We use the sum-product algorithm (SPA) on factor graphs to enable joint detection and decoding over multipath channels in single-carrier modulation formats. We simulate the multipath channels as accurately as possible at the receiver side by discretizing channel parameters, such as channel paths and attenuation coefficients, to compute means of Gaussian observations. We assign these parameters Bernoulli random variables and consider priors of 0.5 for them. We propose a novel implementation strategy to reduce the complexity of the receiver considerably.