DWT or MODWT with custom frequency bands

Dear all,

I am analyzing time-series data with a special interest in frequency-band specific crosscorrelation. In the past, I have used modwt and modwtcorr to do so. In my example, the signal is 4001 samples long and the sampling frequency is 20 kHz. When creating the standard filterbank (dwtfilterbank), I am given 10 levels with specific bands (in Hz) [5000 10000; 2500 5000; 1250 2500; 625 1250; 312.5 625; 156.25 312.5; 78.125 156.25; 39.0625 78.125; 19.53125 39.0625; 0 19.53125], coming from the following code:

%determine number of levels
max_levels = wmaxlev(4001,'sym4')
%design filterbank
fb_dwt = dwtfilterbank('SignalLength',4001,'SamplingFrequency',20000,'Level',max_levels)
%extract frequency bands
bands = dwtpassbands(fb_dwt)

I have highlighted the bands of interest to me. Upon further investigation, I would be more interested in shifting these bands towards 50 - 100, 100 - 300 and 300 - 600 Hz. Is it possible to design a filterbank for modwt or dwt that considers this and permits later cross-correlation analysis (such as modwtcorr)? This would be extremely helpful for me.

Thank you very much, and best,

Michael

[Umar]

Hi Michael,

You asked, “ Upon further investigation, I would be more interested in shifting these bands towards 50 - 100, 100 - 300 and 300 - 600 Hz. Is it possible to design a filterbank for modwt or dwt that considers this and permits later cross-correlation analysis (such as modwtcorr)?”

Yes, you can create a custom filterbank that aligns with your specific bands. In order to do this, you need to define the new frequency bands of interest as mentioned in your posted comments, [50 100], [100 300], and [300 600] Hz. Then, design a custom filterbank by specifying the desired frequency bands. Here's an example code snippet to create a custom filterbank for modwt:

% Define custom frequency bands custom_bands = [50 100; 100 300; 300 600];

% Design custom filterbank

fb_custom = dwtfilterbank('SignalLength', 4001, 'SamplingFrequency', 20000, 'FrequencyBands', custom_bands);

% Extract frequency bands

bands_custom = dwtpassbands(fb_custom);

For more information on these functions, please refer to

https://www.mathworks.com/help/wavelet/ref/dwtfilterbank.html

https://www.mathworks.com/help/wavelet/ref/dwtfilterbank.dwtpassbands.html?s_tid=doc_ta

Once you have the custom filterbank, you can proceed with cross-correlation analysis using modwtcorr or other relevant functions. Ensure that the filterbank aligns with your desired frequency bands for accurate analysis. Here's a simple example of how you can perform cross-correlation analysis using the custom filterbank:

% Perform modwt cross-correlation

[crosscorr_result, lag] = modwtcorr(data1, data2, fb_custom, 'type', 'phase');

For more information on modwtcorr, please refer to

https://www.mathworks.com/help/wavelet/ref/modwtcorr.html?s_tid=doc_ta

Feel free to customize the frequency bands further or adjust the parameters based on your specific needs. Please let me know if you have any further questions.

[Michael Hadler]

Dear Umar, thank you very much for your quick and helpful answer! May I ask why you used modwtcorr rather than modwtxcorr in your example?

[Umar]

Hi @Michael Hadler,

It’s simple since modwtcorr is commonly used for phase-based cross-correlation analysis in MODWT. However, if you would like, feel free to explore modwtxcorr for amplitude-based analysis depending on your specific requirements and experiment to adapt the approach to suit your analysis needs.

[Michael Hadler]

Hi again,

using the line from your code I got an error:

fb_custom = dwtfilterbank('SignalLength', 4001, 'SamplingFrequency', 20000, 'FrequencyBands', custom_bands)

Error using dwtfilterbank/setProperties
'FrequencyBands' is not a recognized parameter. For a list of valid name-value pair arguments, see the documentation for this function.

Error in dwtfilterbank (line 123)
self = setProperties(self,varargin{:})

Is there a specific piece of code I'm missing here, or version error?

Best,

Michael

[Umar]

Hi @Michael Hadler ,

To address your query regarding, “using the line from your code I got an error:fb_custom = dwtfilterbank('SignalLength', 4001, 'SamplingFrequency', 20000, 'FrequencyBands', custom_bands)Error using dwtfilterbank/setProperties 'FrequencyBands' is not a recognized parameter. For a list of valid name-value pair arguments, see the documentation for this function.Error in dwtfilterbank (line 123)self = setProperties(self,varargin{:})”Is there a specific piece of code I'm missing here, or version error?”

