1. Units and Calculations, or 10 log 10 (10 x units ) = x dBunits Mike Davis, SETI Institute Spectrum Management Summer School, Green Bank, 6/2002

2. Logarithmic Scaling A logarithmic factor of 10 is a Bel, in honor of Alexander Graham Bell Bels are rarely seen. However, 1/10 th of a Bel, 10log 10 10 x, the deciBel (dB), is the lingua franca of Engineering 1 0 dB 1010 dB1/10-10 dB 10020 dB1/100-20 dB 10 3 30 dB10 -3 -30 dBetc.

3. Numerical Interlude To better than 1%: 0 dB1.0 3 dB2.0 6 dB4.0 9 dB8.0 10 dB10.0 7 dB5.0 4 dB2.5 1 dB1.25 2 dB  / 2 5 dB  8 dB2  11 dB4   21.5 dB 1 Stellar Magnitude is exactly -4 dB

4. What about the UNITS? dB always give ratios (pure numbers), e.g. –Power: PdB = 10log 10 P/P 0 –Several options for P 0 – 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

5. 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

6. Useful Definitions (cont’d) Power Flux Density: –PFD is radiated power passing through a given area: W/m 2 Spectral Power Flux Density: –PFD per unit bandwidth: W/m 2 /Hz –1 Jansky is 10 -26 W/m 2 /Hz (sum of both polarizations)  -260 dBW/m 2 /Hz

7. Useful Definitions (cont’d) Isotropic Aperture (unity gain in all directions) at a wavelength : A i = 2 /4  [m 2 ] (This is the area of a circle with a circumference of.) The isotropic aperture drops off rapidly with : WavelengthIsotropic Aperture 1 m-11 dBm 2 1 mm-71 dBm 2 Effective Aperture with Gain G: A e = G A i =G 2 /4  [m 2 ]

8. Example Tsys: If you know a room temperature of 290K is –204 dBW/Hz, what is Tsys = 29K in these units? -214 dBW/Hz A i : You are observing at 20 cm. What is your isotropic aperture in dBm 2 ?-25 dBm 2 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/m 2 /Hz