Decibels

Decibels allow us to express large numbers, especially ratios and multiplications. It uses a logarithmic scale.

For example:

$$A_{dB}=20\cdot {\log }_{10}A$$

where $A$ is the magnitude of a signal or function.

For various values of A, we would have:

This is very handy for multiplying gains:

$$\begin{array}{rl}A& =A_1{A}_2{A}_3\\ A_{dB}& =20{\log }_{10}(A_1{A}_2{A}_3)\\ & =20{\log }_{10}{A}_1+20{\log }_{10}{A}_2+20{\log }_{10}{A}_3\\ & =A_{1_{dB}}+A_{2_{dB}}+A_{3_{dB}}\end{array}$$