Decibels

The decibel scale expresses amplitudes and power values logarithmically. The definitions for these differ, but are consistent with each other.

Here

and

represent a reference power and amplitude, respectively. Quantifying power or amplitude in decibels essentially means that we are comparing quantities to a standard or that we want to express how they changed. You will hear statements like "The signal went down by 3 dB" and "The filter's gain in the stopband is

" (Decibels is abbreviated dB.).

The consistency of these two definitions arises because power is proportional to the square of amplitude:

Plugging this expression into the definition for decibels, we find that

Because of this consistency, stating relative change in terms of decibels is unambiguous. A factor of 10 increase in amplitude corresponds to a 20 dB increase in both amplitude and power!

Power Ratio	dB
1	0
	1.5
2	3
	5
4	6
5	7
8	9
10	10
0.1	−10

The accompanying table provides "nice" decibel values. Converting decibel values back and forth is fun, and tests your ability to think of decibel values as sums and/or differences of the well-known values and of ratios as products and/or quotients. This conversion rests on the logarithmic nature of the decibel scale. For example, to find the decibel value for

, we halve the decibel value for 2; 26 dB equals 10 + 10 + 6 dB that corresponds to a ratio of 10 × 10 × 4 = 400. Decibel quantities add; ratio values multiply.

One reason decibels are used so much is the frequency-domain input-output relation for linear systems:

. Because the transfer function multiplies the input signal's spectrum, to find the output amplitude at a given frequency we simply add the filter's gain in decibels (relative to a reference of one) to the input amplitude at that frequency. This calculation is one reason that we plot transfer function magnitude on a logarithmic vertical scale expressed in decibels.