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Understanding the Q-function in BASK, BFSK, and BPSK


Q-function in BASK, BFSK, and BPSK: Theory & Interaction

Understanding the Q-function in BASK, BFSK, and BPSK

1. Definition of the Q-function

The Q-function represents the tail probability of the standard normal (Gaussian) distribution. It calculates the probability that a random noise sample will exceed a specific value.

Q(x) = (1 / √(2Ï€)) ∫x∞ e-t²/2 dt

Interactive Q-Function Visualizer

x = 1.0
Q(x) ≈ 0.1587
Move the slider to see how the probability (red area) drops.

Practical Examples of Q(x)

Q(1) ≈ 0.1587: There is a 15.87% chance that Gaussian noise exceeds 1 standard deviation (1σ).

Q(2) ≈ 0.0228: There is only a 2.28% chance that noise exceeds 2 standard deviations (2σ).

The "Waterfall" Effect: Note how the probability drops drastically from 15% to 2% just by doubling the argument. This explains why the Bit Error Rate (BER) drops exponentially as the Signal-to-Noise Ratio (SNR) increases.

2. Why BPSK uses the Q-function

In BPSK, we use two symbols: +√Eb (Bit 1) and -√Eb (Bit 0). The receiver threshold is 0.

Error Logic: An error occurs if noise pushes the signal across the threshold.
Distance = √Eb | Noise Standard Deviation (σ) = √(N₀/2).
Argument x = Distance / σ = √(2Eb/N₀).

3. Q-function Formulas for Modulation Schemes

BPSK / QPSK: BER = Q( √(2Eb/N₀) )

BASK / OOK: BER = Q( √(Eb/N₀) )

Coherent BFSK: BER = Q( √(Eb/N₀) )

Note: BFSK and BASK are 3dB worse than BPSK because they require twice the power to achieve the same distance-to-noise ratio.

Numerical Walkthrough: The 0 dB Case

Why is BPSK BER 0.078 at 0 dB? If SNR is 0 dB, the ratio is 1.0. Let Eb = 1 and N₀ = 1.

  • Threshold Distance: In BPSK, distance = √Eb = 1.
  • Noise Magnitude (σ): σ = √(N₀ / 2) = √(1 / 2) ≈ 0.707.
  • The Argument (x): x = 1 / 0.707 = 1.414 (√2).
  • The Result: BER = Q(1.414) ≈ 0.0786

The BASK "0 and 1" Mapping Case

If you use amplitudes 0 and 1, your average energy Eb is 0.5. At 0 dB SNR (N₀=1):

Distance to threshold (0.5) is 0.5. Noise σ is 0.707.

x = 0.5 / 0.707 = 0.707. BER = Q(0.707) ≈ 23.9%. Wasted potential!

Further Reading

  1. Detailed BER vs SNR Analysis
  2. Introduction to Digital Communication
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