# Radio Frequency Mathematics

When the topic of RF mathematics is discussed, most people cringe and panic because they expect formulas that have logarithms in them. Fear not. You are about to learn RF math, without having to use logarithms.

If you want to refresh yourself on some of your math skills, then review the following:

• Addition and subtraction using the numbers 3 and 10
• Multiplication and division using the numbers 2 and 10

No, we are not kidding. If you know how to add and subtract using 3 and 10 and if you know how to multiply and divide using 2 and 10, then you have all of the math skills you need to perform RF math. Read on!

### Rule of 10s and 3s

Before you fully delve into the rule of 10s and 3s, it is important to know that this rule may not give you the exact same answers that you would get if you used the logarithmic formulas.

The rule of 10s and 3s provides approximate values, not necessarily exact values. If you are an engineer creating a product that must conform to RF regulatory guidelines, you will need to use logarithms to calculate the exact values.

However, if you are a network designer planning a network for your company, you will find that the rule of 10s and 3s will provide you with the numbers you need to properly plan your network. All of the calculations will be based upon the following four rules:

• If you add 3 to the dBms, you must multiply the mWs by 2.
• If you subtract 3 from the dBms, you must divide the mWs by 2.
• If you add 10 to the dBms, you must multiply the mWs by 10.
• If you subtract 10 from the dBms, you must divide the mWs by 10.

Once you remember these rules, you will be able to quickly perform RF calculations.

The received signal strength indicator (RSSI) is an optional 802.11 parameter with a value from 0 to 255. It is designed to be used by the hardware manufacturer as a relative measurement of the RF power that is received.

The RSSI is one of the indicators that is used by a wireless device to determine if another device is transmitting. Although it is optional, most vendors appear to have implemented RSSI.

The actual range of the RSSI value is from 0 to a maximum value (less than or equal to 255) that each vendor can choose on its own (known as RSSI_Max). The RSSI is also used as one of the factors when a client is determining whether it should roam to another access point.

There are two problems that exist when trying to compare RSSI values between different manufacturers’ wireless cards. The first problem is that the manufacturers may have chosen two different values as the RSSI_Max.

So manufacturer A may have chosen a scale from 0 to 100, whereas manufacturer B may have chosen a scale from 0 to 60. All other things being equal, vendor A may indicate a signal with an RSSI value of 25, whereas vendor B may indicate that same signal with an RSSI value of 15.

If you were to compare these two cards without knowing any additional information, you may think that manufacturer A makes a more sensitive card. The second problem with RSSI is that the manufacturers will take their range of RSSI values and compare them to a different range of dBm values.

So manufacturer A may take its 100 number scale and relate it to dBm values of –110 dBm to –10 dBm, whereas manufacturer B will take its 60 number scale and relate it to dBm values of –95 dBm and –35 dBm. So not only do we have different numbering schemes, but we also have different ranges of values.

Given this information, RSSI values can assist with troubleshooting only if you are comparing information reported by different PCs using the same wireless card. If you attempt to compare values between manufacturers, you are definitely comparing apples and oranges.

### System Operating Margin (SOM)/Link Budget

The system operating margin (SOM), also known as link budget, is the calculation of the amount of RF signal that is received minus the amount of signal required by the receiver. Figure 1 shows all of the components and their effects on the SOM of the receiver, known as the receive sensitivity. Manufacturers determine the receive sensitivity for each speed supported by the wireless card. Different speeds use different modulation techniques and encoding methods and some encoding methods are more susceptible to corruption.

In order to determine the receive sensitivity of a card at a specific speed, data must be transmitted to the client at a high power level. If the bit error rate of the received data is below a predefined threshold, meaning the data was received properly, then this signal level is at or above the receive sensitivity of the receiver.

The manufacturer then decreases the power level and checks the data received again. The decrease of power is repeated until the bit error rate is above the predefined threshold, meaning the data was not received properly.

At this point, the receive sensitivity has been exceeded. The last power level test when the data was received properly is the receive sensitivity for that card at the speed tested. This procedure is repeated for each of the speeds that the card supports.

Remember, the lower the number, the weaker the signal and the more reliable the card. A receive sensitivity chart of a client card may look something like this:

 1 Mbps –94 dBm 2 Mbps –93 dBm 5.5 Mbps –92 dBm 6 Mbps –86 dBm 9 Mbps –86 dBm 11 Mbps –90 dBm 12 Mbps –86 dBm 18 Mbps –86 dBm 24 Mbps –84 dBm 36 Mbps –80 dBm 48 Mbps –75 dBm 54 Mbps –71 dBm

Remember that we are dealing with negative numbers here, so –71 dBm is the highest receive sensitivity on this list. Typically, the faster the speed, the higher the receive sensitivity.

