Ferkans — Interactive Telecom Tutor

From AWGN to Fading: Why MGFs?

In AWGN the signal energy is deterministic and error probability is a clean function of $\text{SNR}$ . Wireless channels are fading — the received energy is itself a random variable $\gamma$ with some distribution $f_\gamma$ . The average error probability is therefore $\bar P_e = \mathbb{E}_\gamma[P_e(\gamma)]$ . Direct numerical integration is feasible but uninformative; the moment generating function (MGF) method converts these averages into one-dimensional integrals of the channel MGF — often producing closed forms and, more importantly, exposing the diversity order directly.

Definition:
Craig's Formula for the Q-function

Craig (1991) showed that the Gaussian tail can be written as a definite integral: $Q(x) \;=\; \frac{1}{\pi}\int_0^{\pi/2} \exp\!\left(-\frac{x^2}{2\sin^2\theta}\right)d\theta, \qquad x \geq 0.$ The generalization $Q^{2}(x) = \frac{1}{\pi}\int_0^{\pi/4} \exp(-x^2/(2\sin^2\theta))\,d\theta$ is also useful for error-floor analysis.

The power of Craig's formula is that its integrand is bounded and finite over the full integration range $[0,\pi/2]$ — unlike the usual $Q(x) = \int_x^\infty \frac{1}{\sqrt{2\pi}}e^{-t^2/2}dt$ , where the infinite upper limit complicates averaging over $x$ .

Theorem: Proof of Craig's Formula

For $x \geq 0$ , $Q(x) = \frac{1}{\pi}\int_0^{\pi/2} \exp(-x^2/(2\sin^2\theta))\,d\theta$ .

Interpret $Q(x)$ as the probability that a 2-D standard Gaussian vector lies outside a half-plane; convert to polar coordinates and the radial integral evaluates in closed form, leaving only an angular integral.

Proof

Set up the 2-D probability

Let $(X_1, X_2) \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ . Then $Q(x) = \Pr(X_1 > x)$ . Since $(X_1,X_2)$ is isotropic, we may equivalently compute $\Pr(X_1 \cos\alpha + X_2 \sin\alpha > x)$ for any fixed $\alpha$ , or indeed $Q(x) = \Pr\bigl((X_1,X_2) \in H_x\bigr)$ where $H_x = \{(u,v) : u > x\}$ .

Polar coordinates

Change to polar coordinates $(r,\theta)$ and note that the half-plane $H_x$ in polar form is $\{r\cos\theta > x\}$ , i.e. $r > x/\cos\theta$ with $\theta \in (-\pi/2, \pi/2)$ . Thus $Q(x) = \frac{1}{2\pi} \int_{-\pi/2}^{\pi/2} \int_{x/\cos\theta}^{\infty} e^{-r^2/2}\,r\,dr\,d\theta.$

Evaluate the radial integral

$\int_{x/\cos\theta}^\infty r e^{-r^2/2}\,dr = e^{-x^2/(2\cos^2\theta)}$ . Substituting, $Q(x) = \frac{1}{2\pi}\int_{-\pi/2}^{\pi/2} e^{-x^2/(2\cos^2\theta)}d\theta = \frac{1}{\pi}\int_0^{\pi/2} e^{-x^2/(2\cos^2\theta)}d\theta$ .

Substitute $\theta \to \pi/2 - \theta$

$\cos^2(\pi/2 - \theta) = \sin^2\theta$ , giving the stated form. $\blacksquare$

Theorem: MGF-Based Averaging of Q-function over Fading

Let the instantaneous symbol SNR be $\gamma$ with MGF $M_\gamma(s) = \mathbb{E}[e^{s\gamma}]$ . Then $\mathbb{E}_\gamma\bigl[Q(\sqrt{2a\gamma})\bigr] \;=\; \frac{1}{\pi}\int_0^{\pi/2} M_\gamma\!\left(-\frac{a}{\sin^2\theta}\right) d\theta.$ For $M$ -PSK with equal priors in fading, the exact average symbol error probability is $\bar P_e^{\text{MPSK}} \;=\; \frac{1}{\pi}\int_0^{(M-1)\pi/M} M_\gamma\!\left(-\frac{\sin^2(\pi/M)}{\sin^2\theta}\right) d\theta.$

Applying Craig's formula inside the expectation and swapping the order of integration turns a "fading-averaged Q-function" into "MGF evaluated at an angle-dependent argument, then integrated over a bounded angle range." Because the MGF of many common fading distributions (Rayleigh, Ricean, Nakagami-m) has a closed form, we get closed-form expressions for $\bar P_e$ .

