Noise, and where the microvolt goes
Thermal, 1/f, amplifier, and electrode noise, added up for a real EEG channel. Why the electrode dominates, why the breadboard was so bad, and how to read a noise floor.
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You measured a noise floorNoise floorWhat your instrument records when the input is shorted or connected to a perfectly quiet source; everything smaller than this is invisible to you. Glossary entry with the inputs shorted, and a worse one with electrodes in the phantom. The difference is the electrodes. Let’s account for every microvolt: where noise comes from, how sources add, and why in a well-designed EEG system the chip is rarely the limit.
Every resistor hisses
A resistor at room temperature produces a random voltage across itself just from the thermal motion of its electrons. The RMS value is √(4 k T R Δf): Boltzmann’s constant, temperature, resistance, bandwidth. For 10 kΩ over a 40 Hz bandwidth at body temperature that is about 80 nV RMS. For 1 MΩ, about 800 nV. This is thermal noiseThermal (Johnson) noiseThe random voltage every resistor produces just by being warm; about 13 nanovolts per root hertz for 10 kilohms, which matters at microvolt levels. Glossary entry and there is no circuit trick that removes it; there is only lowering the resistance or the bandwidth or the temperature.
An electrode with an impedance of 50 kΩ at 10 Hz is, for noise purposes, a 50 kΩ resistor: about 180 nV RMS over the EEG band, per electrode, and there are two in every measurement. A dry electrode at 1 MΩ is 800 nV each. That is already most of a microvolt before the amplifier has done anything.
The amplifier adds two kinds
A voltage noise, quoted in nanovolts per root hertz, which for a good instrumentation amplifier is 8 to 10 nV/√Hz, so about 60 nV RMS over 40 Hz. Negligible against the electrodes. And a current noise, in femtoamps or picoamps per root hertz, which flows through the electrode impedance and becomes a voltage: current noise times impedance. For the INA128’s 0.3 pA/√Hz through a 50 kΩ electrode that is 15 nV/√Hz, comparable to its voltage noise; through a 1 MΩ dry electrode it is 300 nV/√Hz, which dominates everything. This is why chips meant for dry electrodes are chosen for low current noise.
The low-frequency rise
Below a few hertz, every amplifier and every electrode gets noisier, following roughly one over frequency. This 1/f noise1/f noiseNoise that grows at low frequencies, present in amplifiers and electrodes, and the reason sub-hertz EEG is hard. Glossary entry comes from slow fluctuations in materials, in the chip’s transistors and in the electrode’s chemistry. It is why the noise floor you measured is dominated by slow wander, why a 0.5 Hz high-pass makes the trace look so much cleaner, and why sub-hertz EEG (which exists and is interesting) is hard to record.
Adding them up
Independent noise sources add as the square root of the sum of squares, not directly. Two 180 nV electrodes and a 60 nV amplifier: √(180² + 180² + 60²) ≈ 260 nV. The biggest source dominates; halving a small source does almost nothing. This is the arithmetic that tells you where to spend effort: a 1 µV-noise chip replaced by a 0.5 µV chip does nothing if the electrodes contribute 2 µV.
Your shorted-input noise floor is 0.8 µV RMS. With electrodes in the phantom it is 2.1 µV RMS. What are the electrodes contributing?
About 1.9 µV. Independent sources add in quadrature: √(2.1² − 0.8²) = √(4.41 − 0.64) ≈ 1.94 µV. The electrodes are contributing more than twice what the electronics do, which is the normal situation, and it tells you that better electrodes, not a better chip, is the next improvement.
Why the breadboard was so bad
None of the above explains a noise floor of 15 µV, which is what a breadboarded amplifier typically shows. That is not thermal noise; it is pickup. Long, springy, unshielded connections at a high-impedance node act as antennas for the room’s electric field, and the resulting voltage is far larger than any thermal source. The cure is short, soldered, and, for the input node, guarded. That is why the noise floor of the same circuit falls by a factor of five when it moves from breadboard to soldered board, and why the ADS1299 layout rules exist.
Reading a noise floor
Report it as RMS in microvolts over a stated bandwidth, input-referred, with the inputs shorted at the electrode connector. “0.9 µV RMS, 0.5 to 40 Hz, gain 24, 250 SPS” is a complete statement; “low noise” is not. Also report peak-to-peak, which is roughly six times RMS for Gaussian noise and is what you see on the trace. And plot the noise spectrum: its shape tells you which source dominates. A flat floor is thermal or amplifier noise; a rise at low frequency is 1/f; a spike is pickup at that frequency.
Deep dive Noise density versus RMS 3 min
Datasheets quote noise as a density in nV/√Hz because the RMS value depends on bandwidth. To convert, multiply by the square root of the bandwidth: 8 nV/√Hz over 40 Hz is 8 × 6.3 ≈ 50 nV RMS. For a 1/f source the integral is a logarithm and the low cutoff matters; that is why “from 0.1 Hz” and “from 1 Hz” noise figures differ so much.
Deep dive Why gain helps with noise, up to a point 3 min
The converter has its own noise, referred to its input. An amplifier of gain G in front of it divides that noise by G when referred to the electrodes. So more gain makes the converter’s noise matter less, until the signal plus offsets threaten to exceed the converter’s range. That trade sets the gain of 24 in the ADS1299 for EEG. For EMG, whose signals are bigger, a lower gain is used and the converter’s noise contributes more, which is acceptable because EMG is bigger.
Deep dive Averaging and the square root of N 2 min
Noise that is independent from trial to trial shrinks as the square root of the number of trials averaged. That is the whole basis of the event-related potential, which Phase 3 is about: a 5 µV response in 20 µV of noise is invisible in one trial and clear in a hundred. The signal-to-noise ratioSignal-to-noise ratio (SNR)How big the thing you want is compared to everything else, often in decibels; the number every design decision is trying to raise. Glossary entry after averaging is the single-trial ratio times √N.
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