**Name**

**stkwave1** One-dimensional wavelet transform using Starck's algorithm (band-limited scaling function)

**Command Synopsis**

**stkwave1** *np* *in* *out*

np: resolution np

in: input in Fourier domain (the size of signal must be a power of 2)

out: result in Fourier domain, from left to right : details and approximation

**Function Summary**

void stkwave1 (np , in , out )

Fsignal in , out ;

int np ;

**Description**

The module `stkwave1` computes the wavelet transform of a one-dimensional signal according to the work
of Starck *et al.* [SBLP94] who use an overcomplete frequency-domain approach (band-limited wavelet).

As seen in `owave1`, multi-resolution analysis corresponds to considering a scale function and a wavelet used to compute details and approximations of a signal.

() = ()(2^{j})

() = ()(2^{j})

with
() =
and () =

The frequency band is reduced by a factor of 2 while the resolution scales up. We go from a resolution to the following resolution multiplying the filter H and the frequential signal.

The details are obtained filtering the same signal by G.

J.L. Starck uses a

() = *B*_{3}(4)

that is to say
(
(2) = () - (2)

The second difference with the Mallat's algorithm is the non-decimation of the details. Which implies, for a size The wavelet transformation algorithm for a resolution

- Compute by FFT, set
(
*f*)= and initialize j to 1. - Multiply
(
*f*) to*H*gives the approximation for a resolution j : (*f*) - Multiply
(
*f*) to*G*gives the details for a resolution j : (*f*) - If
*j*<*np*, the frequency band of (*f*) is reduced by a factor 2 which corresponds to keep one coefficient out of two in the time space, j is then incremented and we go back to point 2.

In this module, the input signal is assumed to be already in the Fourier domain that is, = .

**See Also**

**Version 1.0**

Last Modification date : Thu Apr 15 08:03:52 2004

**Author**

Claire Jonchery, Amandine Robin