BER and Outage Probability in Atmospheric Turbulence
After modeling turbulence using log-normal, Gamma–Gamma, and exponential distributions, the next step is to understand how turbulence affects bit error rate (BER) and outage probability.
1. What BER and Outage Mean in Turbulent Channels
Bit Error Rate (BER)
BER is the probability that a transmitted bit is detected incorrectly. In free-space optical (FSO) links, turbulence causes random fading of the received intensity, which makes the instantaneous SNR random.
Therefore, BER must be averaged over the turbulence statistics:
Average BER = E_I[ BER(γ(I)) ]
Outage Probability
Outage probability measures the likelihood that the received signal is too weak to maintain reliable communication.
P_out = P( γ < γ_th )
Since SNR is proportional to received intensity:
γ = γ̄ · I
2. Relation Between Intensity, SNR, and Turbulence
For IM/DD FSO systems, the instantaneous SNR can be written as:
γ = (R P I)² / N₀
In normalized form:
γ = γ̄ · I
This shows that BER and outage performance depend directly on the probability distribution of the turbulence-induced intensity I.
3. BER Under Weak Turbulence (Log-Normal)
For on–off keying (OOK), the instantaneous BER is:
P_b(γ) = 0.5 · erfc( √γ )
The average BER is obtained by averaging over the log-normal PDF:
P_b =
∫₀^∞ 0.5 · erfc( √(γ̄ I) ) · f_I(I) dI
Under weak turbulence, intensity variance is small, so BER degradation is mild.
4. BER Under Moderate Turbulence (Gamma–Gamma)
For moderate turbulence, the Gamma–Gamma distribution is used:
P_b =
∫₀^∞ 0.5 · erfc( √(γ̄ I) ) · f_GG(I) dI
Exact solutions involve Meijer-G functions, but a useful high-SNR approximation is:
P_b ∝ γ̄^−min(α, β)
Smaller α and β lead to steeper BER degradation.
5. BER Under Strong Turbulence (Negative Exponential)
When turbulence is very strong, intensity follows a negative exponential distribution:
f_I(I) = e^−I
The average BER has a closed-form expression:
P_b =
0.5 · ( 1 − √( γ̄ / (1 + γ̄) ) )
Strong turbulence causes BER saturation, making diversity techniques essential.
6. Outage Probability Analysis
P_out = ∫₀^{I_th} f_I(I) dI
Weak Turbulence
P_out =
Φ( (ln I_th + σ² / 2) / σ )
Moderate Turbulence
Outage probability is obtained numerically from the Gamma–Gamma CDF.
Strong Turbulence
P_out = 1 − e^−I_th
7. MATLAB Monte Carlo Simulation
Outage Probability
gamma_bar = 20;
gamma_th = 5;
I_th = gamma_th / gamma_bar;
P_out = mean(I < I_th);
BER Simulation (OOK)
gamma = gamma_bar .* I;
BER = mean(0.5 * erfc(sqrt(gamma)));
8. Summary
| Turbulence | BER Behavior | Outage Behavior |
|---|---|---|
| Weak | Slight BER increase | Rare outages |
| Moderate | Strong degradation | Frequent fades |
| Strong | BER saturation | Outage dominated |
Turbulence randomizes SNR, BER must be averaged over fading, and in strong turbulence outage probability becomes the dominant performance metric.