Units and Calculations,
or
10 log10(10x units) = x dBunits
Mike Davis, SETI Institute
Spectrum Management Summer School,
Green Bank, 6/2002
Logarithmic Scaling
• A logarithmic factor of 10 is a Bel, in honor of
Alexander Graham Bell
• Bels are rarely seen. However, 1/10th of a Bel,
10log1010x, the deciBel (dB), is the lingua franca
of Engineering
• 1
0 dB
• 10
10 dB
1/10 -10 dB
• 100
20 dB
1/100 -20 dB
• 103
30 dB
10-3 -30 dB
etc.
Numerical Interlude
To better than 1%:
•
•
•
•
•
•
•
•
0 dB
3 dB
6 dB
9 dB
10 dB
7 dB
4 dB
1 dB
1.0
2.0
4.0
8.0
10.0
5.0
2.5
1.25
•
•
•
•
2 dB
5 dB
8 dB
11 dB
/2

2
4
• 2
1.5 dB
• 1 Stellar Magnitude is
exactly -4 dB
What about the UNITS?
• dB always give ratios (pure numbers), e.g.
– Power: PdB = 10log10P/P0
– Several options for P0 – watts, milliwatts, …
– Append the unit to dB: dBW, dBm,…
• Not limited to Power. For example
– Bandwidth B:
10 MHz  70 dBHz
– Time :
2000 seconds  33 dBs
• Seconds * Hz gives a pure number:
– (B ): (70 dBHz+33dBs)/2 = 51.5 dB
Useful Definitions
• Power P = kTB
– K is Boltzman’s constant, 1.38 10-23
Joules/Kelvin (-228.6 dBW/Hz/K)
– T is absolute temperature in Kelvins
– B is the bandwidth
• At Room Temperature (290 K):
kT dB = -204.0 dBW/Hz
Useful Definitions
(cont’d)
• Power Flux Density:
– PFD is radiated power passing through a given
area: W/m2
• Spectral Power Flux Density:
– PFD per unit bandwidth: W/m2/Hz
– 1 Jansky is 10-26 W/m2/Hz (sum of both
polarizations)  -260 dBW/m2/Hz
Useful Definitions
(cont’d)
• Isotropic Aperture (unity gain in all directions) at a
wavelength  :
Ai = 2/4 [m2]
(This is the area of a circle with a circumference of .)
• The isotropic aperture drops off rapidly with :
Wavelength
1m
1 mm
Isotropic Aperture
-11 dBm2
-71 dBm2
• Effective Aperture with Gain G:
Ae = G Ai =G 2/4 [m2]
Example
• Tsys: If you know a room temperature of 290K is
–204 dBW/Hz, what is Tsys = 29K in these units?
-214 dBW/Hz
• Ai: You are observing at 20 cm. What is your
isotropic aperture in dBm2? -25 dBm2
• Radiometer Equation: You observe for 2000
seconds with a bandwidth of 10 MHz. What is
your T/Tsys = 1/(B )?
-51.5 dB
• What Spectral Power Flux Density arriving in an
isotropic sidelobe equals this noise power?
-240.5 dBW/m2/Hz