Interactive BER vs SNR Simulator for m-ary PSK & QAM

BER vs Eb/N0 Simulation

Modulation: M-ary PSK M-ary QAM (Square) ASK (Binary/OOK) FSK (Binary Coherent) M (Order): 2 (BPSK) 4 (QPSK) 8-PSK/QAM 16-PSK/QAM 64-PSK/QAM

Eb/N0 Range (dB) [start:step:end]:

*Changes update the plot in real-time.

Mathematical Background

The Bit Error Rate (BER) is the probability that a bit is misidentified due to noise. We plot this against $E_b/N_0$ (Energy per bit to noise power spectral density ratio).

1. Energy Conversion

For M-ary modulations, each symbol carries $k = \log_2(M)$ bits. The Symbol Energy ($E_s$) relates to Bit Energy ($E_b$) as:

Es/N0 = (Eb/N0) × log2(M)

2. The Complementary Error Function

Errors in Gaussian noise (AWGN) are calculated using the erfc(x) function. It is related to the Q-function by: Q(x) = ½ erfc(x/√2).

3. Modulation Formulas

M-ary PSK:
BER ≈ (1 / log₂M) × erfc( √(E_s/N₀) × sin(π/M) )

M-ary QAM (Square):
BER ≈ [ 2(1 - 1/√M) / log₂M ] × erfc( √( 1.5 × E_s/N₀ / (M - 1) ) )

Binary ASK (On-Off Keying):
BER = ½ erfc( √(E_b/4N₀) )

Binary FSK (Coherent):
BER = ½ erfc( √(E_b/2N₀) )