Propagation Model

# Introduction#

Path loss calculations are carried out between a transmitter and a receiver using propagation models and other calculations related to radio wave propagation such as shadowing and fading. Propagation models are mathematical representations of the average loss in signal strength over distance. Shadowing and fading margins are added to this average loss to get more precise path loss values.

NetSim is a packet level network simulator, for answering questions such as: application throughputs, packet latencies, probability of packet delivery, rate of packet delivery, etc. For such a simulation it is sufficient and efficient to model the power attenuation over a channel. Sufficient, because the received power determines the probability of packet reception. Efficient, because a symbol-by-symbol model would require a channel model with a complex response function, and modelling of all the propagation paths; this will make the simulation very slow, defeating the very objective of packet-level modelling. Questions such as file transfer throughput, voice and video quality, the effect of network sizing, the comparisons between different wireless access technologies can all be answered by a packet level model, along with power loss models over the communication channels.

Propagation models are used to model power loss (signal attenuation) for all wireless links. These include WLAN – 802.11, Legacy Networks, ZigBee / IOT / WSN – 802.15.4, LTE, Cognitive radio – 802.22 and VANET. The 5G NR propagation models are described in the 5G NR technology library document.

## Propagation Loss#

The three different and mutually independent propagation phenomena influence the power of the received signal are (a) path loss, (b) shadowing, and (c) fading. The different models available in NetSim are:

1. Path loss Models

1. Friis Free Space Propagation (Default option in GUI)
2. Log Distance
3. HATA Suburban
4. HATA Urban
5. COST 231 HATA Suburban
6. COST 231 HATA Urban
7. Indoor Office
8. Indoor Factory
9. Indoor Home
10. No Path Loss
11. Two Ray
12. Patloss Matrix File
13. None

1. None
2. Constant
3. Lognormal

1. None
2. Rayleigh
3. Nakagami
4. Rician

# Path loss#

Path loss is the reduction in power density of an electromagnetic wave as it propagates through space. Path loss may be due to many effects, such as free reflection, aperture-medium coupling loss, and absorption. The general formula by which received power is calculated is

RxPower = TxPower + Gt + GR − PLd0 − 10log Dη

Where η is the path loss exponent, whose value is normally in the range of 2 to 5, Gt is the transmitter antenna gain, and GR is the receiver antenna again. In NetSim, the default value for path loss exponent η, is 2.

D is the distance between transmitter and the receiver, measured in meters. D is assumed to be greater than d0, the far field reference distance. PLdois the path loss at reference distance, d0 (d0 is assumed as 1m). PLdo depends on the protocol and is a user input available in the PHY layer of the radios. For 802.11b the default value of PLd0 is 40dB.

Example: Calculate the received power at node 2 due to node 1’s transmission. The transmit power of node1 is 100mW (20dBm), frequency is 2412 MHz, and Gtand GR are 0. The distance between the nodes is 100m and η is assumed as 2

Rxpower (dBm)= 20 + 0 + 0 − 40 − 10 × 2 × log10(100)

= 20 − 40 − 40=  − 60 dBm

The default value for reference distance d0 and path loss at reference distance PLd0 are

1. 802.11 a / b / g / n / ac / p

2. 2.4 GHz: Default d0= 1m and PLd0 = > 40dB

3. 5 GHz: Default d0 = 1m and PLd= 47 > dB

4. 802.15.4

5. Default d0 = 8m and PLd0 = 58.5dB

6. In LTE and 5G NR the calculations are done for each carrier for uplink and download

7. Default d0 = 1m and PLd0 = 32dB

## Path loss models#

### Friis Free space propagation model#

The free space propagation model is used to predict received signal strength when the transmitter and receiver have a clear, unobstructed line-of-sight path between them. Satellite communication systems and microwave line-of-sight radio links typically undergo free space propagation. The free space power received by a receiver antenna which is separated from a radiating transmitter antenna by distance d, is given by the Friis free space equation.

$$\ P_{r}\ = \ P_{t} + \ G_{t} + \ G_{r} + \ 20\log_{10}{\left( \frac{\lambda}{\left( 4*\pi*d_{0} \right)} \right)\ }\ + \ \left( 10*2*\log_{10}\left( \frac{d_{0}}{d} \right)\ \right)$$

where Ptis the transmitted power, Pr is the received power, Gt is the transmitter antenna gain, Gr is the receiver antenna gain, d is the T-R separation distance in meters and λ is the wavelength in meters.

