g. a 4 times 4 MIMO channel) using a single FPGA.