Ferkans — Interactive Telecom Tutor

The Q-function as Universal Currency

Section 9.1 established the optimal detector structure. Now we compute the exact error probabilities for the constellations introduced in Chapter 8. Every BER/SER expression in AWGN ultimately reduces to the Gaussian Q-function or its close relative, the complementary error function. We also introduce Craig's formula — an alternative Q-function representation that is indispensable for averaging BER over fading distributions in Section 9.4.

Definition:
The Gaussian Q-Function

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

$Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^{\infty} e^{-t^2/2}\, dt = \frac{1}{2}\, \operatorname{erfc}\!\left(\frac{x}{\sqrt{2}}\right)$

Key properties:

$Q(0) = 1/2$ , $Q(\infty) = 0$ , $Q(-\infty) = 1$
Symmetry: $Q(-x) = 1 - Q(x)$
Monotonically decreasing for $x > 0$
Relationship to erfc: $Q(x) = \frac{1}{2}\operatorname{erfc}(x/\sqrt{2})$

Chernoff bound: for $x > 0$ ,

$Q(x) \leq \frac{1}{2} e^{-x^2/2}$

Tighter bounds: for $x > 0$ ,

$\frac{1}{\sqrt{2\pi}}\, \frac{x}{x^2 + 1}\, e^{-x^2/2} \leq Q(x) \leq \frac{1}{\sqrt{2\pi}}\, \frac{1}{x}\, e^{-x^2/2}$

The Q-function is implemented as a built-in function in MATLAB (qfunc), Python/SciPy (from scipy.stats import norm; norm.sf), and most numerical computing environments.

Theorem: BER for BPSK and QPSK in AWGN

The bit error rate for BPSK and QPSK with Gray mapping in an AWGN channel is

$P_b = Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right)$

For QPSK, this result holds because the I and Q channels are independent BPSK detectors, each with energy $E_b$ per bit.

Equivalently, in terms of symbol energy $E_s$ :

$P_b^{\text{BPSK}} = Q\!\left(\sqrt{\frac{2E_s}{N_0}}\right), \qquad P_b^{\text{QPSK}} = Q\!\left(\sqrt{\frac{E_s}{N_0}}\right)$

since $E_s = E_b$ for BPSK and $E_s = 2E_b$ for QPSK.

The argument of the Q-function is $\sqrt{2E_b/N_0} = d_{\min}/\sqrt{2N_0}$ , which is the ratio of half the minimum distance to the noise standard deviation. Larger $d_{\min}$ or smaller $N_0$ gives a larger argument and hence lower $P_b$ .

Proof

BPSK derivation

Signal points: $s_0 = -\sqrt{E_b}$ , $s_1 = +\sqrt{E_b}$ . The ML detector decides $\hat{s} = s_1$ if $r > 0$ .

$P_e = P(r < 0 \mid s_1 \text{ sent}) = P(\sqrt{E_b} + w < 0) = Q\!\left(\frac{\sqrt{E_b}}{\sqrt{N_0/2}}\right) = Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right)$

By symmetry, $P(e \mid s_0) = P(e \mid s_1)$ , so $P_b = Q(\sqrt{2E_b/N_0})$ .

QPSK decomposition

QPSK decomposes into two independent BPSK channels (I and Q), each with bit energy $E_b = E_s/2$ . Each channel has $P_b = Q(\sqrt{2E_b/N_0})$ . Since the bit errors on I and Q are independent, the overall BER is the same:

$P_b = Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right)$

QPSK achieves the same BER as BPSK at twice the spectral efficiency. $\blacksquare$

Theorem: Approximate BER for M-QAM in AWGN

For square $M$ -QAM ( $M = 4, 16, 64, 256, \ldots$ ) with Gray mapping in AWGN, the bit error rate is approximately

$P_b \approx \frac{4}{\log_2 M} \left(1 - \frac{1}{\sqrt{M}}\right) Q\!\left(\sqrt{\frac{3 \log_2 M}{M - 1} \cdot \frac{2E_b}{N_0}}\right)$

This approximation becomes exact as SNR $\to \infty$ (where only nearest-neighbour errors contribute).

