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continuous wavelet transform and inverse

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continuous wavelet transform and inverse


jon erickson (view profile)


22 Jul 2008 (Updated )

Continuous wavelet transform (CWT) and Inverse CWT for reconstructing original signal.

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This collection of files to perform an inverse continuous wavelet transform is an extension of the wavelet software package originally written by Torrence and Compo (
The main functions are:

1. contwt.m: (continuous wavelet transform). This is essentially Torrence and Compo's wavelet.m with a few modifications (more inputs and outputs for easier access)

2. invcwt.m: inverse continuous wavelet transform.

3. example_invcwt.m: Demo/example usage. This is a template for building simple sine wave, setting wavelet parameters, and comparing original and reconstructed signal.

Please see the help in each function for details and usage.

Required Products MATLAB
MATLAB release MATLAB 8.3 (R2014a)
MATLAB Search Path
Other requirements none
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Comments and Ratings (31)
02 Feb 2017 Ramona May

21 Dec 2016 Paul C

Paul C (view profile)

Thank you very much Jon! I'll try that out following your tips. Happy festive.

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18 Dec 2016 jon erickson

jon erickson (view profile)

Hi Paul C. Thanks for your comments. They prompted me to update the example included so that you and others who have asked similar questions can play around a bit. Basically, the answer lies in the wavelet bases--and how parameters are set--relative to the original signal. I've updated the screenshot example to show 20 s of a omega = 2 (f = pi) signal. Previously it was 10 s long, which was essentially too short to have a Morlet wavelet trying to oscillate 6 times within the Gaussian envelope to really properly capture that behavior. With 20 s, one can imagine in the mind's eye how the Morlet wavelet better 'fits', thus the reconstruction is better. Keep in mind also that the parameters set for smallest scale, max number of scales to compute, the mother wavelet all affect how well the original signal can be reconstructed. Probably the best intuition I can off specifically as it relates to the example shown: Given we have a pure sinusoid at a frequency of f_o, we are unlikely to be able to *exactly* reconstruct because we'll have wavelets at various scales (pseudo-freuqencies), none of which exactly match f_o. The average error is indicated by the variable 'dE' in the figure and in the example script. In summary, reconstruction can be better or worse depending on how well the underlying wavelets at various scales describe the original signal. Hope this helps and good luck! -Jon

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15 Dec 2016 Paul C

Paul C (view profile)

Very nice code. Could you please explain a bit why the reconstruction doesn't match the original signal (notably in the example visible here with the Morlet wavelet)?

11 Oct 2016 jon erickson

jon erickson (view profile)

nour yousfi: the output variable wave is a matrix with wavelet coefficients. So you can do imagesc(wave) to see the result in time-frequency space. If you want to see what's happening for a single frequency across time, just select the subset of coefficients that match the frequency (scale) you care about.

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12 Sep 2016 shri nidhi

I have cwt of signal how will i recover the signal which is unknown to me?

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07 Sep 2016 nour yousfi

how can I plot cwt vs time in fixed frequency(natural frequency)

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19 Jun 2016 Johan Ibrahimowich

20 Nov 2015 Death Saurer

That so very good day.
Let me turn to you asking for your kind help in order to get the code in matlab for the fourier transform of continuous-time (FFT) directly and inversely, since my knowledge in matlab are not very good.
I appreciate who I can help and collaborate.
Excuse my bad English and my native language is Spanish

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08 Apr 2015 jon erickson

jon erickson (view profile)

Yousef - Glad you got the example code to run. The limitation you mentioned arises because I adapted Torrence and Compo's original code, which only had a small family of wavelet bases included. A few more could easily be added in the future simply by adding more in wave_bases.m.

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22 Feb 2015 Yousef

Yousef (view profile)

I could run the code with the "example" you posted, but here is my main problem:
Why is this limited to only three family of functions. How about "db10" or "cgau8"?

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22 Feb 2015 Yousef

Yousef (view profile)

I am trying to run the code. But there is an error:
Unable to read file ref.txt
where can I find ref.txt

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15 Oct 2014 Greg

Greg (view profile)

Clean, concise, simple to use, excellent results. Thank you very much! If a few more wavelets could be added in the future, would be much appreciated.

19 May 2013 jon erickson

jon erickson (view profile)

If you want to isolate a single scale (frequency) in the reconstruction, then zero out all coefficients for all scales and all time, except the selected scale. Then reconstruct.

Trevor: Yes, you are correct. Filling in zeros for gaps ensures those terms (time points) ensure the sums remain unchanged.
Just add 'haar' in the list of available wave bases, as you suggest.

