# Log Distance Path Loss Model

While simulating a MISO system, I am trying to consider the effect of varying the distance between the transmitter and receiver on the BER.

My search for the effect of varying distance has pointed me to Log Distance Path Loss.

My query is where should I include this path loss during the Matlab implementation. Should the received signal be convoluted with the path loss?

I am completely clueless as to how this can be included. Any resources or pointers to theory would be appreciated.

There are direct and indirect effects of distance to the Bit Error Rate (BER) between two transceivers.

The direct effect that distance has on the BER is through the attenuation of the signal. As the transmitter get's further away from the receiver, the power of the received signal is diminished which makes the Signal to Noise Ratio lower which makes signal reception more difficult and the likelihood for a misinterpreted symbol higher.

This is the simplest case with the transceivers operating in free space where theoretically there are no other objects or sources of "noise".

The indirect effect of distance on the BER in a realistic environment (i.e. one containing surfaces that the signal can bounce off, such as buildings) is the introduction of multipath propagation. This means that multiple copies of the same signal arrive at the receiver at slightly different times having followed slightly different paths. Depending on the frequency of the signals, these slight delays may lead to destructive interference. In other words, two versions of the same signal cancel out each other. This condition leads, again, to a decrease in the SNR and signal distortions which leads to a higher probability for errors in symbol reception. A similar (indirect) effect is introduced due to the Doppler shift of the frequencies if the velocity of movement of the transmitter is comparable to the wavelength of the used frequencies.

There are multiple ways by which you can include these effects to a simulation.

When simulating the full transceiver stack, i.e. Source, encoder, modulator, channel, demodulator, decoder, Target then Free Space Loss can be applied at the output of the modulator (as part of the "channel" building block) as an attenuation of the produced signal given a distance L.

Multipath propagation can also be applied at the "channel" stage, either by convolving the modulator's output with a suitable impulse response or by generating an "envelope" signal from a Rayleigh distribution.

• For a specific application in my project, I am working with a multiple transmitters and a single receiver (MISO) system. I am trying to analyze the effect of the distance between the various transmitters and the receiver on the BER, which is the reason I have asked about the Log Distance Path loss model. For the purpose of simulation, I intend to use an AWGN channel and include the path loss due to varying distance. For now, I do not intend to use any other channel. Even then, should I apply the log distance path loss at the channel? – smyslov Jun 8 '15 at 10:29
• The short answer is "yes", the long answer depends a lot on the details of the experiment. – A_A Jun 9 '15 at 9:31
• Ideally, including the log distance path loss would mean that I convolve the channel output with the path loss factor in dB. Is this right? – smyslov Jun 9 '15 at 9:33
• No. FSL is a simple multiplication of your signal vector with a single value. Convolution with the impulse response of the channel is something quite different (please see en.wikipedia.org/wiki/Convolution). I hope we don't start arguing about convolving with h=[0.001] (!) :) – A_A Jun 9 '15 at 9:49
• My bad. I meant the multiplication of path loss value in dB with the channel output. – smyslov Jun 9 '15 at 9:53

The log-distance path loss model is a simplified model that tries to model fluctuations in the received signal power. It is generally used in dense urban environments (i.e. a city center or suburbs) or inside a building (where transmitter and receiver are not in the same room). It also assumes large distances (much more than tens of feet).

In this model, the path loss is divided in three separate processes. First, you have the free-space loss, due to the distance between antennas, and which changes with the square of the distance. Then, you have a path loss due to the environment, which changes with a different exponent (not square). The exponent depends on the kind of materials the signal has to travel through. Generally, this exponent is calculated empirically for each propagation environment.

These firt two losses are deterministic. The third loss process is random. In an urban environment, you'll have shadowing, which is caused by the receiver antenna moving behind buildings or trees. The logarithm of this loss is Gaussian distributed. You will also have signal attenuation due to fading caused by multipath; assuming flat, slow fading, this will follow a Rayleigh distribution. You can find the equation in wikipedia (https://en.wikipedia.org/wiki/Log-distance_path_loss_model). This model is also very well explained in "Wireless Communications" by Andrea Goldsmith, in chapter 3 if I remember correctly. The book provides many examples.

Regarding how to implement this model in Matlab: the model allows you to calculate an overall path loss $PL_{dB}$. If your transmit power is $T_{dB}$, then the received power is $R_{dB}=T_{dB}-PL_{dB}$. In general, $PL_{dB}$ is random; you would generate appropriate random numbers in Matlab and change the received signal's power to account for the loss.

Having said all this, if you're running your experiments in a lab, I don't think this model is appropriate. What I would do is to try to set up a fixed environment (no people or objects moving around when you're doing the experiment). If your signal is narrowband, the loss will be a single number that you can determine empirically. This number is random but does not change with time, as would be the case in a dynamic environment.