The log-distance path loss model is a
radio propagation model
Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere.
As a form of electromagnetic radiation, like light waves, radio waves are affected ...
that predicts the
path loss
Path loss, or path attenuation, is the reduction in power density (attenuation) of an electromagnetic wave as it propagates through space. Path loss is a major component in the analysis and design of the link budget of a telecommunication system. ...
a
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
encounters inside a building or densely populated areas over distance.
Mathematical formulation
The model
![Slow fading Log-distance](https://upload.wikimedia.org/wikipedia/commons/a/a3/Slow_fading_Log-distance.png)
Log-distance path loss model is formally expressed as:
:
where
*
is the total
path loss
Path loss, or path attenuation, is the reduction in power density (attenuation) of an electromagnetic wave as it propagates through space. Path loss is a major component in the analysis and design of the link budget of a telecommunication system. ...
in
decibels
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a Power, root-power, and field quantities, power or root-power quantity on a logarithmic scale. Two signals whose ...
(dB).
*
is the transmitted power in
dBm
DBM or dbm may refer to:
Science and technology
* dBm, a unit for power measurement
* DBM (computing), family of key-value database engines including dbm, ndbm, gdbm, and Berkeley DB
* Database Manager (DBM), a component of 1987's ''Extended Edi ...
where
is the transmitted power in
watts
Watts is plural for ''watt'', the unit of power.
Watts may also refer to:
People
*Watts (surname), list of people with the surname Watts Fictional characters
*Watts, main character in the film '' Some Kind of Wonderful''
*Watts family, six chara ...
.
*
is the received power in dBm where
is the received power in watts.
*
is the
path loss
Path loss, or path attenuation, is the reduction in power density (attenuation) of an electromagnetic wave as it propagates through space. Path loss is a major component in the analysis and design of the link budget of a telecommunication system. ...
in decibels (dB) at the reference distance
calculated using the Friis
free-space path loss
In telecommunication, the free-space path loss (FSPL) (also known as Free Space Loss, FSL) is the attenuation of radio energy between the feedpoints of two antennas that results from the combination of the receiving antenna's capture area plus the ...
model.
*
is the length of the path.
*
is the reference distance, usually 1 km (or 1 mile) for a large cell and 1 m to 10 m for a microcell.
*
is the
path loss
Path loss, or path attenuation, is the reduction in power density (attenuation) of an electromagnetic wave as it propagates through space. Path loss is a major component in the analysis and design of the link budget of a telecommunication system. ...
exponent.
*
is a
normal (or Gaussian) random variable with zero
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
, reflecting the attenuation (in decibels) caused by
flat fading
In wireless communications, fading is variation of the attenuation of a signal with various variables. These variables include time, geographical position, and radio frequency. Fading is often modeled as a random process. A fading channel is a ...
. In the case of no fading, this variable is 0. In the case of only
shadow fading or
slow fading
In wireless communications, fading is variation of the attenuation of a signal with various variables. These variables include time, geographical position, and radio frequency. Fading is often modeled as a random process. A fading channel is a ...
, this random variable may have
Gaussian distribution with
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
in decibels, resulting in a
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
of the received power in watts. In the case of only fast fading caused by multipath propagation, the corresponding fluctuation of the signal envelope in volts may be modelled as a random variable with
Rayleigh distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distribut ...
or
Ricean distribution (and thus the corresponding gain in watts
may be modelled as a random variable with
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
).
Corresponding non-logarithmic model
This corresponds to the following non-logarithmic gain model:
:
where
is the average multiplicative gain at the reference distance
from the transmitter. This gain depends on factors such as
carrier frequency
In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has a m ...
, antenna heights and antenna gain, for example due to directional antennas; and
is a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
that reflects
flat fading
In wireless communications, fading is variation of the attenuation of a signal with various variables. These variables include time, geographical position, and radio frequency. Fading is often modeled as a random process. A fading channel is a ...
. In case of only slow fading (shadowing), it may have
log-normal
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
distribution with parameter
dB. In case of only
fast fading due to
multipath propagation
In radio communication, multipath is the propagation phenomenon that results in radio signals reaching the receiving antenna by two or more paths. Causes of multipath include atmospheric ducting, ionospheric reflection and refraction, and reflec ...
, its amplitude may have
Rayleigh distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distribut ...
or
Ricean distribution. This can be convenient, because Power (Watts) is proportional to the square of amplitude. Squaring a Rayleigh-distributed random variable produces an
Exponentially-distributed random variable. In many cases, exponential distributions are computationally convenient and allow direct closed-form calculations in many more situations than a Rayleigh (or even a Gaussian).
Empirical coefficient values for indoor propagation
Empirical measurements of coefficients
and
in dB have shown the following values for a number of indoor wave propagation cases.
[''Wireless communications principles and practices'', T. S. Rappaport, 2002, Prentice-Hall]
See also
*
ITU model for indoor attenuation The ITU indoor propagation model, also known as ''ITU model for indoor attenuation'', is a radio propagation model that estimates the path loss inside a room or a closed area inside a building delimited by walls of any form. Suitable for appliances ...
*
Radio propagation model
Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere.
As a form of electromagnetic radiation, like light waves, radio waves are affected ...
*
Young model
References
Further reading
*
*
{{Radio frequency propagation models
Radio frequency propagation model