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.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))Understanding the Q-function:
The Q-function, Q(x), gives the probability that a standard normal (Gaussian) random variable exceeds x.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.
MATLAB code for theoretical BER vs SNR for BPSK
Output
Figure: Theoretical BER vs SNR for BPSK
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