Understanding Diversity Order (DO)
At high SNR, the Bit Error Rate (BER) behaves like:
BER ≈ k / (SNR^DO)
Where:
kis a constant depending on modulation and channel.DOis the diversity order, indicating how steeply BER decreases with SNR.
Mathematical Concept of DO
Formally, diversity order is defined as:
DO = - lim_{SNR → ∞} (log(BER) / log(SNR))
This means: if you plot log(BER) vs log(SNR), the slope at high SNR is the diversity order. Slopes can be fractional depending on the fading channel statistics.
Case 1: DO = 1
BER ∝ 1 / SNR^1 = 1 / SNR- BER decreases linearly on a log-log scale with SNR.
- Doubling SNR roughly halves BER.
- Steep slope → faster improvement → more robust system.
Case 2: DO = 0.5
BER ∝ 1 / SNR^0.5 = 1 / √SNR- BER decreases more slowly than DO = 1.
- Doubling SNR decreases BER by ~1/√2 ≈ 0.707.
- Flatter slope → slower improvement → less robust system.
Intuition with Numbers
Suppose BER = 0.01 at some SNR, then increasing SNR by 10 times:
- DO = 1 → BER decreases 10 times → new BER ≈ 0.001
- DO = 0.5 → BER decreases √10 ≈ 3.16 times → new BER ≈ 0.0032
Summary
| Aspect | DO = 1 | DO = 0.5 |
|---|---|---|
| BER improvement rate | Faster decrease with SNR | Slower decrease with SNR |
| System robustness | Higher | Lower |
| Error reduction slope | Steeper | Flatter |
Conclusion
DO = 1 means more diversity → BER improves faster with better SNR. DO = 0.5 means less diversity → BER improves slower.