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## continuous wavelet transform and inverse

version 1.4 (742 KB) by

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 (http://paos.colorado.edu/research/wavelets/software.html)
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.

Vrajeshri Patel

jon erickson

### jon erickson (view profile)

Hi Anthony,
Y and DT are the input signal and the sampling period. There is no default per se - it's whatever signal you want to analyze and the period at which it was sampled. Hope this clarifies. Good luck. -Jon

Anthony Windmon

### Anthony Windmon (view profile)

Hi Jon,

Thank you for this code! I am a first semester graduate student and I will be using this to retrieve features from some audio files. My question is regarding the contwt file and these parameters contwt(Y,DT,PAD,DJ,S0,J1,MOTHER,PARAM). What are the typical/default values used for Y and DT?

Trinath Suna

### Trinath Suna (view profile)

Hi Jon,
I have a modis NDVI data of 250m resolution from 2000-2015. I wanted to do CWT for time frequency analysis followed by linear regression to derive slope at each pixel level that will infer the trend during 2000-2015. I am new to MATLAB. Could you please provide your valuable inputs in order to start the matlab program .

Regards
Trinath
trinath.suna@gmail.com

Partha Das

### Partha Das (view profile)

My Problem is solved. Lot of thanks Jon.

jon erickson

### jon erickson (view profile)

Hi Partha Das,

You spotted a bug with 'param(nn)'. The indexing nn should not have been there (vestigial remnant of looping over various paramater values). My apologies for not having caught that bug earlier. I've re-uploaded a corrected version to the file exchange, v1.4. Or you can just delete off the (nn) indexing in the file you have.

The meaning/usage of the 'param' input depends on the mother wavelet you are using. From >>help contwt.m:

% PARAM = the mother wavelet parameter.
% For 'MORLET' this is k0 (wavenumber), default is 6.
% For 'PAUL' this is m (order), default is 4.
% For 'DOG' this is m (m-th derivative), default is 2.

Hope this helps resolve your questions. Good luck! -Jon

Partha Das

### Partha Das (view profile)

In example_invcwt.m at line 21 i.e.
[wave, period, scale, coi, dj, paramout, k] = contwt(x,dt, pad, dj, so, j1, mother, param(nn));

what is param(nn)? & what will be the value of 'nn'.

Ramona May

Paul C

### Paul C (view profile)

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

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

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)?

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.

shri nidhi

### shri nidhi (view profile)

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

nour yousfi

### nour yousfi (view profile)

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

Johan Ibrahimowich

Death Saurer

### Death Saurer (view profile)

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

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.

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"?

Yousef

### Yousef (view profile)

hi,
I am trying to run the code. But there is an error:
where can I find ref.txt

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.

jon erickson

### jon erickson (view profile)

Venkata:
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.

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.

Venkata

### Venkata (view profile)

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.

ernesto

### ernesto (view profile)

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

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.

zahra Khawaja

### zahra Khawaja (view profile)

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)?

Alexandre

### Alexandre (view profile)

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

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.

Hani Ali

### Hani Ali (view profile)

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!

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.

Wayne Liu

### Wayne Liu (view profile)

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...

Jan Sieber

### Jan Sieber (view profile)

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

%% test invcwt(contwt(y))
dt=1e-3;
t=-pi:dt:pi;
y=sin(t);
% default is MORLET according to documentation:
[wave,per,scale,coi,dj,paramout,k]=contwt(y,dt);
yrec=invcwt(wave,'MORLET',scale,paramout,k);
plot(t,y-yrec);

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.

Ravi Rastogi

### Ravi Rastogi (view profile)

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

Ajay

### Ajay (view profile)

I meant does not seem to perform perfect reconstruction.

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.

Jon

### Jon (view profile)

Bogdan:

worked as of this afternoon when I tested it.

Bogdan Hlevca

### Bogdan Hlevca (view profile)

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.