# What is the meaning of |H(jw)| concerning the transfer function

Noob question here but my books talks about the magnitude of the transfer function |H(jw)| a lot but I don't understand the meaning of |H(jw)| itself. Because of this I can't interpret the meaning of Bode plots of filters as it is the Y-axis of these Bode plots.

A transfer function takes a complex number input (or in this case, pure imaginary $j \omega$), and produces a complex number output.

The "magnitude" of $H$ is simply the complex absolute value:

\begin{gather} |H(j \omega)| = \sqrt{\mathrm{real}(H(j \omega))^2 + \mathrm{imag}(H(j \omega))^2} \end{gather}

As far as a physical meaning, consider an input signal with an amplitude $V_0$ and frequency $\omega$. $H(j \omega)$ "transforms" this input signal into an output signal, with final amplitude $V_1 = |H(j \omega)| V_0$, but it also can shift the phase of the output signal with respect to the input signal.

To summarize, the two plots of the Bode plot are:

1. $|H(j \omega)|$, which tells you the ratio of input and output amplitudes (the "gain", or "attenuation")
2. $\tan^{-1}\left(\frac{\mathrm{imag}(H(j \omega))}{\mathrm{real}(H(j \omega))}\right)$, which tells you the phase shift (note: this formulation is only accurate in the 1st quadrant; see atan2 for the definition which extends to the entire complex plane).

$H(j\omega)$ is a complex number, say, $a+jb$, where $a$ and $b$ are functions of frequency, $\omega$. The magnitude of the complex number, $\sqrt {a^2+b^2}$, is the gain of the transfer function at $\omega \: rad\:s^{-1}$. And the phase angle at $\omega \:rad\:s^{-1}$ is: $arctan\large \left(\frac{b}{a}\right)$