Dealing with an Unfriendly Equation

The general equation for the radius of the first Fresnel zone is, unfortunately, not particularly friendly in practical terms. The first unfriendly aspect is that all of the variables (h, d1, d2, and λ) must use the same unit of distance measurement. In practice the radius of the first Fresnel zone is most critical in outdoor RF where it would be most convenient to use feet to measure the radius (h) and miles to measure the distance from the transmitter to the obstruction (d1) and from the obstruction to the receiver (d2). The second unfriendly aspect is that most specifications for 802.11 speak in terms of frequency (2.4 GHz, 5.8 GHz) and not in terms of wavelength (λ).

To make the Fresnel zone equation more usable in the field it is necessary to apply some conversion factors for miles and frequency. The introduction of the conversion factors results in a value, called the constant of proportionality, which is used to calculate the zone radius with mixed units and with frequency instead of wavelength. The derivation of this constant of proportionality follows.

Step 1: Weʼll first convert wavelength to frequency by remembering that they are related by the speed of light so that λ = c/F. Substituting c/F for λ results in the following:

Step 2: We want the radius to be measured in feet and the distances from transmitter and receiver to the obstacle to be measured in miles. Every time thereʼs a variable thatʼs going to be given in miles, we must convert it to feet by multiplying by 5280 (feet in one statute mile), as follows:

Step 3: The denominator under the root can now be factored as follows:

Copyright 2003 - Joseph Bardwell

Step 4: Like terms in the numerator and denominator now cancel:

Step 5: The resulting equation at this step now becomes:

Step 6: Thereʼs another unfriendly aspect to consider now. The basic relationship between wavelength and frequency (F=c/λ) uses cycles/second to measure F. One cycle/second is also called one Hertz. Wireless networks (and many other environments) typically measure frequency in Gigahertz (GHz) where 1 GHz = 1,000,000,000 Hertz. One Hertz is, therefore, 1/1,000,000,000 GHz (1/10-9 GHz) and this conversion factor must be included in the equation as well:

Step 7: The constant terms (c, 10-9, and 5280) can be grouped and separated from the rest of the equation as follows:

Step 8: The speed of light (represented by the lower-case letter “c”) is typically stated as being 300,000,000 meters/second in a vacuum. The most contemporary measurements for the speed of light put it slightly less than this value, with 299,792,458 meters/second being the value given in scientific literature. For the purposes of the calculations that follow, however, the typical value will be used. As it turns out, the most widely published equations for dealing with the Fresnel zones use 300,000,000 meters/second in their inner workings and so, to obtain results that are consistent with the ones youʼll most likely encounter in the 802.11 wireless networking realm, this value will be given to c. There are 1609 meters in one

Copyright 2003 - Joseph Bardwell

mile and, therefore, by dividing, it can be found that the speed of light may be represented as 186,404.8714 miles/second. Moreover, there are 5280 feet in a statute mile and so, by multiplying, the speed of light could also be given as 984,217,720.8898 feet/second. These values will be important because c is included in the constant of proportionality while calculating the radius of the first Fresnel zone. If the radius is going to be measured in feet then the speed of light will have to be represented in feet/second. If the radius is going to be measured in meters then the speed of light is going to have to be represented in meters/ second. In every case the units must all match, feet-for-feet, miles-for-miles, meters-for- meters, and so forth. All distances must be represented using the same length units.

Remember that the units under the left-hand root are now feet, and the units under the right-hand square root are statute miles. To calculate the left-hand square root it will be necessary to represent c in feet/second. Using the value 984,217,720.8898 feet/second for c, the square root of the product c*10-9*5280 = 72.08792941. This value for the constant of proportionality is commonly rounded up to 72.1 and the equation for the radius of the first Fresnel zone becomes:

One More Equation

In literature you will encounter the equation for the maximum radius of the first Fresnel zone when the length of the path between transmitter and receiver is known. For your edification, the derivation is shown here.

Step 1: Start with the general equation for the radius of the first Fresnel zone and assume that the obstacle is in the exact center of the transmission path. At this point the radius of the zone is at its maximum. At this point in the path d1 = d2 and the length of the path (L) is equal to d1+d2. Consequently:

Step 2: Substitute these values into the general equation:

Copyright 2003 - Joseph Bardwell