Equivalent Noise Bandwidth
# Equivalent Noise Bandwidth: Count the Whole Filter Response
Equivalent noise bandwidth (ENBW) is the width of an ideal rectangular filter that passes the same white-noise power as the real filter, after accounting for its reference gain. It translates a shaped frequency response into a useful integration width for noise calculations.
The −3 dB cutoff identifies one point on a response. ENBW depends on the area under the entire squared-gain curve, including attenuation beyond that cutoff and any gain peaking.
## 1. Follow the equal-area construction
*Calculated single-pole response and its equivalent ideal rectangle. The plot ends at 4fc, but the blue tail continues beyond the drawing. Equal area refers to the full integral from zero to infinity, not just the visible portion.*
For a low-pass transfer function H(f) with nonzero DC gain H₀, the one-sided definition is Bn = ∫₀∞ |H(f)|² df / |H₀|². If the input voltage-noise amplitude density en is white, output noise is vout,rms = en |H₀| √Bn. Use en in V/√Hz and Bn in hertz; the result is volts RMS.
For a unity-gain single pole, |H(f)|² = 1/[1 + (f/fc)²], so Bn = (π/2)fc ≈ 1.571fc. Equivalently, an unloaded RC low-pass has Bn = 1/(4RC). The orange point in the diagram is half power at fc; the rectangle extends farther because it must include noise that passes through the real filter’s tail.
## 2. Compare responses, not only pole counts
*Analytical examples, not measurements. Both filters have unity DC gain and the same overall −3 dB cutoff. The miniature curves show normalized power gain; the numerical values come from integrating the complete ideal responses.*
A second-order Butterworth low-pass has |H(f)|² = 1/[1 + (f/fc)⁴], giving Bn = [π/(2√2)]fc ≈ 1.111fc. With fc = 10 kHz and en = 10 nV/√Hz, output noise is about 1.253 µV RMS for one pole and 1.054 µV RMS for the second-order Butterworth response.
Two cascaded, isolated, identical RC poles have a different response. If each pole has corner fp, |H(f)|² = [1 + (f/fp)²]⁻² and Bn = πfp/4. Their overall −3 dB cutoff is about 0.644fp, making the ratio Bn/fc about 1.220. Thus “two poles” alone does not specify the ENBW factor. Unbuffered RC sections can load one another and require analysis of the actual network.
## 3. Use the correct transfer path for each source
A circuit can have several noise sources that see different transfer functions. An input resistor, amplifier voltage noise, amplifier current noise and a later stage’s resistor noise do not necessarily share the signal gain or the same integration bandwidth. Propagate each source to the observation node, then combine independent contributions in mean-square units.
For nonwhite noise, use the full expression vout,rms² = ∫ Sin(f)|H(f)|² df over the relevant band. A single white-noise ENBW factor does not capture flicker noise, spectral lines or frequency-dependent source density. State the lower and upper integration limits when the analysis is band limited.
| Quantity | Meaning | Units |
|---|---|---|
| −3 dB cutoff | Frequency where power gain is half its reference value | Hz |
| Amplitude noise density | Square root of noise power density | V/√Hz |
| Equivalent noise bandwidth | Reference-normalized squared-gain area | Hz |
| Integrated output noise | Square root of filtered noise power | V RMS |
## 4. Verify the filter used in the noise budget
Start from the actual transfer function, including loading, amplifier bandwidth and gain peaking. Integrate squared magnitude with adequate frequency resolution and a sufficiently high upper limit. Confirm that extending the integration range does not materially change the result; a plot with a short frequency axis can hide a meaningful tail.
Keep one-sided versus two-sided conventions consistent, and use hertz rather than angular frequency unless the integration includes the required 2π conversion. For a band-pass response, choose and state the normalization gain instead of dividing by a zero DC gain.
Analog ENBW and the ENBW of an FFT window are related equal-power concepts but refer to different operations. A spectrum analyzer or digital filter may specify its own noise bandwidth; use the relevant response and calibration. In sampled systems, account for aliasing and the complete analog/digital signal chain rather than treating analog ENBW as an antialias guarantee.
The resistor thermal noise density becomes an RMS voltage only after its frequency-dependent path and measurement band are specified. ENBW supplies that integration width for the white-noise case.
## References
- [Analog Devices: Activity — Noise](https://wiki.analog.com/university/labs/noise_adalm2000) — equal-noise brick-wall interpretation.
- [Analog Devices: Understanding and Eliminating 1/f Noise](https://www.analog.com/en/resources/analog-dialogue/articles/understanding-and-eliminating-1-f-noise.html) — one-pole noise bandwidth and nonwhite noise.
- [Analog Devices: Practical Filter Design for Precision ADCs](https://www.analog.com/en/resources/analog-dialogue/articles/practical-filter-design-precision-adcs.html) — filter response and RC noise-bandwidth considerations.