Ferkans — Interactive Telecom Tutor

Why Signals Fluctuate

Chapter 5 treated large-scale effects: path loss decays smoothly with distance, and shadowing varies over tens of metres. But a mobile user also experiences rapid fluctuations — the signal can drop 30–40 dB within half a wavelength ( $\sim$ 15 cm at 1 GHz). These fluctuations are caused by multipath propagation: the transmitted signal arrives at the receiver via multiple paths (reflections, diffractions, scattering), each with a different delay, amplitude, and phase. When these copies add constructively the signal is strong; when they add destructively it nearly vanishes.

Definition:
Multipath Propagation

In a wireless channel, the transmitted signal reaches the receiver via $L$ distinct paths. The received signal is

$r(t) = \sum_{l=0}^{L-1} \alpha_l(t)\, s(t - \tau_l(t))\, e^{j\phi_l(t)}$

where for the $l$ -th path:

$\alpha_l(t)$ is the (real, positive) amplitude
$\tau_l(t)$ is the propagation delay
$\phi_l(t) = 2\pi f_0 \tau_l(t) + \phi_{0,l}$ is the phase

Because $\phi_l$ depends on $\tau_l$ through the carrier frequency $f_0$ , even small changes in path length (fraction of $\lambda$ ) cause large phase changes, leading to rapid fading.

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Definition:
Small-Scale Fading

Small-scale fading refers to the rapid fluctuation of the received signal amplitude, phase, or multipath delays caused by small changes (on the order of a wavelength) in the spatial separation between transmitter and receiver.

It is characterised by:

Spatial scale: varies over distances $\sim \lambda/2$
Temporal scale: varies over times $\sim 1/(2 f_D)$ where $f_D$ is the maximum Doppler shift
Frequency scale: varies over bandwidths $\sim 1/\sigma_\tau$ where $\sigma_\tau$ is the RMS delay spread

Historical Note: Early Observations of Multipath Fading

Multipath fading was first systematically studied in the 1950s–60s for military and early mobile radio systems. R. H. Clarke (1968) developed the foundational statistical model for multipath fading in a mobile environment, showing that for many scatterers the envelope follows a Rayleigh distribution. W. C. Jakes extended this work in his influential 1974 book, which established the simulation methodology still used today.

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Example: Phase Variation Over Half a Wavelength

A mobile at $f_0 = 2$ GHz moves $\Delta d = \lambda/2$ .

(a) What is $\lambda$ ?

(b) If a reflected path has a path-length difference that changes by $\Delta d$ , what is the resulting phase change?

(c) Explain why this can cause a deep fade.

Solution

Wavelength

$\lambda = c/f_0 = 3 \times 10^8 / 2 \times 10^9 = 0.15$ m $= 15$ cm.

Phase change

$\Delta\phi = 2\pi \Delta d / \lambda = 2\pi \times 0.5 = \pi$ radians ( $180°$ ).

Deep fade explanation

A phase change of $\pi$ flips the sign of the reflected component. If the direct and reflected paths had similar amplitudes and were adding constructively, after the mobile moves $\lambda/2$ they add destructively, causing a deep fade (potentially 20–40 dB drop). $\blacksquare$

Multipath Phasor Summation

Watch how multipath phasors rotate and sum as a mobile moves through half a wavelength. The total received signal transitions from constructive to destructive interference, illustrating why small-scale fading causes 30--40 dB fluctuations over short distances.

Phasor diagram showing five multipath components (coloured arrows) and their vector sum (white arrow). As the mobile moves

\lambda/2

, the phases rotate and the sum nearly cancels.

Common Mistake: Confusing Large-Scale and Small-Scale Fading

Mistake:

Treating all signal variation as the same phenomenon and using a single model for everything.

Correction:

The total received signal variation has three independent components that operate at different scales:

Path loss: deterministic, varies over km ( $\propto d^{-n}$ )
Shadowing: log-normal, varies over 10–100 m
Small-scale fading: Rayleigh/Rice, varies over $\lambda/2$

In practice, one separates these by spatial averaging: local mean (over $\sim 10\lambda$ ) gives path loss + shadowing, and deviations from the local mean give small-scale fading.

Large-Scale vs Small-Scale Fading

Property	Large-scale fading	Small-scale fading
Cause	Distance, obstacles (buildings, hills)	Multipath interference
Spatial scale	Tens to hundreds of metres	Half wavelength (~cm)
Distribution	Log-normal (dB)	Rayleigh, Rice, Nakagami
Time scale	Seconds (pedestrian)	Milliseconds (vehicular)
Modelled as	Mean path loss + random offset	Complex multiplicative coefficient
Chapter	5	6 (this chapter)

Quick Check

A mobile at 900 MHz moves 7.5 cm. Approximately how many wavelengths has it moved?

$0.225\lambda$

$0.45\lambda$

$0.5\lambda$

$\lambda/4$

Correction:

0.225\lambda

Correct. $\lambda = c/f_0 = 3 \times 10^8 / 900 \times 10^6 = 0.333$ m $= 33.3$ cm. $\Delta d / \lambda = 7.5 / 33.3 \approx 0.225$ .

Multipath Propagation

The phenomenon where a transmitted signal reaches the receiver via multiple paths due to reflection, diffraction, and scattering, causing small-scale fading.

Small-Scale Fading

Rapid fluctuations in signal amplitude and phase caused by constructive/destructive interference of multipath components. Varies over distances of $\lambda/2$ .

Multipath Propagation: Physics and Consequences