Ferkans — Interactive Telecom Tutor

Why Transformations of Random Variables Matter

In telecommunications we rarely observe a random variable "as is." If $X$ represents Gaussian noise at the receiver, the signal power is $|X|^2$ , the envelope is $|X|$ , and the log-amplitude is $\ln|X|$ . Each of these is a function of $X$ , and its distribution is generally different from that of $X$ itself.

Consider the complex baseband received signal in a narrowband channel with no line-of-sight component: $Z = X + jY,$ where $X, Y \sim \mathcal{N}(0, \sigma^2)$ are independent (by the central limit theorem applied to many scattered paths). The envelope $R = |Z| = \sqrt{X^2 + Y^2}$ turns out to follow a Rayleigh distribution; the instantaneous power $P = R^2$ is exponential; and if a deterministic line-of-sight component is present the envelope becomes Ricean. The sum of $k$ squared independent standard normals yields the chi-squared distribution $\chi^2_k$ , which appears in hypothesis testing and in the analysis of diversity combiners.

All of these distributions are derived from the Gaussian via elementary transformations. This section develops the general machinery for finding the distribution of $Y = g(X)$ given the distribution of $X$ , introduces moment-generating and characteristic functions as bookkeeping devices for moments, and catalogues the fading distributions that arise in wireless channels.

Theorem: CDF Method for Functions of a Random Variable

Let $X$ be a continuous random variable with known CDF $F_X$ and PDF $f_X$ , and let $Y = g(X)$ where $g : \mathbb{R} \to \mathbb{R}$ is a measurable function. Then the CDF of $Y$ is $F_Y(y) = P(Y \leq y) = P\!\bigl(g(X) \leq y\bigr),$ and, wherever $F_Y$ is differentiable, $f_Y(y) = \frac{d}{dy} F_Y(y).$

The idea is conceptually the simplest possible: translate the event $\{Y \leq y\}$ into an equivalent event involving $X$ , compute its probability using $F_X$ , and differentiate. The difficulty lies entirely in characterizing the set $\{x : g(x) \leq y\}$ , which depends on the shape of $g$ .

Show Hint

Write $F_Y(y) = P(g(X) \leq y)$ and identify the region in $x$ -space.

For monotone increasing $g$ , the region is $\{x \leq g^{-1}(y)\}$ .

Proof

Proof for strictly increasing $g$

Assume $g$ is strictly increasing and differentiable, so that $g^{-1}$ exists and is also strictly increasing. Then $g(X) \leq y \iff X \leq g^{-1}(y),$ and therefore $F_Y(y) = P(X \leq g^{-1}(y)) = F_X\!\bigl(g^{-1}(y)\bigr).$ Differentiating with respect to $y$ via the chain rule: $f_Y(y) = f_X\!\bigl(g^{-1}(y)\bigr)\,\frac{d}{dy}g^{-1}(y).$ Since $g$ is strictly increasing, $dg^{-1}/dy > 0$ , so the absolute value is superfluous here. For strictly decreasing $g$ the inequality flips, introducing a minus sign that the absolute value absorbs. $\blacksquare$

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Theorem: PDF of a Monotone Transformation

Let $X$ be a continuous random variable with PDF $f_X$ , and let $Y = g(X)$ where $g$ is strictly monotone (either increasing or decreasing) and differentiable on the support of $X$ . Then $f_Y(y) = f_X\!\bigl(g^{-1}(y)\bigr)\, \left|\frac{d}{dy}g^{-1}(y)\right|$ for $y$ in the range of $g$ , and $f_Y(y) = 0$ otherwise.

The PDF is a density: it measures probability per unit length. When $g$ stretches or compresses the $x$ -axis, the density must be rescaled by the reciprocal of the local stretching factor $|g'(g^{-1}(y))|$ . The absolute value ensures the density remains nonnegative regardless of whether $g$ is increasing or decreasing.

Show Hint

Apply the CDF method from Theorem thm-cdf-method.

Handle the increasing and decreasing cases separately, then unify with the absolute value.

