Interactive BER Calculator
Calculate theoretical BER for m-ary PSK and QAM systems instantly.
M-ary PSK
M-ary QAM
Theoretical Error Probability of Digital Modulation
1. Bit Error Rate (BER) Formulas
BER indicates the end-to-end performance of a data link. For higher-order modulation like QAM, we use approximations based on Gray Coding (where adjacent symbols differ by only one bit).
Binary Phase Shift Keying (BPSK)The baseline for performance; very robust against noise.
\[ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \] Quadrature Phase Shift Keying (QPSK)Carries 2 bits per symbol but maintains the same BER as BPSK.
\[ P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \] M-ary PSK (e.g., 8-PSK) \[ P_b \approx \frac{2}{\log_2(M)} Q\left( \sqrt{\frac{2E_s}{N_0}} \sin\left(\frac{\pi}{M}\right) \right) \] Rectangular M-ary QAM (16, 64, 256-QAM)As M increases, the distance between points decreases, making them more sensitive to noise.
16-QAM: \[ P_b \approx \frac{3}{4} Q\left( \sqrt{\frac{4}{5}\frac{E_b}{N_0}} \right) \]
64-QAM: \[ P_b \approx \frac{7}{12} Q\left( \sqrt{\frac{2}{7}\frac{E_b}{N_0}} \right) \]
256-QAM: \[ P_b \approx \frac{15}{32} Q\left( \sqrt{\frac{8}{85}\frac{E_b}{N_0}} \right) \]
2. Symbol Error Rate (SER) Formulas
SER is useful for designing the physical layer and hardware. One symbol error usually results in only one bit error if Gray Coding is used.
BPSK & QPSKBPSK: \[ P_s = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]
QPSK: \[ P_s \approx 2Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \]
General M-QAM Symbol Error RateA symbol is "lost" if the noise pushes it outside its decision boundary.
16-QAM: \[ P_s \approx 3Q\left( \sqrt{\frac{4}{5}\frac{E_b}{N_0}} \right) \]
64-QAM: \[ P_s \approx \frac{7}{2} Q\left( \sqrt{\frac{2}{7}\frac{E_b}{N_0}} \right) \]
Summary of Parameters
- \(E_b\): Energy required to send one bit.
- \(E_s\): Energy required to send one symbol (\(E_s = E_b \cdot \log_2 M\)).
- \(N_0\): Noise power density.
- AWGN Channel: These formulas assume a static environment with only Gaussian noise (no fading).
- Gray Coding: A technique where adjacent constellation points differ by only one bit to minimize BER.
Performance Comparison (in MATLAB)
.png)
| Technique | Bits/Symbol |
|---|---|
| BPSK | 1 |
| QPSK | 2 |
| 8-PSK | 3 |
| 16-QAM | 4 |
| 64-QAM | 6 |
Modulation Scheme and Bandwidth Requirement
| Modulation | | Bits/Symbol | | Bandwidth |
|---|---|---|
| BPSK | 1 | Rb |
| QPSK | 2 | Rb / 2 |
| 8-PSK | 3 | Rb / 3 |
| 16-QAM | 4 | Rb / 4 |
Read more about Modulation Scheme & Bandwidth Requirement
QPSK vs. 4-QAM: Why They Should Have Same BER?
In digital communications, QPSK and 4-QAM exhibit identical Bit Error Rate (BER) performance. Although they are categorized differently, they share the same constellation geometry and power efficiency.