In terms of symbol energy:

$P_s \approx 4\left(1 - \frac{1}{\sqrt{M}}\right) Q\!\left(\sqrt{\frac{3E_s}{(M-1)N_0}}\right)$

The factor $(1 - 1/\sqrt{M})$ accounts for the fraction of nearest neighbours: corner points have 2 neighbours, edge points have 3, and interior points have 4. The average number of neighbours weighted by position gives $4(1 - 1/\sqrt{M})$ per symbol.

Proof

SER decomposition

Square $M$ -QAM $= \sqrt{M}$ -PAM $\times$ $\sqrt{M}$ -PAM. The SER for $\sqrt{M}$ -PAM is

$P_{\sqrt{M}} = \frac{2(\sqrt{M}-1)}{\sqrt{M}} Q\!\left(\sqrt{\frac{3E_s}{(M-1)N_0}}\right)$

The symbol is correct iff both I and Q are correct: $P_s = 1 - (1 - P_{\sqrt{M}})^2 \approx 2P_{\sqrt{M}}$ at high SNR.

SER to BER with Gray mapping

With Gray mapping, BER $\approx$ SER $/\log_2 M$ :

$P_b \approx \frac{P_s}{\log_2 M} = \frac{4(\sqrt{M}-1)}{\sqrt{M}\log_2 M} Q\!\left(\sqrt{\frac{3\log_2 M}{M-1} \cdot \frac{2E_b}{N_0}}\right)$

using $E_s = E_b \log_2 M$ . $\blacksquare$

Definition:
Craig's Formula (Alternative Q-Function Representation)

Craig's formula expresses the Q-function as a single integral with finite limits:

$Q(x) = \frac{1}{\pi} \int_0^{\pi/2} \exp\!\left(-\frac{x^2}{2\sin^2\phi}\right) d\phi$

More generally, the square of the Q-function is

$Q^2(x) = \frac{1}{\pi} \int_0^{\pi/4} \exp\!\left(-\frac{x^2}{2\sin^2\phi}\right) d\phi$

Craig's formula is essential for computing average BER over fading channels because:

The SNR $\gamma$ appears only in the exponent
The expectation $E_\gamma[\cdot]$ can be taken inside the integral, yielding the MGF of $\gamma$
The resulting integral over $\phi$ has finite limits and can often be evaluated in closed form

Substituting $\gamma = x^2/2$ :

$Q(\sqrt{2\gamma}) = \frac{1}{\pi} \int_0^{\pi/2} \exp\!\left(-\frac{\gamma}{\sin^2\phi}\right) d\phi$

Craig's formula was published in 1991 and immediately became the standard tool for performance analysis of digital communications over fading channels. Before this result, averaging BER over fading required numerical integration of nested integrals.

Example: Computing BER at Specific $E_b/N_0$

Compute the BER for BPSK, QPSK, 16-QAM, and 64-QAM at $E_b/N_0 = 10$ dB in AWGN.

Solution

Convert to linear

$E_b/N_0 = 10$ dB $= 10$ (linear).

BPSK and QPSK

$P_b = Q(\sqrt{2 \times 10}) = Q(\sqrt{20}) = Q(4.47) \approx 3.87 \times 10^{-6}$

16-QAM

$P_b \approx \frac{4}{4}\left(1 - \frac{1}{4}\right) Q\!\left(\sqrt{\frac{3 \times 4}{15} \times 20}\right) = \frac{3}{4} Q(\sqrt{16}) = 0.75\, Q(4) \approx 0.75 \times 3.17 \times 10^{-5}$

$P_b \approx 2.38 \times 10^{-5}$

64-QAM

$P_b \approx \frac{4}{6}\left(1 - \frac{1}{8}\right) Q\!\left(\sqrt{\frac{3 \times 6}{63} \times 20}\right) = \frac{7}{12} Q(\sqrt{5.71}) = 0.583\, Q(2.39)$

$P_b \approx 0.583 \times 8.42 \times 10^{-3} \approx 4.91 \times 10^{-3}$

At 10 dB, 64-QAM has BER $\sim 5 \times 10^{-3}$ , far too high for uncoded transmission but correctable with FEC. $\blacksquare$

Example: Comparing Modulation Schemes

Find the $E_b/N_0$ required for BER $= 10^{-6}$ for BPSK, QPSK, 16-QAM, and 64-QAM. What is the power penalty (in dB) of each scheme relative to BPSK?

