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Working with amplitude and the decibel scale.

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

power s in decibels 10 10 logbase --> power s power s 0
amplitude s in decibels 20 10 logbase --> amplitude s amplitude s 0

Here power s 0 and 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 prefix "deci" implies a tenth; a decibel is a tenth of a Bel. Who is this measure named for?

Alexander Graham Bell. He developed it because we seem to perceive physical quantities like loudness and brightnesslogarithmically. In other words, percentage , not absolute differences, matter to us. We use decibels today because common valuesare small integers. If we used Bels, they would be decimal fractions, which aren't as elegant.

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

power s amplitude s 2
Plugging this expression into the definition for decibels, we find that
10 10 logbase --> power s power s 0 10 10 logbase --> amplitude s 2 amplitude s 0 2 20 10 logbase --> amplitude s amplitude s 0
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!

Decibel table

Power Ratio dB
1 0
2 1.5
2 3
10 5
4 6
5 7
8 9
10 10
0.1 -10
Common values for the decibel. The decibel values for all but the powers of ten are approximate, but are accurate to a decimalplace.

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 2 , 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 : 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.

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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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