db, dbm, dbw db 10 log (x) where x is unitless! For example, amplifier gain is a unitless value!
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1 2/15/2005 db.doc 1/9 db, dbm, dbw Decibel (db), is a specific function that operates on a unitless parameter: db log (x) where x is unitless! Q: A unitless parameter! What good is that!? A: Many values are unitless, such as ratios and coefficients. For example, amplifier gain is a unitless value! E.G., amplifier gain is the ratio of the output power to the input power: P P out in = G ( ) Gain in db = log G G db
2 2/15/2005 db.doc 2/9 Q: Wait a minute! I ve seen statements such as:. the output power is 5 dbw. or. the input power is 17 dbm. Of course, Power is not a unitless parameter!?! A: True! But look at how power is expressed; not in db, but in dbm or dbw. Q: What the heck does dbm or dbw refer to?? A: It s sort of a trick! Say we have some power P. Now say we divide this value P by one 1 Watt. The result is a unitless value that expresses the value of P in relation to 1.0 Watt of power. For example, if P = 2500mW, then P 1W = This simply means that power P is 2.5 times larger than one Watt! Since the value P 1W is unitless, we can express this value in decibels!
3 2/15/2005 db.doc 3/9 Specifically, we define this operation as: ( ) log P dbw P 1 W For example, P = 0 Watts can alternatively be expressed as P dbw =+ dbw. Likewise, P = 1mW can be expressed as ( ) 20 ( ) 30 P dbw = dbw. Q: OK, so what does dbm mean? A: This notation simply means that we have normalized some power P to one Milliwatt (i.e., P 1mW ) as opposed to one Watt. Therefore: ( ) log P dbm P 1 mw For example, P = 0 Watts can alternatively be expressed as P dbm =+ dbm. Likewise, P = 1mW can be expressed as ( ) 50 ( ) 0 P dbm = dbm. Make sure you are very careful when doing math with decibels!
4 2/15/2005 db.doc 4/9 Standard db Values Note that log ( ) = db Therefore an amplifier with a gain G = is likewise said to have a gain of db. Now consider an amplifier with a gain of 20 db Q: Yes, yes, I know. A 20 db amplifier has gain G=20, a 30 db amp has G=30, and so forth. Please speed this lecture up and quit wasting my valuable time making such obvious statements! A: NO! Do not make this mistake! Recall from your knowledge of logarithms that: log n = n log [ ] = n
5 2/15/2005 db.doc 5/9 Therefore, if we express gain as G = n, we conclude: ( ) n G = G db = n In other words, G =0 = 2 (n =2) is expressed as 20 db, while 30 db (n =3) indicates G = 00 = 3. Likewise 0 mw is denoted as 20 dbm, and 00 Watts is denoted as 30 dbw. Note also that mw = -3 mw is denoted as 30 dbm. Another important relationship to keep in mind when using decibels is log [ 2] 3.0. This means that: log n 2 = n log [ 2] 3 n Therefore, if we express gain as G = 2 n, we conclude: ( ) n G = 2 G db 3n As a result, a 15 db (n =5) gain amplifier has G = 2 5 = 32. Similarly, 1/8 = 2-3 mw (n =-3) is denoted as 9 dbm.
6 2/15/2005 db.doc 6/9 Multiplicative Products and Decibels Other logarithmic relationship that we will find useful are: log [ xy] = log [ x] + log [ y] and its close cousin: log x = log [ x ] log [ y ] y Thus, the relationship P out = G P in is written in decibels as: log log out in Pout GPin = 1mW 1mW Pout GPin = log 1mW 1mW Pout Pin = log [ G ] + log 1mW 1mW P ( dbm) = G( db) + P ( dbm) out P = G P out It is evident that deebees are not a unit! The units of the result can be found by multiplying the units of each term in a summation of decibel values.
![and its close cousin: log x = log [ x ] log [ y ] y Thus, the relationship P out = G P in is written in decibels as: log log out in Pout](/docs-images/46/21551047/images/page_6.jpg "GPin = 1mW 1mW Pout GPin = log 1mW 1mW Pout Pin = log [ G ] + log 1mW 1mW P ( dbm) = G( db) + P ( dbm) out P = G P out It is evident that")
7 2/15/2005 db.doc 7/9 For example, say some power P 1 = 6dBm is combined with power P 2 = dbm. What is the resulting total power TP = P1 + P2? Q:This result really is obvious of course the total power is: ( ) = ( ) + ( ) TP dbm P1 dbm P2 dbm = 6dBm + dbm = 16 dbm A: NO! Never do this either! Logarithms are very helpful in expressing products or ratios of parameters, but they are not much help when our math involves sums and differences! log [ x + y ] =???? So, if you wish to add P 1 =6 dbm of power to P 2 = dbm of power, you must first explicitly express power in Watts: P 1 = dbm = mw and P 2 =6 dbm = 4 mw
8 2/15/2005 db.doc 8/9 Thus, the total power P T is: TP = P1 + P2 = 40. mw + 0. mw = mw Now, we can express this total power in dbm, where we find: mw PT ( dbm) = log = dbm. mw The result is not 16.0 dbm!. We can mathematically add 6 dbm and dbm, but we must understand what result means (nothing useful!). 4mW mw 6 dbm + dbm = log + log 1mW 1mW 2 40 mw = log 2 1mW 2 = 16 db relative to 1 mw Thus, mathematically speaking, 6 dbm + dbm implies a multiplication of power, resulting in a value with units of Watts squared!
9 2/15/2005 db.doc 9/9 A few more tidbits about decibels: db db n ndB (can you show why?) I wish I had a nickel for every time my software has crashed-oh wait, I do!
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