Solution

BPSK and QPSK

$10^{-6} = Q(\sqrt{2E_b/N_0})$

$Q^{-1}(10^{-6}) = 4.753$

$\sqrt{2E_b/N_0} = 4.753 \implies E_b/N_0 = 11.3 = 10.53$ dB

BPSK and QPSK require the same $E_b/N_0 = 10.53$ dB.

16-QAM

Solving $10^{-6} = 0.75\, Q(\sqrt{3 \times 4 \times 2(E_b/N_0)/15})$ numerically:

$E_b/N_0 \approx 14.5$ dB.

Power penalty over BPSK: $14.5 - 10.5 = 4.0$ dB.

64-QAM

Solving for 64-QAM numerically:

$E_b/N_0 \approx 18.8$ dB.

Power penalty over BPSK: $18.8 - 10.5 = 8.3$ dB.

Each doubling of spectral efficiency costs roughly 4 dB in $E_b/N_0$ . $\blacksquare$

BER Curves in AWGN

Compare the bit error rate of standard modulation schemes as a function of $E_b/N_0$ . Toggle modulation formats on/off and optionally show the Chernoff and high-SNR approximations. Note the 3 dB gap between each doubling of $M$ for QAM.

Parameters

Show BPSK

Show QPSK

Show 16-QAM

Show 64-QAM

Show approximations

Q-Function and Its Approximations

Compare the exact Q-function with the Chernoff bound $Q(x) \leq \frac{1}{2}e^{-x^2/2}$ and Craig's integral form. The plot shows $Q(x)$ on a log scale. Notice that the Chernoff bound is within 3 dB for $x > 3$ , while Craig's formula is exact (the curves coincide).

Parameters

Exact

Q(x)

Chernoff bound

Craig formula

Max

x

6

Quick Check

Rank the following modulation schemes from lowest to highest BER at $E_b/N_0 = 12$ dB in AWGN: (A) BPSK, (B) 16-QAM, (C) 64-QAM, (D) QPSK.

A = D < B < C

A < D < B < C

C < B < D < A

A < D < C < B

Correction:

A = D < B < C

BPSK and QPSK have identical BER: $P_b = Q(\sqrt{2E_b/N_0})$ . 16-QAM requires more $E_b/N_0$ than BPSK/QPSK for the same BER, and 64-QAM requires even more. Therefore, at fixed $E_b/N_0$ , BER increases from BPSK/QPSK to 16-QAM to 64-QAM.

Common Mistake: Confusing $E_b/N_0$ with SNR

Mistake:

Using $E_b/N_0$ and SNR interchangeably in BER calculations.

Correction:

$E_b/N_0$ and SNR are related but not the same:

$\text{SNR} = \frac{E_s}{N_0 W} = \frac{E_b \log_2 M}{N_0 W} = \frac{E_b}{N_0} \cdot \frac{\log_2 M}{W T_s}$

For Nyquist signaling ( $W = 1/T_s$ ):

$\text{SNR} = \frac{E_b}{N_0} \cdot \log_2 M = \frac{E_b}{N_0} \cdot \eta$

At the same $E_b/N_0$ , higher-order modulations have higher SNR (more energy per symbol) but also more constellation points to distinguish. The BER depends on the ratio $d_{\min}^2/N_0 \propto E_s/(M-1)N_0$ , which decreases with $M$ .

Common Mistake: Using BER Formulas Outside Their Validity Range

Mistake:

Applying the approximate BER formula $P_b \approx N_{\min}/\log_2 M \cdot Q(d_{\min}/\sqrt{2N_0})$ at low SNR where multiple error events contribute significantly.