Proof

Case 1: $g$ strictly increasing

From the CDF method, $F_Y(y) = F_X(g^{-1}(y))$ . Differentiating: $f_Y(y) = f_X(g^{-1}(y))\,\frac{d}{dy}g^{-1}(y).$ Since $g$ is increasing, $g^{-1}$ is also increasing, so $dg^{-1}/dy > 0$ and $|dg^{-1}/dy| = dg^{-1}/dy$ .

Case 2: $g$ strictly decreasing

When $g$ is strictly decreasing, $g^{-1}$ is also strictly decreasing, so $g(X) \leq y \iff X \geq g^{-1}(y)$ . Thus $F_Y(y) = P(X \geq g^{-1}(y)) = 1 - F_X(g^{-1}(y)).$ Differentiating: $f_Y(y) = -f_X(g^{-1}(y))\,\frac{d}{dy}g^{-1}(y).$ Since $g^{-1}$ is decreasing, $dg^{-1}/dy < 0$ , so $-dg^{-1}/dy = |dg^{-1}/dy|$ .

Unified formula

Combining both cases: $f_Y(y) = f_X\!\bigl(g^{-1}(y)\bigr)\, \left|\frac{d}{dy}g^{-1}(y)\right|.$ This completes the proof. $\blacksquare$

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Example: Affine Transformation of a Gaussian

Let $X \sim \mathcal{N}(\mu, \sigma^2)$ and $Y = aX + b$ with $a \neq 0$ . Find the distribution of $Y$ .

Solution

Step 1: Identify the transformation

The function $g(x) = ax + b$ is strictly monotone (increasing if $a > 0$ , decreasing if $a < 0$ ). Its inverse is $g^{-1}(y) = (y - b)/a$ , and $dg^{-1}/dy = 1/a$ , so $|dg^{-1}/dy| = 1/|a|$ .

Step 2: Apply the monotone-transform formula

$f_Y(y) = f_X\!\left(\frac{y - b}{a}\right)\,\frac{1}{|a|}.KATEXPLACEHOLDER0ENDf_Y(y) = \frac{1}{\sqrt{2\pi}\,|a|\sigma} \exp\!\left(-\frac{\bigl(\frac{y-b}{a} - \mu\bigr)^2}{2\sigma^2}\right) = \frac{1}{\sqrt{2\pi}\,|a|\sigma} \exp\!\left(-\frac{(y - (a\mu + b))^2}{2a^2\sigma^2}\right).$ $

Step 3: Identify the result

This is the PDF of $\mathcal{N}(a\mu + b,\; a^2\sigma^2)$ . Therefore $Y = aX + b \sim \mathcal{N}(a\mu + b,\; a^2\sigma^2).$ The Gaussian family is closed under affine transformations: shifting changes the mean, scaling changes both mean and variance (the variance scales by $a^2$ , not $a$ ).

Definition:
Chi-Squared Distribution

Let $Z_1, Z_2, \ldots, Z_k$ be independent standard normal random variables, $Z_i \sim \mathcal{N}(0, 1)$ . The random variable $Q = \sum_{i=1}^{k} Z_i^2$ has the chi-squared distribution with $k$ degrees of freedom, written $Q \sim \chi^2_k$ . Its PDF is $f_Q(q) = \frac{1}{2^{k/2}\,\Gamma(k/2)}\,q^{k/2 - 1}\,e^{-q/2}, \qquad q > 0,$ where $\Gamma(\cdot)$ is the gamma function.

Key moments:

$E[Q] = k$ .
$\mathrm{Var}(Q) = 2k$ .

The chi-squared distribution is a special case of the gamma distribution: $\chi^2_k = \mathrm{Gamma}(k/2, 2)$ .

In detection theory, the test statistic for an energy detector is a sum of squared samples. Under the null hypothesis (noise only), each sample is Gaussian, so the test statistic is $\chi^2$ distributed. The number of degrees of freedom $k$ equals the time-bandwidth product, connecting the chi-squared distribution directly to spectrum sensing in cognitive radio.

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Definition:
Rayleigh Distribution

Let $X, Y$ be independent random variables with $X, Y \sim \mathcal{N}(0, \sigma^2)$ . The envelope of the complex signal $Z = X + jY$ is $R = |Z| = \sqrt{X^2 + Y^2}.$ The random variable $R$ has the Rayleigh distribution with parameter $\sigma$ , whose PDF is $f_R(r) = \frac{r}{\sigma^2}\, \exp\!\left(-\frac{r^2}{2\sigma^2}\right), \qquad r \geq 0.$

Key moments:

$E[R] = \sigma\sqrt{\pi/2}$ .
$E[R^2] = 2\sigma^2$ .
$\mathrm{Var}(R) = \bigl(2 - \pi/2\bigr)\sigma^2$ .