### Log distance#

The average received power logarithmically decreases with distance, whether in outdoor or indoor radio channels. The average large-scale path loss for an arbitrary T-R separation is expressed as a function of distance by using path loss exponent n.

$$\ P_{r}\ = \ P_{t} + \ G_{t} + \ G_{r} + \ 20\log_{10}{\left( \frac{\lambda}{\left( 4*\pi*d_{0} \right)} \right)\ }\ + \ \left( 10*\eta*\log_{10}\left( \frac{d_{0}}{d} \right)\ \right)$$

Where η is path loss exponent. NetSim allows users to set 2.0  ≤ η≤ 5.0

d0 is the reference distance, and the model is applicable only for d > d0

d is the Transmitter Receiver separation distance

λ is the wavelength and is equal to $\frac{c}{f}\$where c is the speed of light and f is the frequency

Default settings in NetSim

Gt = Gr = 0 dB

d0 = 1 m

### Hata Urban#

The hata model is an empirical formulation of the graphical path loss data provided by Okumura. Hata presented the urban area propagation loss as a standard formula and supplied correction equations for applications to other situations. The standard formula for median path loss in urban areas is given by

Pr = [Pt] − L50 (dB)

L50(dB)= 69.55 + 26.16 × log (fc)– 13.82 × log (hte)– a(hre)+ (44.9 –6.55×log(hte)) × log(d)

Where

L50 (dB) = 50th percentile (median) value of path loss

fc  = Frequency in MHz

hte = Transmitter antenna height (Range 30m to 200m, default 30m)

hre = Receiver antenna height (Range 1m to 10m, default 1m)

d = Separation distance in km. Since the input is in meters, it is divided by 1000 to convert to km.

a (hre) = correction factor for effective mobile antenna height which is a function of the size of coverage area.

a (hre)= 8.29 × (log10(1.54×hre) )2 – 1.1 in dB for fc\< 300 MHz

a (hre)= 3.2× (log10(11.74×hre) )2 – 4.97 in dB for fc≥  300 MHz

### Hata Suburban#

To obtain path loss in suburban area, the standard Hata urban formula is modified as

Pr = [Pt] − L50 (dB)

$$L_{50}\ (dB) = \ L_{50}(urban)(dB)–\ 2\left\lbrack \frac{\log f_{c}}{28} \right\rbrack^{2}–\ 5.4$$

### COST231 Hata Urban and COST231 Hata Suburban#

The European Co-operative for Scientific and Technical Research (EURO-COST formed COST231 working committee to develop an extended version of the Hata model COST231 proposed the following formula to extend Hata’s model. The proposed model for path loss is

Pr = [Pt] − L50 (dB)

L50(dB)= 46.3 + 33.9log(fc) – 13.82log(hte) – a(hre)+ (44.9 –6.55log(hte))log(d) + CM

$$Where\ C_{M} = \left\{ \begin{matrix} 3\ dB\ for\ Urban \\ \ \\ \ \ \ \ \ \ \ 0\ dB\ for\ Suburban \\ \end{matrix} \right.\ \$$

### Indoor office and Indoor factory#

$$Pr\ = \ \lbrack Pt\rbrack + \ \lbrack Gt\rbrack + \ \lbrack Gr\rbrack + 20\log_{10}\left( \frac{\lambda}{\left( 4 \times \pi \times d_{0} \right)} \right)\ \ + \ \left( 10 \times \eta \times \log_{10}\left( \frac{do}{d} \right)\ \right)\$$

$$Where\ \eta = \left\{ \begin{matrix} 2.6\ \ for\ Indoor\_ office\ \\ \ \\ \ \ \ 2.1\ \ for\ Indoor\_ factory \\ \end{matrix} \right.\$$

### Indoor home#

$$Pr\ = \ \lbrack Pt\rbrack + \ \lbrack Gt\rbrack + \ \lbrack Gr\rbrack + 20\log_{10}\left( \frac{\lambda}{\left( 4 \times \pi \times d_{0} \right)} \right)\ \ + \ \left( 10 \times \eta \times \log_{10}\left( \frac{do}{d} \right)\ \right)\$$

Where η = 3

### Two Ray#

The Two-Rays Ground Reflected Model is a radio propagation model which predicts the path losses between a transmitting antenna and a receiving antenna when they are in LOS (line of sight). Generally, the two antenna each have different height. The received signal having two components, the LOS component and the multipath component formed predominantly by a single ground reflected wave. The standard formula for Two-ray model is

Pr = [Pt] + [Gt] + [Gr] − 40log10(d) + 10log10(G×ht2×hr2)

Where

GtGr= Transmit and receive antenna gains

ht = z coordinate of the transmitter plus transmitter antenna height

d = Distance between transmitter and receiver

### Range Based#

The propagation loss depends only on the distance (range) between transmitter and receiver. There is a single Range attribute that determines the path loss. This is not a real-world loss model but a theoretical model useful for experimentation.

