Yess....the equations are available with Product specialist of each manufacturer. Pls contact the technical team.

There could be changes in the equation, depending upon software.

Dear M. Farooq,

If you would model the noise as a white Gaussian noise ( as done in the simplest and most probably almost anywhere in literature, depending on the application ) then denote:

n(t) (noise) to be a Gaussian random variable (process here since it is a function of t, or stochastic too ) of zero mean and variance sigma^{2}.

Now let's say that you would like to collect N samples of your analog signal :

y(t) = x(t) + n(t)        (1) 

Sampling at fs = 20 MHz (for example) then the N samples are spaced by (!/fs = Ts = 50 nsec), in order to collect these samples, you sample at time instants:

t = ts + nTs (depending on index n = 0 .. N -1 is index n in eqn (1) ) where ts is your first sample. 

OK now whatever your sampling frequency is (provided you have a sufficient number of data samples N) , your signal-to-noise ratio (SNR) is consistent because the power of a signal is the variance of this signal i.e. var(x(n)) (average over the square of your samples). You would of course get a better estimate as N increases but that's not the point here.

So, SNR = (var(x)/var(n)) ind of your sampling rate.

However, if one chooses to OVERSAMPLE, then you would reduce the noise.

HOW:

Well, in time domain, if you're oversampling say by a factor of 2 (or from fs = 20MHz to fs= 40 MHz) you would for example choose generate the following sequence.

Given at 20 MHz the samples of your signal are {y(1),y(2),y(3), ... } .

At 40 MHz, the samples of your signal are {y(1),0,y(2),0,y(3),0,...}, followed by a low (or band)pass filter from [-40MHz/2,+40MHz/2] (because Nyquist says for a sampling frequency fs , your information is bounded between -fs/2 and +fs/2) BUT YOUR INFORMATION IS FOUND BETWEEN [-20/2MHz, +20/2 MHz] since you did not add any new information to the signal SO YOU FILTER AT [-20/2MHz, +20/2 MHz] .

In that way, noise power is reduced by a factor of 1/M ( M=2 upsampling factor).

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WHY?

In time domain: Roughly speaking, signal power is the same, noise is somehow zero on the zeros padded between 2 consecutive samples .

In frequency domain: Precisely, after upsampling and before filtering. The noise has spread from the original band [-20/2 MHz , +20/2 MHz] to [-40MHz/2,+40MHz/2]. 

So the only information in the band [+ 20/2 MHz, + 40/2 MHz] and it's negative symmetry is noise :-). SO NOISE HAS BEEN SPREAD with its same power sigma^2.

Because you would filter at [-20/2 MHz , +20/2 MHz], you would also (in that way) filter the noise spread. So you end up with less noise, mathematically it is :

Let Pn_new be the new power of your noise after upsampling by a factor of M.

Let Pn_old be the power of your noise (which is sigma^2). Then:

Pn_new = Pn_old/M.

Hope this helps.

It is not really meaningful to compare detector noise at different sampling rates when dealing with chromatography. In order to produce a signal with a good signal-to noise ratio, the detector measures e.g. light almost 100 per cent of the time. In modern detectors analog-to-digital conversion is part of the detector. For older detectors A/D conversion was part of the chromatography data system or "integrator". In both cases, the A/D converter must be constructed is such a way that it integrates the signal  between consecutive sampling. If not, the performance will deteriorate seriously as the sampling rate is lowered. Now, if the detector actually is measuring all of the time, then reducing the sampling rate, will reduce the noise (e.g. measured as Root Mean Square of signal variance, RMS noise) and increasing the sampling rate will increase the noise. But this is not a big deal, as long as the number of datapoints over the peak is sufficient. Because what counts at the end is how well the chromatographic peak is measured (normally, the peak area is used). And a good chromatography data system (software with algorithms used for peak detection and integration) will do the job: It eliminates fast noise if this was not already done by using a lower sampling rate.

"Time constant" comes from old analog detectors, which could have an electric capacitor connected to dampen fast noise (which was not really optimal). For modern detectors (or chromatography data systems) the "time constant" is a measure of the digital sampling rate reduction, and should be proportional to the expected chromatogrphic peak width. 

