How do we calculate BER vs. SNR for real systems?

 

How do we calculate BER vs. SNR for real systems?

In real systems such as BPSK over an AWGN channel, bits are transmitted as signals (e.g., +√P or −√P). Due to noise, the received signal becomes:

y = √P · x + n

where n ∼ CN(0, σ²) is complex Gaussian noise, and x ∈ {−1, +1}.

At the receiver, we perform detection (e.g., using sign detection). The probability of a bit error is given by the Gaussian Q-function:

BER = Q(√(2P / σ²)) = Q(√(2 · SNR))

where SNR = P / σ². This gives the theoretical BER vs. SNR curve.

BER vs. SNR from Gaussian Q factor

Linear Equalization (e.g., Zero Forcing)

When you're using an equalizer like Zero Forcing (ZF), you get:

• Effective noise enhancement depending on the channel matrix H.
• The diagonal elements of W = (H^HH)⁻¹ (in ZF) tell you how the noise variance is scaled after equalization.

So the BER for the i-th stream becomes:

BER_i = Q( √(2 · Ω_s / (σ² · W_ii)) )

Where:

• Ω_s = symbol power
• W_ii = noise amplification factor from the equalizer

 

The probability of error can be calculated as  

The above example is for a block of size k after zero-forcing equalization. Ωs is the average power, and Wii is the i-th diagonal element of the noise enhancement matrix W. W denotes the inverse Gram matrix used to estimate the symbol error covariance.

 

Channel Knowledge in Zero-Forcing (ZF) Equalization

In Zero-Forcing (ZF) equalization, the receiver must know the channel matrix H in order to compute (H^HH)⁻¹H^H, which is the ZF equalizer matrix.

Channel Estimation is typically performed at the receiver using:

• Pilot symbols
• Training sequences

Once the receiver estimates H, it can compute the ZF equalizer.

 

Further Reading

1. Theoretical and simulated BER vs. SNR for ASK, FSK, and PSK
2. BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ..