BER vs Eb/N0
The bit error rate of a coherent digital link in additive white Gaussian noise (AWGN) follows a Q-function "waterfall" in Eb/N0. Pick a modulation and an Eb/N0 to get the BER, or set a target BER to read off the required Eb/N0. Higher-order constellations pack more bits per symbol but need several more dB to hit the same error rate — the curve makes that trade explicit.
Equations & Parameters ▸
\(Q(x)=\tfrac12\,\mathrm{erfc}\!\left(\tfrac{x}{\sqrt2}\right),\quad \gamma=\tfrac{E_b}{N_0}\ \text{(linear)}\)
\(\text{BPSK/QPSK: } P_b=Q\!\big(\sqrt{2\gamma}\big)\qquad M\text{-PSK: } P_b\approx\tfrac{2}{k}Q\!\big(\sqrt{2k\gamma}\,\sin\tfrac{\pi}{M}\big)\)
\(M\text{-QAM: } P_b\approx\tfrac{4}{k}\Big(1-\tfrac{1}{\sqrt M}\Big)Q\!\Big(\sqrt{\tfrac{3k}{M-1}\gamma}\Big),\quad k=\log_2 M\)
\(\text{BPSK/QPSK: } P_b=Q\!\big(\sqrt{2\gamma}\big)\qquad M\text{-PSK: } P_b\approx\tfrac{2}{k}Q\!\big(\sqrt{2k\gamma}\,\sin\tfrac{\pi}{M}\big)\)
\(M\text{-QAM: } P_b\approx\tfrac{4}{k}\Big(1-\tfrac{1}{\sqrt M}\Big)Q\!\Big(\sqrt{\tfrac{3k}{M-1}\gamma}\Big),\quad k=\log_2 M\)
| Modulation | Constellation family and order M. |
| Eb/N0 | Energy per bit to noise density (dB). |
| Target BER | Optional — solves for the Eb/N0 that achieves this bit error rate. |
| Pb | Approximate bit error probability (Gray-coded, AWGN, coherent detection). |
Reference: J. G. Proakis & M. Salehi, Digital Communications, 5th ed., 2008. Q-function via the Numerical-Recipes erfc approximation.
Inputs
Scheme & order
dB
Operating pointe.g. 1e-6
Results
At Eb/N0
Bits per symbol—
Bit error rate Pb—
Es/N0—
For target BER
Required Eb/N0—
Margin at operating pt—
Diagram