There is also a purely mathematical anser to your question, with litlle relevance to chromatography: 

There is no general exact relationship between noise and sampling rate, because the characteristics of the noise (the noise spectrum in the frequency domain) may vary. In the case when the noise is so-called white noise (detecting radioactive decay which is totally random would give you white noise), then the relationship is simple: Noise is directly proportional to the square root of the data collection frequency.

Dr. Anhnoff, I'd rather agree with you now that perhaps there exists no mathematical relationship between sampling frequency and the "white noise". All my recent searches to find something relevant have gone in vain. Yet everyone talks about it qualitatively. When you say "if the detector actually is measuring all of the time, then reducing the sampling rate, will reduce the noise (e.g. measured as Root Mean Square of signal variance, RMS noise) and increasing the sampling rate will increase the noise." how do you quantify this increase or decrease as a function of sampling frequency. I think this is the gist of my question. Practicing chromatographers can easily see that increasing sampling frequency the baseline becomes very noisy. When you say that noise is proportional to the square root of the data collection frequency, as far as I understand, this goes for shot noise limited cases where the light levels are very low e.g. in fluorescence and Raman spectroscopy and the statistics is governed by "counting".

I agree with you. While Dr. Ahnoff's analogy is valid for counting statistics (not just radionuclides; it goes for electron multipliers as well) the conventional view of "noise" in a light-based detector doesn't seem to hold. I think that part of the issue is a hardware issue; in order to model a light-based detector you have to fully control the light source and that is simply not done in practice. Anyone who's done any appreciable amount of spectroscopic detection knows that at low levels the gain is not proportional to the peak area - sure would be nice it is was! Since the gain is a reflection of the detection circuitry in physical terms it means that there is inherent instability in the absorption/detection circuit that can't be accounted for in the signal. I think your quest is noble and necessary; I also think that it has to be dealt with in theoretical terms, not in terms of actual instrumental output unless you can build an incredibly stable emitter/detector system.

There is a recent article in Analytica Chimica Acta "The Myth of Data Acquisition Rate". It is an interesting thing to read (one may not fully agree though). Here is the link:

http://www.sciencedirect.com/science/article/pii/S0003267014013518

I think that the discussion around your questions is a little confusing. Maybe because neither of the terms Sampling Rate and Noise has been defined at the start of the dicussion.

Sampling rate: A sample of the signal could either be the collected signal over a time period of fixed length (e.g. 1 ms) independent of sampling rate, or it could be the collected signal over (more or less) the entire time period between two consecutive samples. These two cases will of course be quite different when it comes to how sampling rate affects noise (or rather signal-to-noise)

Noise: When you ask for a relation between noise and sampling frequency there might be a silent assumption that less noise is better and more noise is worse. But it is not that easy.  Noise consists of variations in the signal with different frequencies. Variations with very low frequency are seen as "baseline drift" and can be handled prety well by chromatography software. Variations with a frequency comparable to that of the peak (the reciprocal of the peak width) will probably affect the measurement of your chomatographic peak. Variations with frequency much higher than that of the chromatographic peak can be filtered out (before or after data collection) and is therefore not really a problem. The higher sampling rate you have, the more you will see of high frequency noise, and if you calculate a RMS-noise value, this will definitely be higher. If you want to know how much higher, then you need to have a frequency spectrum of the noise.

A note on raw continuous analogue signals such as the current from a spectrophotometric detector (often a reverse-biased photodiode).

The current or voltage is amplified, the frequency response of the amplifier being chosen to pass all the frequency components of the sharpest likely peak, and attenuate components of higher frequency. The amplified signal is fed into an analogue-to-digital convertor (ADC) that (probably) has a fixed conversion frequency which ideally should be so rapid that the digital output changes little from one sampling to the next.

I don't know the current details, but the manufacturer may choose to provide the digital output from the instrument as the average of ADC values each so-many milliseconds (bunching). It may be possible for the user can adjust this at the level of the instrument, but usually one configures the data system to bunch the intitial bunches into longer ones. If the detector is properly designed (fast ADC...), the intermediate and final bunches should be equivalent to the normalised  integral of the continuous signal over this bunching interval.

The bunching process helps separate peaks from the higher frequency components of the noise; it's a kind of lag (low-pass) filter like an analogue R-C network, that will cause the peaks to tail. You choose a fast bunch rate for best precision when the S/N ratio is high and a slower one (< 10 bunches/peak) for trace analytes.

