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# Documentation

## One-Dimensional Fractional Brownian Motion Synthesis

This section takes you through the features of One-Dimensional Fractional Brownian Motion Synthesis using one of the Wavelet Toolbox™ specialized tools.

For the examples in this section, switch the extension mode to symmetric padding, using the command

```dwtmode('sym')
```

The toolbox requires only one function to generate a fractional Brownian motion signal: wfbm. You'll find full information about this function in its reference page.

In this section, you'll learn how to

• Generate a fractional Brownian motion signal

• Look at its main properties

• Save the synthesized signal

Since you can perform the generation either from the command line or using the graphical interface tools, this section has subsections covering each method.

A fractional Brownian motion (fBm) is a continuous-time Gaussian process depending on the Hurst parameter 0 < H < 1. It generalizes the ordinary Brownian motion corresponding to H = 0.5 and whose derivative is the white noise. The fBm is self-similar in distribution and the variance of the increments is given by

```Var(fBm(t)-fBm(s)) = v |t-s|^(2H)
```

where v is a positive constant.

### Fractional Brownian Motion Synthesis Using the Command Line

According to the value of H, the fBm exhibits for H > 0.5, long-range dependence and for H < 0.5, short or intermediate dependence.

Let us give an example of each situation using the wfbm file, which generates a sample path of this process.

```% Generate fBm for H = 0.3 and H = 0.7

% Set the parameter H and the sample length
H = 0.3; lg = 1000;
% Generate and plot wavelet-based fBm for H = 0.3
fBm03 = wfbm(H,lg,'plot');
```
```% Generate and plot wavelet-based fBm for H = 0.7
fBm07 = wfbm(H,lg,'plot');

% The last step is equivalent to
% Define wavelet and level of decomposition
% w = ' db10'; ns = 6;
% Generate
% fBm07 = wfbm(H,lg,'plot',w,ns);
```

It appears that fBm07 clearly exhibits a stronger low-frequency component and has, locally, a less irregular behavior.

### Fractional Brownian Motion Synthesis Using the Graphical Interface

1. Start the Fractional Brownian Motion Synthesis Tool.

From the MATLAB® prompt, type

```wavemenu
```

The Wavelet Toolbox Main Menu appears. Click Fractional Brownian Generation 1-D to display the One-Dimensional Fractional Brownian Motion Synthesis Tool.

2. Generate fBm.

From the Fractal Index edit button, type 0.3 and from the Seed frame, select the item State and set the value to 0. Next, click the Generate button.

The synthesized signal exhibits a locally highly irregular behavior.

3. Now let us try another value for the fractal index. From the Fractal Index edit button, type 0.7 and from the Seed frame, select the item State and set the value to 0. Next, click the Generate button.

The synthesized signal clearly exhibits a stronger low-frequency component and has locally a less irregular behavior. These properties can be investigated by clicking the Statistics button.

### Saving the Synthesized Signal

The Fractional Brownian Motion Synthesis Tool lets you save the synthesized signal to disk. The toolbox creates a MAT-file in the current folder with a name you choose.

To save the synthesized signal from the present selection, use the option File > Save Synthesized Signal. A dialog box appears that lets you specify a folder and filename for storing the signal. After saving the signal data to the file fbm07.mat, load the variables into workspace.

```load fbm07
whos
```
NameSizeBytesClass
FBM_PARAMS1x1668struct array
fbm071x10008000double array

```FBM_PARAMS

FBM_PARAMS =
SEED: [2x1 double]
Wav: 'db10'
Length: 1000
H: 0.7000
Refinement: 6
```

The synthesized signal is given by fbm07. In addition, the parameters of the generation are given by FBM_PARAMS, which is a cell array of length 5.