How to find psi function in Haar continuous wavlelet transform:CWT

Dear all, I has a problem when i try to use Haar wavelet to reconstruct an impact force-F if I know response (strain-e). Lets me describe:

I have impact force (F is a matrix 1x1024) and response (strain e: matrix 1x1024). After using CWT in matlab with scale m=2,4,8,16,32,64,128,256,512,1024, i got a matrix of coefficient (1024x10). And I know that CWT is actually a convolution between F and Psi(Haar wavelet function).

I would like to ask How can i get Psi(Haar wavelet func...) which corresponds to each scale of m?

Thank you so much for your help

Answer 1

[Wayne King]

Hi Hai Tran, If you are using dyadic scales as you do above, I think you would be better to use the MODWT (introduced in R2015b, rather than the CWT). With the MODWT, you do not lose time resolution just like with the CWT but if you are willing to restrict yourself to the dyadic scales, the MODWT has a lot of advantages (energy preservation -- perfect reconstruction just to name a couple important ones).

That said, here is a way to get an idea what the "daughter" wavelets look like at each scale.

scales = 2.^(1:10);
delta = zeros(1024,1);
% Only provides daughter wavelets at one translation
delta(500) = 1;
dwaves = cwt(delta,scales,'haar');

Now each row of dwaves will actually be the time-reverse of the Haar wavelet at that scale.

For scale 32 (2^5)

   w5 = fliplr(dwaves(5,:));

[Hai Tran]

Dear Wayne King, I have to say thank you so much for your help. Your answer is very nice answer and i also checked CWT and MODWT(as you suggested) and that is what i am finding now. Additionally, I would like to understand some things: 1. Why dwaves matrix has 11 rows? 11th row is a approximation data? 2. Is Each row of dwaves (correspond to 1 scale) also a results when we use convolution between F and Haar wavelet at that scale? [I mean use conv(f,w1 or w2...,w10)]. Why I ask this aspect because you said ' Only provides daughter wavelets at one translation'. For example if i use scale m=2, 'daughter wavelet' will be w2(1:2:1023)=-1/sqrt(4) and w2(2:2:1024)=1/sqrt(4). Is this right? please see attach image (i perform haar wavelet func:psi at scale m=5,6,8,9) Thank you so much for your considering of my question

<<

Answer 2

[Wayne King]

Hi Hai, For the CWT you will get 10 rows. For the MODWT, the final row is the scaling coefficients as you guessed.

Yes, when the scale is equal to 2^1 or 2, the Haar wavelet will have magnitude 1/2 or 1/sqrt(4).

Because the impulse was only at one translation, 500, in the case above, you only get the Haar wavelet centered at that position.

[Hai Tran]

Dear Wayne King, Thank you so much for your answer

[Hai Tran]

Dear Dr. Wayne King, What is difference between cD1(row1) in MODWT(f,scale,'haar') and CONV(f,haar) (2), where haar =[1/sqrt(2) -1/sqrt(2)]? Thank so much