The Rayleigh distribution models the envelope of the received signal in a non-line-of-sight (NLOS) multipath fading channel. By the central limit theorem, the superposition of many independent scattered paths produces in-phase and quadrature components that are approximately Gaussian with zero mean and equal variance, making the envelope Rayleigh.

The instantaneous power $P = R^2$ is exponentially distributed with mean $2\sigma^2$ : this is the basis of the exponential fading model used throughout wireless link budget analysis.

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Definition:
Ricean Distribution

Let $X \sim \mathcal{N}(\nu\cos\theta, \sigma^2)$ and $Y \sim \mathcal{N}(\nu\sin\theta, \sigma^2)$ be independent, where $\nu \geq 0$ is the amplitude of a deterministic line-of-sight (LOS) component and $\theta$ is its phase. The envelope $R = \sqrt{X^2 + Y^2}$ follows the Ricean (or Rician) distribution with PDF $f_R(r) = \frac{r}{\sigma^2}\, \exp\!\left(-\frac{r^2 + \nu^2}{2\sigma^2}\right)\, I_0\!\left(\frac{r\nu}{\sigma^2}\right), \qquad r \geq 0,$ where $I_0(\cdot)$ is the modified Bessel function of the first kind of order zero.

The Ricean $K$ -factor is defined as $K = \frac{\nu^2}{2\sigma^2},$ the ratio of LOS power to scattered (diffuse) power.

Special cases:

$K = 0$ ( $\nu = 0$ ): the Ricean reduces to the Rayleigh.
$K \to \infty$ : the channel becomes deterministic (AWGN).

The Ricean distribution models fading in channels with a dominant LOS path plus weaker scattered components — common in satellite links, millimeter-wave urban microcells, and short-range indoor channels. The $K$ -factor is often quoted in dB; typical values range from $K \approx 3\;\mathrm{dB}$ (moderate LOS) to $K > 10\;\mathrm{dB}$ (strong LOS).

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Definition:
Nakagami- $m$ Distribution

The Nakagami- $m$ distribution has PDF $f_R(r) = \frac{2m^m}{\Gamma(m)\,\Omega^m}\,r^{2m-1}\, \exp\!\left(-\frac{m\,r^2}{\Omega}\right), \qquad r \geq 0,$ where $\Omega = E[R^2]$ is the mean power and $m \geq 1/2$ is the fading severity parameter (or shape parameter), defined by $m = \frac{\bigl(E[R^2]\bigr)^2} {\mathrm{Var}(R^2)} = \frac{\Omega^2}{E\bigl[(R^2 - \Omega)^2\bigr]}.$

Key moments:

$E[R^2] = \Omega$ .
$E[R] = \frac{\Gamma(m + 1/2)}{\Gamma(m)} \left(\frac{\Omega}{m}\right)^{1/2}$ .

Relation to gamma distribution: If $R \sim \mathrm{Nakagami}(m, \Omega)$ , then the power $P = R^2 \sim \mathrm{Gamma}(m, \Omega/m)$ .

Special cases:

$m = 1$ : Nakagami- $m$ reduces to Rayleigh (with $\Omega = 2\sigma^2$ ).
$m = 1/2$ : one-sided Gaussian.
$m \to \infty$ : deterministic (no fading).
The Nakagami- $m$ can approximate the Ricean for appropriate $m = (K+1)^2 / (2K+1)$ .

The Nakagami- $m$ distribution was proposed by Minoru Nakagami in 1960 as a flexible model that fits empirical fading data better than Rayleigh or Ricean in many scenarios. Its main advantage is analytic tractability: the gamma-distributed power leads to closed-form expressions for error rates and outage probabilities with maximal-ratio combining (MRC).