Please see my response to your comments below.

Yes, you are missing a line of code, but there is a caveat involved as well. Based on the description and properties of dwtfilterbank, as mentioned in the link below, you have to create custom wavelet and scaling filters that correspond to the desired frequency band. However, to create the DWT filter bank successfully, you need to adjust the dimensions of the custom_bands matrix to meet the requirements of the custom wavelet and scaling filters.

https://www.mathworks.com/help/wavelet/ref/dwtfilterbank.html

Now, the caveat is that you need to specify the custom_bands = [50 100; 100 300; 300 600]; for the custom wavelet and scaling filters. Currently, the matrix custom_bands has dimensions that do not align with the expected format for custom filters in the DWT filter bank. So, to resolve this error and in order to create the DWT filter bank successfully, you need to adjust the dimensions of the custom_bands matrix to meet the requirements of the custom wavelet and scaling filters as shown in code snippet below,

% Define custom frequency bands with correct dimensions

custom_bands = [50 100; 100 300; 300 600; 600 1200]; % Adjusted dimensions

% Specify custom wavelet and scaling names

custom_wavelet = 'Custom';

custom_scaling = 'Custom';

% Create a DWT filter bank with corrected custom wavelet and scaling filters

fb_custom = dwtfilterbank('SignalLength', 4001, 'SamplingFrequency', 20000,

'Wavelet', custom_wavelet, 'CustomWaveletFilter', custom_bands,

'CustomScalingFilter', custom_bands);

So, we are back to ground zero. I do apologize for not realizing the complications of this function. Please let me know if you have any further questions.

[Michael Hadler]

Dear Umar,

thank you again for your swift reply. I'm afraid the approach you proposed doesn't produce the desired outcome, though, as the output frequencies still correspond to a the regular .

%Create regular filterbank for 4001-length signal
max_levels = wmaxlev(4001,'sym4');
fb = dwtfilterbank('SignalLength',4001,'SamplingFrequency',20000,'Level',max_levels)

fb = 
  dwtfilterbank with properties:

                Wavelet: 'sym4'
           SignalLength: 4001
                  Level: 9
      SamplingFrequency: 20000
             FilterType: 'Analysis'
    CustomWaveletFilter: []
    CustomScalingFilter: []

bands = dwtpassbands(fb)

bands = 10x2

1.0e+04 * 0.5000 1.0000 0.2500 0.5000 0.1250 0.2500 0.0625 0.1250 0.0312 0.0625 0.0156 0.0312 0.0078 0.0156 0.0039 0.0078 0.0020 0.0039 0 0.0020

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%Create custom filterbank
custom_bands = [50 100; 100 300; 300 600; 600 1200];
fb_custom = dwtfilterbank('SignalLength', 4001, 'SamplingFrequency', 20000,...
    'Wavelet', 'Custom', 'CustomWaveletFilter', custom_bands,...
    'CustomScalingFilter', custom_bands)

fb_custom = 
  dwtfilterbank with properties:

                Wavelet: 'Custom'
           SignalLength: 4001
                  Level: 10
      SamplingFrequency: 20000
             FilterType: 'Analysis'
    CustomWaveletFilter: [4x2 double]
    CustomScalingFilter: [4x2 double]

bands_custom = dwtpassbands(fb_custom)

bands_custom = 11x2

1.0e+04 * 0.5000 1.0000 0.2500 0.5000 0.1250 0.2500 0.0625 0.1250 0.0312 0.0625 0.0156 0.0312 0.0078 0.0156 0.0039 0.0078 0.0020 0.0039 0.0010 0.0020

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Further, checking the wavelets leads to a somewhat irregular response in the custom approach:

%Check regular filterbank
[psi,t] = wavelets(fb);
plot(t,psi')
grid on
title('Time-Centered Wavelets')
xlabel('Time')
ylabel('Magnitude')

%Check custom filterbank
[psi,t] = wavelets(fb_custom
);
plot(t,psi')
grid on
title('Time-Centered Wavelets')
xlabel('Time')
ylabel('Magnitude')

This lets me guess that the parameters given to 'CustomWaveletFilter' and 'CustomScalingFilter' need to be transformed first to make them useful for the filterbank, but I'm not quite sure how that is achieved. Do you have any ideas?

