Hi Euis,

I'm not an expert in signal processing, but my interpretation of the methods is as follows:

1.Residuals usually represent differences between a prediction and observed values. I suspect the prediction here is something like the baseline of your signal. Where the residual is high, peaks occur. Unfortunately, it is possible that this approach picks up noise rather than actual peaks.

2. Derivatives apply to mathematical functions and allow to identify stationary points such as local maxima and local minima. For detecting upward peaks, one should look for x-values where the first derivative is zero and the second derivative is smaller than zero. Again, the presence of noise may result in false detections and this is why the signal, or derivatives of the signal, should be smoothed first.

3. Deconvolution is often used to remove a specific type of signal (such as noise) from the observed signal. If you specify a peak (such as the Gaussian one) to deconvolute your signal, you will be able to calculate where this peak has a good fit, and thus where the "real" peaks in your data occur.

I think in general deconvolution is the most suitable method in signal processing, as the other approaches are more general concepts in the field of mathematics.

I never used the residual method so I will address my view on the other two.

Second Derivative: works well for small peaks and shoulders. Local minimum is a good indication for peak maximum.

Deconvolution; can be used for totally hidden peaks. For example you have a big broad peak but you know that your molecule should have 3 very close peaks, so you can deconvolute the broad peak into the 3 components.

Deconvolution is used a lot and I an not a big fan because you can generate peaks that are truly not there. You can get 3 peaks or 5 or 7. The sum of them will give you the initial big boad peak....So it boils down to researching the theory and the molecule and then deconvolution becomes a powerful technique.

And finally a summary of the 3 methods according to SigmaPlot:

1. The Residuals procedure initially places peaks by finding local maxima in a smoothed data stream. Hidden peaks are then optionally added where peaks in the residuals occur.

2. The Second Derivative procedure searches for local minima within a smoothed second derivative data stream. These local minima often reveal hidden peaks.

3. The Deconvolution procedure uses a Gaussian response function with a Fourier deconvolution/ filtering algorithm. A successfully deconvolved spectrum will consist of “sharpened” peaks of equivalent area. The goal is to enhance the hidden peaks so that each represents a local maximum.

1. Analysis of the residues is a good way to automate the process of finding the peaks. It avoids the possibility of including fictitious peaks, but it is not very sensitive to small peaks.

2. Analysis of second derivatives is very sensitive to the errors of experimental data procedure. It may be used for the preliminary detection of suspected peaks, but then this hypothesis must be confirmed by other methods, for example, by the analysis of residues.

3. Deconvolution can remove noise, and it will facilitate the selection of small peaks. However, small peaks can also be filtered by deconvolution procedure.

Let's be honest: there is no such thing as the (true) deconvolution procedure. Simply because the convolution operation, generally, has no inverse. And here is why:

Your observed signal is a convolution of the true signal with "window" function, which is usually unknown and therefore has to be approximated somehow. It is true that

F(S*W) = F(S).F(W), where F is Fourier transform, S means signal, and W describes window function. A star, "*" means convolution operation, while dot "." is ordinary multiplication. Therefore it seems obvious that F(S)=F(S*W)/F(W), so it should be enough to get the inverse Fourier transform of the result obtained this way. But ... F(W) may change its sign, many times within the domain of interest, so you will have to perform division by zero. This may be avoided, if W itself is a Gaussian shape (so F(W) is also Gaussian, everywhere positive). Remember, the rectangular window (i.e. equal to some constant positive value at a given section of a real line and zero otherwise) has Fourier transform proportional to sin(x)/x.

Numerical differentiation, on the other hand, doubles the noise level (double differentiation - quadruples). For this reason reliable results may only be obtained from high quality data. Smoothing the raw results before differentiation only makes things worse, as it is - more or less - a convolution procedure making all peaks more broad (while keeping their area), thus decreasing their amplitudes.

In summary: I recommend the approach given by Vladimir Bakhrushin.

By the way: why not to give a try to all the available options of your software? Personal experience is priceless.

Can someone pls tell the steps to do deconvolution using peak fit