What is Decibels; Bode diagram?

The gain in decibels is $G_{\mathrm{dB}} = 20\log_{10}G$: a factor $10$ is $20\,\mathrm{dB}$, a factor $2$ is $6\,\mathrm{dB}$, $1/\sqrt2$ is $-3\,\mathrm{dB}$. The Bode diagram plots $G_{\mathrm{dB}}$ and $\varphi$ against $\log_{10}\omega$ (or $f$). The cutoff frequency is where $G$ falls to $G_{\max}/\sqrt2$ ($-3\,\mathrm{dB}$ from the maximum); the passband is the range of frequencies within $3\,\mathrm{dB}$ of the maximum. A decade is a factor $10$ in frequency, an octave a factor $2$.