Wireless 101
M08 · L03
Module 8 · Lesson 3

A Hundred Copies, Arriving at Once

Reflection, diffraction and scattering each deliver a copy of your signal by a different route. The receiver adds them as vectors — and the difference between adding up and cancelling out can be 40 dB and half a footstep apart.

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Wireless 101
M08 · L03
Superposition again — M2-L3

Two Copies: +6 dB or Nothing

The multipath sum
r(t) = \sum_i a_i e^{-j\theta_i} s(t-\tau_i), \;\; \theta_i = 2\pi f_c \tau_i

Equal copies in phase double the field, so power is +6.0 dB. In anti-phase the field is zero. Half a wavelength of differential path swings you between them — and at 2 GHz that is 7.5 cm.

02 / 11
Wireless 101
M08 · L03
Rayleigh — nothing dominates

How Often It Falls Apart

Fade depth, Rayleigh channel
P(\text{fade} > x\,\text{dB}) = 1 - e^{-10^{-x/10}}
  • 10 dB: 1 − e−0.1 = 9.5% of the time
  • 20 dB: 1 − e−0.01 = 1.0%
  • 30 dB: 1 − e−0.001 = 0.10%
  • Each extra 10 dB is ten times rarer
03 / 11
Wireless 101
M08 · L03
Rician — one path wins

The K-Factor

K = A²/2σ² is dominant power over scattered power. K → ∞ is the free space of M8-L1; K = 0 is exactly Rayleigh. Indoor line of sight runs 5–10 dB, open rural higher.

Rayleigh
9.5%
K = 5 dB
2.5%
K = 10 dB
0.07%

Chance of a 10 dB fade. K, not transmit power, is what buys reliability.

04 / 11
Wireless 101
M08 · L03
Delay spread → coherence bandwidth

Fading Across Frequency

50%-correlation form — state which you use
B_c \approx \tfrac{1}{5\sigma_\tau}

Indoors στ ≈ 50 ns → 4 MHz. Urban στ ≈ 1 µs → 200 kHz. Wider than Bc and your own spectrum fades unevenly: frequency-selective fading, and multipath ISI that no transmit filter can fix (M6-L2).

05 / 11
Wireless 101
M08 · L03
Try it — 2 GHz, 200 ms window

Fading Explorer

Raise K to calm the trace; raise the speed to compress the fades in time.

K K −20 dB · Rayleigh
Speed 100 km/h
fd 185.2 Hz Tc 2.28 ms deepest −30.9 dB 9.5% below −10 dB · deep fading
Interactive: build a multipath channel tap by tap and watch it fade — AI-generated
06 / 11
Wireless 101
M08 · L03
Doppler → coherence time

Fading Across Time

Maximum shift, and coherence time
f_d = \tfrac{v f_c}{c} = \tfrac{v}{\lambda} \;\Big|\; T_c \approx \tfrac{0.423}{f_d}
  • 2 GHz → λ = 0.15 m
  • 100 km/h = 27.8 m/s → fd = 27.8/0.15 = 185 Hz, Tc = 0.423/185 = 2.3 ms
  • 5 km/h = 1.39 m/s → fd = 9.3 Hz, Tc = 46 ms
  • Doppler spread is two-sided, about 2fd
07 / 11
Wireless 101
M08 · L03
Fast or slow, small or large

Four Words That Get Mixed Up

Fading is fast when Ts > Tc — below about 440 symbols/s at Tc = 2.3 ms, so nearly every real link is slow. And small-scale multipath rides on top of large-scale shadowing, which is M8-L4.

Small scale
cm, ms
Shadowing
tens of m
Fast below
440 Bd
08 / 11
Wireless 101
M08 · L03
A deep fade is a coincidence

How Engineers Fight Back

  • Diversity — antennas > λ/2 apart, carriers > Bc apart, repeats > Tc apart
  • Equalisation — measure the channel and invert it
  • Interleaving + coding — spread a codeword past Tc
  • OFDM and MIMO — Module 10
  • Fade margin — 20 dB buys 1%, 30 dB buys 0.1% (M9-L4)
09 / 11
Wireless 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

10 / 11
Wireless 101
M08 · L03
Recap

What you learned

  • Multipath is a phasor sum: +6 dB or a null
  • Rayleigh: 9.5% at 10 dB, 1.0% at 20, 0.10% at 30
  • Rician K: ∞ is free space, 0 is Rayleigh
  • Bc ≈ 1/(5στ) — 4 MHz indoors, 200 kHz urban
  • fd = v/λ = 185 Hz, Tc = 2.3 ms at 100 km/h
Up next in Module 8
Indoor vs Outdoor →
All lessons Read article
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