Decibels

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

Here $\mathrm{power}(s_0)$ and $\mathrm{amplitude}(s_0)$ represent a reference power and amplitude, respectively. Quantifying power or amplitude in decibelsessentially means that we are comparing quantities to a standard or that we want to express how they changed. You will hearstatements like "The signal went down by 3 dB" and "The filter's gain in the stopband is $-60$ " (Decibels is abbreviated dB.).

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

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 productsand/or quotients. This conversion rests on the logarithmic nature of the decibel scale. For example, to find the decibelvalue for $\sqrt{2}$ , we halve the decibel value for $2$ ; $26$ dB equals $10+10+6$ dB that corresponds to a ratio of $10\times 10\times 4=400()$ . Decibel quantities add; ratio values multiply.

One reason decibels are used so much is the frequency-domain input-output relation for linear systems : $Y(f)=X(f)H(f)$ . Because the transfer function multiplies the input signal's spectrum, to find the output amplitude at a given frequency wesimply add the filter's gain in decibels (relative to a reference of one) to the input amplitude at that frequency. This calculationis one reason that we plot transfer function magnitude on a logarithmic vertical scale expressed in decibels.