Proof

Apply Craig's formula

$Q(\sqrt{2a\gamma}) = \frac{1}{\pi}\int_0^{\pi/2} \exp(-a\gamma/\sin^2\theta)\,d\theta$ . Take expectation over $\gamma$ : $\mathbb{E}\bigl[Q(\sqrt{2a\gamma})\bigr] = \frac{1}{\pi}\int_0^{\pi/2} \mathbb{E}\bigl[e^{-a\gamma/\sin^2\theta}\bigr] d\theta.$

Recognize the MGF

$\mathbb{E}[e^{-a\gamma/\sin^2\theta}] = M_\gamma(-a/\sin^2\theta)$ by the definition of the MGF. Swapping expectation and integration is justified by Fubini (non-negative integrand).

M-PSK case

For M-PSK the exact conditional SER is $P_e(\gamma) = \frac{1}{\pi} \int_0^{(M-1)\pi/M} e^{-\gamma\sin^2(\pi/M)/\sin^2\theta}\,d\theta$ . Averaging over $\gamma$ gives the stated formula. $\blacksquare$

Example: BPSK in Rayleigh Fading

In Rayleigh fading, $\gamma \sim \text{Exp}(\bar\gamma)$ with mean $\bar\gamma$ . Compute the average BPSK error probability $\bar P_e = \mathbb{E}_\gamma[Q(\sqrt{2\gamma})]$ , and find its high-SNR slope.

Solution

Rayleigh SNR MGF

For $\gamma \sim \text{Exp}(\bar\gamma)$ , $M_\gamma(s) = 1/(1 - s\bar\gamma)$ for $s < 1/\bar\gamma$ .

Plug into MGF formula

$\bar P_e = \frac{1}{\pi}\int_0^{\pi/2} \frac{1}{1 + \bar\gamma/\sin^2\theta}\,d\theta = \frac{1}{\pi}\int_0^{\pi/2} \frac{\sin^2\theta}{\sin^2\theta + \bar\gamma}d\theta$ .

Evaluate in closed form

Using the standard integral $\int_0^{\pi/2}\frac{\sin^2\theta\,d\theta} {\sin^2\theta + b} = \frac{\pi}{2}\bigl(1 - \sqrt{b/(1+b)}\bigr)$ , we obtain $\bar P_e = \tfrac{1}{2}\Bigl(1 - \sqrt{\bar\gamma/(1+\bar\gamma)}\Bigr).$

High-SNR slope

Taylor expand around $\bar\gamma \to \infty$ : $\sqrt{\bar\gamma/(1+\bar\gamma)} = 1 - \tfrac{1}{2\bar\gamma} + O(\bar\gamma^{-2})$ , hence $\bar P_e \approx 1/(4\bar\gamma)$ . The error decays as $\bar\gamma^{-1}$ — diversity order 1 — in stark contrast to the exponential AWGN decay $Q(\sqrt{2\bar\gamma}) \sim e^{-\bar\gamma}$ .

Theorem: Diversity Order from the MGF

Suppose $M_\gamma(s) \sim c|s|^{-L}$ as $|s|\to\infty$ for some constants $c, L > 0$ . Then for any constellation at high SNR, $\bar P_e(\bar\gamma) \;=\; O(\bar\gamma^{-L}), \qquad \bar\gamma \to \infty.$ The exponent $L$ is called the diversity order of the channel. Rayleigh fading has $L=1$ ; $L$ -branch MRC or a Nakagami- $m$ channel with parameter $m=L$ has diversity order $L$ .

The fading-averaged BER at high SNR is governed by the behaviour of the MGF near $-\infty$ ; this behaviour is controlled by how quickly the fading distribution puts mass near $\gamma = 0$ (deep-fade probability). Diversity techniques reduce the mass near zero, raising $L$ .

Proof

Leading-order of the integrand

At high $\bar\gamma$ , the argument $-a/\sin^2\theta$ of the MGF diverges (after rescaling), so $M_\gamma(-a\bar\gamma/\sin^2\theta) \sim c(a\bar\gamma/\sin^2\theta)^{-L}$ .

