Ferkans — Interactive Telecom Tutor

When the Channel Is a Single Complex Number

When the signal bandwidth $W$ is much smaller than the coherence bandwidth $B_c$ (i.e., $W \ll B_c$ ), all frequency components experience the same fading. The channel reduces to a single complex multiplicative coefficient $h(t) = \alpha(t)\,e^{j\phi(t)}$ . This is flat fading or narrowband fading. The key question becomes: what is the distribution of the fading envelope $r = |h|$ and power $|h|^2$ ?

Theorem: Rayleigh Fading Distribution

If the channel consists of a large number of independent scattered paths with no dominant line-of-sight component, then by the Central Limit Theorem the in-phase and quadrature components are i.i.d. Gaussian:

$h = X + jY, \quad X, Y \sim \mathcal{N}(0, \sigma^2)$

The envelope $r = |h| = \sqrt{X^2 + Y^2}$ follows a Rayleigh distribution:

$f_r(r) = \frac{r}{\sigma^2} \exp\!\left(-\frac{r^2}{2\sigma^2}\right), \quad r \geq 0$

The instantaneous power $\gamma = r^2 = |h|^2$ is exponentially distributed:

$f_\gamma(\gamma) = \frac{1}{\bar{\gamma}} \exp\!\left(-\frac{\gamma}{\bar{\gamma}}\right), \quad \gamma \geq 0$

where $\bar{\gamma} = 2\sigma^2$ is the mean power.

Each scatterer contributes a phasor $a_l e^{j\phi_l}$ with uniformly random phase. Summing many such phasors gives a circularly symmetric complex Gaussian (CLT). The magnitude of a complex Gaussian is Rayleigh, and its squared magnitude is exponential.

Proof

CLT application

Write $h = \sum_{l=1}^{L} a_l e^{j\phi_l}$ where $\phi_l \sim \text{Uniform}[0, 2\pi)$ and $a_l$ are i.i.d.

$X = \text{Re}(h) = \sum_l a_l \cos\phi_l$ , $Y = \text{Im}(h) = \sum_l a_l \sin\phi_l$ .

For large $L$ : $X \sim \mathcal{N}(0, \sigma^2)$ , $Y \sim \mathcal{N}(0, \sigma^2)$ with $\sigma^2 = \frac{1}{2}\sum_l E[a_l^2]$ .

Rayleigh PDF

Using the polar transformation $(X, Y) \to (r, \theta)$ with Jacobian $r$ :

$f_{r,\theta}(r, \theta) = f_{X,Y}(r\cos\theta, r\sin\theta) \cdot r$ $= \frac{r}{2\pi\sigma^2} e^{-r^2/(2\sigma^2)}$ .

Integrating over $\theta \in [0, 2\pi)$ :

$f_r(r) = \frac{r}{\sigma^2} e^{-r^2/(2\sigma^2)}$ . $\blacksquare$

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Definition:
Properties of Rayleigh Fading

For a Rayleigh-distributed envelope $r$ with parameter $\sigma$ :

Mean: $E[r] = \sigma\sqrt{\pi/2} \approx 1.253\,\sigma$
Mean power: $\Omega = E[r^2] = 2\sigma^2$
Variance: $\text{Var}(r) = (2 - \pi/2)\sigma^2$
Median: $r_{\text{med}} = \sigma\sqrt{2\ln 2} \approx 1.177\,\sigma$
CDF: $F_r(r) = 1 - e^{-r^2/(2\sigma^2)}$

The probability of a deep fade below threshold $r_0$ :

$P(r \leq r_0) = 1 - e^{-r_0^2 / (2\sigma^2)} \approx \frac{r_0^2}{2\sigma^2} \quad \text{for } r_0 \ll \sigma$

This means deep fades (below $-20$ dB relative to mean) occur with probability $\approx 1$ %.

Theorem: Ricean Fading Distribution

When a dominant line-of-sight (LOS) component with amplitude $A$ is present alongside scattered components, the channel is

$h = A\,e^{j\theta_0} + X + jY$

where $X, Y \sim \mathcal{N}(0, \sigma^2)$ . The envelope $r = |h|$ follows a Ricean distribution:

$f_r(r) = \frac{r}{\sigma^2} \exp\!\left(-\frac{r^2 + A^2}{2\sigma^2}\right) I_0\!\left(\frac{rA}{\sigma^2}\right), \quad r \geq 0$

where $I_0(\cdot)$ is the modified Bessel function of the first kind, order zero. The Ricean K-factor is

$K = \frac{A^2}{2\sigma^2} = \frac{\text{LOS power}}{\text{scattered power}}$

The K-factor measures how dominant the LOS component is. $K = 0$ ( $A = 0$ ): pure Rayleigh (no LOS). $K \to \infty$ : the channel becomes deterministic (AWGN-like). Typical values: $K = 3$ – $10$ dB for outdoor LOS, $K = 0$ – $6$ dB for indoor.

Proof

Non-central chi distribution

$h = (A + X) + jY$ (absorbing the LOS phase into $\theta_0 = 0$ without loss of generality).

$|h|^2 = (A + X)^2 + Y^2$ where $A + X \sim \mathcal{N}(A, \sigma^2)$ .

