Library / Circuits, electronics and noise / ELN-49
Circuits, electronics and noise
Noise bandwidth of a first-order low-pass filter
Equivalent noise bandwidth of a single-pole, unity-gain low-pass filter with a −3 dB frequency fc.
- Ref
- ELN-49
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- Bn [Hz]
A first-order low-pass filter passes slightly more white noise than a brick-wall filter at the same −3 dB frequency, because its response falls slowly. Integrating |H|² = 1/(1 + (f/fc)²) over all frequencies gives the equivalent noise bandwidth Bn = (π/2) fc ≈ 1.571 fc. Use Bn, not fc, in noise formulas such as Johnson noise. Higher-order filters are closer to 1 times fc.
Equivalent noise bandwidth of a single-pole, unity-gain low-pass filter with a −3 dB frequency fc. It is the width of the ideal brick-wall filter that passes the same white noise power, and is larger than fc.
Calculator
- Ratio of noise bandwidth to cut-off, π/21.571
- Noise bandwidth of a 2nd-order Butterworth filter, 1.111 fc1,111 Hz
- Johnson noise of 10 kΩ at 300 K in this bandwidth5.101×10−7 V
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A first-order filter with a cut-off of 1,000 Hz passes the noise of a 1,571 Hz wide ideal filter.
- Using fc instead of Bn would underestimate the noise voltage by 20.2 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| fc Cutoff (−3 dB) frequency | 1 to 1,000,000 Hz | 1.571 | 1,571,000 | 1,571,000 | +10% | +10 % |
Sweep of Cutoff (−3 dB) frequency
| Cutoff (−3 dB) frequency (Hz) | Noise bandwidth (Hz) |
|---|---|
| 1 | 1.571 |
| 5.623 | 8.833 |
| 31.62 | 49.67 |
| 177.8 | 279.3 |
| 1,000 | 1,571 |
| 5,623 | 8,833 |
| 31,620 | 49,670 |
| 177,800 | 279,300 |
| 1,000,000 | 1,571,000 |
Sweep one variable
Variables and how each one contributes
Noise bandwidth
What the equation gives you. Everything else feeds it.
Cutoff (−3 dB) frequency
The −3 dB cut-off frequency, in Hz. Raises the noise bandwidth in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 1,571 to 1,728 Hz.
Worked example with the values above
- Cut-off frequency: fc = 1,000 Hz.
- Multiply by π/2 = 1.5708.
- Bn = 1,570.8 Hz.
Assumptions and limits
- White noise with a uniform spectral density over the stated one-sided bandwidth.
- Independent, uncorrelated noise sources, so powers (not voltages) add.
Common mistakes
- Using the −3 dB bandwidth as the noise bandwidth.
- Using this value for a filter of higher order.
- Using a two-sided bandwidth.
Statistics and uncertainty
Noise values are RMS statistics: the voltage at any moment is random, and its peak-to-peak value is about six times the RMS for a Gaussian distribution. Measured noise also depends on the bandwidth and on flicker noise at low frequencies.
Where it is used
- Estimating the noise floor, resolution and signal-to-noise ratio of sensor and measurement circuits.
- Choosing amplifiers for a given source resistance.
Maths you need
- Spectral density: power or voltage squared per hertz
- RMS: the root of the mean square
- Boltzmann’s constant: 1.380649 × 10⁻²³ J/K