Integrate

$\bar P_e \sim \frac{c}{\pi a^L}\bar\gamma^{-L} \int_0^{\pi/2}\sin^{2L}\theta\,d\theta$ . The angular integral is a finite constant (expressible via beta functions), so $\bar P_e = \Theta(\bar\gamma^{-L})$ . $\blacksquare$

MGF-Based SER in Rayleigh and Nakagami Fading

Compute the exact fading-averaged SER via Craig's integral for M-PSK under Rayleigh and Nakagami- $m$ fading and compare with the AWGN baseline. The diversity-order slope $-L$ (dB/decade) emerges on the log-log plot.

Parameters

M-PSK Constellation

Channel model

Max SNR (dB)40

Why This Matters: MGF Method in Research

The MGF method is the workhorse of performance analysis in wireless communications research. Every major textbook on fading channels (Simon and Alouini's Digital Communication over Fading Channels, 2005) tabulates MGFs for dozens of fading distributions. The method extends to diversity combining (MRC, EGC, SC), cooperative relaying, RIS-assisted channels, and cell-free massive MIMO — a single formula covers all of them once the SNR MGF is derived.

See full treatment in Chapter 4

Common Mistake: Sign Conventions for the MGF

Mistake:

Some books define $M_X(s) = \mathbb{E}[e^{sX}]$ while others use $M_X(s) = \mathbb{E}[e^{-sX}]$ (the Laplace transform of the pdf), flipping the sign of $s$ . Applying a formula from a mismatched source silently gives a wrong answer.

Correction:

We adopt $M_\gamma(s) = \mathbb{E}[e^{s\gamma}]$ throughout, which is the statistics-textbook convention. Under this convention, the MGF formula for fading-averaged $Q$ evaluates $M_\gamma$ at negative arguments. If you import a formula, verify the sign convention by checking the Rayleigh case against the closed-form result $\bar P_e^{\text{BPSK},\text{Rayleigh}} = \tfrac{1}{2}(1 - \sqrt{\bar\gamma/(1+\bar\gamma)})$ .

Historical Note: Craig's 1991 Insight

1991-1998

James Craig introduced his alternative Q-function representation in a 1991 MILCOM paper titled "A New, Simple and Exact Result for Calculating the Probability of Error for Two-Dimensional Signal Constellations." The formula had been hiding in plain sight for decades but no one had recognized its value for fading analysis. Simon and Alouini popularized the MGF method built around Craig's formula in a series of papers through the mid-1990s, culminating in their 2005 textbook which tabulates error formulas for essentially every fading distribution and modulation of practical interest.

🎓CommIT Contribution(2023)

MGF Tools for ISAC Performance Analysis

G. Caire, F. Liu — IEEE Transactions on Wireless Communications

In recent work, Caire and collaborators at TU Berlin extended the MGF-based error-analysis framework from pure communications to integrated sensing and communication (ISAC) scenarios. The key observation: both the detection probability (sensing) and the symbol-error probability (communication) can be expressed via MGFs of the same effective channel gain, enabling a unified Pareto-front analysis of the sensing-communication tradeoff in generalized fading environments.

mgfisacfadingtradeoffView Paper →

Quick Check

A receiver uses 3-branch MRC with i.i.d. Rayleigh fading per branch. What is the diversity order $L$ of the combined SNR?

1

2

3

6

Correction:

3

The combined SNR is gamma-distributed with shape 3, so MGF decays as $|s|^{-3}$ : diversity order 3.

MGF-Based Error Analysis and Craig's Formula

From AWGN to Fading: Why MGFs?

Definition: Craig's Formula for the Q-function

Theorem: Proof of Craig's Formula

Set up the 2-D probability

Polar coordinates

Evaluate the radial integral

Substitute $\theta \to \pi/2 - \theta$

Theorem: MGF-Based Averaging of Q-function over Fading

Apply Craig's formula

Recognize the MGF

M-PSK case

Example: BPSK in Rayleigh Fading

Rayleigh SNR MGF

Plug into MGF formula

Evaluate in closed form

High-SNR slope

Theorem: Diversity Order from the MGF

Leading-order of the integrand

Integrate

MGF-Based SER in Rayleigh and Nakagami Fading

Parameters

Why This Matters: MGF Method in Research

Common Mistake: Sign Conventions for the MGF

Historical Note: Craig's 1991 Insight

MGF Tools for ISAC Performance Analysis

Quick Check

Definition:
Craig's Formula for the Q-function