Binary transmissions over Gaussian wiretap channel under soft/hard decision decoding

We study the secrecy capacity of the Gaussian wiretap channel with binary input under either soft or hard decision decoding. A closed-form expression of the secrecy capacity is derived by assuming that the eavesdropper uses the same decoding method as the legitimate receiver. An upper bound of the loss of secrecy capacity is provided whilst a smart eavesdropper choosing the best decoding method and a legitimate receiver employing an insufficient decoding method. Simulations show that in the low SNR (of the main channel) region, the secrecy capacity of the binary-input Gaussian wiretap channel under soft decision decoding is larger than the one under hard decision decoding; as the SNR increases, the secrecy capacity under hard decision decoding tends to overtake.