Hi Yousef,

it might depend on the scale (resp. frequency) resolution that you have chosen in  your DWT analysis whether or not you may achieve a sparse representation (or at least approximation) of your monocomponent signal by means of the DWT. Can you specify both signal and DWT resolution chosen, please?

Best regards

Peter

Thank Peter for your consideration...

I have decomposed my simple sine wave signal of frequency 1/(2*pi) Hz using db10 basis function in five level. Graphical representation is shown in the attached picture. I have no trouble with CWT, because it is similar to STFT. But I have difficulty figuring out the sparse representation of monotonic signals while using DWT.

Yousef,

1. From the figure, I think you are using dyadic DWT. And it seems using symmetric extension or just zero padding. This induces high response near boundary. Have you try 'per'  Periodization etc? Since the signal is of finite number of sample, we have to do something at the boundary. Choose one your really want.

2. The frequency response of d1 is essentially the upper half of the full spectrum; d2 is the upper half of the lower half of the full spectrum and etc. From your plot, the dwt coefficients have high response in d5 & a5. It is actually sparse in scale (frequency)!  And show you the response of discontinuities near boundaries.

Why not plot more levels? and do try frequency varying signals e.g. chirp.

Dear Tai-Chiu,

1. You are right I was using symmetric extension, but I just changed the mode for "Per" status. Edge distortion still exists, however for a large number of samples it can be overlooked.

2. As you recommended I tried "chirp signal". It is actually a nice example to see Time-Frequency Points Localization in TF domain. Frequency components are disjoint in transformed domain. I also used matlab "wavedec" command that yields a vector. The results were consistent.

Thanks,