Download Appendix B: Decibel Arithmetic

Decibel Arithmetic
Appendix B
The decibel, often called dB, is widely used in radar system analysis and
design. It is a way of representing the radar parameters and relevant quantities
in terms of logarithms. The unit dB is named after Alexander Graham Bell,
who originated the unit as a measure of power attenuation in telephone lines.
By Bell’s definition, a unit of Bell gain is
P
log  -----0
 Pi 
(B.1)
where the logarithm operation is base 10, P 0 is the output power of a standard
telephone line (almost one mile long), and P i is the input power to the line. If
voltage (or current) ratios were used instead of the power ratio, then a unit Bell
gain is defined as
V0 2
log  -----
 Vi 
I0 2
log  ----
 Ii 
or
(B.2)
–1
A decibel, dB, is 1 ⁄ 10 of a Bell (the prefix “deci” means 10 ). It follows
that a dB is defined as
V 2
P
I 2
10 log  -----0 = 10 log  -----0 = 10 log  ---0-
 Pi 
 Vi 
 Ii 
(B.3)
The inverse dB is computed from the relations
P 0 ⁄ P i = 10
dB ⁄ 10
V 0 ⁄ V i = 10
dB ⁄ 20
I 0 ⁄ I i = 10
dB ⁄ 20
© 2000 by Chapman & Hall/CRC
503
(B.4)
Decibels are widely used by radar designers and users for several reasons.
Perhaps the most important of them all is that utilizing dBs drastically reduces
the dynamic range that a designer or a user has to use. For example, an incoming radar signal may be as weak as 0.000000001V , which can be expressed in
dBs as 10 log ( 0.000000001 ) = – 90dB . Alternatively, a target may be located
at range R = 1000000m = 1000Km which can be expressed in dBs as
60dB .
Another advantage of using dB in radar analysis is to facilitate the arithmetic
associated with calculating the different radar parameters. The reason for this
is the following: when using logarithms, multiplication of two numbers is
equivalent to adding their corresponding dBs, and their division is equivalent
to subtraction of dBs. For example,
250 × 0.0001
------------------------------- =
455
(B.5)
[ 10 log ( 250 ) + 10 log ( 0.0001 ) – 10 log ( 455 ) ]dB = – 42.6dB
In general,
A×B
10 log  ------------- = 10 log A + 10 log B – 10 log C
 C 
(B.6)
q
10 log A = q × 10 log A
(B.7)
Other dB ratios that are often used in radar analysis include the dBsm (dB squared meters). This definition is very important when referring to target
RCS, whose units are in squared meters. More precisely, a target whose RCS is
2
2
2
σ m can be expressed in dBsm as 10 log ( σ m ) . For example, a 10m tar2
get is often referred to as 10dBsm target, and a target with RCS 0.01m is
equivalent to a – 20dBsm .
Finally, the units dBm and dBW are power ratios of dBs with reference to
one milliwatt and one Watt, respectively.
P
dBm = 10 log  -------------
1mW
(B.8)
P
dBW = 10 log  --------
1W
(B.9)
To find dBm from dBW, add 30 dB, and to find dBW from dBm, subtract 30
dB.
© 2000 by Chapman & Hall/CRC