---
title: "Frequency selective fading and Rayleigh distribution"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:frequency-selective-rayleigh"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/frequency-selective-rayleigh
site: BCT Engineering
---

# Frequency selective fading and Rayleigh distribution

Wireless Communication · Chapter 3: Radio Wave Propagation in Mobile Networks

- Asked in: 2080 Chaitra, 2+4 marks
- Video: [What are Flat Fading and Frequency Selective Fading?](https://www.youtube.com/watch?v=KiKPFT4rtHg) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/frequency-selective-rayleigh
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

What is frequency selective fading? Explain Rayleigh distribution in brief.

## Short answer

Frequency selective fading occurs when Bs>Bc or Ts<sigma_tau, so different frequency components fade unequally and cause ISI. Rayleigh distribution p(r)=(r/sigma^2)exp(-r^2/2sigma^2) describes the envelope of Gaussian in-phase and quadrature components with no LOS; mean is 1.253 sigma, median 1.177 sigma, RMS is sqrt(2) sigma, and phase is uniform.

## Answer

#### Frequency selective fading
**Frequency selective fading** occurs when the channel has a constant gain and linear phase over a bandwidth smaller than the signal bandwidth, i.e. when

```text
Bs > Bc   and   Ts < στ
```
Different frequency components of the signal fade by different amounts, so the received signal is distorted. In time, multiple delayed copies of each symbol overlap the next symbols, causing **inter-symbol interference (ISI)**. It needs equalizers, RAKE receivers or OFDM. A common rule: the channel is frequency selective if Ts < 10στ.

#### Rayleigh distribution
In a mobile channel with many scattered paths and **no dominant (LOS) path**, the in-phase and quadrature components of the received signal are independent Gaussian random variables with zero mean and variance σ². The envelope r = √(I² + Q²) then follows the **Rayleigh distribution**:

```text
p(r) = (r/σ²)·exp(−r²/(2σ²)),  0 ≤ r ≤ ∞
p(r) = 0,                       r < 0
```
- σ = rms value of received voltage before envelope detection; σ² = time-average power.
- CDF: P(R ≤ r) = 1 − exp(−r²/(2σ²))

| Quantity | Value |
|---|---|
| Mean | σ√(π/2) = 1.2533σ |
| Variance | (2 − π/2)σ² = 0.4292σ² |
| Median | 1.177σ |
| RMS value | √2·σ |

```text
p(r)
 |   .-.
 |  /   '.
 | /      '.
 |/         '--.____
 +-----σ--------------- r
   (peak at r = σ)
```

The phase is uniformly distributed over 0 to 2π. Rayleigh fading is used to model flat fading in dense urban areas. When a strong LOS path exists, the envelope follows the **Ricean** distribution instead.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Frequency selective fading occurs when:
   A. Bs < Bc and Ts > στ
   B. Bs > Bc and Ts < στ
   C. Bs = 0
   D. Ts is much larger than στ
   Answer: B. Signal bandwidth exceeds coherence bandwidth (symbol period shorter than delay spread), so frequency components fade differently. (Key point: Condition for frequency selective fading)

2. What is a main effect of frequency selective fading in time?
   A. Inter-symbol interference from delayed copies
   B. Only a uniform amplitude drop
   C. Frequency shift of the carrier
   D. Perfect, undistorted reception
   Answer: A. Delayed copies of a symbol overlap following symbols, causing ISI, so equalizers, RAKE or OFDM are used. (Key point: Effect on the signal)

3. The envelope follows a Rayleigh distribution when:
   A. A strong line-of-sight path dominates
   B. There are many scattered paths and no dominant LOS path
   C. There is only one path
   D. The receiver is stationary with a single reflector
   Answer: B. I and Q are zero-mean Gaussian with no dominant path; with strong LOS the envelope is Ricean. (Key point: When Rayleigh applies)

4. Which statement about Rayleigh fading is correct?
   A. The phase is Gaussian distributed
   B. The pdf peaks at r = 2σ
   C. The CDF is P(R ≤ r) = exp(−r^2/(2σ^2))
   D. The phase is uniform over 0 to 2π and the pdf peaks at r = σ
   Answer: D. The envelope pdf peaks at σ, the CDF is 1 − exp(−r^2/2σ^2), and the phase is uniform. (Key point: Rayleigh properties)

## Related in Wireless Communication

- [Short note: Rayleigh and Ricean fading distribution](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Arayleigh-ricean-distribution-note) — Wireless Communication, chapter 3
- [Rayleigh versus Rician fading channel](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Arayleigh-vs-rician-fading) — Wireless Communication, chapter 3
- [Numerical: rms delay spread; flat or selective for AMPS/GSM](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Arms-delay-spread-amps-gsm) — Wireless Communication, chapter 3

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- [Gumbel's distribution and frequency factor derivation](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Agumbel-distribution-frequency-factor) — Engineering Hydrology, chapter 6