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BER Derivation from Constellation Points


BER vs SNR: Constellation Analysis

BER Derivation from Constellation Points

The probability of a bit error in an AWGN channel is determined by the minimum Euclidean distance (\(d_{min}\)) between symbols in a constellation. The generic formula for Bit Error Rate (BER) is:

\[ BER = Q \left( \frac{d_{min}}{\sqrt{2N_0}} \right) \]

1. Binary PSK (BPSK)

BPSK is antipodal. The points are located at \(+\sqrt{E_b}\) and \(-\sqrt{E_b}\). Because the carrier is always "on" at full strength, the Peak Power equals the Average Power.

  • Distance (\(d_{min}\)): \(2\sqrt{E_b}\)
  • BER Formula: \(Q(\sqrt{2E_b/N_0})\)
  • At 0 dB SNR: \(Q(\sqrt{2}) \approx 0.078\)

2. Orthogonal FSK

FSK symbols are perpendicular in signal space. Like PSK, FSK has a constant envelope, meaning the power never fluctuates regardless of which frequency is sent.

  • Distance (\(d_{min}\)): \(\sqrt{2E_b}\)
  • BER Formula: \(Q(\sqrt{E_b/N_0})\)
  • At 0 dB SNR: \(Q(1) \approx 0.158\)

3. Amplitude Shift Keying (ASK/OOK)

ASK is unique because the power is not constant. One symbol is "Off" (0 energy) and the other is "On" (Amplitude \(A\)). This creates two ways to calculate SNR:

The Peak Power vs. Average Power Distinction:
  • Average Power: Since the signal is off 50% of the time, the average energy \(E_b = A^2/2\).
  • Peak Power: The transmitter must be capable of hitting amplitude \(A\). If we define SNR based on this peak (\(A^2/N_0\)):
\[ BER_{ASK(Peak)} = Q \left( \sqrt{\frac{A^2}{2N_0}} \right) \]

If Peak SNR is 0 dB (meaning \(A^2/N_0 = 1\)), the formula becomes \(Q(\sqrt{0.5})\).
Result: BER ≈ 0.239 (24%).

Summary Table at 0 dB SNR

Scheme Power Characteristic Distance (\(d_{min}\)) BER at 0 dB
BPSK Constant \(2\sqrt{E_b}\) 7.8%
FSK Constant \(\sqrt{2E_b}\) 15.8%
ASK (Peak) Variable \(A\) 24.0%

Conclusion: ASK performs the worst at 0 dB Peak SNR because for half of the transmission time (the "0" bit), there is no signal energy at all to fight the noise, whereas BPSK and FSK use their full power for every single bit.

Read More: Learn how the Q-function is used to derive theoretical BER curves for various digital modulation schemes in AWGN channels.

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