Spike timing dependent plasticity is one of the synaptic plasticity rules that plays a key role in brain learning. This brief presents a set of piecewise linear approximations in order to produce a digital neuromorphic realization of the pair-based and triplet-based spike timing dependent plasticity rules. The proposed models are validated by simulation and an field-programmable gate array implementation demonstrates how the proposed approximations can alter synaptic weights according to the timing differences between a set of different patterns of spikes.