Wavelets

Does anybody here know about them? I'm particularly interested in them in the context of filtering.

Looked at them a while back. What are you looking for?

I'm interested in their use as filters. How would you say that they compare in terms of properties to, say, a Kalman filter?

Originally posted by Palynka
I'm interested in their use as filters. How would you say that they compare in terms of properties to, say, a Kalman filter?

From what I can find, Kalman is about system modeling. I may have had another term for it at uni.

I looked at them more in terms of Wavelet Transform. Similar to Fourier Transform, but you are not using sine waves, but rather wavelets. The FT is great for time invariant signals, but does not show discontinuities (e.g. signal starting). WT copes better with discontinuities. But if you want to design a low pass filter, then the sine representation of the FT is more appropriate.

Originally posted by gezza
From what I can find, Kalman is about system modeling. I may have had another term for it at uni.

I looked at them more in terms of Wavelet Transform. Similar to Fourier Transform, but you are not using sine waves, but rather wavelets. The FT is great for time invariant signals, but does not show discontinuities (e.g. signal starting). WT copes better w ...[text shortened]... ou want to design a low pass filter, then the sine representation of the FT is more appropriate.

Thanks, that's exactly one of the things I was looking for.

Do you know how they would differ in terms of frequency domain?

Originally posted by Palynka
Thanks, that's exactly one of the things I was looking for.

Do you know how they would differ in terms of frequency domain?

Wavelets are more akin to pulses, a one time affair. Frequency domain is about recurring waves, a sine wave, for instance. A single cycle of a sine wave could be analyzed by wavelets better than frequency domains. What frequency area are you interested in filtering? I have an interest in hi Q low frequency filters myself, related to brain wave activity, frequencies on the order of 1 to 20 Hz, how to implement one in a simple op amp circuit. I need to have 4 or 5 bands, so 5 separate filters each reacting to a specific band, say 1 to 5 Hz, 6 to 10, 11 to 15, 16 to 20, etc.

Originally posted by sonhouse
Wavelets are more akin to pulses, a one time affair. Frequency domain is about recurring waves, a sine wave, for instance. A single cycle of a sine wave could be analyzed by wavelets better than frequency domains. What frequency area are you interested in filtering? I have an interest in hi Q low frequency filters myself, related to brain wave activity, fre ...[text shortened]... arate filters each reacting to a specific band, say 1 to 5 Hz, 6 to 10, 11 to 15, 16 to 20, etc.

I'm interested in filtering recurring waves which can have pulses, so to speak. The idea is to take into account possible pulses (e.g deep short-lived recessions) in macroeconomic time-series.

These series have a long-term (think trend) and a short-term component (think business cycle). Filtering is often used to try to extract what are long-run very low frequency movements from short-term fluctuations around this trend.

Also if there are occasional downward surges which are not standard business cycle movements then this could lead to estimation problems in standard band-pass or kalman filters. I was wondering if wavelets could deal with that, for example.

I thought wavelets allowed for flexibility in some trade-off between the time and frequency domain (isn't that the point of the separation into father and mother wavelets?)

Originally posted by Palynka
I'm interested in filtering recurring waves which can have pulses, so to speak. The idea is to take into account possible pulses (e.g deep short-lived recessions) in macroeconomic time-series.

These series have a long-term (think trend) and a short-term component (think business cycle). Filtering is often used to try to extract what are long-run very low ...[text shortened]... and frequency domain (isn't that the point of the separation into father and mother wavelets?)

According to this Wiki article, the father wavelet is made up so as to eliminate the need for integration, and it talks about mother wavelets and a lot more, maybe you can get some use out of it:
http://en.wikipedia.org/wiki/Wavelet

Originally posted by Palynka
I'm interested in filtering recurring waves which can have pulses, so to speak. The idea is to take into account possible pulses (e.g deep short-lived recessions) in macroeconomic time-series.

These series have a long-term (think trend) and a short-term component (think business cycle). Filtering is often used to try to extract what are long-run very low ...[text shortened]... and frequency domain (isn't that the point of the separation into father and mother wavelets?)

Not sure if this is going to help you, but I'll give it a go. This is how I used these things.

The standard fourier transform (FT) is a one-off affair - sample your signals for ever, then run an FT once to get to a Frequency domain plot.
There is then a sliding window FT - sample for e.g. 1024 samples, plot the result, shift a sample and repeat. You then get an Frequency domain plot, together with how it varies in time. You are actually dealing with sine waves, cut to the size of the window.
The trouble is that a signal starting from nothing will have frequency components all over the place.

Enter the Wavelet Transform (WT). Similar to sliding window FT, the different times are dealt with by translation, and the different "frequencies" by scale. There are a bunch of different wavelets. Some are orthogonal. Some are not. It depends what you need which is useful.

In either case you end up with a representation of your signal as it varies in time, as a 3-D plot.