Power Levels
The term decibel has its origin in the fact that power and audio levels are related on logarithmic basis.
For standardization, the bel (\(\textrm{B}\)) is defined by the following equation relating two power levels, \(P_1\) and \(P_2\).
\(\boxed{G = \log_{10}\left(\frac{P_2}{P_1}\right)}\quad\textrm{B}\)
The decibel (\(\textrm{dB}\)) is defined such that \(10\;\textrm{decibels} = 1\;\textrm{bel}\).
\(\boxed{G_\mathrm{dB} = 10\log_{10}\left(\frac{P_2}{P_1}\right)}\quad\textrm{dB}\)
The terminal rating of electronic communication equipment (amplifiers, microphones, etc.) is commonly in decibels. For a specified terminal (output) power (\(P_2\)) there must be a reference power level \(P_1\). The reference level is generally accepted to be \(1\;\textrm{mW}\). The resistance associated with the \(1\;\textrm{mW}\) power level is \(600\;\Omega\), chosen because it is the characteristic impedance of audio transmission lines. When the \(1\;\textrm{mW}\) level is employed as the reference level, the decibel symbol frequently appears as \(\textrm{dBm}\).
\(\boxed{G_\mathrm{dBm} = \left.10\log_{10}\left(\frac{P_2}{1\;\mathrm{mW}}\right)\right|_{600\,\Omega}}\quad\textrm{dBm}\)
The decibel gain should more correctly be referred to as the voltage or current gain in decibels to differentiate it from the common usage of decibel as applied to power levels.
\(\displaystyle G_\mathrm{dB} = 10\log_{10}\left(\frac{P_2}{P_1}\right) = 10\log_{10}\left(\frac{\left.V_2\right.^2 / R_i}{\left.V_1\right.^2 / R_i}\right) = 10\log_{10}\left(\frac{\left.V_2\right.^2}{\left.V_1\right.^2}\right)\)
\(\boxed{G_\mathrm{dB} = 20\log_{10}\left(\frac{V_2}{V_1}\right)}\quad\textrm{dB}\)
Cascaded Stages
The magnitude of the overall voltage gain of a cascaded system is given by
\(\displaystyle \left|A_{v_T}\right| = \left|A_{v_1}\right|\cdot\left|A_{v_2}\right|\cdot\left|A_{v_3}\right|\cdots\left|A_{v_n}\right|\)
Applying the proper logarithmic relationship results in
\(\displaystyle G_v = 20\log_{10}\left|A_{v_T}\right| = 20\log_{10}\left|A_{v_1}\right| + 20\log_{10}\left|A_{v_2}\right| + 20\log_{10}\left|A_{v_3}\right| + \cdots + 20\log_{10}\left|A_{v_n}\right|\)
The equation states that the decibel gain of a cascaded system is simply the sum of the decibel gains of each stage.
\(\boxed{G_{\mathrm{dB}_T} = G_{\mathrm{dB}_1} + G_{\mathrm{dB}_2} + G_{\mathrm{dB}_3} + \cdots + G_{\mathrm{dB}_n}}\quad\textrm{dB}\)