# Analysis of data and curves

# Fast Fourier Transform

This function can be accessed by the command &fft-cmd;. It can be found in the Analysis Menu when a table or a plot is selected. The Fourier transform decomposes a signal in its elementary components by assuming that the signal x(t) can be describe as a sum:

## Fourier equation

in which are the frequencies, an are the amplitudes of each frequency and are the phase corresponding frequency. &appname; will compute these parameters and build a new plot of the amplitude as a function of the frequency.

## An example of a inverse FFT.

FFT performed on a curve to extract the characteristic frequencies. The signal is on the bottom plot, while the amplitude-frequency plot is on the top layer. In this example, the amplitude curve has been normalized, and the frequencies have been shifted to obtain a centered x-scale.

Some parameters of the FFT can be modified in the FFT dialog.

# Correlation

This function can be accessed by the command &correlate-cmd;. It can be found in the Analysis Menu when a table is selected. The correlation function, also known as the covariance function is used to test the similarity of two signals x(t) and y(t). It is computed by:

## Covariance function of two signals x(t) and y(t)

in which and are the mean values of the signals x(t) and y(t) respectively.

If the number of points is N, the function will be computed between -N/2 and N/2. The abscissae are therefore point numbers and nott values.

## An example of a correlation between two sinus functions.

The first plot shows the two signals, the second one is the correlation function between the two signal which shows that there are correlations, and the third one is the Fourier transform which is done to extract the caracteristic frequencies of the correlation function.

The correlation of a signal with itself can also be used in spectral analysis (it is then called autocorrelation or autocovariance function).

# Convolution

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# Deconvolution

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# The Fit Wizard

This function can be accessed by the command &non-linear-curve-fit-cmd;. It can be found in the Analysis Menu when a table is selected.

## The results of the &non-linear-curve-fit-cmd;.

The results are shown in the log window, the curve is plotted in the active window, and a table is created to store the fit.

# Fitting to specific curves

&appname; include quick access to the most usefull functions for fitting.

## Fitting to a line

This command is used to fit a curve which has a linear shape.

### The results of a &fit-linear-cmd;.

The results will be given in the Log panel:

## Fitting to a polynome

This command is used to fit a curve which has a linear shape. The results will be given in the Log panel

### The results of a &fit-polynomial-cmd;, showing the initial data, the curve added to the plot, and the results in the log panel.

## Fitting to a Bolzmann function

This command is used to fit a curve which has a sigmoidal shape. The function used is:

### Bolzmann equation

in which A1 is the low Y limit, A2 is the high Y limit, x0 is the inflexion point and dx is the width.

### The results of a &fit-bolzmann-cmd;.

When the X axis is set to a logarithmic scale, the &fit-bolzmann-cmd; command uses the Logistical equation for fitting:

### Logistic dose response equation

where A1 is the initial Y value, A2 is the final Y value, x0 is the inflexion point (center) and p is the power.

## Fitting to a Gauss function

This command is used to fit a curve which has a bell shape. The function used is:

### Gauss equation

in which A is the height, w is the width, xc is the center and y0 is the Y-values offset.

### The results of a &fit-gaussian-cmd;.

## Fitting to a Lorentz function

This command is used to fit a curve which has a bell shape. The function used is:

### Lorentz equation

in which A is the area, w is the width, xc is the center and y0 is the Y-values offset.

### The results of a &fit-lorentzian-cmd;.

# Multi-Peaks fitting

This kind of fitting allows to fit your data points to a sum of N gaussian or lorentzian functions.

The first step is to specify the number of peaks. Then you must define the position of each peak on the curve. This is done by clicking on the plot, then validate your choice for each peak with the ENTER key.

## The results of a &fit-multipeak-gaussian-cmd;.

# Filtering of data curves

In this section, it will be assumed that you have the following data curve:

This signal has a power spectrum with high and low frequencies. We can analyze this by doing a FFT on the data curve, this leads to the following figure:

The newt sections will show the influence of the different filters on this data curve.

## FFT low pass filter

This filter allows to cut the high frequencies of a signal. You just have to select the cut-off frequency of the filter. Let us assume that we want to keep the frequencies below 1 Hz, we will obtain:

### Signal after a FFT low pass filter

The power spectrum of this new signal shows that the frequencies below 1 Hz have been kept.

## FFT high pass filter

This filter allows to cut the low frequencies of a signal. You just have to select the cut-off frequency of the filter. Let us assume that we want to keep the frequencies above 1 Hz, we will obtain:

### Signal after a FFT high pass filter

The power spectrum of this new signal shows that the frequencies above 1 Hz have been kept.

## FFT band pass filter

This filter allows to cut the low and high frequencies of a signal. You just have to select the high and low cut-off frequencies of the filter. Let us assume that we want to keep the frequencies between 1.5 and 3.5 Hz, we will obtain:

### Signal after a FFT band pass filter

The power spectrum of this new signal shows that only the frequencies at 2 and 3 Hz have been kept.

## FFT block band filter

This filter allows to keep the low and high frequencies of a signal. You just have to select the high and low cut-off frequencies of the filter. Let us assume that we want to remove the frequencies between 1.5 and 3.5 Hz, we will obtain:

### Signal after a FFT block band filter

The power spectrum of this new signal shows that only the frequencies below 1.5 Hz and above 3.5 Hz have been kept.

# Interpolation

The interpolation command will create a new data curve with a high number of points by interpolation of your data. The dialog box allows to define this number of points (default value = 1000). Then the method used for interpolation, the interval of X-values and the color of the interpolated curve can be chosen. In addition to the new curve in the active plot, a new table will be created.

The simplest interpolation method is the linear method. In this case, a linear variation is used to compute the data points between two values. The cubic method will use the Cubic Splines method (in this case at least 4 points are needed). The last method Akima is a polynomial interpolation. You can refer to the corresponding section of the GNU Scientific Library for more details.

## Comparison of the three methods of interpolation