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Theoretical BER vs SNR for m-ary PSK and QAM


Relationship Between Bit Error Rate (BER) and Signal-to-Noise Ratio (SNR)

The relationship between Bit Error Rate (BER) and Signal-to-Noise Ratio (SNR) is a fundamental concept in digital communication systems. Here’s a detailed explanation:

  • BER (Bit Error Rate): The ratio of the number of bits incorrectly received to the total number of bits transmitted. It measures the quality of the communication link.
  • SNR (Signal-to-Noise Ratio): The ratio of the signal power to the noise power, indicating how much the signal is corrupted by noise.

Relationship

The BER typically decreases as the SNR increases. This relationship helps evaluate the performance of various modulation schemes.


BPSK (Binary Phase Shift Keying)

  • Simple and robust.
  • BER in AWGN channel: BER = 0.5 × erfc(√SNR)
  • Performs well at low SNR.

QPSK (Quadrature Phase Shift Keying)

  • Transmits 2 bits per symbol.
  • BER: BER = 0.5 × erfc(√(SNR))
  • More spectrally efficient than BPSK, slightly higher BER at same SNR.

M-ary PSK (Phase Shift Keying)

  • Encodes multiple bits using M phases.
  • General BER expression is complex and depends on cos and sin terms.
  • Higher-order schemes (e.g., 8-PSK, 16-PSK) have higher BER for same SNR.

M-ary PSK is a modulation technique where each symbol represents log₂(M) bits by shifting the phase of a carrier. The phases are spaced evenly in a circle, and the distance between points decreases with higher M, making it more error-prone at low SNRs.

BER (approximate for large M and high SNR in AWGN):

BER ≈ (2 / log₂(M)) × erfc(√SNR × sin(Ï€ / M))
  • As M increases, spectral efficiency improves but BER performance degrades.
  • Gray coding is typically used to minimize BER.
  • Exact BER uses summation expressions, but approximations are common for analysis.

M-ary QAM (Quadrature Amplitude Modulation)

  • Uses both amplitude and phase variations.
  • BER: BER = (log₂(M)/2) × (1 - 1/√M) × erfc(√((3SNR)/(M - 1)))
  • Higher-order QAM (e.g., 16-QAM, 64-QAM) is more spectrally efficient, but BER increases with order.

Practical Considerations

  • AWGN Channel: Additive White Gaussian Noise affects the signal; SNR is key to system performance.
  • Simulation: SNR is varied to analyze BER behavior using tools like MATLAB.
  • Error Correction: Techniques like Forward Error Correction (FEC) help reduce BER.

Example

For a BPSK system in an AWGN channel:

Formula: BER = 0.5 × erfc(√SNR)

At SNR = 0 dB: BER ≈ 0.078649603525


Conclusion

Understanding the BER vs. SNR relationship is essential for designing efficient digital communication systems. Different modulation schemes provide trade-offs between spectral efficiency and BER performance.


Further Reading

  1. MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSk, ...
  2. Theoretical vs. simulated (using MATLAB) BER vs. SNR for ASK, FSK, and PSK

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