If you have done a lot of wireless the below is bread and butter and appears in many text books in various guises, but for those that need a quick summary or someone looking for a quick “in” to further reading and practise the below might clear a few things up.

**First Logarithms**

Why? (trust me here, this is brief and worthwhile). Well Logarithms are useful to represent the ratio between two values (i.e. values that can be measured), **and we use ratios all the time in Wireless. **This is because it is so so much easier than using real numbers with many decimal places when looking at signal strength, gain and power levels. So what are they?

Logarithms are actually fairly easy at a basic level.

For example – how many 2s do I need to multiply to get 8?

2 x 2 x 2 = 8

So I had to multiply three 2s to get 8, so the Log is 3 – and the way you write this is below:

The base is the little 2, so log base 2 of 8 = 3. We are essentially using three numbers here: The number we are multiplying (2 in this case), how many times it is multiplied (3 in this case), and the number we want to end up with (8 in this example).

Ok, one more just so we are clear.

Work out the following:

so 6 x 6 x 6 x 6 = 1296, so we need four of the 6s multiplied to get 1296 so the answer is:

Incidentally this also tells you the exponent – so 6 to the power 4 = 1296.

All well and good and a refreshing trip down memory lane but what about wireless?

Well log base 10 is used **a lot** in wireless, particularly when it comes to dB.

What is dB (decibel)? Well it is a ratio, and what is good at representing ratios? Well logarithms of course. The decibel is actually a unit of measure that came out of Bell Telephone Labs, and was very useful around the attenuation of audio frequency signal down a mile long telephone cable.

The decibel (dB) therefore is a good way to express the ratio of two power levels. A couple of equations are coming up, nothing too difficult, but hold on for the result as that is where the trick comes in to impress your friends (most people have friends who are impressed by RF conversations right?).

When you express a power ratio in decibels then it is 10 times the 10-based logarithm of the ratio. What on earth does that mean?

We are trying to find a ratio, so for 1 Bel (B)

*Ratio*_{B} = log_{10}(*P*_{1} / *P*_{0})

To take this a step further 1 Bel is 10 decibels, so that is where you get your 10 x from. Therefore for dB:

*Ratio*_{dB} = 10 x log_{10}(*P*_{1} / *P*_{0})

So this gives you a handy way to work out a ratio between two real power values, and in the wireless world you would typically use this when looking at **Gain** (how well an antenna converts input power into radio waves headed in a specified direction.)

Let’s have a look where this is useful.

*G*_{dB} = 10* *log_{10}(*P*_{2} /* P*_{1})

P_{2} *is the power level.*

P_{1} *is the referenced power level.*

G_{dB} *is the power ratio or gain in dB.*

So for the gain in dB for a system with input power of 10W and output power of 20W then

*G*_{dB} = 10 log_{10}(*P _{output}*/

*P*) = 10 log

_{input}_{10}(20W/10W) = 3.01dB

**Now remember that figure 3.01dB**

The first of 2 decibel values an RF engineer has etched into their head is **3dB**, because as you have just seen, this is a ratio of 2 (yes we know it is actually 3.01dB, but that is close enough for RF design) – 20/10 is 2 – voila.

So this is great to work out power ratios in your head. If you know the power level has doubled then you have a 3dB gain, if the signal level is four times higher at the output than the input you have a 6dB gain. Equally if you have -3db then the ratio is 1/2 or a half.

The second figure to have hardwired into your brain is **10dB**. Remember dB is a ratio, and 10dB is handily a ratio of 10 🙂 Equally -10dB is 1/10 or 1 tenth.

For example, if the signal level at the output is 10 times higher than the input then you have a 10 times ratio (i.e. 10W input and 100W at output, a 10x gain) which is **10dB.**

Even more usefully you can now combine the two. Say you have an amplifier with a gain ratio of 20 ( 20 times or 10 x 2), then the gain value is 10 + 3 which is 13dB. (3dB is a 2 x ratio).

Got it?

So now say I want to calculate the power ratio of a given value….

P_{2} is equal to the reference power P_{1} times 10 raised by the gain in G_{dB} divided by 10.

*P*_{2} = *P*_{1}_{ }*⋅ *10^{(G}dB^{ / 10)}

P_{2} *is the power level.*

P_{1} *is the referenced power level.*

G_{dB}* is the power ratio or gain in dB.*

Basically a positive gain means there is more power at the output than the input and a negative gain means less power at the output than the input. Now consider a 40dB gain, well that is 10000 times more power at the output than the input, whereas a 20 dB negative gain is 100 times less power at the output than the input. You can see now where these factors of 10 and logarithms can be useful for quick calculations.

**dBm**

So we have established that dB is a ratio, but what about dBm? Well this is also a ratio but a ratio to a real value, i.e. the ratio of a power level relative to 1mW (or 1 thousandth of a Watt – 0.001W). **dBm therefore is a way to express absolute power**. 10 dBm then is 10mW, or 10 x 1 mW. 20 dBm is 100 mW. Remember the factor of 10 earlier? So 20dBm references 1mW x 10 x 10 = 100mW.

Finally you need to be careful when expressing the difference between 2 power levels. 20 dBm – 10dBm = 3**dB** and not dB**m** because again, here we are expressing a ratio between 2 values as dictated by decibels.

So why use ratios and logarithms at all? Well look at the table below of typical 802.11 power levels and signal strength. At -90dBm you are talking 0.000000001 mw. Calculations based on a ratio or logarithm become much easier to compare than saying “the signal strength is 0.0000something as opposed to 0.00000something so we know the power has decreased by? and the negative gain is?”. In essence it provides meaning to the low power levels involved in wireless communication and makes working out real-world designs a whole lot easier.

So there you have it a few handy basic wireless tips.

A couple more tips to keep up your sleeve is knowing that 5db is a ratio of 3 and -5dB is 1/3 or a third. Not exactly of course, but close enough to work things out.

**dBi**

**dBi** is another measure around antennas that is handy to keep in mind, as this is the gain with reference to an antenna that transmits evenly around a 360 degree sphere. This is the perfect antenna which radiates power in all directions equally. Of course real antennas like this do not exist, but it can be a useful reference point when looking at antenna gain in relation to the theoretical perfect antenna.

One of the most common antennas that you will come across is the dipole antenna, or you may have heard it called a rubber duck antenna. This has a doughnut shaped radiation pattern (toroidal). It is handy to know that this has a gain of 2.15dB over an isotropic antenna, so if you had a different type of antenna with a gain of 5dB, you now know this is 2.85dB of gain higher than a common dipole – 2.15dB + 2.85dB (sounds a bit like we are bird-watching now).

And what is 5dB again? Well it is a ratio, and coincidentally ratio of around 3 times the power gain over an isotropic antenna. See, I told you knowing the 5dB ratio was handy.

So remember, 3dB, 10dB, (maybe 5dB) and 2.15dBi and you can work things out like a wireless design bod in no time at all:-)

Hopefully these very basic wireless tips will send you on your way into the magical world of RF with a little less confusion when dB is thrown around like confetti.