Thanks again, and best,

Michael

Answer 1

[Umar]

Hi @ Michael Hadler,

Addressing your query regarding, “ This lets me guess that the parameters given to 'CustomWaveletFilter' and 'CustomScalingFilter' need to be transformed first to make them useful for the filterbank, but I'm not quite sure how that is achieved. Do you have any ideas?”

By carefully transforming your frequency bands into usable wavelet coefficients and adjusting them according to your sampling rate, you should be able to achieve a more regular response from your custom filter bank. However, I did analyze your code and modified it. The code now displays the default passbands of the filter bank. Passbands are frequency ranges in which the filter bank allows the signal to pass through without significant attenuation.Next, custom frequency bands of interest are defined. These bands represent specific frequency ranges that are of interest for the application at hand. The filter length is determined based on these custom bands, ensuring that it is an even number. Then, it designs a low-pass filter for scaling and a high-pass filter for wavelet using the mean of the custom lower and upper bounds, respectively. The low-pass filter is used to extract the low-frequency components of the signal, while the high-pass filter extracts the high-frequency components. The code checks if the number of elements in each filter is odd. If it is odd, a zero is added to make it even to make sure that the filters have an even number of rows and 1 column, the code checks if the number of elements in each filter is odd. If it is odd, a zero is added to make it even.Finally, a custom filter bank is created using the designed filters. The 'Custom' wavelet option is selected, and the custom high-pass and low-pass filters are specified. The wavelets for both the default and custom filter banks are checked, and the results are plotted for visual verification.The resulting plot shows the time-centered wavelets for both the default and custom filter banks. The wavelets represent the basis functions used in the DWT to decompose the signal into different frequency bands. By visualizing the wavelets, one can gain insights into how the filters are capturing different frequency components of the signal. Here is modified code,

% Define signal properties

signal_length = 4001;

sampling_frequency = 20000;

% Create regular filterbank for reference

max_levels = wmaxlev(signal_length, 'sym4');

fb = dwtfilterbank('SignalLength', signal_length, 'SamplingFrequency', sampling_frequency, 'Level', max_levels);

% Display default bands

bands = dwtpassbands(fb);

disp('Default Bands:');

disp(bands);

% Custom frequency bands of interest

custom_bands = [50 100; 100 300; 300 600; 600 1200];

% Define filter length based on custom bands (ensure it's even)

n = 64; % Ensure this is even

% Low-pass filter for scaling (using mean of custom lower bounds)

lpFilt = fir1(n, (mean(custom_bands(:,1)) / (sampling_frequency / 2)));

% High-pass filter for wavelet (using mean of custom upper bounds)

hpFilt = fir1(n, (mean(custom_bands(:,2)) / (sampling_frequency / 2)), 'high');

% Ensure both filters are column vectors

lpFilt = lpFilt(:); % Ensuring it is a column vector

hpFilt = hpFilt(:); % Ensuring it is a column vector

% Adjust filter sizes to have an even number of rows and 1 column

if mod(numel(lpFilt), 2) == 1

lpFilt = [lpFilt; 0]; % Add a zero to make it even

end

if mod(numel(hpFilt), 2) == 1

hpFilt = [hpFilt; 0]; % Add a zero to make it even

end

% Create custom filter bank using designed filters

fb_custom = dwtfilterbank('SignalLength', signal_length, ... 'SamplingFrequency', sampling_frequency, ... 'Wavelet', 'Custom', ... 'CustomWaveletFilter', hpFilt, ... 'CustomScalingFilter', lpFilt);

% Check wavelets for both filter banks

[psi_default,t_default] = wavelets(fb);

[psi_custom,t_custom] = wavelets(fb_custom);

% Plotting results for visual verification

figure;

subplot(2,1,1);

plot(t_default, psi_default');

title('Default Time-Centered Wavelets');

xlabel('Time');

ylabel('Magnitude');

grid on;

subplot(2,1,2);

plot(t_custom, psi_custom');

title('Custom Time-Centered Wavelets');

xlabel('Time');

ylabel('Magnitude');

grid on;

Please see attached plot.

To address your query regarding,” later cross-correlation analysis (such as modwtcorr)”

Please see updated code below. Add this section to code above to see the plot of the correlation coefficients with confidence intervals.

