Theoretical Bit Error Rate (BER) vs Signal-to-Noise Ratio (SNR) for BPSK in AWGN Channel
Let’s simplify the explanation for the theoretical Bit Error Rate (BER) versus Signal-to-Noise Ratio (SNR) for Binary Phase Shift Keying (BPSK) in an Additive White Gaussian Noise (AWGN) channel.
Key Points
BPSK Modulation
Transmits one of two signals: +√Eb or −√Eb, where Eb is the energy per bit. These signals represent binary 0 and 1.
AWGN Channel
The channel adds Gaussian noise with zero mean and variance N₀/2 (where N₀ is the noise power spectral density).
Receiver Decision
The receiver decides if the received signal is closer to +√Eb (for bit 0) or −√Eb (for bit 1).
Bit Error Rate (BER)
The probability of error (BER) for BPSK is given by the Q-function, which measures the tail probability of the normal distribution — i.e., the probability that a Gaussian random variable exceeds a certain value.
Understanding the Q-function
The Q-function, Q(x), gives the probability that a standard normal (Gaussian) random variable exceeds x. In this context, it gives the probability that noise pushes the received signal across the wrong decision boundary, resulting in a bit error.
For BPSK, bits ‘0’ and ‘1’ map to +1 and −1, respectively. The probability of error is the probability that noise exceeds a threshold, depending on the signal’s distance from zero.
Calculate the Probability of Error using Q-function
For a Gaussian noise with mean = 0 and variance = N₀/2, the probability of error is:
Pb = Q(1/σ)
where σ = √(N₀/2)
So, Pb = Q(√(2/N₀))
Since SNR = Eb/N₀, we get:
Pb = Q(√(2 × SNR)) or equivalently Q(√(2Eb/N₀)).
Formula for BER
BER = Q(√(2Eb/N₀))
Here, Eb/N₀ is the energy per bit to noise power spectral density ratio, also known as the bit SNR.
Simplified Steps
- Calculate the SNR: γb = Eb/N₀
- Find the Q-function value: BER = Q(√(2γb))
Intuition
For High SNR (γb is large):
The argument of the Q-function √(2γb) becomes large, Q(x) is small ⇒ fewer errors. Result: BER is low.
For Low SNR (γb is small):
The argument of the Q-function √(2γb) is small, Q(x) is larger ⇒ more errors. Result: BER is higher.
Approximation for High SNR
For large SNR, the BER can be approximated using the complementary error function (erfc):
Q(x) ≈ ½ erfc(x/√2)
Thus, BER ≈ ½ erfc(√γb)
So, the final formula for BPSK in AWGN is:
BER = Q(√(2Eb/N₀))
Higher SNR ⇒ lower BER ⇒ better performance and fewer errors.
MATLAB Code: Theoretical BER vs SNR for BPSK
% The code is written by SalimWireless.Com
clc;
clear;
close all;
snrdb = 0:1:10;
snrlin = 10.^(snrdb./10);
tber = 0.5 .* erfc(sqrt(snrlin));
semilogy(snrdb, tber, '-bh')
grid on
title('BPSK with AWGN');
xlabel('Signal to noise ratio');
ylabel('Bit error rate');
Output