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Definition:
Log-Normal Distribution

A random variable $X$ is log-normally distributed if $\ln X \sim \mathcal{N}(\mu, \sigma^2)$ , i.e., $X = e^Y$ where $Y \sim \mathcal{N}(\mu, \sigma^2)$ . The PDF is $f_X(x) = \frac{1}{x\,\sigma\sqrt{2\pi}}\, \exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \qquad x > 0.$

Key moments:

$E[X] = e^{\mu + \sigma^2/2}$ .
$\mathrm{Var}(X) = \bigl(e^{\sigma^2} - 1\bigr)\,e^{2\mu + \sigma^2}$ .

In wireless, the parameter $\sigma$ is often expressed in dB: $\sigma_{\mathrm{dB}} = \sigma \cdot 10/\ln 10 \approx 4.343\,\sigma$ . Typical values of $\sigma_{\mathrm{dB}}$ for outdoor shadowing range from 4 to 12 dB.

The log-normal distribution models large-scale shadow fading (also called log-normal shadowing). As a signal propagates through the environment, it encounters many obstructions, each imposing a multiplicative attenuation. In dB, these attenuations become additive, and by the CLT the total path loss (in dB) is approximately Gaussian — hence the received power (in linear scale) is log-normal. This should be contrasted with small-scale fading (Rayleigh/Ricean), which operates on a much shorter spatial scale.

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Definition:
Moment-Generating Function (MGF)

The moment-generating function of a random variable $X$ is $M_X(s) = E[e^{sX}] = \int_{-\infty}^{\infty} e^{sx}\,f_X(x)\,dx,$ defined for all $s \in \mathbb{R}$ (or $s \in \mathbb{C}$ ) for which the expectation exists.

Key properties:

Moment extraction: The $n$ -th moment of $X$ is obtained by differentiating $n$ times at $s = 0$ : $E[X^n] = M_X^{(n)}(0) = \left.\frac{d^n}{ds^n}M_X(s) \right|_{s=0}.$
Uniqueness: If $M_X(s)$ exists in an open interval containing $s = 0$ , it uniquely determines the distribution of $X$ . That is, $M_X(s) = M_Y(s)$ for all $s$ in some open interval implies $F_X = F_Y$ .
Normalization: $M_X(0) = 1$ .

The MGF is a powerful tool because it converts sums of independent random variables into products (see the next theorem). However, the MGF may not exist for heavy-tailed distributions (e.g., Cauchy). The characteristic function $\Phi_X(\omega)$ , defined below, always exists and serves as a universal replacement.

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Theorem: MGF of a Sum of Independent Random Variables

Let $X$ and $Y$ be independent random variables whose MGFs $M_X(s)$ and $M_Y(s)$ both exist in a neighborhood of $s = 0$ . Then $W = X + Y$ has MGF $M_W(s) = M_X(s)\,M_Y(s).$ More generally, if $X_1, \ldots, X_n$ are mutually independent, $M_{X_1 + \cdots + X_n}(s) = \prod_{i=1}^{n} M_{X_i}(s).$

Sums of independent random variables become products in the MGF domain. This is the probabilistic analog of the convolution theorem: the PDF of a sum is the convolution of the individual PDFs, and the MGF (a two-sided Laplace transform of the PDF) converts convolution to multiplication.

Show Hint

Write $M_W(s) = E[e^{s(X+Y)}] = E[e^{sX} e^{sY}]$ .

Use independence to factor the expectation.

Proof

$M_W(s) = E\bigl[e^{s(X+Y)}\bigr] = E\bigl[e^{sX}\,e^{sY}\bigr].KATEXPLACEHOLDER0ENDM_W(s) = E\bigl[e^{sX}\bigr]\,E\bigl[e^{sY}\bigr] = M_X(s)\,M_Y(s).$ $The extension to$ n $independent random variables follows by induction.$ \blacksquare$

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Definition:
Characteristic Function

The characteristic function (CF) of a random variable $X$ is $\Phi_X(\omega) = E[e^{j\omega X}] = \int_{-\infty}^{\infty} e^{j\omega x}\,f_X(x)\,dx, \qquad \omega \in \mathbb{R},$ where $j = \sqrt{-1}$ .