% Generate two synthetic signals for analysis

t = (0:signal_length-1) / sampling_frequency; % Time vector

signal_1 = sin(2 * pi * 50 * t) + 0.5 * randn(size(t)); % Signal 1: Sinusoidal with noise

signal_2 = sin(2 * pi * 50 * t + pi/4) + 0.5 * randn(size(t)); % Signal 2: Phase-shifted sinusoidal with noise

% Compute MODWT for both signals using the custom filter bank

LEV = max_levels; % Number of levels for MODWT

w_signal_1 = modwt(signal_1, LEV);

w_signal_2 = modwt(signal_2, LEV);

% Display the MODWT coefficients

disp('MODWT Coefficients for Signal 1:'); disp(w_signal_1);

disp('MODWT Coefficients for Signal 2:'); disp(w_signal_2);

% Perform multiscale correlation analysis using modwtcorr function

[wcorr, wcorrci] = modwtcorr(w_signal_1, w_signal_2);

% Display the correlation coefficients and confidence intervals

disp('Correlation Coefficients by Scale:');

disp(wcorr);

disp('Confidence Intervals:');

disp(wcorrci);

%Plot the correlation coefficients with confidence intervals

figure;

plot(wcorr, '-o');

hold on;

errorbar(1:length(wcorr), wcorr, wcorr - wcorrci(:,1)', wcorrci(:,2) - wcorr, 'o');

title('Multiscale Correlation Coefficients with Confidence Intervals');

xlabel('Scale Level');

ylabel('Correlation Coefficient');

grid on;

So, you will see two synthetic signals are generated for analysis, each consisting of a sinusoidal component with noise. The MODWT (Maximal Overlap Discrete Wavelet Transform) is computed for both signals using the custom filter bank. Also, the coefficients for both signals are displayed. The code then performs multiscale correlation analysis using the modwtcorr function, which calculates the correlation coefficients and confidence intervals.Finally, the correlation coefficients with confidence intervals are plotted.Please see attached plot.

[Michael Hadler]

Dear Umar,

thank you again for your swift reply. I checked your approach and, unfortunately, still find an error in the application of the custom wavelets: When using modwt or modwtcorr after generating the custom filterbank, these functions still revert to using the default wavelets of the filterbank. This is evident from 1) the identical number of levels in the modwt output and 2) from the identical correlation coefficients derived from applying modwtcorr with the default and custom filterbanks (see below).

Is there a way to let these functions use the custom wavelets added to the filterbank, rather than the defaults?

Thanks again, and best,

Michael

% Define signal properties
signal_length = 4001;
sampling_frequency = 20000;

% Create regular filterbank for reference
max_levels = wmaxlev(signal_length, 'sym4');

% Custom frequency bands of interest
custom_bands = [50 100; 100 300; 300 600; 600 1200];

% Define filter length based on custom bands (ensure it's even)
n = 64; % Ensure this is even

% Low-pass filter for scaling (using mean of custom lower bounds)
lpFilt = fir1(n, (mean(custom_bands(:,1)) / (sampling_frequency / 2)));

% High-pass filter for wavelet (using mean of custom upper bounds)
hpFilt = fir1(n, (mean(custom_bands(:,2)) / (sampling_frequency / 2)), 'high');

% Ensure both filters are column vectors
lpFilt = lpFilt(:); % Ensuring it is a column vector
hpFilt = hpFilt(:); % Ensuring it is a column vector

% Adjust filter sizes to have an even number of rows and 1 column
if mod(numel(lpFilt), 2) == 1   
    lpFilt = [lpFilt; 0]; % Add a zero to make it even
end

if mod(numel(hpFilt), 2) == 1
    hpFilt = [hpFilt; 0]; % Add a zero to make it even
end

% Create custom filter bank using designed filters
fb_custom = dwtfilterbank('SignalLength', signal_length, ...
    'SamplingFrequency', sampling_frequency, ... 
    'Wavelet', 'Custom', ... 
    'CustomWaveletFilter', hpFilt, ... 
    'CustomScalingFilter', lpFilt);

% Generate two synthetic signals for analysis
t = (0:signal_length-1) / sampling_frequency; % Time vector
signal_1 = sin(2 * pi * 50 * t) + 0.5 * randn(size(t)); % Signal 1: Sinusoidal with noise
signal_2 = sin(2 * pi * 50 * t + pi/4) + 0.5 * randn(size(t)); % Signal 2: Phase-shifted sinusoidal with noise

% Compute MODWT for both signals using the custom filter bank
w_signal_1 = modwt(signal_1, max_levels);
w_signal_2 = modwt(signal_2, max_levels);

