# Definition of Bandwidth as angular frequency

I am searching for a Formal Definition of bandwidth given the transfer function of a system, or the bode plot.

Is it the frequency $$\omega_b$$ in which $$20log|H(j\omega_b)| = -3\text{dB}$$ or $$20log|H(j\omega_b)| = 20log|H(j0)| - 3\text{dB}$$

And if there is a definition without dB, I would like to know about it. Thanks in advance!

EDIT-homework: Application of definition: In which interval is $$\omega_b$$ contained? • What is the application? Where will this number be used?
– AJN
Jan 19, 2022 at 11:07
• @AJN I want to determine whether $\omega_b > 8.7$ or it will be approximately 5rad/s Jan 19, 2022 at 11:11
• However I still want to learn the definition Jan 19, 2022 at 11:12
• According to Wikipedia there appears to be multiple definitions. "*One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a specified level of performance. ... In the case of frequency response, degradation could, for example, mean more than 3 dB below the maximum value or it could mean below a certain absolute value. As with any definition of the width of a function, many definitions are suitable for different purposes. *".Without further context, I would guess 5r/s.
– AJN
Jan 19, 2022 at 11:21
• It depends on context. The bode plot above shows a plot with DC gain, so typ. use -3dB from the DC level. Jan 19, 2022 at 13:15

Observation: Bandwidth is defined as the frequency range $$[\omega_1 \ \omega_2]$$ over which the control of the system is "effective". Usually, $$\omega_1 = 0$$ and then, by definition, $$\omega_2 = \omega_B$$ is the bandwidth.
Definition: The (closed-loop) bandwidth, $$\omega_b$$, is the frequency where the norm of the sensitivity function, $$|S(j\omega)|$$, first crosses the $$-3\text{dB}$$ line from below. This means that: $$\big|S(j\omega_B)\big| = \frac{1}{\sqrt{2}} = 0.707 \approx -3\text{dB}$$
The bandwidth in terms of the closed loop transfer function $$T(s)$$, $$\omega_{BT}$$, is the highest frequency at which the norm of the closed loop transfer function, $$|T(j\omega)|$$, crosses the $$-3\text{dB}$$ line from above. However, this is usually a poor indicator of performance and is mathematically stated as: $$\big|T(j\omega_{BT})\big| = \frac{1}{\sqrt{2}} = 0.707 \approx -3\text{dB}$$
Τhe gain crossover frequency, $$\omega_c$$, is the frequency where the norm of the loop transfer function, $$|L(j\omega)|$$, first crosses the $$0\text{dB}$$ line from above. And mathematically: $$\big|L(j\omega_c)\big| = 1 = 0\text{dB}$$ These three frequencies, and for systems which have a phase margin of the order $$PM < 90^o$$, are related with one another by the following inequality: $$\omega_B < \omega_c < \omega_{BT}$$ All of the above definitions and frequencies of the system can be used as specifications for controller design and especially when using the loop shaping design method, where the designer tries to impose certain behaviour to the loop transfer function $$L(s)$$. As a last comment, I would like to write down the relations between the functions mentioned: $$L(s), S(s), T(s)$$. Suppose we have a plant described by $$G(s)$$ and which is controlled using the controller $$K(s)$$. Then:
$$L(s) = G(s)\cdot K(s) \rightarrow \text{Loop Transfer Function}$$ $$S(s) = \frac{1}{1+L(s)} \rightarrow \text{Sensitivity Function}$$ $$T(s) = \frac{L(s)}{1+L(s)} \rightarrow \text{Closed Loop Transfer Function}$$ $$P(s) = 1 + L(s) \rightarrow \text{System's Characteristic Polynomial}$$ Note that for the loop transfer function the multiplication should be $$G(s)\cdot K(s)$$ because when dealing with multivariable systems (MIMO) this is how you get the proper loop transfer matrix. Only if both matrices $$G(s), K(s)$$ are diagonal then the multiplications $$G(s)\cdot K(s) = K(s)\cdot G(s)$$ produce the same results. However, for single-input single-output systems (SISO), it doesn't matter.
• Thank you, I don't really get everything you said, if you could provide me with a link of the proof of the inequality $\omega_c < \omega_{BT}$ I would be grateful. Also the answer to the homework depends on the function we plot. If it is the open loop transfer function then from the plot we can only see $\omega_c$. Hence $\omega_b >8.7$.If it is the closed loop tf then $|H(j\omega_b)| = \frac{1}{\sqrt(2)} |H(j0)|$ Jan 20, 2022 at 12:19