Key properties:

Always exists: Since $|e^{j\omega x}| = 1$ , the integral converges absolutely for every $\omega$ and every distribution.
Relation to MGF: $\Phi_X(\omega) = M_X(j\omega)$ whenever the MGF exists.
Moment extraction: $E[X^n] = \frac{1}{j^n}\,\Phi_X^{(n)}(0)$ when the $n$ -th moment exists.
Uniqueness: The CF uniquely determines the distribution (inversion theorem).
Fourier transform of the PDF: $\Phi_X(\omega) = \mathcal{F}\{f_X\}(\omega)$ , the Fourier transform of the density. Hence $f_X(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \Phi_X(\omega)\,e^{-j\omega x}\,d\omega$ .
Independence and sums: If $X$ and $Y$ are independent, then $\Phi_{X+Y}(\omega) = \Phi_X(\omega)\,\Phi_Y(\omega)$ .

The characteristic function is the Fourier transform of the PDF and always exists, unlike the MGF. In modern probability theory, the CF is the standard tool for proving limit theorems such as the central limit theorem (the CF of the standardized sum converges pointwise to $e^{-\omega^2/2}$ , the CF of a standard normal).

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Example: MGF of the Gaussian and Sum of Independent Gaussians

(a) Derive the MGF of $X \sim \mathcal{N}(\mu, \sigma^2)$ .

(b) Use the result to prove that the sum of independent Gaussian random variables is itself Gaussian.

Solution

Part (a): Derive $M_X(s)$

$M_X(s) = E[e^{sX}] = \int_{-\infty}^{\infty} e^{sx}\, \frac{1}{\sqrt{2\pi}\sigma}\, e^{-(x-\mu)^2/(2\sigma^2)}\,dx.KATEXPLACEHOLDER0ENDsx - \frac{(x - \mu)^2}{2\sigma^2} = -\frac{1}{2\sigma^2}\bigl[(x - \mu)^2 - 2\sigma^2 s x\bigr].KATEXPLACEHOLDER1END(x - \mu)^2 - 2\sigma^2 s x = \bigl(x - (\mu + \sigma^2 s)\bigr)^2 - 2\mu\sigma^2 s - \sigma^4 s^2.KATEXPLACEHOLDER2ENDM_X(s) = e^{\mu s + \sigma^2 s^2/2}\, \underbrace{\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma}\, e^{-(x - (\mu + \sigma^2 s))^2/(2\sigma^2)}\,dx}_{= 1} = e^{\mu s + \sigma^2 s^2 / 2}.KATEXPLACEHOLDER3ENDM_X(s) = \exp\!\left(\mu s + \frac{\sigma^2 s^2}{2}\right).$ $

Part (b): Sum of independent Gaussians

Let $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ for $i = 1, \ldots, n$ , mutually independent. Define $W = X_1 + X_2 + \cdots + X_n$ . By the product rule for MGFs of independent sums: $M_W(s) = \prod_{i=1}^{n} M_{X_i}(s) = \prod_{i=1}^{n} \exp\!\left(\mu_i s + \frac{\sigma_i^2 s^2}{2}\right) = \exp\!\left(\left(\sum_{i=1}^{n}\mu_i\right) s + \frac{\left(\sum_{i=1}^{n}\sigma_i^2\right) s^2}{2}\right).$ This is the MGF of $\mathcal{N}\!\left(\sum_{i=1}^{n}\mu_i,\; \sum_{i=1}^{n}\sigma_i^2\right)$ . By uniqueness of the MGF, $W \sim \mathcal{N}\!\left(\sum_{i=1}^{n}\mu_i,\; \sum_{i=1}^{n}\sigma_i^2\right).$ The Gaussian family is therefore closed under addition of independents: the sum of independent Gaussians is Gaussian, with the means and variances adding.

