Theoretical BER vs SNR for BPSK

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

Fig. 1: Constellation Diagrams of BASK, BFSK, and BPSK [↗]

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:

P_b = Q(1/σ)

where σ = √(N₀/2)

So, P_b = Q(√(2/N₀))

Since SNR = Eb/N₀, we get:

P_b = 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

1. Calculate the SNR: γb = Eb/N₀
2. 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

Figure: Theoretical BER vs SNR for BPSK

Also Read

1. Theoretical BER vs SNR for Binary ASK and FSK
2. Theoretical BER vs SNR for M-ary PSK and QAM
3. Comparison of Theoretical and Simulated BER vs SNR for ASK, FSK, and PSK
4. MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ...