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.

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