Answer
b) sounds very logical, but it is to pessimistic because it does not appreciate
what dithering can do to improve the resolution of an ADC.
The
reasoning followed in answer b) is valid when the LSB value is larger
than the total peak-to-peak noise level (background EEG + electrode +
amplifier noise). In this case, the ADC indeed is only able to capture
signals larger than the LSB value. However, when one or more LSBs are
smaller than the noise, these LSB are constantly toggling between values
due to the noise input -in technical terms:these bits are dithered- and
the capturing of small signals becomes a purely statistical effect. It
is amazing to see that even the technical departments of some well know
manufacturers do not seem to understand this and they keep speaking of
LSB values in terms of "minimum
resolvable signal" Consider for example a system with 1 uV LSB
value. Now suppose that a small 0.1 uV ERP signal is riding on a 10 uVpp
total noise floor. In 9 out of 10 sweeps, the large 10uVpp total noise
signal fully determines the ADC output and the 0.1 uV signal won't be
able to drive the ADC to its next level. However, in 1 out of 10 sweeps
there will occur the situation where the 0.1uV signal is enough to change
the LSB of the ADC. After averaging a large number of sweeps, a perfect
0.1 uV signal will appear on screen, even though the original sweeps are
captured with an 1 uV resolution. The 1 uV LSB value is used in our previous
16 bit ActiveOne system, and our
customers have indeed been able to measure very clean small ERP signals
with the system, see this ERP example.
The extra 5 LSBs in the ActiveTwo (LSB is
31.25 nV) makes the averaging process a little more precise, resulting
in minor improvements of SNR of the averages, but there is not an essential
difference with the 1 uV example given above. |
So,
signals buried in noise are quantized in all the LSBs below the noise level.
However, it is important to realize that most of the information is already
captured by the "most significant noise bit" (the first bit below
the amplifier noise level). Even more important is that the LSB value does
not impose any limit on the smallest detail in the signal that can be recovered
with averaging. In other words: the ActiveTwo system is well capable of
recovering signal details smaller than 31.25 nV. Some
competing system with adjustable gain are
able to select even smaller LSB values than we use in our new ActiveTwo
system. Customers often ask us how much effect adding more LSBs buried
in noise actually has. As an example, we can calculate the difference
in performance between ActiveTwo's 31.25 nV LSB (with DC amplifier
and 524 mVpp input range), and the 7 nV LSB of typical competing AC
design on it's highest gain setting on it's highest gain setting (with
DC measurements eliminated, and a mere 400 uV input range).
The
input noise caused by the analog amplifier circuitry of both the competing
sytem and the ActiveTwo system is approx. 0.5 uVrms for a 100 Hz bandwidth
(approx. 2 uVpp, all contemporary systems are near this value). The digitalization
process generates quantization noise, this is extra noise caused by the
introduced "steps" in the signal (the smooth signal becomes
like a staircase, the steps represent extra noise). The amount of quantization
noise is proportional with the LSB value (large steps add much noise):
noise theory predicts that the quantization noise has a magnitude of LSB/Ö12
(approx. 0.3 LSB). So, the ActiveTwo adds 31.25/Ö12
= 9 nVrms quantization noise, whereas the competing system adds 7/Ö12
= 2 nVrms. The amplifier noise and quantization noise are independent
noise sources, so the powers of these noise sources should be added to
find the total noise level. See formula given above. Lets see how this
works out when we do this calculation for both the ActiveTwo, and for
the competing system on its highest gain setting:
Competitor:
total noise = Ö{
(amp noise)^2 + (quan noise)^2 } = Ö{
(0.5)^2 + (0.002)^2 } = 0.500004 uVrms |
ActiveTwo:
total noise = Ö{
(amp noise)^2 + (quan noise)^2 } = Ö{
(0.5)^2 + (0.009)^2 } = 0.50008 uVrms |
In
a real-world ERP measurement, this difference in noise would even be smaller
than this 0.02% because the noise level is not determined by the relatively
small 2 uVpp amplifier noise, but by the 10-100 uV of background EEG.
The effect described above can be visualized with a simple LabVIEW demonstration
program which can be downloaded from here.
You need the LabVIEW Run
Time Engine to be able to run this program. When you experiment with
various settings for noise amplitude, signal amplitude and LSB, the program
will be very educational showing the averaging of ERPs which are drowned
in noise. The demonstration shows 5 effects: |
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Click
on the picture to download demonstration program |
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1) |
The
averaging process only works properly so long as one or more LSBs
are dithered by noise. For a given LSB value, decreasing the noise
will actually lead to a too low noise level, resulting in corruption
of the average results. |
2) |
If the LSB is not dithered, meaningful average results can only be
calculated for signals much larger than the LSB. |
3) |
Provided
the LSB is dithered, signals smaller than the LSB can be averaged
out of the noise without any problem. The LSB is no "magic border"
whatsoever, there is a gradually decrease of SNR when the signal
amplitude is decreased below the LSB value. |
4) |
Provided
the LSB is dithered, SNR of the average increases with the square
root of the number of sweeps. Because of the square root relation,
the first few sweeps make the most difference, after many sweeps,
it takes a lot of time to see improvements. Recovering a small signal
from a lot of noise is always possible, but it might take a lot of
sweeps (and time). |
5) |
Decreasing
the LSB below approx. 1/10th of the peak-to-peak noise, has no significant
effect on the SNR of the average result. |
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So,
the strange thing about noise is that you do not want too much of
it (because the averaging process would take too long), but that you
also do not want too less of it (because the LSB has to be dithered). |
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The
demonstration program allows you to simulate the LSB values and average
results of several systems from BioSemi and it's competitors. All
commercially available systems have an input noise in the order of
3uVpp (100 Hz bandwidth) and have their LSB small enough to be sufficiently
dithered. Consequently you won't see much difference between the systems
in terms of ERP result. Your choice between the systems therefore
has to be based on other features. |
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