# Bandwidth from a transfer function

Currently I'm having problem wrapping my head around the following.

Suppose you have a dynamical system described by the transfer function $$G(s)=\frac{as}{(s+b)(s+c)}$$ depending on the variables $a$, $b$ and $c$. In order to calculate the frequency response of the system $s=i\omega$. With that one is now able to draw the Bode plot wherein the magnitude specified by $$M(\omega)=20\log_{10}\lvert G(i\omega)\rvert$$ is plotted over $\omega$.

The bandwidth of this frequency response should now be defined (as far as I understood it) as the differences of frequencies for which the maximal magnitude is lowered by 3 dB.

Is this the correct definition? Because people mainly show example in which the maximal magnitude is 0 dB and the bandwidth then is calculated as the difference of frequencies for which the magnitude crosses 3 dB.

Would it then be correct to solve $$20\log_{10}\lvert G(i\omega)\rvert = 20\log_{10}G_\text{max}-3 \text{ dB}$$ for $\omega$ with \begin{align} 20\log_{10}G_\text{max}&=\frac{d}{d\omega}\left(20\log_{10}\lvert G(i\omega)\rvert\right)\\ G_\text{max}&=\frac{d}{d\omega}\lvert G(i\omega)\rvert \end{align} If so, could someone help me with that? I should be able to do that but seem to be slow on the uptake.

Thanks

The bandwidth of this frequency response should now be defined (as far as I understood it) as the differences of frequencies for which the maximal magnitude is lowered by 3 dB.

Both definitions are used. But I believe this also depends a bit on the field of study. For control engineering we use the cross-over frequency, the bandwidth is the frequency at which the magnitude is equal to 1, i.e., 0 [dB]. This is done because it is related to the stability at the system. Because if the magnitude response $|G(jw)|$ crosses at 0 [dB] and the phase $\angle G(jw)$ is larger then -180 degrees, you know your system is stable, assuming the magnitude does not cross 0 [dB] afterwards anymore. This has to do with Nyquist stability criterion.

However in electrical engineering, specific to filter design, The frequency defined at -3 [dB] is being used as the bandwidth. Because this is the magnitude at which the filter is not active anymore. From wikipedia

https://en.wikipedia.org/wiki/Bandwidth_(signal_processing)#X-dB_bandwidth

In basic electric circuit theory, when studying band-pass and band-reject filters, the bandwidth represents the distance between the two points in the frequency domain where the signal is $\frac{1}{\sqrt{2}}$ of the maximum signal amplitude (half power).

• Thanks for clearing that up. So, regarding my example, if $a=b=c=1$ the magnitude is always below -5 dB. Hence, Matlab's bandwidth command results in Inf. – Pascal May 27 '16 at 12:30
• From what I understands from Matlabs bandwidth command, nl.mathworks.com/help/control/ref/bandwidth.html, yes that is the reason it results in Inf. – WG- May 29 '16 at 20:40
• ps. @Pascal if the answer, answered your question, could you mark it as such that people know your question has been answered. – WG- Jun 2 '16 at 15:34