Fading Distributions in Wireless Channels

Property	Rayleigh	Ricean	Nakagami- $m$	Log-Normal
PDF	$\frac{r}{\sigma^2} e^{-r^2/(2\sigma^2)}$	$\frac{r}{\sigma^2} e^{-(r^2+\nu^2)/(2\sigma^2)} I_0\!\bigl(\frac{r\nu}{\sigma^2}\bigr)$	$\frac{2m^m}{\Gamma(m)\Omega^m} r^{2m-1} e^{-mr^2/\Omega}$	$\frac{1}{x\sigma\sqrt{2\pi}} e^{-(\ln x - \mu)^2/(2\sigma^2)}$
Parameters	$\sigma > 0$	$\nu \geq 0$ , $\sigma > 0$ (or $K$ -factor)	$m \geq 1/2$ , $\Omega > 0$	$\mu \in \mathbb{R}$ , $\sigma > 0$
Physical model	Many scattered paths, no LOS	LOS + scattered paths	General fading severity	Multiplicative obstructions (shadowing)
Mean power	$2\sigma^2$	$\nu^2 + 2\sigma^2$	$\Omega$	$e^{2\mu + \sigma^2}(e^{\sigma^2} - 1) + e^{2\mu+\sigma^2}$
Power distribution	Exponential	Non-central $\chi^2$ (2 dof)	Gamma $(m, \Omega/m)$	Log-normal
When it applies	Dense urban NLOS, rich scattering	Suburban, satellite, mmWave with LOS	Flexible fit; indoor, land-mobile	Shadowing on dB-scale (large-scale fading)
Relation to Gaussian	Envelope of zero-mean complex Gaussian	Envelope of nonzero-mean complex Gaussian	Via gamma; approximates Ricean	$e^{\mathcal{N}}$ (exponentiated Gaussian)
Closed-form BER?	Yes (simple)	Series involving $I_0$	Yes (via gamma integral)	No (numerical / approximation)
Key reference	Rayleigh (1880)	Rice (1948)	Nakagami (1960)	Gudmundson (1991)

From Gaussian to Rayleigh to Ricean: Fading Distribution Genesis

Starting from two i.i.d. Gaussian components, watch the Rayleigh envelope emerge. Then see how adding a line-of-sight component smoothly transitions the distribution to Ricean as the

K

-factor increases from 0 to 8.

The Rayleigh distribution arises as the envelope of a zero-mean complex Gaussian. Adding a deterministic LOS component shifts the distribution to Ricean, with the

K

-factor controlling the balance between specular and scattered power.

Why This Matters: Why Rayleigh Fading Arises in Non-LOS Channels

Consider a narrowband wireless channel where the transmitted signal reaches the receiver via $N$ independent scattered paths, with no dominant line-of-sight component. The received complex baseband signal is $Z = \sum_{i=1}^{N} a_i\,e^{j\phi_i},$ where $a_i$ and $\phi_i$ are the amplitude and phase of the $i$ -th path. If $N$ is large and the paths are independent with no single dominant term, then by the central limit theorem the real and imaginary parts of $Z$ are each approximately Gaussian: $\mathrm{Re}(Z) \approx \mathcal{N}(0, \sigma^2), \qquad \mathrm{Im}(Z) \approx \mathcal{N}(0, \sigma^2),$ and they are approximately independent (the phases $\phi_i$ being uniformly distributed ensures zero cross-correlation, and for Gaussian random variables, uncorrelated implies independent).

The envelope $R = |Z| = \sqrt{\mathrm{Re}(Z)^2 + \mathrm{Im}(Z)^2}$ therefore follows the Rayleigh distribution. The instantaneous received power $P = R^2$ is exponentially distributed with mean $2\sigma^2$ , and in dB the power fluctuations are approximately $\pm 10{-}20$ dB around the local mean.

This model is remarkably accurate in dense urban environments (e.g., the classic Jakes model) and forms the baseline against which all wireless system designs are tested.

Historical Note: Lord Rayleigh and the Random Superposition of Waves

In 1880, John William Strutt, 3rd Baron Rayleigh, published "On the resultant of a large number of vibrations of the same pitch and of arbitrary phase" (Philosophical Magazine, vol. 10). He showed that when many sinusoidal waves of equal amplitude but random phases are superimposed, the amplitude of the resultant follows what we now call the Rayleigh distribution, and its intensity (squared amplitude) is exponentially distributed.

Rayleigh's original interest was in acoustics and optics — the scattering of sound and light. Nearly a century later, his result found its most influential application in wireless communications, where it describes the envelope fluctuations caused by multipath propagation. The fact that a model from 19th-century wave physics applies directly to 21st-century 5G channels illustrates the remarkable universality of the Gaussian mechanism: whenever many independent contributions add up, the CLT produces Gaussianity, and the envelope becomes Rayleigh.