Correction:

The nearest-neighbour approximation is accurate only at high SNR (typically $E_b/N_0 > 5$ dB for QAM). At low SNR:

Second and third nearest neighbours contribute non-negligibly
The union bound overestimates $P_e$ significantly
The Gray mapping approximation BER $\approx$ SER $/\log_2 M$ breaks down because errors to distant symbols become probable

For low-SNR analysis, use the exact SER expression (summing over all decision region boundaries) or Monte Carlo simulation.

⚠️Engineering Note

Numerical Computation of the Q-Function

Computing $Q(x)$ for large $x$ (high SNR) requires care. The naive formula $Q(x) = 0.5 \cdot \operatorname{erfc}(x/\sqrt{2})$ suffers from catastrophic cancellation in floating-point arithmetic when $Q(x) < 10^{-15}$ (roughly $x > 8$ ).

Recommended implementations:

Python/SciPy: Use scipy.special.erfc or scipy.stats.norm.sf, which use extended-precision internal routines. Alternatively, scipy.special.log_ndtr computes $\ln Q(x)$ directly, avoiding underflow for $x$ up to 37.
MATLAB: qfunc(x) handles large arguments correctly.
C/C++: Use erfc() from <math.h> (IEEE 754 compliant), or the logarithmic form $\ln Q(x) \approx -x^2/2 - \ln(x\sqrt{2\pi})$ for asymptotic regimes.

For Monte Carlo BER simulations, avoid computing $Q(x)$ and instead count errors directly. To reliably estimate $P_e = 10^{-k}$ , collect at least $100 / P_e = 10^{k+2}$ transmitted symbols.

Practical Constraints

•
Double-precision floating point: accurate for Q(x) down to approximately 1e-308
•
Monte Carlo: 100/P_e symbols needed for 10% relative accuracy

Key Takeaway

The core message of this section in three points:

The Q-function is the universal building block for error probability in AWGN: every BER and SER expression reduces to $Q(\cdot)$ .
BER improves exponentially with SNR in AWGN: doubling $E_b/N_0$ (3 dB) dramatically reduces BER, and the improvement accelerates at higher SNR.
Craig's formula rewrites $Q(\sqrt{2\gamma})$ with $\gamma$ only in the exponent, enabling closed-form BER averaging over fading via the MGF — the key technique in Section 9.4.

Q-Function

The tail probability of the standard Gaussian distribution: $Q(x) = P(Z > x)$ for $Z \sim \mathcal{N}(0,1)$ . It is the universal function for expressing error probabilities in AWGN channels. Related to erfc by $Q(x) = \frac{1}{2}\operatorname{erfc}(x/\sqrt{2})$ .

Bit Error Rate (BER)

The probability that a transmitted bit is decoded incorrectly at the receiver. For uncoded systems in AWGN, BER depends on the modulation scheme and $E_b/N_0$ . With Gray mapping, BER $\approx$ SER $/\log_2 M$ at high SNR.

Error Probability in AWGN

The Q-function as Universal Currency

Definition: The Gaussian Q-Function

Theorem: BER for BPSK and QPSK in AWGN

BPSK derivation

QPSK decomposition

Theorem: Approximate BER for M-QAM in AWGN

SER decomposition

SER to BER with Gray mapping

Definition: Craig's Formula (Alternative Q-Function Representation)

Example: Computing BER at Specific Eb/N0E_b/N_0Eb​/N0​

Convert to linear

BPSK and QPSK

16-QAM

64-QAM

Example: Comparing Modulation Schemes

BPSK and QPSK

16-QAM

64-QAM

BER Curves in AWGN

Parameters

Q-Function and Its Approximations

Parameters

Quick Check

Common Mistake: Confusing Eb/N0E_b/N_0Eb​/N0​ with SNR

Common Mistake: Using BER Formulas Outside Their Validity Range

Numerical Computation of the Q-Function

Key Takeaway

Q-Function

Bit Error Rate (BER)

Definition:
The Gaussian Q-Function

Definition:
Craig's Formula (Alternative Q-Function Representation)

Example: Computing BER at Specific $E_b/N_0$

Common Mistake: Confusing $E_b/N_0$ with SNR