This is not always the case in instances where we compare different technologies, such as 802.11b at 11 Mbps (direct sequence spread spectrum, or DSSS) and 802.11g at 6 Mbps (Orthogonal Frequency Division Multiplexing, or OFDM).

You may be wondering why these numbers are negative when up till now most of the dBm numbers you have worked with have been positive. Figure 3.3 shows a simple summary of the gains and losses in an office environment.

Until now you have worked primarily with calculating the IR and EIRP. It is the effect of free space path loss that makes the values negative, as you will see in the calculations based upon Figure 1.

The link budget is equal to the received signal minus the receive sensitivity. In this example, the received signal is the sum of all components, which is

20 dBm + 5 dBi – 73.98 dB + 2.14 dBi = –46.84 dBm

If the receive sensitivity of the laptop’s radio is –71 dBm, then the link budget is

–46.84 dBm – (–71 dBm) = 24.16 dBm

Let’s look at the SOM of a point-to-point wireless network, as seen in Figure 2. In this case, the two antennas are 10 kilometers apart. In addition to the effects of the antennas and cables, there are also lightning arrestors. Assume that the receiver sensitivity is –80 dBm. In this configuration, the calculation for the link budget is as follows:

 Transceiver 10 dBm 10' LMR 600 cable –.44 dB Lightning arrestor –.1 dB 50' LMR 600 cable –2.21 dB Parabolic antenna +25 dBi FSPL –120 dB Parabolic antenna +25 dBi 50' LMR 600 cable –2.21 dB Lightning arrestor –.1 dB 10' LMR 600 cable –.44 dB Total signal –65.5 dBm

So the SOM is: –65.5 dBm – (–80 dBm) = 14.5 dBm

Fade margin is a level of desired signal above what is required. A good way to explain fade margin is to think of it as a comfort zone. If a receiver has a receive sensitivity of –80 dBm, then as long as the signal received is greater than –80 dBm, the transmission will be successful.

The problem is that the signal being received fluctuates due to many outside influences. In order to accommodate for the fluctuation, it is a common practice to add 10 to 20 dBs to the receive sensitivity value.

The additional value that is added is known as the fade margin. Let’s say that a receiver has a sensitivity of –80 dBm and a signal is typically received at –76 dBm. Then under normal circumstances, this communication is successful.

However, due to outside influences, the signal may fluctuate by ± 5 dBm. This means that most of the time, the communication is successful, but on those occasions that the signal has fluctuated to –81 dBm, the communication will be unsuccessful.

By adding a fade margin of 10 dBm, you are now stating that for your needs, the receive sensitivity is –70 dBm, and you will plan your network so that the received signal is greater than –70 dBm. If the received signal fluctuates, you have already built in some padding, in this case 10 dBm.

In some environments where RF performance is well documented, different fade margin values are associated with service levels and uptime statistics. If you look back at Figure 3 and added a fade margin of 10 dBm to the receive sensitivity of –80 dBm, then the amount of signal required for the link would be –70 dBm. Since the signal is calculated to be received at –65.5 dBm, you will have a successful communication. However if you chose a fade margin of 15 dBm, the amount of signal required would be –65 dBm, and based upon the configuration in Figure 3.4, you would not have enough signal to satisfy the link budget plus the 15 dBm fade margin.

Since RF communications can be affected by many outside influences, it is common to have a fade margin to provide a level of link reliability. By increasing the fade margin, you are essentially increasing the reliability of the link.

### Inverse Square Law

Earlier you learned about the 6 dB rule, which states that a +6 dB change in signal will double the usable distance of a signal and a –6 dB change in signal will halve the usable distance of a signal. This rule and these numbers are actually based upon the Inverse Square Law, originally developed by Isaac Newton.

This law states that the change in power is equal to 1 divided by the square of the change in distance. What this means is that if you are receiving a signal at a certain power level and a certain distance (D) and you were to double the distance (2 × D), then the new power level will change by 1 ÷ (2 × D)2.

If at a distance of 1 feet (call this D) you were receiving a signal of 4 mW, then at a distance of 2 feet (2 × D) the power would change by 1 ÷ 22, which is 1/4. So the power at 2 feet is 4 mW × 1/4, which is equal to 1 mW.

To see how this relates to the 6 dB rule, using the rule of 10s and 3s, consider that to change from 4 mW to 1 mW, you would need to divide the mW column by 2 twice. This would require you to subtract 3 twice from the dBm column, giving you a –6 dBm change caused by the doubling of the distance of the signal. 