Wavelet denoising on low pass filtered signal

Hello,

I'm having an issue with the wdenoise function.

I'm sampling essensially white noise, hardware filtered to 200 MHz, at 2.5 GS/s with an oscilloscope.

I'm then denoising this data by using the following function:

wdenoise(x, 15,'Wavelet', 'sym8', 'DenoisingMethod','UniversalThreshold', ...
    'ThresholdRule','Soft', 'NoiseEstimate','LevelIndependent');

It works great:

The next step is to first filter the 200 MHz signal using a digital low pass filter to 20 MHz:

df = designfilt('lowpassfir','FilterOrder',110,'CutoffFrequency',20e6/(2.5e9/2));
D = mean(grpdelay(df));
y = filter(df,[x zeros(D,1)']);  
y = y(D+1:end);

The filter works great:

Now comes the problem - I'm trying to wdenoise the filterd signal, and instead of getting at least the same picture as with the 200 MHz after denoising, the signal after LPF and the signal after denoising said filtered signal are literally the same waveform.

I'm not a DSP expert and not hugely familiar in the inner workings of each function, so I'm most likely missing something. Please help.

[Mathieu NOE]

hello

I just wonder what is the target of this filtering ? and why you need wavelet denoising ?

a LP Butterworth (of whatever order) would not suffice ?

[Egor Liokumovitch]

The idea is to sample instantanious pulses consisting of frequencies of up to 200 kHz. Wavelets are perfect for this, as they are filtering all the white noise and leaving the desired pulse.

For example, the following plot is showing a signal with a bandwidth of 5 MHz which should include a short pulse at a peak frequency of 5 kHz, before denoising and after

For the system I'm prototyping, the filter in the FPGA will be either a 5 MHz or 1 MHz (TBD), so if the upper frequency for a specific recording is known, say 200 kHz, it would be beneficial to filter the whole signal to around that number, and then use denoising. Wavelets should have an easier time picking a weaker pulse out of a weaker noise.

My thinking was since currently I'm fixed with the filter in the FPGA, that I should be able to do the same filtering in MATLAB before further processing, but after an LPF wdenoise completely breaks, and I can't figure out why.

Performing an FFT on the original signal and the LPF signal show similar behaviour up to 20 MHz:

[Mathieu NOE]

so if I understand correctly the matlab wavelet denoising is not giving you the expected results (as I can see in your original post)

[Egor Liokumovitch]

Yes. It works great on the 200 MHz signal, but stops working when the same 200 MHz signal is first low pass filtered to 20 MHz, and then denoised. It just provides the same low passed signal. Literally a copy.

I'm running successive denoising on the 200 MHz signal, and on the signal after LPF:

xd = wdenoise(S200,22,'Wavelet', 'sym8', 'DenoisingMethod','UniversalThreshold', ...
    'ThresholdRule','Soft', 'NoiseEstimate','LevelIndependent');
xd2 = wdenoise(S200LPF,22,'Wavelet', 'sym8', 'DenoisingMethod','UniversalThreshold', ...
    'ThresholdRule','Soft', 'NoiseEstimate','LevelIndependent');

Everything is the same, but the result:

And if I plot the filtered signal and the denoised signal, its exactly the same, sample for sample:

Answer 1

[Egor Liokumovitch]

Problem has been solved. I talked to an experienced DSP engineer and he explained the process behind wavelet denoising, more or less. To my understanding, wavelet denoising starts by dividing the singal's frequency components into an LPF and HPF. It filters each section, then divides again to an LPF and HPF, like a tree, and so on.

When I use an LPF, I have already removed all the HPF frequencies that the wavelet denoising expects from the sampling rate, so there's nothing for it to do, so I just receive my signal after LPF and denoising without change.

What I need to do to fix this is to desample the signal and lower the sampling frequency by the same ratio I'm lowering the bandwidth.

After an LPF I downsampled the signal by the same ratio as the filtering and now the wavelet works again.