Tool · Fiber and telecom

BER Calculator

What bit error ratio does a Q factor or an OSNR give? The calculator converts between Q, Q in dB and BER for a two-level signal, and gives the BER of coherent BPSK, QPSK, 16-QAM and 64-QAM at a given OSNR and symbol rate, with the OSNR each needs to reach the common FEC thresholds. Background: bit error rate, OSNR and forward error correction.

Q factor and BER

Start from

Q factor

Q (dB), 20 log₁₀ Q

Bit error ratio

Q is the eye opening in units of the noise: (μ₁ − μ₀)/(σ₁ + σ₀) for a two-level signal. Type a BER as 1e-12.

From OSNR (coherent formats)

Modulation

Symbol rate (GBd)

OSNR (dB)

OSNR reference bandwidth (GHz)

Polarizations

The usual reference bandwidth is 12.5 GHz, 0.1 nm at 1550 nm. The result is for ideal transmitter and receiver in white Gaussian noise; real transceivers need 1 to 3 dB more OSNR.

Presets

Readouts

BER against OSNR

At the chosen symbol rate and polarization. Gray, from left: BPSK, QPSK, 16-QAM, 64-QAM; the current format is blue. Dashed lines are the HD-FEC and KP4 thresholds.

BER against Q

Each extra decibel of Q near BER 10⁻¹² buys about two decades of BER.

Learn with it

Two short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The Q-to-BER conversion, its inverse and the format BER curves are compared with closed-form values.

Check	Expected	Computed	Tolerance

The expected values follow BER = ½ erfc(Q/√2), the Gray-coded square-QAM expression and SNR = 2B_refOSNR/(pR_s) from Essiambre et al. (2010). The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

A two-level signal with means $μ_{1}, μ_{0}$ and Gaussian noise $σ_{1}, σ_{0}$ , sampled at the best threshold, has

Q = \frac{μ_{1} - μ_{0}}{σ_{1} + σ_{0}}, B E R = \frac{1}{2} e r f c ⁣ (\frac{Q}{\sqrt{2}})

with Q in decibels written as $20 \log_{10} Q$ . For coherent formats in white Gaussian noise the SNR per symbol follows from the OSNR measured in a reference bandwidth $B_{r e f}$ , with $p$ polarizations at symbol rate $R_{s}$ :

S N R = \frac{2 B_{r e f}}{p R_{s}} O S N R

Gray-coded square $M$ -QAM then has, in the nearest-neighbor approximation that is exact for QPSK,

B E R \approx \frac{2}{\log_{2} M} (1 - \frac{1}{\sqrt{M}}) e r f c \sqrt{\frac{3 S N R}{2 (M - 1)}}

and BPSK has $\frac{1}{2} e r f c \sqrt{S N R}$ . The required OSNR is found by inverting the curve numerically. Two pre-FEC thresholds are shown: 2.4 × 10⁻⁴ for the RS(544,514) “KP4” code of IEEE 802.3, and 3.8 × 10⁻³ for the 7 % hard-decision codes of ITU-T G.975.1. The results are back-to-back limits for ideal transceivers: implementation penalties, laser phase noise, nonlinear interference and non-Gaussian noise all raise the OSNR a real link needs, typically by 1 to 3 dB at the transceiver alone.

Worked example

Q = 7, or 16.90 dB, gives $B E R = \frac{1}{2} e r f c (7 / \sqrt{2}) = 1.280 \times 10^{- 12}$ . Closing the eye by 1 dB, to Q = 6.237, raises the BER about 170-fold to $2.225 \times 10^{- 10}$ . A 100G DP-QPSK signal at 32 GBd reaches the 3.8 × 10⁻³ threshold at an OSNR of 12.61 dB in 12.5 GHz; DP-16QAM at the same symbol rate needs 19.27 dB.

Related: the OSNR calculator gives the OSNR after a chain of amplifiers, the how to measure OSNR article covers the measurement, and the QAM and eye diagram entries explain the formats and Q.

References: S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell System Technical Journal 52, 843–874 (1973). R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). J. G. Proakis and M. Salehi, Digital Communications, 5th ed., McGraw-Hill (2008), chapter 4. ITU-T G.975.1 (2004). IEEE 802.3-2022, clause 119.