Theoretical BER vs SNR for BPSK

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 a function called the Q-function. The Q-function Q(x) measures the tail probability of the normal distribution, i.e., the probability that a Gaussian random variable exceeds a certain value x. 

Understanding the Q-function:

The Q-function, Q(x), gives the probability that a standard normal (Gaussian) random variable exceeds x.

In the above context, he Q-function gives the probability that noise pushes the received signal across the wrong decision boundary, resulting in a bit error.

For the BPSK case, suppose we map the binary bits '0' and '1' to +1 and -1, respectively. If we transmit binary bit '0' (mapped to +1), but additive AWGN noise causes the received signal to fall below 0 (i.e., $1 + \text{noise} < 0$, where the threshold is 0), the receiver wrongly detects it as bit '1'. Similarly, if we transmit bit '1' (mapped to -1), but noise makes the received signal exceed 0 (i.e., $-1 + \text{noise} > 0$), the receiver incorrectly detects it as bit '0'. Therefore, we need to find the probability of error, which corresponds to the probability that noise exceeds a certain value. In this case, the noise standard deviation is given by $\sigma = \sqrt{\frac{N_0}{2}}$assuming the signal power is 1, the noise power is $\frac{N_0}{2}$, and the SNR is $\frac{1}{\sigma^2}$.​

Calculate the Probability of Error using Q-function

In either case, the noise is Gaussian with mean = 0 and variance = N0/2.
The probability of noise exceeding ±1 can be calculated with the Q-function:

P_b = Q(1/σ)

Where:

σ = √(N₀/2)

So:

P_b = Q(1/√(N₀/2)) = Q(√(2/N₀))

Since:

SNR = E_b/N₀

We get:

P_b = Q(√(2 × SNR)) 

or,  P_b = 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) for large x is small, meaning fewer errors.

Result: BER is low.

For Low SNR (γb​ is small):

The argument of the Q-function √(2γb) is small.

Q(x) for small x is larger, meaning more errors.

Result: BER is higher.

Approximation for High SNR

For large SNR values, the BER can be approximated using the complementary error function (erfc):

Q(x)≈1/2erfc(x/√(2))

Thus,

BER≈1/2erfc(√(γb))


So, BER Formula for BPSK in AWGN is:

BER=Q(√2Eb/N0) 

Higher SNR leads to lower BER, meaning better performance and fewer errors.

 

Copy the MATLAB code for 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 about

1. Theoretical BER vs SNR for binary ASK and FSK  
2. Comparison of theoretical and simulated BER vs SNR for ASK, FSK, and PSK