Wireless Channel Fading - moerjielovecookie

A typical characteristic of wireless channels is the fading phenomenon: fluctuations in signal amplitude over time and frequency. Additive noise is the most common source of signal degradation, while fading is another source that causes non-additive signal disturbances in wireless channels. Fading can also be caused by multipath propagation or obstruction by obstacles.
Fading is roughly divided into two types: large-scale fading and small-scale fading. Large-scale fading occurs when mobile devices move over long distances (cell size distances). It is caused by path loss of the signal (a function of distance) and shadowing caused by large obstacles. Small-scale fading is the rapid fluctuation of signal levels caused by constructive or destructive interference from multiple paths when mobile devices move over shorter distances.
Fading Channel.png

1 Large-Scale Fading#

1.1 General Path Loss Model#

The free space propagation model is used to predict the strength of signal reception in line-of-sight environments. It is commonly used in satellite communication systems.

P_r(d)=\frac{P_t G_t G_r \lambda^2}{(4\pi)^2d^2L}

Where:
$P_t$ is the transmit power, in units: W
$d$ represents the distance between the transmitter and receiver, in units: m
For isotropic antennas, the transmit antenna gain is $G_t$, and the receive antenna gain is $G_r$
$L$ is the system loss independent of the propagation environment, representing the overall attenuation in the actual hardware system
Assuming no loss in the system hardware, the free space path loss is:

PL_F(d)[dB]=10\log_{10}(\frac{P_t}{P_r})=-10\log_{10}( \frac{G_t G_r \lambda^2}{(4\pi)^2d^2})

The free space path loss varies with distance and antenna gain as shown in the figure:
Free Space Loss.png
If there is no antenna gain, the formula can be simplified to:

PL_F(d)[dB]=10\log_{10}(\frac{P_t}{P_r})=20\log_{10}(\frac{4\pi d}{\lambda})

Log-Distance Path Loss Model is as follows:

PL_{LD}(d)[dB]=PL_F(d_0)+10n\log_{10}(\frac{d}{d_0})

Where $d_0$ is a reference distance, which varies in different propagation environments. For example, in a wide-area coverage cellular system (radius greater than 10 km), $d_0$ is typically set to $1 km$; for a cell radius of $1 km$ or microcell systems with very small radii, it can be set to $100 m$ or $1 m$, respectively.
The path loss exponent $n$ is determined by the propagation environment.

Environment	Path Loss Exponent (n)
Free Space	2
Urban Cellular	2.7~3.5
Urban Cellular Shadow	3-5
Indoor Line-of-Sight Transmission	1.6-1.8
Indoor Obstacle Grouping	4-6
Factory Obstacle Blocking	2-3
The log-distance path loss model diagram is as follows:

Since the surrounding environment can change with the actual position of the receiver, even if the distance between the transmitter and receiver is the same, each path can have different path losses. When designing a more realistic environment, the log-normal shadowing model is more practical. Let $X_{\sigma}$ represent a Gaussian random variable with a mean of $0$ and a standard deviation of $\sigma$, the log-normal shadow fading model is:

PL(d)[dB]=\overline{PL}(d)+X_{\sigma}=PL_F(d_0)+10\log_{10}(\frac{d}{d_0})+X_{\sigma}

At this time, the schematic diagram is:
Log-Normal Shadow.png

1.2 Okumura/Hata Model#

Among all models predicting path loss in urban areas, the Okumura model is the most widely adopted, primarily suitable for mobile communication systems with carrier frequencies ranging from 500-1500 MHz, cell radii of 1-100 km, and antenna heights of 30-1000 m. In the Okumura model, path loss can be expressed as:

\mathrm{PL}_{\mathrm{Ok}}\left(d\right)\left[\mathrm{dB}\right]=\mathrm{PL}_{\mathrm{F}}+A_{\mathrm{MU}}\left(f,d\right)-G_{\mathrm{Rx}}-G_{\mathrm{Tx}}+G_{\mathrm{AREA}}

Where $A_{\mathrm{MU}}\left(f,d\right)$ is the medium fluctuation attenuation factor at frequency $f$, and $G_{AREA}$ is the propagation environment gain for the specific area.
The HATA model extends the Okumura model to various propagation environments, including urban, suburban, and open areas, making it the most commonly used path loss model today. For a transmit antenna height of $h_{Tx}[m]$, carrier frequency of $f_c[MHz]$, and distance of $d[m]$, the path loss in urban areas according to the Hata model is:

\begin{array}{c} \mathrm{PL}_{\mathrm{Hata,U}}\left(d\right)\left[\mathrm{dB}\right] = 69.55+26.16\log_{10}f_{\mathrm{c}}-13.82\log_{10}h_{\mathrm{Tx}}-\\ C_{\mathrm{Rx}}+\left(44.9-6.55\log_{10}h_{\mathrm{Tx}}\right)\mathrm{log}_{10}d \end{array}

Where $C_{Rx}$ is a coefficient related to the receive antenna, depending on the size of the coverage area.
For moderate-sized coverage areas, $C_{Rx}$ is given by:

C_{Rx}=0.8+(1.1\log_{10}f_c-0.7)\cdot h_{Rx}-1.56\lg f_c

Where $h_{Rx}[m]$ is the height of the receive antenna.
For large coverage areas, $C_{Rx}$ depends on the carrier frequency:

C_{\mathrm{Rx}}=\begin{cases}8.29\Big(\log_{10}\Big(1.54h_{\mathrm{Rx}}\Big)\Big)^2-1.1,&150\:\mathrm{MHz}\leqslant f_{\mathrm{c}}\leqslant200\:\mathrm{MHz}\\3.2\Big(\log_{10}\Big(11.75h_{\mathrm{Rx}}\Big)\Big)^2-4.97\:,&200\:\mathrm{MHz}\leqslant f_{\mathrm{c}}\leqslant1500\:\mathrm{MHz}\end{cases}

For suburban areas, the Hata model is:

\mathrm{PL}_{\mathrm{Hata,SU}}\left(d\right)\left[\mathrm{dB}\right]=\mathrm{PL}_{\mathrm{Hata,U}}\left(d\right)-2\left(\log_{10}\frac{f_{\mathrm{c}}}{28}\right)^{2}-5.4

For open areas, the model is:

\mathrm{PL}_{\mathrm{Hata,O}}\left(d\right)\left[\mathrm{dB}\right]=\mathrm{PL}_{\mathrm{Hata,U}}\left(d\right)-4.78\left(\log f_{\mathrm{c}}\right)^{2}+18.33\log_{10}f_{\mathrm{c}}-40.97