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Wavelet Decomposition/ FFT

Started by clutchfft March 6, 2013
I have managed to decompose a signal via wavelet transform.  I have a
signal with 256 points.  The transform produced 256 wavelet coefficients. 
In FFT, we use the odd/even samples and combine them through a butterfly
operation using the twiddle factor = e^-i*2PI*kn/N.  

The paper I am implementing says that we replace the odd/even samples with
the wavelet coefficients and the twiddle factor would be the frequency
response of the wavelet filters.  I am not sure what they mean when they
say frequency response of the wavelet filters.

Thanks
On Wednesday, March 6, 2013 8:39:46 AM UTC-6, clutchfft wrote:
> I am not sure what they mean when they say frequency response of the wavelet filters.
Niether do we if we don't know to which paper you refer.
On Wednesday, March 6, 2013 10:06:58 AM UTC-6, maury wrote:
> On Wednesday, March 6, 2013 8:39:46 AM UTC-6, clutchfft wrote: > I am not sure what they mean when they say frequency response of the wavelet filters. Niether do we if we don't know to which paper you refer.
Are you refering to Guo and Burrus paper?
>On Wednesday, March 6, 2013 10:06:58 AM UTC-6, maury wrote: >> On Wednesday, March 6, 2013 8:39:46 AM UTC-6, clutchfft wrote: > I am
not sure what they mean when they say frequency response of the wavelet filters. Niether do we if we don't know to which paper you refer.
> >Are you refering to Guo and Burrus paper? >
Yes, Guo and Burrus paper. I think the frequency response is the DFT of the wavelet filter.
On Wed, 06 Mar 2013 08:39:46 -0600, clutchfft wrote:

> I have managed to decompose a signal via wavelet transform. I have a > signal with 256 points. The transform produced 256 wavelet > coefficients. In FFT, we use the odd/even samples and combine them > through a butterfly operation using the twiddle factor = e^-i*2PI*kn/N.
If by odd/even samples you mean adjoining pairs of samples, then the butterfly operations get carried out on a much more rich set of samples than that (at least if you're doing the whole bit-reversed ordering thing). And I would hardly call something as systemic and organized as the values of the Euler identity at evenly spaced locations on the circle as a "twiddle factor". -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
On Wednesday, March 6, 2013 8:39:46 AM UTC-6, clutchfft wrote:
> I have managed to decompose a signal via wavelet transform. I have a signal with 256 points. The transform produced 256 wavelet coefficients. In FFT, we use the odd/even samples and combine them through a butterfly operation using the twiddle factor = e^-i*2PI*kn/N. The paper I am implementing says that we replace the odd/even samples with the wavelet coefficients and the twiddle factor would be the frequency response of the wavelet filters. I am not sure what they mean when they say frequency response of the wavelet filters. Thanks
I have not had a chance to read the paper yet, but if you look at Figure 1, and compare that with Figure 4, I think you should be able to determine what Burris means by _frequency response of the wavelet filters_
On Wednesday, March 6, 2013 10:22:22 AM UTC-6, Tim Wescott wrote:
> On Wed, 06 Mar 2013 08:39:46 -0600, clutchfft wrote: > I have managed to decompose a signal via wavelet transform. I have a > signal with 256 points. The transform produced 256 wavelet > coefficients. In FFT, we use the odd/even samples and combine them > through a butterfly operation using the twiddle factor = e^-i*2PI*kn/N. If by odd/even samples you mean adjoining pairs of samples, then the butterfly operations get carried out on a much more rich set of samples than that (at least if you're doing the whole bit-reversed ordering thing). And I would hardly call something as systemic and organized as the values of the Euler identity at evenly spaced locations on the circle as a "twiddle factor". -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
Hi Tim, Burris uses the term _twiddle factor_ in the paper.
On Wed, 06 Mar 2013 11:03:15 -0800, maury wrote:

> On Wednesday, March 6, 2013 10:22:22 AM UTC-6, Tim Wescott wrote: >> On Wed, 06 Mar 2013 08:39:46 -0600, clutchfft wrote: > I have managed >> to decompose a signal via wavelet transform. I have a > signal with 256 >> points. The transform produced 256 wavelet > coefficients. In FFT, we >> use the odd/even samples and combine them > through a butterfly >> operation using the twiddle factor = e^-i*2PI*kn/N. If by odd/even >> samples you mean adjoining pairs of samples, then the butterfly >> operations get carried out on a much more rich set of samples than that >> (at least if you're doing the whole bit-reversed ordering thing). And I >> would hardly call something as systemic and organized as the values of >> the Euler identity at evenly spaced locations on the circle as a >> "twiddle factor". -- Tim Wescott Control system and signal processing >> consulting www.wescottdesign.com > > Hi Tim, > Burris uses the term _twiddle factor_ in the paper.
Well, my issue is with him, then. To me a "twiddle factor" is a constant associated with a kludge that you use to iron out minor differences between theory and reality, or a calibration constant that you use to iron out differences between one unit and the next. It is (in my mind) about the farthest thing from the appropriate term to use for the elegant and self-consistent math that you see in any of the various Fourier transforms. I suspect (particularly if it's a tutorial paper) that his intent was to show the similarities between wavelets and the FFT (which, it is my understanding, is one of the ways that you can view the FFT). But it still sticks sideways in my craw. -- My liberal friends think I'm a conservative kook. My conservative friends think I'm a liberal kook. Why am I not happy that they have found common ground? Tim Wescott, Communications, Control, Circuits & Software http://www.wescottdesign.com
On Wednesday, March 6, 2013 2:03:00 PM UTC-6, Tim Wescott wrote:
> On Wed, 06 Mar 2013 11:03:15 -0800, maury wrote: > On Wednesday, March 6, 2013 10:22:22 AM UTC-6, Tim Wescott wrote: >> On Wed, 06 Mar 2013 08:39:46 -0600, clutchfft wrote: > I have managed >> to decompose a signal via wavelet transform. I have a > signal with 256 >> points. The transform produced 256 wavelet > coefficients. In FFT, we >> use the odd/even samples and combine them > through a butterfly >> operation using the twiddle factor = e^-i*2PI*kn/N. If by odd/even >> samples you mean adjoining pairs of samples, then the butterfly >> operations get carried out on a much more rich set of samples than that >> (at least if you're doing the whole bit-reversed ordering thing). And I >> would hardly call something as systemic and organized as the values of >> the Euler identity at evenly spaced locations on the circle as a >> "twiddle factor". -- Tim Wescott Control system and signal processing >> consulting www.wescottdesign.com > > Hi Tim, > Burris uses the term _twiddle factor_ in the paper. Well, my issue is with him, then. To me a "twiddle factor" is a constant associated with a kludge that you use to iron out minor differences between theory and reality, or a calibration constant that you use to iron out differences between one unit and the next. It is (in my mind) about the farthest thing from the appropriate term to use for the elegant and self-consistent math that you see in any of the various Fourier transforms. I suspect (particularly if it's a tutorial paper) that his intent was to show the similarities between wavelets and the FFT (which, it is my understanding, is one of the ways that you can view the FFT). But it still sticks sideways in my craw. -- My liberal friends think I'm a conservative kook. My conservative friends think I'm a liberal kook. Why am I not happy that they have found common ground? Tim Wescott, Communications, Control, Circuits & Software http://www.wescottdesign.com
I can certainly understand your being upset. As an aside, many years ago, there was a _serious_ paper published by a physicist explaining (and defining) the _fudge factor_, _kluge factor_, and _twiddle factor_. If you can still find it, it was a bit humourous.
>On Wednesday, March 6, 2013 8:39:46 AM UTC-6, clutchfft wrote: >> I have managed to decompose a signal via wavelet transform. I have a
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>at Burris means by _frequency response of the wavelet filters_ >
It seems like the frequency response of the wavelet filter is the DFT. So for example, after wavelet transform/decomposition, we have wavelet coefficients from the low pass filter = X and wavelet coefficients from the high pass filter = Y. I think the frequency response would be DFT(X) and DFT(Y). Does that make sense?