Units and Calculations, or 10 log10(10x units) = x dBunits

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Units and Calculations,or10 log10(10x units)
x dBunits

• Mike Davis, SETI Institute
• Spectrum Management Summer School,
• Green Bank, 6/2002

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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.

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Numerical InterludeTo better than 1

• 0 dB 1.0
• 3 dB 2.0
• 6 dB 4.0
• 9 dB 8.0
• 10 dB 10.0
• 7 dB 5.0
• 4 dB 2.5
• 1 dB 1.25

• 2 dB ? / 2
• 5 dB ?
• 8 dB 2 ?
• 11 dB 4 ?
• ?2 1.5 dB
• 1 Stellar Magnitude is exactly -4 dB

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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 dBHz33dBs)/2 51.5 dB

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Useful Definitions

• Power P kTB
• K is Boltzmans 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

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Useful Definitions (contd)

• 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

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Useful Definitions(contd)

• 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 Isotropic Aperture
• 1 m -11 dBm2
• 1 mm -71 dBm2
• Effective Aperture with Gain G
• Ae G Ai G ?2/4? m2

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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