The data system should provide digital filtering, of which the realisation is a cleverly weighted running average of the bunches. Unlike a lag filter, the output at a given time takes account of the signal at both past and future times. That way, it's more discriminating than low-pass filtering, and symmetrical peaks don't become unsymmetrical. It can be difficult for a user to optimise bunch rates and digital filter parameters.

I mentioned elsewhere that if noise is limiting you can run the chromatogram more slowly and reduce the bunch rate in proportion. With spectrophotometric detectors the baseline is the maximum of the primitive signal (light intensity), and as another contributor mentioned, it would be unwise to assume normally-distributed noise. Sometimes you see intermittent noise due to the discharge lamp flickering ('flicker noise' is in fact a term used in signal processing circles). In such cases, it may be advantageous to average several chromatograms rather than run one chromatogram slowly, because the S/N improvement should then be better than the square root of N. I havn't seen any literature on that.

   Understanding the effects of sampling rate, etc., in chromatographic systems is not necessarily well  done using the typical simplistic math models and visual examination of data sets. Many modern instrument systems have hidden in them some sophisticated data massaging algorithms, and the fact is that theory and simplistic assumptions such as the form of noise distribution and range of the that form do not match what occurs or what  is needed from real chromatographic systems.

Performance studies involve both an understanding of the raw data error and also the performance in analysis.  While theory is fun for simplistic discussions, the performance of  analytical systems needs to be grounded in actual data..

A quite good approach starting from the raw data is to define carefully the error distribution function. It is often/usually non Gaussian, and very often an  assymetric operator. This reality make all the arguments based on the common statistical assumptions rather baseless. 

The next step is to ascertain if this error function is constant over the analytical range of interest. This is often a second place which provides surprises.

Once these aspects are sorted out, the accuracy and reliability of a system can be understood.

Farooq, that was a most interesting read! As you say, I don't agree with everything they say, but it certainly is well conceived.

I think that Christopher, Scott and I are all stating essentially the same thing. None of us have confidence in the algorithms inherent in digital systems (give me an analog system any day!). I agree with both of them; the sampling errors inherent in the A/D process are likely to badly skew the statistics and what you end up modeling may be very instrument dependent.

I have found that theoretical modeling is a great academic exercise of but modest value (at best). Measurement systems have all sorts of non ideal things going on: nonlinear sensors, sensors sensing stray fields, drift in zero and drift in gain, and a long list of stuff like that. Some instrument companies hide the non ideal performance in their data treatment 'enhancements' and electronic conditioning of the signals.

This  problem is present in analog and digital systems.

Thank you all for your interesting views. 

That noise is white noise, if you handle context of noise,use multivariate technique ,principal component analysis is noise reduction tool.

The main idea with all detectors and measurement systems is the comparison of signal and noise. That is the essence of import.  The kinds of noise in an instrument can be fast, slow, and can be dependent on  the signal in some way. The time distribution function of the noise has no single general distribution function, but the important thing is that it differs from that of the signal..  

I tend to agree with Scot, and take the view that practical systems used in analytical laboratories don't lend themselves to mathematical analysis. I we suppose the design has been optimised, the digitisation frequency will be set to match or slightly over-sample with respect to the analogue frequency response, so the effect of changing the sampling frequency (or bunching in the acquisition system) may be unpredictable, smaller or greater than with (unfiltered) white noise.

Incidentally, one contribution to noise from discharge lamps is flicker noise - the discharge flickers literally and unpredictably like a candle. This noise isn't Gaussian and the double beam arrangement doesn't compesnate perfectly. You can often see flicker noise on the baseline (remember with spectrophotometry zero signal = maximum power transmission). Also, noise contributions due to turbulence and pressure fluctuations caused by pump pulsations are regular. If either or both of these sources are significant, the improvement obtained by averaging replicate runs should be better than root(n).

Yes, of course the SNR is determined by the sampling rate. The noise current is square root proportional to the noise spectral width. And the noise spectral width is related to the sampling rate by the time-bandwidth relation. When the sampling rate increases, the noise spectrum is wider, the noise is more. The comment of Martin Ahnoff is mainly correct.