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

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 = 10^y

Log(A x B) = Log(A) + Log(B)
Hence: Log(x^N) = 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

But to make numbers more convenient, scale by a factor of 10 (the deciBel or dB)

Then, G = 10Log(P_out / P_in) 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.

P_R = P_T + 20 - 1 + 30 - 2 - 204 + 30 -1 + 60 = P_T - 68 dB

For converting from a power ratio to dB, first work out powers of 10, e.g:

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.

Examples of converting from dB to a Ratio (or more generally, ratio = 10^dB/10):

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, V_RMS

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(x^N) = NLog(x)

TIP: Take care with "Voltage gain in dB" which is usually a power gain, i.e 20Log(V₂ / V₁)

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