Below is the derivation for Bit Error Rate (BER) and Symbol Error Rate (SER) at 0 dB \(E_b/N_0\).
1. The 16-QAM Case (\(E_b/N_0 = 0\) dB)
16-QAM is treated as two independent 4-PAM signals on the Real and Imaginary axes. At 0 dB, the linear ratio \(\gamma_b = 1\).
Bit Error Rate (BER) Calculation:
\[P_b \approx \frac{3}{4} Q\left( \sqrt{\frac{4}{5} \frac{E_b}{N_0}} \right)\] \[P_b \approx 0.75 \times Q(\sqrt{0.8}) = 0.75 \times Q(0.8944)\] \[P_b \approx 0.75 \times 0.1855 = \mathbf{0.139} \approx \mathbf{0.14}\]Symbol Error Rate (Per Dimension) Calculation:
For 4-PAM (one axis of 16-QAM):
\[P_{pam} = \frac{3}{2} Q\left( \sqrt{\frac{4}{5} \frac{E_b}{N_0}} \right)\] \[P_{pam} = 1.5 \times Q(0.8944) = 1.5 \times 0.1855 = \mathbf{0.278} \approx \mathbf{0.28}\]Logic: Because of Gray coding, each symbol error typically results in only 1 bit error. Thus, \(BER \approx SER_{dim} / 2 \text{ bits per dim} = 0.28 / 2 = 0.14\).
2. The QPSK Case (\(E_b/N_0 = 0\) dB)
For QPSK, the Bit Error Rate formula is identical to BPSK. However, if the simulation uses Symbol SNR (\(E_s/N_0\)) instead of Bit SNR (\(E_b/N_0\)), the result shifts.
True BER (Energy per Bit):
\[P_b = Q\left( \sqrt{2 \frac{E_b}{N_0}} \right) = Q(\sqrt{2}) = Q(1.414)\] \[P_b = \mathbf{0.0786} \approx \mathbf{0.079}\]The "0.15" Case (Energy per Symbol):
If 0 dB is applied to the Symbol Power (\(E_s\)), then \(E_b = E_s/2\), which means \(E_b/N_0 = 0.5\):
\[P_b = Q\left( \sqrt{2 \times 0.5} \right) = Q(1)\] \[P_b = \mathbf{0.158} \approx \mathbf{0.15}\]Conclusion: If your QPSK simulation yields 0.15 BER at 0 dB, it is accurately calculating the error for a system where the noise power equals the total symbol power.
Comparison Table
| Modulation | SNR Definition | Math Value | Simulated BER |
|---|---|---|---|
| QPSK | \(E_b/N_0 = 0 \text{dB}\) | \(Q(1.414)\) | 0.078 |
| QPSK | \(E_s/N_0 = 0 \text{dB}\) | \(Q(1)\) | 0.15 |
| 16-QAM | \(E_b/N_0 = 0 \text{dB}\) | \(0.75 Q(0.894)\) | 0.14 |
Understanding Bits per Dimension
A complex digital signal consists of two independent physical components: the In-phase (I) axis and the Quadrature (Q) axis. These are the two "dimensions" of the signal.
| Modulation | Total Bits (\(k\)) | Bits per Dimension | Levels per Axis |
|---|---|---|---|
| BPSK | 1 | 1 (Uses 1D only) | 2 (\(-1, 1\)) |
| QPSK | 2 | 1 | 2 (\(-1, 1\)) |
| 16-QAM | 4 | 2 | 4 (\(-3, -1, 1, 3\)) |
In your 16-QAM simulation at 0 dB, you observed a Symbol Error Rate (SER) per dimension of 0.28.
Since there are 2 bits per dimension, the bit error rate is simply:
BER = SER_dim / Bits_per_dim = 0.28 / 2 = 0.14