## Free Space Path Loss Diagrams

April 1, 2018

François Vergès

0

​Free Space Path Loss (or FSPL) represents the amount of energy that a given radio wave loses as it travels through the air away from its source.  Understanding FSPL will help us understand how far a Wi-Fi signal can go. It is also widely used by Wi-Fi survey tools to predict Wi-Fi signal propagation.

It is something that we can calculate by applying this mathematical formula (d in km and f in GHz):
FSPL is not specific to Wi-Fi waves, it can be applied to any other waves using other frequencies.
In a Wi-Fi environment, FSPL refers to the amount of power a Wi-Fi signal losses as it travels away from the transmitter (it can be an AP or a station).

This loss is relative to 2 main components:

• Frequency
• Distance
The following diagram shows exactly how much energy a Wi-Fi wave will lose as it travels away from an AP:

### FREQUENCY

Let’s start with frequency. Here, we are talking about the frequency of the radio wave.  As you can see in the formula above, the higher the frequency is, the higher the loss will be. In Wi-Fi, this means that a 2.4GHz signal is less prone to loss than a 5GHz signal. We commonly say that 2.4GHz travels further away than 5GHz.
Another factor is affecting « how far » a Wi-Fi signal is travelling. And this is related to a specific receive characteristic. Any dual band receivers will need 1 antenna to listen to 2.4GHz signals and another one to listen to 5GHz signals. Because the wavelength of a 2.4GHz signal (12 cm) is longer that the one of a 5GHz signal (6 cm), the size of the antenna needs to be bigger. This means that the 2.4GHz receiver will have a larger receive aperture and will be able to better “hear” the incoming signal.
These phenomenas might be even more accentuated with 6GHz where the wavelength of the signal is about 5cm.

The FSPL formula in Wi-Fi defines that, in the first meter:

• A 2.4GHz signal is losing about 40dB
• Calculation for channel 1: 20log(0.001) + 20log(2.412) + 92.45 = 40.09dB
• Calculation for channel 11: 20log(0.001) + 20log(2.462) + 92.45 =  40.28dB
• Calculation for channel 13: 20log(0.001) + 20log(2.472) + 92.45 = 40.31dB
• A 5GHz signal is losing about 47dB
• Calculation for channel 36: 20log(0.001) + 20log(5.180) + 92.45 = 46.74dB
• Calculation for channel 52: 20log(0.001) + 20log(5.260) + 92.45 = 46.87dB
• Calculation for channel 100: 20log(0.001) + 20log(5.500) + 92.45 = 47.26dB
• Calculation for channel 149: 20log(0.001) + 20log(5.745) + 92.45 = 47.64dB
• Calculation for channel 165: 20log(0.001) + 20log(5.825) + 92.45 = 47.76dB
• A 6GHz signal is losing about 48.5dB
• Calculation for channel 1: 20log(0.001) + 20log(5.955) + 92.45 = 47.94dB
• Calculation for channel 49: 20log(0.001) + 20log(6.195) + 92.45 = 48.29dB
• Calculation for channel 97: 20log(0.001) + 20log(6.435) + 92.45 = 48.62dB
• Calculation for channel 109: 20log(0.001) + 20log(6.495) + 92.45 = 48.70dB
• Calculation for channel 129: 20log(0.001) + 20log(6.595) + 92.45 = 48.83dB
• Calculation for channel 181: 20log(0.001) + 20log(6.855) + 92.45 = 49.17dB
• Calculation for channel 201: 20log(0.001) + 20log(6.955) + 92.45 = 49.30dB
• Calculation for channel 253: 20log(0.001) + 20log(7.115) + 92.45 = 49.49dB

### DISTANCE

Here, we are talking about the distance travelled by the Wi-Fi wave away from the source. Common sense tells us that it would make sense if the signal was to lose power as it travels away from its source.
The FSPL formula tells us exactly by how much following the inverse square law.

The inverse square law tells us that as the distance from the source doubles, the energy is spread out over 4 times the area. This results in the signal losing 4 times it’s original amplitude. In other words, the Wi-Fi signal is losing 6dB every time the distance from the source is doubled (This is represented in red in the diagram above).

So, if we know the power of our signal leaving the access point, we can calculate how far this signal will go. This is assuming there is no source of attenuation between the transmitter and the receiver.

What happens when we introduce a wall between the transmitter and the receiver?

### INTRODUCING A WALL

As expected, the wall will attenuate the signal. The FSPL will be applied to the signal as it travels to the wall and away from the wall as explained on this following diagram. The amount of attenuation will depend on the structure of the wall.
If we know the amount of power that the Wi-Fi signal will have leaving the AP (Equivalent Isotropically Radiation Power or EIRP), we can calculate the size of our desired cell.
Here is an example if we have an AP using an EIRP of 14dBm (or 25mW) for both frequency bands:
It is now easy for us to see that the 2.4GHz cell will be twice as big at the 5GHz cell if the same EIRP is used.
​In order to obtain the same cell size for both frequency bands, we can offset the EIRP of the 2.4GHz by 7dB. The way we can do it is by decreasing the 2.4GHz transmit power by 7dB. Most Wi-Fi system will only let us increase or decrease transmit power by an increment of 3dB. Therefore, the best we can do it decreasing the 2.4GHz transmit power by 6dB:

### RESSOURCES ON FSPL ##### Post by fverges@semfionetworks.com

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