Quick Check

Let $X \sim \mathcal{N}(0, 1)$ and $Y = X^2$ . Using the CDF method, what distribution does $Y$ follow?

$\mathcal{N}(0, 1)$

Exponential with rate $1/2$

$\chi^2_1$ (chi-squared with 1 degree of freedom)

Rayleigh

Correction:

\chi^2_1

(chi-squared with 1 degree of freedom)

By definition, $\chi^2_1$ is the distribution of the square of a single standard normal. Alternatively, by the CDF method: $F_Y(y) = P(X^2 \leq y) = P(-\sqrt{y} \leq X \leq \sqrt{y}) = 2\Phi(\sqrt{y}) - 1$ for $y > 0$ , and differentiating yields $f_Y(y) = \frac{1}{\sqrt{2\pi y}}\,e^{-y/2}$ , which is the $\chi^2_1$ PDF.

Quick Check

In a wireless channel, the Ricean $K$ -factor is measured to be $K = 0$ . Which fading distribution does the channel follow?

Log-normal

Nakagami- $m$ with $m = 2$

Rayleigh

AWGN (no fading)

Correction:

Rayleigh

When $K = 0$ , the LOS power $\nu^2 = 0$ , so there is no line-of-sight component. The Ricean distribution with $K = 0$ reduces exactly to the Rayleigh distribution. This is the canonical non-LOS fading model.

Quick Check

Let $X \sim \mathcal{N}(\mu, \sigma^2)$ . What is $M_X(s)$ ?

$e^{\mu s}$

$e^{\mu s + \sigma^2 s^2 / 2}$

$\frac{1}{1 - \sigma^2 s}$

$e^{j\mu s}$

Correction:

e^{\mu s + \sigma^2 s^2 / 2}

The MGF of a Gaussian $\mathcal{N}(\mu, \sigma^2)$ is $M_X(s) = \exp(\mu s + \sigma^2 s^2/2)$ . Setting $s = 0$ gives $M_X(0) = 1$ (normalization). The first derivative at $s = 0$ gives $E[X] = \mu$ , and the second derivative gives $E[X^2] = \mu^2 + \sigma^2$ .

Chi-Squared Distribution

The distribution of the sum of squares of $k$ independent standard normal random variables: $Q = \sum_{i=1}^{k} Z_i^2 \sim \chi^2_k$ . A special case of the gamma distribution with shape $k/2$ and scale $2$ . Mean $k$ , variance $2k$ .

Rayleigh Distribution

The distribution of the envelope $R = \sqrt{X^2 + Y^2}$ when $X, Y \sim \mathcal{N}(0, \sigma^2)$ are independent. PDF: $f_R(r) = (r/\sigma^2)\exp(-r^2/(2\sigma^2))$ for $r \geq 0$ . Models small-scale fading in non-line-of-sight wireless channels.

Ricean $K$ -Factor

The ratio $K = \nu^2 / (2\sigma^2)$ of line-of-sight (LOS) power to diffuse (scattered) power in a Ricean fading channel. $K = 0$ corresponds to Rayleigh fading (no LOS); $K \to \infty$ corresponds to a purely deterministic channel. Typically expressed in dB.

Nakagami- $m$ Distribution

A two-parameter distribution for fading envelopes, parameterized by the shape (fading severity) $m \geq 1/2$ and mean power $\Omega$ . Reduces to Rayleigh when $m = 1$ . The power $R^2 \sim \mathrm{Gamma}(m, \Omega/m)$ , yielding closed-form error-rate expressions.

Moment-Generating Function (MGF)

For a random variable $X$ , the MGF is $M_X(s) = E[e^{sX}]$ . It encodes all moments via $E[X^n] = M_X^{(n)}(0)$ , uniquely determines the distribution (when it exists in a neighborhood of $s = 0$ ), and converts sums of independents to products: $M_{X+Y}(s) = M_X(s) M_Y(s)$ .

Characteristic Function

The characteristic function of $X$ is $\Phi_X(\omega) = E[e^{j\omega X}]$ , the Fourier transform of the PDF. Unlike the MGF, it always exists for every distribution. It uniquely determines the distribution and shares the product property for sums of independents: $\Phi_{X+Y}(\omega) = \Phi_X(\omega)\Phi_Y(\omega)$ .

