BPSK and QPSK Error Rates

Bit and symbol error rates of BPSK, Gray-coded QPSK and OQPSK versus Eb/N0, with spectral efficiency, bandwidth, and the shot-noise-limited coherent receive power and margin.

Model: Textbook approximation

Inputs

Modulation

Eb/N0 (dB)Data Rate (Gbps)Wavelength (nm)Photodiode Quantum Efficiency η

Coherent Receiver

RX Power (dBm)

Results

BER3.87e-6

Spectral Efficiency1 bit/s/Hz

Symbol Rate1.00 Gbaud

Required BW (Nyquist)1.00 GHz

Shot-noise N₀-158.9 dBm/Hz

Photons per Bit at P_req10.0

Required RX Power-58.9 dBm

Eb/N0 at RX power18.9 dB

BER at that Eb/N0< 10⁻¹⁵

Power Margin8.9 dB

Required Eb/N0 for Target BER

BER 1e-36.8 dB

BER 1e-59.6 dB

BER 1e-610.5 dB

BER 1e-912.5 dB

BPSK BER: P_e = ½ erfc(√(E_b/N₀))

QPSK / OQPSK (Gray-coded): same BER as BPSK; symbol error rate P_s = 1 − (1 − P_e)²

P_req: (E_b/N₀)·N₀·R at the local-oscillator shot-noise limit, N₀ = hν/η (heterodyne or intradyne) or hν/2η (single-quadrature homodyne, BPSK only). At η = 1 a BER of 10⁻⁹ needs 18 photons per bit (heterodyne) or 9 (homodyne). Thermal noise, LO intensity noise and amplifier ASE come on top: enter a lower η or a higher E_b/N₀ for them.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Exact error rates for ideal coherent detection in AWGN. Required power at the LO shot-noise limit, N₀ = hν/η (heterodyne) or hν/2η (homodyne BPSK); no thermal noise, LO RIN or ASE.

Proakis J. G., Salehi M. (2008). Digital Communications, 5th ed. McGraw-Hill. Binary antipodal signalling and Gray-coded QPSK.
Ip E., Lau A. P. T., Barros D. J. F., Kahn J. M. (2008). Coherent detection in optical fiber systems. Opt. Express 16(2), 753–791. Table 1: SNR per symbol of shot-noise-limited heterodyne and homodyne receivers. doi:10.1364/OE.16.000753

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