The magnitude of a complex Gaussian with non-zero mean follows the Rice distribution, which is the square root of a non-central chi-squared with 2 degrees of freedom and non-centrality parameter $\lambda = A^2/\sigma^2$ . $\blacksquare$

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Definition:
Nakagami-m Fading

The Nakagami- $m$ distribution generalises both Rayleigh and approximates Rice:

$f_r(r) = \frac{2m^m r^{2m-1}}{\Gamma(m)\,\Omega^m} \exp\!\left(-\frac{m r^2}{\Omega}\right), \quad r \geq 0$

where $\Omega = E[r^2]$ and $m \geq 1/2$ is the fading parameter:

$m = 1$ : Rayleigh
$m = (K+1)^2/(2K+1)$ : approximates Ricean with K-factor $K$
$m \to \infty$ : no fading (AWGN)

The Nakagami- $m$ is useful because the power $\gamma = r^2$ follows a Gamma distribution, which has a tractable MGF enabling closed-form BER analysis.

Rayleigh Fading from Random Phasor Sum

Visualise the Central Limit Theorem at work: as the number of multipath components

L

increases from 1 to 32, the head-to-tail phasor sum converges to a circularly symmetric complex Gaussian, whose envelope follows the Rayleigh distribution.

Random walk of

L

phasors in the complex plane. For large

L

, the sum

h = \sum a_l e^{j\phi_l}

is approximately

\mathcal{CN}(0, 2\sigma^2)

by the CLT.

Rayleigh vs Ricean Fading Envelope

Adjust the K-factor to see how the fading envelope distribution transitions from Rayleigh ( $K = 0$ ) to near-deterministic ( $K \to \infty$ ). The left panel shows the PDF; the right panel shows a sample time series.

Parameters

K-factor (dB)6

Scatter RMS

\sigma

1

Example: Outage Due to Rayleigh Fading

A Rayleigh fading channel has mean SNR $\bar{\gamma} = 20$ dB. The receiver requires $\gamma_{\min} = 10$ dB for acceptable performance.

(a) What is the outage probability?

(b) By how much must $\bar{\gamma}$ increase to achieve 1% outage?

Solution

Outage probability

For exponentially distributed power: $P_{\text{out}} = P(\gamma < \gamma_{\min}) = 1 - e^{-\gamma_{\min}/\bar{\gamma}}$ .

In linear: $\bar{\gamma} = 100$ , $\gamma_{\min} = 10$ .

$P_{\text{out}} = 1 - e^{-10/100} = 1 - e^{-0.1} = 1 - 0.905 = 9.5$ %.

Required mean SNR for 1% outage

$0.01 = 1 - e^{-10/\bar{\gamma}}$ $\Rightarrow e^{-10/\bar{\gamma}} = 0.99$ $\Rightarrow \bar{\gamma} = -10/\ln(0.99) = 10/0.01005 = 995$ $= 30.0$ dB.

Need $\bar{\gamma} = 30$ dB for 1% outage — a 10 dB increase over the original 20 dB. This illustrates the severe penalty of Rayleigh fading: reducing outage from 10% to 1% requires 10 dB more power. $\blacksquare$

Common Mistake: Assuming Rayleigh Fading Everywhere

Mistake:

Assuming Rayleigh fading for all wireless channels, including LOS scenarios.

Correction:

Rayleigh fading assumes no dominant LOS component. In LOS environments (open rural, elevated BS, indoor near-LOS), the channel is Ricean with a non-zero K-factor. Using Rayleigh in a Ricean channel overestimates outage and underestimates performance, leading to over-provisioned systems.

Quick Check

What fading distribution does the Ricean distribution reduce to as $K \to 0$ ?

Gaussian

Rayleigh

Exponential

Uniform

Correction:

Rayleigh

Correct. When $K = 0$ , there is no LOS component ( $A = 0$ ), so $I_0(0) = 1$ and the Rice PDF reduces to the Rayleigh PDF.

Why This Matters: From Scalar Fading to MIMO Channels

This chapter models fading as a scalar complex coefficient $h(t)$ or a tapped-delay line. With multiple antennas at transmitter and receiver, the channel becomes a matrix $\mathbf{H}(f; t) \in \mathbb{C}^{N_r \times N_t}$ , where each entry fades according to the distributions in this chapter (Rayleigh, Ricean, Nakagami). The spatial correlation between entries depends on antenna spacing and the angular spread of multipath. The MIMO book develops the full spatial channel model, capacity analysis, and beamforming techniques that build directly on the scalar fading foundations established here.

Rayleigh Fading

Fading model for NLOS channels with many scatterers. The envelope $r \sim \text{Rayleigh}(\sigma)$ and the power $\gamma \sim \text{Exp}(1/\bar{\gamma})$ .

Ricean K-Factor

$K = A^2/(2\sigma^2)$ : ratio of LOS power to scattered power. $K = 0$ is Rayleigh; $K \to \infty$ is no fading.

Nakagami-m Distribution

A generalised fading distribution with parameter $m$ . Includes Rayleigh ( $m=1$ ) and approximates Ricean fading. The power follows a Gamma distribution.

Narrowband Fading: Rayleigh and Ricean