Equivalent Noise Bandwidth

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What is Equivalent Noise Bandwidth (ENBW or NBW)

Given a system with a transfer function $H(s)$, the spectrum of any input noise will be shaped by it. Under white noise (with equal power over all frequencies), it is convenient to replace the transfer function by a "brickwall" filter, in which the noise power is the same up to a certain frequency $\Delta f$ and after that is zero. That frequency is defined so that the total output noise power is the same for both the system and the brickwall filter. Thus, the area under this rectangle must be the same as the area under the original $H(s)^2$ system (the square means the input and output are powers instead of amplitudes).

$$ A_{H(s)} = A_{rectangle}$$ $$ \int_0^\infty \left|H(j\omega)\right|^2 \partial \omega = H_{max}^2 2\pi\Delta f$$ where $H_{max}$ is the maximum value of $H(s)$. $$ \Delta f = \frac{1}{2\pi}\int_0^\infty \frac{\left|H(j\omega)\right|^2}{H_{max}^2} \partial \omega $$

For example, for the 1st order low-pass filter $$H(j\omega)=\frac{H_{max}}{1+j\frac{\omega}{\omega_{3dB}}}$$ $$ \left|H(\omega)\right| = \frac{H_{max}}{\sqrt{1+\left(\frac{\omega}{\omega_{3dB}}\right)^2}}$$ Solving for $\Delta f$: $$\Delta f = \frac{1}{2\pi}\int_0^\infty \frac{1}{1+\left(\frac{\omega}{\omega_{3dB}}\right)^2} \partial \omega $$ $$\Delta f = \frac{1}{2\pi}\int_0^\infty \frac{1}{1+x^2} \partial x\omega_{3dB}\qquad\qquad\left(x = \frac{\omega}{\omega{3db}}\right)$$ $$\Delta f = \frac{\omega_{3dB}}{2\pi}\int_0^\infty \frac{1}{1+x^2} \partial x $$ $$\Delta f = \frac{\omega_{3dB}}{2\pi} \left.\tan^{-1}\frac{\omega}{\omega_{3db}}\right|_0^\infty $$ $$\Delta f = f_{3dB}\frac{\pi}{2} $$

There are some tabulated values for when the systems are low-pass filters of different orders.

Order$\Delta f$
11.57 $f_{3dB}$
21.22 $f_{3dB}$
31.15 $f_{3dB}$
41.13 $f_{3dB}$
51.11 $f_{3dB}$
Higher order filters will be closer to the "brickwall" filter, thus the coefficient is closer to 1.