Advanced M-ary Modulation Simulator: Constellation, min dist, Efficiency, SER, EVM (RMS)

Advanced M-ary Communication Lab

Analytical & Statistical Performance of Digital Modulation

Theoretical Probability of Error (\(P_s\))

\[ P_s = Q\left(\sqrt{\frac{2 E_b}{N_0}}\right) \]

Modulation (M-ary) BPSK (M=2) QPSK (M=4) 8-PSK (M=8) 16-QAM (M=16) 64-QAM (M=64) 256-QAM (M=256)

SNR (\(E_b/N_0\)): 12 dB

Efficiency2 bps/Hz

Min Dist (\(d_{min}\))1.41

Symbol Error1.2e-5

EVM (RMS)0.0%

Constellation Diagram

Noise PDF & Decision Tail

1. Geometric Mapping

Each point on the grid represents a unique bit sequence. As M increases, points are packed tighter.

2. Noise Contribution

The cloud around each point is AWGN. If the cloud overlaps a decision boundary, a bit flip occurs.

3. Error Probability

The Q-function measures the area of the tail that crosses into the wrong decision territory.

Minimum Euclidean Distance (d_min) Calculation

In the simulator logic, the variable d_{min_norm} represents the normalized minimum distance. It defines the gap between a signal point and the nearest decision boundary as a function of SNR (γ_b = E_b/N₀).

BPSK / QPSK

√(2 γ_b)

8-PSK

√(6 γ_b) sin(π/8)

16-QAM

√(0.8 γ_b)

64-QAM

√((4/21) γ_b)

256-QAM

√((4/85) γ_b)

Normalized SNR (γ_b)

The code converts the dB slider value to a linear ratio: 10^{(SNR_dB / 10)}. This represents the energy per bit (E_b) over noise spectral density (N₀).

The Decision Threshold

The d_{min_norm} value determines how much noise a symbol can tolerate before an error occurs. In the Noise PDF chart, the red error area begins exactly at this distance from the center.

Relation to Probability of Error

The Symbol Error Rate (SER) is calculated by passing this distance into the Gaussian Q-function: SER ≈ Coefficient × Q(d_{min_norm})

Note: For M-QAM, the distance is derived from the constellation's average energy. For example, 16-QAM uses √(0.8 γ_b) because d = √(4/5 E_b/N₀) for that geometry.