Short note: Noise equivalent bandwidth

The noise equivalent bandwidth (NEB) $B_N$ of a filter is the bandwidth of an ideal rectangular filter, with the same maximum gain $\lvert H(0)\rvert$, that passes the same noise power as the real filter when both are driven by white noise:

$$B_N=\frac{1}{\lvert H(0)\rvert^2}\int_0^{\infty}\lvert H(f)\rvert^2df$$

• Output noise power can then be written simply as $N=N_0B_N\lvert H(0)\rvert^2$ (or $kTB_N$).
• It is used in noise figure, noise temperature and receiver SNR calculations.
• Example: RC low-pass filter, $B_N=\frac{1}{4RC}=\frac\pi2f_{3dB}\approx1.57f_{3dB}$.
• For sharp-cut-off filters $B_N\approx$ 3 dB bandwidth; for gentle roll-off filters $B_N$ is larger.