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. BER vs SNR for ASK, FSK, and PSK Online Simulator 
3. Understanding the Q-function in BASK, BFSK, and BPSK
4. Theoretical BER vs SNR for M-ary PSK and QAM
5. Comparison of Theoretical and Simulated BER vs SNR for ASK, FSK, and PSK
6. MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ...