Q-Function's Role in Rayleigh Fading
Analyzing AWGN vs. Rayleigh Fading using the General Error Rule
1. The "General" Rule for the Q-Function
Regardless of the modulation, the process for Rayleigh fading always follows this template:
- Identify the AWGN Error Probability: \( P_e(\gamma) = A \cdot Q(\sqrt{B\gamma}) \)
- Average it over Rayleigh:
\[ \int_{0}^{\infty} P_e(\gamma) \cdot p_{Rayleigh}(\gamma) d\gamma \]
BPSK Analysis
The Rayleigh closed-form solution used in the code: \[ P_{b, Rayleigh} = \frac{1}{2} \left( 1 - \sqrt{\frac{\bar{\gamma}}{1 + \bar{\gamma}}} \right) \] is the exact analytical result of integrating \( Q(\sqrt{2\gamma}) \) over the Rayleigh distribution.
BER Analysis Simulator
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The Q-function is calculated based on the selected scheme, then averaged across the fading PDF.
2. How it changes for other Modulations
To use other schemes, you only need to change the argument inside the Q-function and the resulting integral.
BFSK (Binary Frequency Shift Keying)
Smaller distance between signals; performs worse than BPSK.
- AWGN: \( P_b = Q(\sqrt{\gamma}) \)
- Rayleigh: \( P_{b, Rayleigh} = \frac{1}{2} (1 - \sqrt{\frac{\bar{\gamma}}{2 + \bar{\gamma}}}) \)
M-QAM (e.g., 16-QAM, 64-QAM)
Formula becomes a scaling of the Q-function because points are packed closer.
- General AWGN Form: \( P_b \approx k \cdot Q(\sqrt{C \cdot \gamma}) \)
- (k and C are constants based on constellation size M)
Summary Table
| Modulation | AWGN Q-Function Argument | Rayleigh Performance |
|---|---|---|
| BPSK / QPSK | \( \sqrt{2\gamma} \) | Best (requires least SNR) |
| BFSK | \( \sqrt{\gamma} \) | 3 dB worse than BPSK |
| 16-QAM | \( \sqrt{0.4\gamma} \) (approx) | Much worse; requires higher SNR |