Receivers at or within Range meters see a 0 dB pathloss. Hence received power equals transmit power. Receivers beyond Range see a 1000 dB pathloss. Hence received power will be close to -1000 dBm i.e., zero in linear units.

This is a link-level property and would apply to all devices connected to that point-to-multipoint or multi-point-to-multipoint link. Thus, users can have AP1 and associated Wireless nodes set to Range based pathloss, and AP2 and its wireless nodes set to a different pathloss model.

This pathloss is applied not just for transmissions but for Carrier sensing and Interference calculations also. For example, as shown in Fig 2‑1, consider a scenario with A transmitting to B and C transmitting to D. Let B be within range of A, D be within range of C and B also be within range of C. In this case, if there is a simultaneous transmission from A and C, the transmission from A will fail at B due to interference of the C to D transmission.

Fig ‑:Transmission failure due to interference from neighbouring transmissions. In this case A > B transmission fails due to interfering signal from C, when C transmits to D. This occurs because B is within range of C.

Similarly let us consider another example as shown in Fig 2‑2 where B is within the range of A, D is within the range of C. Further, A and C are within range of one another. In this case if there is a transmission from A, C detects the medium to be busy and vice versa which leads to carrier sense blocking at transmitters.

Fig ‑: Carrier sense (CS) blocking due to neighbouring transmissions. If A transmits, then C is CS blocked and if C transmits then A is CS blocked.

Given below in Table 2‑1 are the lower bound, upper bound, and default values for the Range (m) parameter in the GUI for different protocols.

Network Range (m)
Min Max Default
Wi-Fi (Internetworks, MANET, VANET), TDMA, Pure Aloha, Slotted Aloha, Cognitive Radio 1 1000 50
WSN, IoT 1 500 20
UWAN 100 50000 10000

Table ‑: Min, Max and Default values of Range(m) for Range Based pathloss model

### Pathloss Matrix File#

With this option users can define the pathloss for the wireless links. The name of the trace file generated should be PathlossMatrix\<Wireless Link Id>.txt and it should be per the NetSim Pathloss Matrix File format.

The propagation loss is fixed for each pair of nodes and does not depend on their actual positions. This model should be useful for synthetic tests. By default the propagation loss is assumed to be symmetric. The value of pathloss for each pair of nodes is read from a file.

The name of the trace file generated should be kept as PathlossMetrics\<Wireless Link Id>.txt and it should be in the NetSim Pathloss Metrics File format. The NetSim Pathloss Matrix File format is as follows

Step 1: Open node (Wireless_Link) properties -> select pathloss model as PATHLOSS_MATRIX_FILE and click on Configure Pathloss metrics.

Step 2: Inside the text file and write the code in format shown below

# Commented lines

# Empty lines will be ignored

# Format for writing this file is

# SNR value must be in increasing order

# time(sec),tx,rx,loss(dB)

time \<Time_in_Secs>, \<Tx_Node_ID>, \<Rx_Node_ID, \<Pathloss(dB)>

### Default value of pathloss exponent#

The default value of path loss exponent for all path loss models in NetSim are as shown below Table 2‑2.

Path loss model Path loss exponent (default)
Friis free space 2
Log distance 2
COST231 Urban -
COST231 Hata Suburban -
Hata Urban -
Hata Suburban -
Indoor Office 2.6
Indoor Factory 2.1
Indoor Home 3

Table 2‑2: Default value of path loss exponent for path loss models

Pathloss models predict the mean path loss as a function of transmission and reception parameters such as frequency, antenna heights, and distance, etc. Therefore, the predicted path loss between a transmitter and a receiver is constant, in a given environment and for a given distance.

Measurements have shown that at any value of d, the path loss PL (d) at a particular location is random and distributed log-normally about the mean distance-dependent value i.e.

$$PL(d)\lbrack dB\rbrack = \ PL_{d_{0}}\ \ + \ 10 \times \eta \times log\left( \frac{d}{d_{0}} \right)\ + \ \chi$$

Where χ is a zero-mean Gaussian distributed random variable (in dB) with standard deviation σ (in dB). The default value for σ is 5 dB, and the range of σ (in dB) is 5 ≤ σ ≤ 12.

Fading is caused by interference between two or more versions of transmitted signal which arrive at the receiver at slightly different times. These waves, called multipath waves, combine at the receiver antenna to give a resultant signal which can vary widely in amplitude and phase, depending on the distribution of the intensity and relative propagation time of the waves and the bandwidth of the transmitted signal.

