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. Theoretical BER vs SNR for m-ary PSK and QAM
   
3. Comparison of theoretical and simulated BER vs SNR for ASK, FSK, and PSK