Input Referred Noise
# Input-Referred Noise: Compare Amplifiers at the Signal Input
Input-referred noise is the equivalent noise at a chosen input reference plane that would produce the observed output noise through the circuit’s signal transfer function. It puts noise contributions from different locations on a common scale, making amplifier and signal-chain comparisons easier to interpret.
The reference plane matters: the sensor terminals, an amplifier pin and a complete circuit’s input port can have different gains and impedances between them. State which one is being used.
## 1. Build the equivalent model
*Conceptual noise-equivalent model of a linear system. The lower amplifier retains the same signal transfer but its internal noise is represented at the selected input plane. The noise-source icon is symbolic and does not imply a sine wave.*
For voltage amplitude spectral densities, ein,eq(f) = eout(f)/|G(f)|, where G(f) is the actual voltage transfer from the selected input to the output. In power-density units, Sin,eq(f) = Sout(f)/|G(f)|². Dividing a power density by gain rather than gain squared gives the wrong result.
This is a mathematical reference change; it does not move the physical noise source. If the output measurement includes source noise, downstream instrument noise or interference, those contributions are also present in the derived result unless they have been separated appropriately.
## 2. Normalize before comparing
*Synthetic values illustrating gain normalization, not measurements or product claims. Both gains are flat across the same measurement band. The output-noise bars use one common scale and the input-noise bars use a separate common scale.*
Amplifier A has 20 µV RMS at its output and gain magnitude 10, giving 2 µV RMS referred to input. Amplifier B has 100 µV RMS at its output and gain magnitude 100, giving 1 µV RMS referred to input. B’s larger output number does not mean it has the larger input-equivalent noise.
Dividing an integrated RMS number by one gain value is valid when that gain is effectively constant over the relevant band. For frequency-dependent gain, refer the spectrum first and then integrate: vin,eq,rms² = ∫ Sout(f)/|G(f)|² df over the stated band. Near a transfer zero, input referral may become unbounded or unhelpful; specify a meaningful passband.
## 3. Distinguish circuit input noise from op-amp en
A data-sheet op-amp voltage-noise density en models one internal source. The complete circuit’s signal-input-referred noise can be different because that source reaches the output through noise gain, while the external signal follows signal gain.
For an ideal unity inverter, signal gain is −1 and noise gain is 2. An op-amp en of 5 nV/√Hz contributes 10 nV/√Hz at the output. Referred to the external signal input, that contribution is also 10 nV/√Hz, not the 5 nV/√Hz data-sheet source value. Resistor noise and input current noise add further contributions through their own transfer paths.
## 4. Combine stages and sources consistently
For two cascaded stages with flat gains, uncorrelated equivalent input voltage-noise densities e1 and e2, and first-stage gain G1, the chain’s amplifier-added input noise is √[e1² + (e2/|G1|)²] under consistent bandwidth and impedance assumptions. If e1 = 3 nV/√Hz, e2 = 20 nV/√Hz and G1 = 10, the result is approximately 3.61 nV/√Hz.
Early gain reduces the input-referred importance of later-stage noise. It also consumes output headroom and can change bandwidth, distortion and overload recovery. Increasing gain is therefore part of a signal-chain tradeoff, rather than a universal noise fix.
| Reported quantity | Meaning | Required context |
|---|---|---|
| Input voltage-noise density | Amplitude density at a stated input plane | V/√Hz, frequency, source conditions |
| Integrated input noise | RMS after input referral and integration | Band limits and transfer function |
| Amplifier-added noise | Noise attributed to the amplifier or circuit | Source/instrument treatment |
| Source-inclusive noise | Total including the signal source’s noise | Source impedance and temperature |
For independent contributions, add power densities or mean-square voltages. Include correlation terms where sources are correlated. If a measured baseline is removed, subtract independent contributions in power units and account for uncertainty; subtracting RMS amplitudes directly is generally invalid.
## 5. Verify the comparison
Use the same reference plane, bandwidth, gain setting, source impedance and loading. Distinguish a spectral density at one frequency from integrated RMS noise, and record whether the result includes the source and measurement instrument. For a current-input circuit, the appropriate input referral may be current noise obtained by dividing output voltage noise by transimpedance in ohms.
The equivalent noise bandwidth helps integrate white noise through a known response; it does not replace the gain normalization that defines the input reference plane.
## References
- [Analog Devices: Practical Input-Referred Calculations in Precision Systems](https://www.analog.com/en/resources/analog-dialogue/articles/practical-input-referred-calculations-in-precision-systems.html) — referring contributions to a common input plane.
- [Analog Devices: Low Noise Amplifier Selection Guide](https://www.analog.com/en/resources/app-notes/an-940.html) — noise-source models and signal/noise-gain distinctions.
- [Analog Devices: Low Noise Instrumentation Amplifiers](https://www.analog.com/en/resources/technical-articles/low-noise-instrumentation-amplifiers.html) — input- and output-stage contributions in instrumentation amplifiers.