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24 Jan 2013 trevor

trevor (view profile)

Hi Jon,

Thanks for this fantastic code!
A question regarding time series with gaps:
As I understand, the Haar wavelet will ignore gaps, as long as they are replaced by zeros and the time series is zero-meaned. If this is correct, could you include (either here or in the wave_bases.m file) code for the Haar wavelet?

Thanks again.

22 Nov 2012 Venkata

The example you have provided works fine. But, when I apply to my data, the reconstructed signal seems to be scaled down.

Secondly, could you plz tell me how to reconstruct only a particular periodic signal (frequency) from the wavelet coefficients..?

Thank you.

13 Nov 2011 ernesto

hey friends Im working with myoelectric signal and i need to use the wavelet transform for processing the signal and i need some like a tutorial for get some knowledge

12 Oct 2011 jon erickson

jon erickson (view profile)

Alexandre: MATLAB did not used to have the inverse cwt function in their wavelet toolbox as of 2008. Hence, at that time, it was filling a gap--that matlab hadn't yet implemented. I have not used the newer version of matlab wavelet toolbox, but my guess is that it is the same concept, but less full-fledged.

Zahra: CWT does not share the same notion of details and approximations--that is DWT only.

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16 Aug 2011 zahra Khawaja

When I decompose the signal with the CWT, how can i reconstruct the details coefficients and the approximations coefficients, like in the discrete wavelet when we use the matlab function (wrcoef)?

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03 Aug 2011 Alexandre

Could you please explain how this differ from the Wavelet Toolbox (cwt.m function) ? Thanks.

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02 Apr 2011 jon erickson

jon erickson (view profile)

If by "instantaneous frequency" you mean the "equivalent Fourier frequency" (i.e. sine wave in infinite time domain at oscillating at a single frequency), then here's the answer. The CWT "pseudofrequency" depends on the mother wavelet you use for the transform. Each mother wavelet has a corresponding "center frequency", and the relation is given as:

f = centerfrq(mother)/(a*delta).
a = scale
delta = sampling period.

I strongly encourage you to read up on wavelet theory, in order to become a competent user of this code.

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01 Apr 2011 Hani Ali

Thank you very much Jon for the great files. Could you please explain how some one could obtain the instantaneous frequency from the transformed wavlet. I found a lot of literature on how to do so for a signal but not its transformed wavlet... I suspect it is rather easy, but i am somewhat lost... Thanks in advance!

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25 Feb 2011 jon erickson

jon erickson (view profile)

Please note that the reconstructed signal is always centered. That is, the reconstruction process does not/cannot account for any 'dc offset' in the original signal.

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24 Jan 2011 Wayne Liu

For people who cannot reconstruct the signal, please check your downloaded 'invcwt.m'. In Line 34, if it is calling the function 'wave_bases', change it to 'wave_bases_rec'. Rerun your program and see if it works...

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30 Nov 2010 Jan Sieber

requested example (what's missing there?) Could be a documentation problem?

%% test invcwt(contwt(y))
% default is MORLET according to documentation:

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07 Jun 2010 jon erickson

jon erickson (view profile)

Ajay and Ravi,
I have never experienced similar issues.
Please provide more specifics on your problem, and I'll try to respond. E.g., please copy/paste the commands you used. It would also be helpful if you posted an image of your original signal and the (erroneous) output.

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08 Mar 2010 Ravi Rastogi

Hi Jon, I am also having trouble reconstructing the signal back. I get a vector having all values NAN.

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07 Jan 2010 Ajay

Ajay (view profile)

I meant does not seem to perform perfect reconstruction.

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07 Jan 2010 Ajay

Ajay (view profile)

The inverse continuous transform does seem to perform a perfect reconstruction. Is there a way to retrieve the original signal back.

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20 May 2009 Jon

Jon (view profile)

The link above is correct if you remove the extra ')' which was accidentally made part of the hyperlink. Sorry about that.

worked as of this afternoon when I tested it.

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02 Nov 2008 Bogdan Hlevca

It would be nice if you had more documentation for the software and attached the data file for the test function.
The link you provided does not work.

25 Feb 2011 1.1

-Added example usage.
-Added screen shot.
-Reloaded updated files, which should resolve the issues using wave_bases.m vs wave_bases_rec.m

14 Oct 2014 1.2

Default number of scales computed is the minimum sufficient number for accurate reconstruction. Previously, depending on the number of samples in the original signal, the continuous wavelet transform computed used one scale to few.

18 Dec 2016 1.3

LAST UPDATED DEC 19 2016: added more extensive example in example_invcwt.m
This may be used as a template to explore how parameters settings influence reconstruction quality.

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