Bit Error Rate (BER) Calculator

Calculate bit error rate from Eb/N0 SNR for BPSK, QPSK, 16-QAM, 64-QAM, and 256-QAM modulation. AWGN-channel theoretical performance.

Calculator Electronics Updated Apr 23, 2026
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How to Use
  1. Enter Eb/N0 in dB.
  2. Pick modulation scheme.
  3. Tool returns theoretical BER in an AWGN channel.
Input
dB
Presets
BER vs Eb/N0
BER
—
Quality
—
Shannon Δ
—dB
Errors/Gbit
—

Show Work

Enter values.

Formulas

BPSK BER
Q(√(2·Eb/N0))
Q = Gaussian tail.
QPSK BER
Q(√(2·Eb/N0))
Same as BPSK (gray coded).
16-QAM
(3/4)·Q(√(4/5·Eb/N0))
Approximation.
M-QAM
(4/k)·(1-1/√M)·Q(√(3k·Eb/N0/(M-1)))
k=log₂(M)
Shannon
Eb/N0 min ≈ -1.6 dB
Asymptotic limit.
Q function
Q(x) = 0.5·erfc(x/√2)
Gaussian Q.

History of BER Analysis

Claude Shannon's 1948 paper established the theoretical minimum Eb/N0 for reliable communication at any bit rate - the Shannon limit. Actual modulation schemes needed 8-10 dB more Eb/N0 than Shannon's bound for practical BER. Modern error-correcting codes (Viterbi 1967, Turbo 1993, LDPC 1963/1996, Polar 2008) have closed the gap to within 1 dB of Shannon, enabling satellite links, 5G cellular, and DVB broadcasting at near-theoretical limits.

About This Calculator

Enter Eb/N0 (bit-energy-to-noise-power-spectral-density) in dB and pick modulation. The tool computes theoretical BER for AWGN channel using closed-form BPSK/QPSK formulas and M-QAM approximations.

Real-world BER is worse due to multipath, interference, phase noise, and sync error. Modern coded systems (LDPC, Turbo, Polar) can operate within 1 dB of Shannon limit, dramatically reducing required Eb/N0. Everything runs client-side.

Frequently Asked Questions

Eb/N0 vs SNR?

Eb/N0 = bit-energy-to-noise-spectral-density ratio. Normalized measure of link quality, independent of modulation bandwidth. Related to SNR by Eb/N0 = SNR · (B/R) where B is bandwidth, R is bit rate.

BER target?

Cellular typical 1e-3 (with FEC drops to 1e-6 effective). Optical fiber 1e-12 raw. For error-free data, target 1e-9 after error correction.

Shannon limit?

Theoretical lower bound: -1.6 dB Eb/N0 for vanishing BER. Real modulations are 1-10 dB worse. Higher-order QAM gets worse with increasing bits/symbol.

Common Use Cases

WiFi OFDM

Eb/N0 = 15 dB, 64-QAM: BER ~1e-3 raw, FEC to 1e-9.

Deep Space

Eb/N0 = 2 dB BPSK, turbo coding: BER 1e-6 data.

Satellite QPSK

Eb/N0 = 8 dB: BER 1e-5 raw, LDPC to 1e-12.

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