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What is the error probability of BPSK?

Published in Digital Modulation 4 mins read

The exact error probability of Binary Phase Shift Keying (BPSK) for coherent detection in an Additive White Gaussian Noise (AWGN) channel is a fundamental metric for evaluating digital communication system performance.

The Exact Error Probability Formula for BPSK

The bit error probability ($P_b$) for coherent BPSK modulation in an ideal AWGN channel is precisely given by:

$P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)$

Where:

  • $P_b$: Represents the Bit Error Probability (BER), which is the ratio of incorrectly received bits to the total number of transmitted bits.
  • $Q(\cdot)$: Denotes the Gaussian Q-function, a mathematical function that describes the tail probability of the standard normal distribution. Specifically, $Q(x) = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} e^{-t^2/2} dt$.
  • $E_b$: Is the average energy per bit of the transmitted signal, measured in Joules.
  • $N_0$: Is the one-sided power spectral density of the Additive White Gaussian Noise, typically measured in Watts/Hz (or Joules).

Understanding BPSK Modulation

Binary Phase Shift Keying (BPSK) is a robust digital modulation technique where binary data is conveyed by shifting the phase of a sinusoidal carrier wave. For example, a binary '0' might be represented by a phase of 180 degrees (π radians), while a binary '1' is represented by a phase of 0 degrees. This direct phase manipulation makes BPSK one of the most power-efficient modulation schemes, offering excellent resilience against noise, especially in challenging low Signal-to-Noise Ratio (SNR) environments.

Components of the Error Probability Formula Explained

The formula fundamentally links the error probability to the Signal-to-Noise Ratio per bit ($E_b/N_0$), which is a critical performance indicator.

  • Signal Energy Per Bit ($E_b$): This quantity measures the total energy contained within a single transmitted bit. A higher $E_b$ signifies a stronger signal relative to noise, thereby increasing the likelihood of correct bit detection at the receiver. If the BPSK signal has an amplitude $A$ and a bit duration $T_B$, the energy per bit is calculated as $E_b = A^2 T_B / 2$.
  • Noise Power Spectral Density ($N_0$): This term quantifies the distribution of noise power across different frequencies. A smaller $N_0$ indicates less pervasive noise, leading to clearer signal reception and improved performance.
  • The Q-function: The argument of the Q-function, $\sqrt{\frac{2E_b}{N_0}}$, is a direct measure of the signal's strength relative to the noise's standard deviation at the point of decision in the receiver. A larger value for this argument results in a smaller output from the Q-function, signifying a lower (better) bit error probability. This relationship demonstrates that the error probability is directly determined by how discernible the signal is from the background noise.

Factors Influencing BPSK Error Probability

While the formula provides an ideal theoretical probability, several practical factors can influence the actual BER in a real-world BPSK system:

  • Signal-to-Noise Ratio (SNR): The most dominant factor. An increase in the $E_b/N_0$ ratio dramatically decreases the bit error rate.
  • Channel Impairments: The formula assumes an ideal AWGN channel. Real-world channels may introduce other issues like fading, interference, and non-linearities, which will degrade performance beyond this theoretical prediction.
  • Synchronization Errors: Inaccurate timing or carrier phase recovery at the receiver can lead to increased errors.
  • Receiver Imperfections: Non-ideal filters, amplifiers, and analog-to-digital converters in the receiver can introduce distortions, resulting in a higher BER than theoretically calculated.

Practical Considerations and Applications

Understanding BPSK's error probability is essential for the design and analysis of various digital communication systems:

  • Link Budget Design: Engineers use this formula to calculate the minimum transmit power required to achieve a target BER over a given channel.
  • System Performance Evaluation: It serves as a benchmark for comparing the efficiency of different modulation schemes and receiver architectures.
  • Optimizing System Parameters: The formula helps in making informed decisions about modulation type, coding schemes, and power allocation in applications such as satellite communications, deep-space probes, and wireless data networks.
Parameter Description Typical Unit
$P_b$ Bit Error Probability (Dimensionless)
$E_b$ Average energy per bit Joules (J)
$N_0$ One-sided noise power spectral density Watts/Hz or J
$Q(x)$ Gaussian Q-function (Dimensionless)

Conclusion

The error probability of BPSK, given by the Q-function of $\sqrt{2E_b/N_0}$, is a fundamental characteristic that highlights its performance in noisy communication environments. This precise mathematical relationship underscores the critical interplay between signal energy, noise power, and the reliability of digital data transmission in a vast array of communication technologies.