In built-up urban areas, fading occurs because the height of the mobile antennas is well below the height of surrounding structures, so there is no single line-of-sight path to the base station.

The default values of Fading parameters in NetSim are as shown below Table 4‑1.

Rayleigh Scale Parameter 1
Nakagami Shape parameter 1
Scale Parameter 1
Rician Shape parameter 1
Scale Parameter 1

Table 4‑1: Default values of Fading parameters

The Nakagami distribution is related to the gamma distribution. In particular, given a random variable

$Y ∼ γ(KӨ){\displaystyle Y\,\sim {\textrm {Gamma}}(k,\theta )}$

it is possible to obtain a random variable ${\displaystyle X\,\sim {\textrm {Nakagami}}(m,\Omega )}$

by setting $k = m,$\theta = \frac{\Omega}{m}${\displaystyle k=m} {\displaystyle \theta =\Omega /m}$ and taking the square root of ${\displaystyle Y}Y$

$$X\ = \ \sqrt{y}$$

In mobile radio channels, the Rayleigh distribution is commonly used to describe the statistical time varying nature of the received envelope of a flat fading signal, or the envelope of an individual multipath component. It is well known that the envelope of the sum of two quadrature Gaussian noise signals obeys a Rayleigh distribution. The Rayleigh distribution has a probability density function (pdf) is given by

$$\ P(r)\ = \ \frac{r}{\sigma^{2}}exp\left( - \frac{r^{2}}{{2\sigma}^{2}} \right)\ \ \ \ \ 0 \< = r \< = \infty$$

P(r) = 0           r \< 0

Where σ = rms value of the received voltage signal before envelope detection

σ2 = time-average power of the received signal before envelope detection

The probability that the envelope of the received signal does not exceed a specified value R is given y corresponding cumulative distribution function (CDF) is given by

$$P(R)\ = \Pr(r \leq R) = \ \int_{0}^{R}{p(r)dr} = 1 - exp\left( - \frac{R^{2}}{{2\sigma}^{2}} \right)$$

The mean value rmean of the Rayleigh distribution is given by

$$r_{mean}\ = \ E\lbrack R\rbrack = \ \int_{0}^{\infty}{rp(r)dr = \ \sigma\sqrt{\frac{\pi}{2}\ }} = 1.2533\sigma$$

And the variance of the Rayleigh distribution is given by σr2 which represents the ac power in the signal envelope

$$\sigma_{r}^{2} = \ E\left\lbrack r^{2} \right\rbrack –\ E^{2}\lbrack r\rbrack = \ \int_{0}^{\infty}{r^{2}p(r)dr - \frac{\sigma^{2}\pi}{2}} = \ \sigma^{2}\left( 2 - \frac{\pi}{2} \right)\ \$$

In communications theory, Rician distributions are used to model scattered signals that reach a receiver by multiple paths. Depending on the density of the scatter, the signal will display different fading characteristics. Rician distributions model fading with a stronger line-of-sight. The Rician distribution has a probability density function (pdf) is given by

$$f\left( x \middle| \nu,\sigma \right) = \frac{x}{\sigma^{2}}\exp\left( \frac{- \left( x^{2} + \nu^{2} \right)}{2\sigma^{2}} \right)I_{0}\left( \frac{x\nu}{\sigma^{2}} \right)$$

Where,

ν= distance between the reference point and the center of the bivariate distribution,

The mean value of the Rician distribution is given by

$$mean = \sigma\sqrt{\frac{\pi}{2}}\ L_{\frac{1}{2}}\left( \frac{{- \nu}^{2}}{2\sigma^{2}} \right)$$

And the variance of the Rician distribution is given by

$$Variance = 2\sigma^{2} + \nu^{2} - \frac{\pi\sigma^{2}}{2}L_{\frac{1}{2}}^{2}\left( \frac{{- \nu}^{2}}{2\sigma^{2}} \right)$$

# SINR Calculation#

Analogous to the SNR used often in wired communications systems, the SINR is defined as the power of a certain signal of interest divided by the sum of the interference power (from all the other interfering signals) and the power of some background noise. The interference power is the difference between the total power received by the receiver and the power received from one particular transmitter. NetSim models an ideal receiver whose noise figure (NF) is zero.

The background thermal noise in dBm at room temperature is given by:

P (dBm) =  − 174 + 10 × log10(Δf)

$${P(mW) = \ 10}^{\left( \frac{P\ (dBm)}{10} \right)}$$

Where Δf is the Bandwidth in Hz.

