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Fractional Brownian motion synthesis

`FBM = wfbm(H,L)`

FBM = wfbm(H,L,'plot')

FBM = wfbm(H,L,NS,W)

FBM =
wfbm(H,L,W,NS)

wfbm(H,L,'plot',NS)

wfbm(H,L,'plot',W)

wfbm(H,L,'plot',NS,W)

wfbm(H,L,'plot',W,NS)

`FBM = wfbm(H,L)`

returns a fractional Brownian
motion signal `FBM`

of the Hurst parameter `H`

(```
0
< H < 1
```

) and length `L`

, following
the algorithm proposed by Abry and Sellan.

`FBM = wfbm(H,L,'plot')`

generates and plots
the `FBM`

signal.

`FBM = wfbm(H,L,NS,W)`

or ```
FBM =
wfbm(H,L,W,NS)
```

returns the `FBM`

using `NS`

reconstruction
steps and the sufficiently regular orthogonal wavelet `W`

.

`wfbm(H,L,'plot',NS)`

or `wfbm(H,L,'plot',W)`

or `wfbm(H,L,'plot',NS,W)`

or `wfbm(H,L,'plot',W,NS)`

generates
and plots the `FBM`

signal.

`wfbm(H,L)`

is equivalent to `WFBM(H,L,6,'db10')`

.

`wfbm(H,L,NS)`

is equivalent to `WFBM(H,L,NS,'db10')`

.

`wfbm(H,L,W)`

is equivalent to `WFBM(H,L,W,6)`

.

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.

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. This example
shows 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');

H = 0.7; % 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);

`fBm07`

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

Abry, P.; F. Sellan (1996), "The wavelet-based synthesis
for the fractional Brownian motion proposed by F. Sellan and Y. Meyer:
Remarks and fast implementation," *Appl. and Comp.
Harmonic Anal.*, 3(4), pp. 377–383.

Bardet, J.-M.; G. Lang, G. Oppenheim, A. Philippe, S. Stoev, M.S. Taqqu (2003), "Generators of long-range dependence processes: a survey," Theory and applications of long-range dependence, Birkhäuser, pp. 579–623.

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