Common Mistake: Rayleigh Distribution vs. Rayleigh Quotient

Mistake:

"The Rayleigh distribution and the Rayleigh quotient are related because they are both named after Lord Rayleigh."

Correction:

Despite sharing a name, these are entirely different concepts:

The Rayleigh distribution is a probability distribution for the envelope of complex Gaussian noise: $f_R(r) = (r/\sigma^2)\exp(-r^2/(2\sigma^2))$ . It arises from Lord Rayleigh's work on random superposition of waves (1880).
The Rayleigh quotient is the ratio $R(\mathbf{A}, \mathbf{x}) = \frac{\mathbf{x}^H \mathbf{A} \mathbf{x}}{\mathbf{x}^H \mathbf{x}}$ for a Hermitian matrix $\mathbf{A}$ . It arises in eigenvalue problems and is used in beamforming optimization and principal component analysis.

Confusing the two — for instance, citing a "Rayleigh fading channel" when one means a "Rayleigh quotient optimization" — can lead to serious miscommunication in technical writing.

Key Takeaway

Every major fading distribution in wireless communications is derived from the Gaussian distribution via simple transformations:

Rayleigh: envelope of a zero-mean complex Gaussian ( $R = |X + jY|$ , both components i.i.d. $\mathcal{N}(0, \sigma^2)$ ).
Ricean: envelope of a nonzero-mean complex Gaussian (add a deterministic LOS phasor).
Chi-squared: sum of squared Gaussians ( $Q = \sum Z_i^2$ ).
Nakagami- $m$ : generalization whose power is gamma-distributed; encompasses Rayleigh ( $m = 1$ ) and approximates Ricean.
Log-normal: exponentiated Gaussian ( $X = e^Y$ ); models large-scale shadowing.

The CDF method and the monotone-transform formula are the two workhorses that connect these distributions back to the Gaussian. The moment-generating function and characteristic function then provide algebraic shortcuts — especially the product rule for sums of independents, which lets us prove closure properties (e.g., sum of Gaussians is Gaussian) without computing convolution integrals.

Understanding these derivations — rather than memorizing the PDFs — gives the engineer a unified mental model: Gaussian in, transformation out, fading distribution obtained.

Functions of Random Variables and Moment-Generating Functions

Why Transformations of Random Variables Matter

Theorem: CDF Method for Functions of a Random Variable

Proof for strictly increasing $g$

Theorem: PDF of a Monotone Transformation

Case 1: $g$ strictly increasing

Case 2: $g$ strictly decreasing

Unified formula

Example: Affine Transformation of a Gaussian

Step 1: Identify the transformation

Step 2: Apply the monotone-transform formula

Step 3: Identify the result

Definition: Chi-Squared Distribution

Definition: Rayleigh Distribution

Definition: Ricean Distribution

Definition: Nakagami-mmm Distribution

Definition: Log-Normal Distribution

Definition: Moment-Generating Function (MGF)

Theorem: MGF of a Sum of Independent Random Variables

Proof

Definition: Characteristic Function

Example: MGF of the Gaussian and Sum of Independent Gaussians

Part (a): Derive $M_X(s)$

Part (b): Sum of independent Gaussians

Fading Distributions in Wireless Channels

From Gaussian to Rayleigh to Ricean: Fading Distribution Genesis

Why This Matters: Why Rayleigh Fading Arises in Non-LOS Channels

Historical Note: Lord Rayleigh and the Random Superposition of Waves

Quick Check

Quick Check

Quick Check

Chi-Squared Distribution

Rayleigh Distribution

Ricean KKK-Factor

Nakagami-mmm Distribution

Moment-Generating Function (MGF)

Characteristic Function

Common Mistake: Rayleigh Distribution vs. Rayleigh Quotient

Key Takeaway

Definition:
Chi-Squared Distribution

Definition:
Rayleigh Distribution

Definition:
Ricean Distribution

Definition:
Nakagami- $m$ Distribution

Definition:
Log-Normal Distribution

Definition:
Moment-Generating Function (MGF)

Definition:
Characteristic Function

Ricean $K$ -Factor

Nakagami- $m$ Distribution