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# ### How Big Is A dB?

Introduction To The dB

Describing Power Signal stages are cascaded, so powers are multiplied by gain or loss. This yields a lot of multiplications. This suggests the need for a logarithmic representation of power. A logarithmic scale is used to
• Condense wide range of numbers
• Ease multiplication
Logarithms
Log(x) = power to which base must be raised to give x. The base is chosen to be 10.
Log(x) = y means that  x = 10y
Log(A x B) = Log(A) + Log(B)
Hence: Log(xN) = N x Log(x)
Some example logarithm values:
• Log(100) = 2 because 10= 100.
• Log(1000) = 3 because 103 = 1000.
• Log(1000000) = 6 because 106 = 1000000.
• Log(10) = 1 because anything to the power of 1 is itself.
• Log(1) = 0 because anything to the power of 0 is 1.
• Log(1/10) = -1 because 10-1 = (1/10)
• Log(1/1000) = -3
The deciBel
Represent gains or attenuations logarithmically (base 10) (the Bel)
But to make numbers more convenient, scale by a factor of 10 (the deciBel or dB)
Then, G = 10Log(Pout / Pin) in dB
Examples:
• An amplifier has a power gain of 1000. What is this in dB?G = 10Log(1000) = 10 x 3 = 30 dB
• An attenuator has its output power 1/10th of its input. What is its transfer function in dB?G = 10Log(1/10) = 10 x -1 = -10 dB. (Note - dB can be negative)
Since Log(A x B) = Log(A) + Log(B) we can add gains and losses. PR = PT + 20 - 1 + 30 - 2 - 204 + 30 -1 + 60 = PT - 68 dB
For converting from a power ratio to dB, first work out powers of 10, e.g:
 Ratio dB 1000 = 103 30 dB 1 = 100 0 dB 1/1000000 = 106 -60 dB
Then note the smaller factors:
• Factor of 2 is 3 dB (remember this!)
• Factor of 4 = 2 x 2 is 3 + 3 = 6 dB etc.
 Ratio dB 20 2 x 10 is 3 + 10 13 dB 1/400 4 x 100 is 6 + 20 -26 dB
Examples of converting from dB to a Ratio (or more generally, ratio = 10dB/10):
 dB Ratio 23 3 + 20 is 2 x 100 200 -3 1/2 -63 -60 - 3 is 1/106 x 1/2 1/2000000 -160 10-16 -167 -170 + 3 is 10-17 x 2 2 x 10-17 7 10 - 3 is 10/2 5 9 3 + 3 + 3 is 2 x 2 x2 8 1 10-9 is 10/8 1.25
Applying dB to Other Units
By default, dB is a power ratio. But it can be other things, for example, dB banana = dB relative to 1 banana.
dBW = dB relative to 1 watt, so:
• 3 dBW = 2 W
• -30 dBW = 1/1000 W = 1 mW (1 milli-watt) = 0 dBm (m here - milliwatt)
• -60 dBW = 1 µW (1 micro-watt) = -30 dBm
Bandwidth in Hz can be expressed in dB-Hz
• 1 MHz = 60 dB-Hz
Similarly, Noise Temperature:
• 200 K =  23 dB-K
By default, with dBs we are dealing with power.
P = V/ R where  V is the root mean square voltage, VRMS
Thus a change in power (e.g. due to amplification) can be represented by:
10Log(P2 / P1) = 10Log(V22 / V12) = 20Log(V2 / V1) since Log(xN) = NLog(x)
TIP: Take care with "Voltage gain in dB" which is usually a power gain, i.e 20Log(V2 / V1)
How Big Is A dB?
Examples of BER vs. Eb/No in dB: • 1dB is approximately 25% change in power
• 1 dB is approximately the smallest detectable audio power change
• 0.1 dB is a practical measurement limit
•  But 1 dB is significant in digital demodulation