Introduction To The dB

 Satellite School / Introduction To The dB Date Submitted: 19 November 2002 Written by Tim Tozer. Website: http://www.elec.york.ac.uk/comms/ ExcellentGoodAveragePoorVery Poor   Rate This Article A short article explaining what the dB is and why it is used. Article contains examples. Comment On This Article
Printable Version
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 102 = 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 = V2 / 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