Binary Transmissions over Additive Gaussian Noise: A Closed-Form Expression for the Channel Capacity

We derive a closed-form expression for the capacity of the additive-Gaussian-noise memoryless channel with binary input. The capacity expression that we obtain contains an infinite series. Truncation of the series -for computation reasonsat an odd or even number of terms results in a lower or upper, respectively, bound on the channel capacity; we prove that the tightness of the bound increases monotonically with the number of terms that are included in the truncated series. Numerical results confirm that, as long as at least the first six terms of the series are calculated, the corresponding lower or upper bound becomes indistinguishable from the exact channel capacity. Therefore, with small computational complexity we obtain accurately the capacity of the Gaussian channel with binary input. Index Terms — Channel capacity, decoding, discreteinput channels, encoding, information rates, Gaussian channels, Gaussian noise.