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How to smooth and filter the signal?

Posted 9 years ago
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Hello

I have a many data points (intensity vs. wavelength). The signal is noisy so I would like to smooth it. The signal is also sumperimposed with another sinusodial function which I try to get rid of. There are some steps where I need you help.

First, here is my long list of data:

Fresnel1 = {{1.549`, 0.004039`}, {1.549002`, 
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As you can see the data is quite noisy and the signal is superimposed with another sinus:

Show[
 ListPlot[Fresnel1, AxesLabel -> {"wavelength [\[Mu]m]", "Intensity"}],
 ListLinePlot[Fresnel1, PlotStyle -> {Thin, PointSize[0.1]}, 
  AxesLabel -> {"wavelength [\[Mu]m]", "Intensity"}]
 ]

enter image description here

What I want to do is:

1.) Filter the sinus with smaller frequency out of my signal

2.) Smooth my signal by filtering the noise

As far as I understand the FFT I have to convert my signal from wavelength into frequency. Then I can apply the Mathematica functions Fourier[] and InverseFourier[] to filter my signal. But I think there is one important think to keep in mind and this is why I need your help.

If I convert the wavelength signal to frequency, the step size changes (is not equal anymore: delta lambda compared to delta frequency). And I have heard that this is important. First of all, is this actually correct? If yes, can you explain me why or where I can find more information about it? I think I have to define a wavelength array. How does this look like?

Fi = First[Dimensions [Fresnel1]];           
FresnelFrequency1 = 
  Table[{(3*10^8)/Fresnel1[[t, 1]]*10^6, Fresnel1[[t, 2]]}, {t, 
    Fi}];       

Show[
 ListPlot[FresnelFrequency1, 
  AxesLabel -> {"Frequency [Hz]", "Intensity"}],
 ListLinePlot[FresnelFrequency1, PlotStyle -> {Thin, PointSize[0.1]}, 
  AxesLabel -> {"Frequency [Hz]", "Intensity"}]
 ]

enter image description here

Because I think I have to deal with this sort of problem more often in the future, how would you solve step (1) and step (2) ?

Hope you can help me.

Peter

POSTED BY: Peter Parker
7 Replies

Mr. Parker (and anyone else with a long chunk of data to include in a post),

In the future, please try to place the data in a file that you attach to your post.
The file can be a Mathematica notebook or plain text, whichever is more convenient.

Having to scroll wa-a-a-a-a-ay down is inconvenient.

Moderation Team

POSTED BY: Moderation Team
Posted 9 years ago

Rather than doing an interpolation followed by an FFT, I encourage you to investigate the Lomb-Scargle approach. This will give you a periodogram without interpolation and the additional errors that involves. Google should help. Also, this book isn't bad: "Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica Support".

POSTED BY: Kevin McCann

Hello everybody,

thank you for all your answers. They made me think and I can tell you now more precise what I really need. Lets go back to my first post. There you can find my data (intensity vs. frequency). The frequency points are not spaced equally. They become closer to higher frequencies. This is why I can not use the FFT!!! This is were the problem begins.

This is my solution (if you have a better idea please tell me):

  1. I'll try to interpolate my data (intensity vs. frequency). This data consist of N data points unequally spaced.
  2. Then I arrange N equally spaced points over the interpolated function.
  3. For this new points I can finally apply FFT. Doing so, I can then chop all the frequencies that I would like to get rid of.
  4. Afterwards I use InverseFourier to get back to the frequency domain (intensity vs. frequency).
  5. Then I transform my spectrum back from frequency domain to wavelength domain (intensity vs. wavelength).

I have to consider one very important fact: I have to make sure to not distort the data!!! This is important. I don't want to modify my data by accident.

If you have a better solution, let me please know!

I think I need your help to actually do all this.

