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

1. Definition of the Q-function

The Q-function is the tail probability of the standard normal distribution:

Q(x) = (1 / √(2π)) ∫_x^∞ e^-t²/2 dt

What is Q(1)?

Q(1) ≈ 0.1587

This means there is about a 15.87% chance that a Gaussian random variable exceeds 1 standard deviation above the mean.

What is Q(2)?

Q(2) ≈ 0.0228

This means there is only a 2.28% chance that a Gaussian value exceeds 2 standard deviations above the mean.

Difference Between Q(1) and Q(2)

Even though the argument changes from 1 to 2 (a small increase), the probability drops drastically:

• Q(1) = 0.1587 → errors fairly likely
• Q(2) = 0.0228 → errors much rarer

This shows how fast the tail of the Gaussian distribution decays. It’s also why BER drops dramatically as SNR increases.

BPSK uses symbols +1 and –1. The confusion comes from thinking that the Q-function uses those symbol values directly. It does not. Below is the simple explanation.

1.1. BPSK Symbols

In normalized BPSK:

• Bit 1 → +1
• Bit 0 → –1

1.2. Why the Q-function does not use +1 or –1 directly

The Q-function input x is not the transmitted bit value. Instead, it represents how far the signal is from the decision boundary in units of noise.

Error occurs only if noise pushes the received sample across 0. So BER means:

BER = probability(noise is large enough to cross the threshold)

1.3. Noise is Gaussian → Q-function applies

The noise has variance N₀/2, so its standard deviation is:

σ = √(N₀ / 2)

1.4. The value of x in Q(x) in BPSK

The Q-function needs a standard normal variable. This requires dividing the signal distance by the noise standard deviation.

Distance from +1 to decision threshold (0):

Distance = 1

Noise std dev:

σ = √(N₀ / 2)

So the Q-function argument is:

x = 1 / √(N₀ / 2) = √(2 / N₀)

In general, using bit energy Eb:

x = √(2E_b / N₀)

1.5. Final Answer

The value of x in the Q-function for BPSK is:

x = √(2E_b / N₀)

This is not 0, 1, +1, or –1. It is the distance between the signal and the decision threshold, measured in noise units.

2. Q-function for Other Modulation Schemes

The Q-function is widely used to express BER in common digital modulation schemes.

2.1 BPSK BER

BER = Q( √(2E_b/N₀) )

2.2 QPSK BER

BER_QPSK = Q( √(2E_b/N₀) )

2.3 BASK / OOK (On-Off Keying)

For coherent detection of BASK (OOK):

BER_BASK = Q( √(E_b/N₀) )

2.4 BFSK (Binary Frequency Shift Keying)

Coherent BFSK:

BER_{coherent BFSK} = Q( √(2E_b/N₀) )

Non-coherent BFSK:

BER_{non-coherent BFSK} = 0.5 · exp( −E_b / (2N₀) )

2.5 Square M-QAM BER

BER ≈ (4/log₂M)(1 − 1/√M) Q( √[ 3 log₂M / (M−1) · (E_b/N₀) ] )

2.6 M-PSK SER

SER ≈ 2 Q( √(2E_b/N₀) · sin(π/M) )

Further Reading

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