• 802.15.4, Δf = 2 MHz

• 802.11a, b, g, Δf = 20 MHz

• 802.11n, Δf = 20 MHz or 40 MHz

• 802.11 ac, Δf= 20 / 40 / 80 / 160 MHz

Therefore, SINR is calculated as:

$SINR\ \lbrack dB\rbrack = {10 \times log}_{10}\left( \frac{Received\ power\lbrack mW\rbrack}{Interference\ Noise\ \lbrack mW\rbrack\ + \ \ Thermal\ Noise\ \lbrack mW\rbrack} \right)$

Note: Floating numbers may lose precision when converting from dbm to mw or viceversa (Ref: https://msdn.microsoft.com/en-us/library/c151dt3s.aspx). Hence

• If the received power (in mw) is less than 0.0001 then it is assumed > to be zero.

• If the received power (in mw) is 0 then dbm value is -10000.0 not -∞

• While adding two powers, decimal points after fifth digit is > ignored. Ex: 2.0000005+3.0000012 = 5.0

# Bit Error Rate (BER) Calculation#

Receiver sensitivity is the minimum signal strength that a receiver can detect. In NetSim, if the received signal power is higher than the receiver sensitivity, then the signal is decoded with its BER calculated from the SINR-BER formulas/tables given in this section.

Note that the BER source codes are not open for user modification. If a user wishes to change the BER then they can comment NetSim’s BER function call and write their own function. This can be written in C or it can be written in MATLAB (and a call made to MATLAB from NetSim).

SINR (dB) = RxPower(dBm) − (Noise+Interference)(dBm)

Noise(mW) = kTB

$$SINR(linear) = 10^{\frac{SINR\ (dB)}{10}}$$

$$\frac{E_{b}}{N_{0}} = \frac{\left( \frac{P \times T_{s}}{M} \right)}{\left( \frac{N + I}{W} \right)} = \frac{(SINR \times W)}{\left( \frac{M}{T_{s}} \right)} = SINR(Linear) \times \frac{ChannelBandwidth(Hz)}{ChannelDataRate(bps)}$$

## BER Calculation for QAM#

Computation of the exact bit error rate (BER) for square M-ary QAM (8, 16, 32, 64, 128 and 256 QAM)

$$P_{b} = \frac{1}{\log_{2}\sqrt{M}}\sum_{k = 1}^{\log_{2}\sqrt{M}}{P_{b}(k)}$$

where

$$P_{b} = \frac{1}{\sqrt{M}}\sum_{j = 0}^{\left( 1 - 2^{- k} \right).\sqrt{M} - 1}\left\lfloor ( - 1)^{\left\lfloor \frac{{j.2}^{k - 1}}{\sqrt{M}} \right\rfloor}.\left( 2^{k - 1} - \left\lfloor \frac{{j.2}^{k - 1}}{\sqrt{M}} - \frac{1}{2} \right\rfloor \right).erfc\left( (2.j + 1)\sqrt{\frac{3\left( \log_{2}M \right).r}{2(M - 1)}} \right) \right\rfloor$$

and

$$r = \ \frac{E_{b}}{N_{0}}$$

## BER Calculation for DQPSK, O-QPSK and QPSK#

$$BER = 0.5 \times ERFC\ \left( 0.5 \times \frac{E_{b}}{N_{0}} \right)^{\frac{1}{2}}$$

Note: The 802.15.4 2003 and 802.15.4 2006 standards used the formula

$$BER = \ \frac{8}{15} \times \frac{1}{16} \times {\sum_{k}^{}{{( - 1)}^{k}\binom{16}{k}}e}^{- 20 \times SINR \times \left( \frac{1}{k} - 1 \right)}\ \$$

for O-QPSK which has since been changed

## BER Calculation for DBPSK and BPSK#

$$BER = 0.5 \times ERFC\ \left( \frac{E_{b}}{N_{0}} \right)^{\frac{1}{2}}$$

## BER Calculation for LTE#

In the case of LTE an SNR-BER table is looked up for each MCS.

## SINR-PER Curves for 802.11g#

Fig ‑: PER – SINR Combined Curve for all PHY rates

We get the BER by looking up the SINR-PER curves. Then PER = 1  − (1−BER)L where L = 1512 × 8. The Fig 6‑1 plot was obtained from the NetSim function used to calculate BER in IEEE 802.11 Phy.c

Calculate_ber_by_calculation (double sinr, PHY_MODULATION modulation,double dataRate_mbps, double bandwidth_mHz)

Fig ‑: 802.11g/n PER – SINR Individual Curves for each PHY rate (MCS)

# References#

1. Ronell B. Sicat, "Bit Error Probability Computations for M-ary Quadrature Amplitude Modulation", EE 242 Digital Communications and Codings, 2009.