Abstract: A design method for the bit-stream Hopfield neural network (HNN) based on Σ-Δ modulation is presented. The signals from the input and output of each neuron are represented by Σ-Δ modulated single-bit streams. This single-bit representation alleviates the fan-in and fan-out issues typical of distributed systems. A parallel distributed network with 12 neurons is designed, and the whole HNN is implemented on a field programmable gate array and backend design. Several key modules of the systems are optimized and the area, power are reduced by using digital backend design. Moreover, the simultaneous learning algorithm (SLA) is used to train the HNN. The SLA is an on-chip learning algorithm and is implemented on board. The bit-stream HNN achieves rather precise character recognition and memorization after the training.