Comparison of M-PSK, M-QAM, and M-PAM Modulation Schemes
1. Key Parameter Relationships
- Bits per symbol: \( k = \log_2 M \)
- Symbol/bit energy: \( E_s = k E_b \)
- SNR per bit: \( \gamma_b = E_b / N_0 \)
- SNR per symbol: \( \gamma_s = E_s / N_0 = k \gamma_b \)
- Spectral efficiency: \( \eta = k \) bits/s/Hz
2. General BER Equations (AWGN, Gray Coding)
| Modulation | BER (in \(E_b/N_0\)) |
|---|---|
| M-PSK | \( \frac{2}{k} Q\left(\sqrt{2k\gamma_b}\sin\frac{\pi}{M}\right) \) |
| M-QAM | \( \frac{4}{k}\left(1-\frac{1}{\sqrt{M}}\right) Q\left(\sqrt{\frac{3k\gamma_b}{M-1}}\right) \) |
| M-PAM | \( \frac{2(M-1)}{Mk} Q\left(\sqrt{\frac{6k\gamma_b}{M^2-1}}\right) \) |
- \(Q(x)\): Q-function.
- \(k = \log_2 M\), \(\gamma_b = E_b / N_0\).
3. Side-by-Side Numerical Comparison: QAM vs PSK
| M | \(k\) | \(E_s/E_b\) | Spectral Efficiency | M-QAM (BER in \(E_b/N_0\)) | M-PSK (BER in \(E_b/N_0\)) |
|---|---|---|---|---|---|
| 2 | 1 | 1 | 1 | — (not used) | \( Q\left(\sqrt{2\gamma_b}\right) \) |
| 4 | 2 | 2 | 2 | \( Q\left(\sqrt{4\gamma_b}\right) \) | \( Q\left(\sqrt{4\gamma_b}\right) \) |
| 16 | 4 | 4 | 4 | \( \frac{3}{4} Q\left(\sqrt{12\gamma_b/15}\right) \) | \( \frac{1}{2} Q\left(\sqrt{8\gamma_b}\sin\frac{\pi}{16}\right) \) |
Notes
- For \(M=2,4\): QAM and PSK have identical BER and SNR terms.
- For \(M=16\): QAM is more power-efficient than PSK at the same spectral efficiency.
- As \(M\) increases, PSK becomes less efficient than QAM (needs more SNR for same BER).
4. Additional Parameter Reference (M-PAM)
| M | \(k\) | \(E_s/E_b\) | BER (in \(E_b/N_0\)) |
|---|---|---|---|
| 4 | 2 | 2 | \( \frac{3}{4} Q\left(\sqrt{12\gamma_b/15}\right) \) |
| 16 | 4 | 4 | \( \frac{15}{64} Q\left(\sqrt{24\gamma_b/255}\right) \) |
5. Summary
- Spectral Efficiency: Identical for given \(M\).
- Power Efficiency: QAM \(>\) PSK \(>\) PAM (for same \(M\)).
- Equation form: Always use the relationships \( E_s = k E_b \) and \( \gamma_s = k \gamma_b \) for conversions.
For more detail on derivations and constellation effects, see textbooks like Proakis, Goldsmith, or Haykin.
_Last updated: June 06, 2025