palmar <runge_kuta@hotmail.com> wrote in message <99843a13703f4848bdc7318e2184306f@33g2000yqj.googlegroups.com>...
> My current work is about analyzing vibration is a signal from an
> accelerometer in a moving vehicle. After some math operations we get a
> signal that represents the vertical displacement of the vehicle versus
> travel distance. No big deal.
>
> We want to locate ithe intervals where excess vibration occurred, in a
> certain frequency band (loose), and have applied Discrete Wavelet
> Transform to that effect. Analyzing the coefficients in that frequency
> band we can locate the intervals where the vibration was higher. The
> thing seems to be working as planned, with actually pretty good and
> motivating results.
>
> Now come our doubts: why look at the coefficients (which time
> resolution has been reduced, depending on the level you are) and not
> reconstruct the signal corresponding to that coefficients, therefore
> gaining the original time resolution and squaring it (to get energy,
> likewise with the coefficients). So, the question is basically should
> we reconstruct the signal and square it to get energy, or use the
> wavelet coefficients which represent energy, anyway.
>
> Thanks in advance for your input.
>
> Palmar
Hi Palmar, often people look at the proportion of the signal's energy represented by wavelet coefficients at a given scale. This gives you a picture of how the signal's energy is distributed across those scales and therefore gives a picture of what scales are most important. The energy is conserved in the decimated wavelet transform as you have indicated:
reset(RandStream.getDefaultStream)
x = randn(1024,1);
[C,L] = wavedec(x,5,'haar');
norm(x,2)
norm(C,2)
% now look at the allocation of this energy by scale
[Ea,Ed] = wenergy(C,L)
The problem with interpreting the above in terms of what scales contributed most significantly is that the number of coefficients at each level are different so you need to scale it accordingly. Doing that you arrive at an estimate of the wavelet variance.
The nice thing about working with the wavelet coefficients and not reconstructing an approximation to the signal based on a given scale as you have indicated is that for many processes, the DWT is a decorrelating transform. In other words, while correlation may exist (and usually does) in the reconstructed waveform (even from a single branch), the DWT coefficients can be treated as uncorrelated random variables, which facilitates all sorts of statistical tests.
The other thing you can do is look at the estimating the wavelet variance by scale for the undecimated wavelet transform. Of course there you don't have the decorrelation property of the DWT, but you also don't have the problem with shift variance and the simple loss of coefficients as you go deeper in the transform. For an example, see the paper by Professor Percival at
http://faculty.washington.edu/dbp/PDFFILES/wavevar.pdf
Wayne
