BER vs Eb/N0 Calculator (AWGN)

For the classic binary schemes over an additive white Gaussian noise channel, the probability of a bit error has an exact closed form built on the Gaussian Q-function — the tail probability of the standard normal distribution. Coherent BPSK achieves Pb = Q(√(2Eb/N0)); Gray-coded QPSK, remarkably, achieves exactly the same per-bit error rate, because its two bits ride on independent quadrature carriers and Gray coding makes each symbol error cost one bit. Coherent orthogonal BFSK gives up 3 dB: Pb = Q(√(Eb/N0)).

The input is Eb/N0 — energy per information bit over the one-sided noise power spectral density — not SNR. The distinction matters: Eb/N0 is normalized per bit, which is what makes curves comparable across schemes and data rates. Use the SNR / Eb/N0 converter to translate from a receiver's measured SNR.

Plotted against Eb/N0 in dB with BER on a log axis, the result is the signature waterfall curve of digital communications: gently sloping at low Eb/N0, then plunging — around 9.6 dB, BPSK crosses 10⁻⁵, and each further dB buys roughly an order of magnitude. This calculator evaluates the exact expression via a documented erfc approximation with relative error below 1.2×10⁻⁷ (accurate deep into the tail), shows every conversion step, and plots your operating point on the curve.

These are uncoded, coherent-detection results with perfect synchronization — the textbook baseline against which coding gain and implementation loss are measured.