Step one: First I tried it using InterpolatingPolynomial. This was not very succesfully. Then I tried this:

Result = FresnelFrequency1[[All, 2]];

n = Length[Result]; (*Number of y-values*)
ftres = FourierDCT[Result]; (*DFT*)
len = 80; (*Number of frequencies that I would like to consider*)
ftres = PadRight[Take[ftres, len], Length[Result], 0];
out = FourierDCT[ftres, 3];
fsum = 1/Sqrt[n]*(ftres[[1]] + 
     2*Sum[ftres[[r]]*Cos[Pi/n*(r - 1) (x - 1/2)], {r, 2, 
        Length[ftres]}]);

Show[Plot[fsum, {x, 0, 1000}, PlotRange -> All], 
 ListLinePlot[ Result,  PlotStyle -> Red]]

Problem here is, that the x-axis is not the same anymore. The function is plotted versus "pixel position" and not versus "frequency".

Step 2: I have an idea, but it is not exactly what I need. Assuming I know my underlying function. Let's simplify and say it is a sinus function. Then I will sample N data points over this function. Like here:

NumberOfPoints = 100;

Punkte = Table[Sin[{t}], {t, 0, NumberOfPoints}];
Punkte2 = Table[{t, Punkte[[t + 1, 1]]}, {t, 0, NumberOfPoints}];
Show[
 ListPlot[Punkte2],
 ListLinePlot[Punkte2]
 ]

But again, I change the x-axis. This is the problem.

Can you please help me solving this problem. Maybe somebody of you has a very beautiful solution or a much better idea in general.

Cheers,

Peter

POSTED BY: Peter Parker
Posted 9 years ago

So, first a question: what do you know about the data? For example, is it a sinusoidal signal + exponential + noise? (Just a guess) If you know enough about the signal, you can do a least-squares fit to a model with parameters, e.g. frequency of the sinusoid, exponential coefficient, and then retrieve the coefficients. I would assume that the "noise" other than the sinusoid is Gaussian, but you may know otherwise.

BTW, a Fourier Transform is in fact a least-squares fit to the data by a sum of sinusoids.

POSTED BY: Kevin McCann

What you're looking to do is de-trend your data. I haven't seen that done with a Fourier transform. I'm not sure how that'd work.

There are many different ways to detrend data. A really basic way to detrend is to use the EstimatedBackground function:

http://reference.wolfram.com/language/ref/EstimatedBackground.html

Once you find the background you can substract it from the signal to detrend it.

data[[All, 2]] - EstimatedBackground[data[[All, 2]], 20]
POSTED BY: Sean Clarke

The workflow for filtering with a fourier transform is:

  1. Apply Fourier to a time series
  2. Remove small frequency components
  3. Apply InverseFourier to get back a time series

The code below assumes your dataset is called data.

We just want the "y" values for the time series, so we'll be using data[[All,2]]. We don't really need the "x" values - the data points should be and are evenly spaced.

Step One:

fdata = Fourier[data[[All, 2]]];

Step Two:

fdataFiltered = Threshold[fdata, {"Hard", 0.0005}]

or

fdataFiltered = (fdata /. x_ /; Abs[x] < 0.0005 -> 0) 

Step Three:

smoothedData = InverseFourier[fdataFiltered]

Fourier transforms aren't the only way to filter time series. There are a lot of different options for different purposes. WeinerFilter works fine for many cases:

WienerFilter[data[[All, 2]], 3, 0.1]
POSTED BY: Sean Clarke

This is really great. Now I know and understand how to filter the signal from the noise. Thank you very much!

But what is about the other problem I have mentioned? As you can see, my signal is superimposed by an other sinus which has a slightly longer frequency. I would like to filter this out. As I mentioned, I have heard that I need to transform my wavelength-signal into a frequency-signal. Only the frequency signal can be later used for the Fourier and InverseFouier functions. Otherwhise I will change the results. I have transformed my data to frequency in my first post by simply using the equation c=lambda*frequency. But in the next step I have to apply some kind of array in order to correct the differente step sizes. As far as I know the FFT accepts data sets of any length, but they must be sampled regulary (same step size)!!! Do you know what I mean? This is what other usually do, as you can see here on page 3: https://www.phys.ksu.edu/personal/washburn/Teaching/Class%20Files/NQO/Tutorials/Tutorial8_FFT.pdf I don't really understand this and I don't know how to apply this to my problem.

What would you do?

Thanks in advance,

Peter

POSTED BY: Peter Parker
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