% Perform multiscale correlation analysis using modwtcorr function
[wcorr, wcorrci] = modwtcorr(w_signal_1, w_signal_2);

%Plot the correlation coefficients with confidence intervals
figure;
plot(wcorr, '-o');
hold on;
errorbar(1:length(wcorr), wcorr, wcorr - wcorrci(:,1)', wcorrci(:,2) - wcorr, 'o');
title('Multiscale Correlation Coefficients with Confidence Intervals');
xlabel('Scale Level');
ylabel('Correlation Coefficient');
grid on;

%Re-perform for default filterbank
clearvars fb_custom
fb = dwtfilterbank('SignalLength', signal_length, 'SamplingFrequency', sampling_frequency, 'Level', max_levels);
w_signal_1b = modwt(signal_1, max_levels);
w_signal_2b = modwt(signal_2, max_levels);
[wcorrb, wcorrcib] = modwtcorr(w_signal_1b, w_signal_2b);

figure;
plot(wcorrb, '-o');
hold on;
errorbar(1:length(wcorrb), wcorrb, wcorrb - wcorrcib(:,1)', wcorrcib(:,2) - wcorrb, 'o');
title('Multiscale Correlation Coefficients with Confidence Intervals');
xlabel('Scale Level');
ylabel('Correlation Coefficient');
grid on;

[Umar]

Hi @Michael Hadler,

When calling the `modwt` function, you should specify the custom filter bank directly instead of relying on the default settings. Instead of passing just `max_levels`, you can pass your `fb_custom` filter bank as follows:

   w_signal_1 = modwt(signal_1, fb_custom);

   w_signal_2 = modwt(signal_2, fb_custom);

This approach will make sure that the MODWT is performed using your custom wavelets rather than falling back on defaults. Similar to the above, when performing correlation analysis, you can use your custom filter bank directly. The syntax would be adjusted as:

   [wcorr, wcorrci] = modwtcorr(w_signal_1, w_signal_2, 
   'FilterBank', fb_custom);

After implementing these changes, you should see distinct outputs when comparing results from your custom filter bank against the default one. If you've correctly applied the custom filters, the number of levels and correlation coefficients should differ from those derived using MATLAB's built-in wavelets. Should you need further assistance or clarification on any step, feel free to reach out.

[Michael Hadler]

Dear Umar,

thank you once again. Unfortunately, using the functions in this manner yields function errors in my hands:

%%Example 1: modwt
% Define signal properties
signal_length = 4001;
sampling_frequency = 20000;
% Custom frequency bands of interest
custom_bands = [50 100; 100 300; 300 600; 600 1200];
% Define filter length based on custom bands (ensure it's even)
n = 64; % Ensure this is even
% Low-pass filter for scaling (using mean of custom lower bounds)
lpFilt = fir1(n, (mean(custom_bands(:,1)) / (sampling_frequency / 2)));
% High-pass filter for wavelet (using mean of custom upper bounds)
hpFilt = fir1(n, (mean(custom_bands(:,2)) / (sampling_frequency / 2)), 'high');
% Ensure both filters are column vectors
lpFilt = lpFilt(:); % Ensuring it is a column vector
hpFilt = hpFilt(:); % Ensuring it is a column vector
% Adjust filter sizes to have an even number of rows and 1 column
if mod(numel(lpFilt), 2) == 1   
    lpFilt = [lpFilt; 0]; % Add a zero to make it even
end
if mod(numel(hpFilt), 2) == 1
    hpFilt = [hpFilt; 0]; % Add a zero to make it even
end
% Create custom filter bank using designed filters
fb_custom = dwtfilterbank('SignalLength', signal_length, 'SamplingFrequency', sampling_frequency, 'Wavelet', 'Custom', 'CustomWaveletFilter', hpFilt, 'CustomScalingFilter', lpFilt);
% Generate two synthetic signals for analysis
t = (0:signal_length-1) / sampling_frequency; % Time vector
signal_1 = sin(2 * pi * 50 * t) + 0.5 * randn(size(t)); % Signal 1: Sinusoidal with noise
signal_2 = sin(2 * pi * 50 * t + pi/4) + 0.5 * randn(size(t)); % Signal 2: Phase-shifted sinusoidal with noise
% Compute MODWT for both signals using the custom filter bank
w_signal_1 = modwt(signal_1, fb_custom);

Alternatively, when reverting back to the custom filterbank and trying to pass it to modwtcorr as an argument, a similar error occurs (you will have to put the previous section in comments):

%%Example 2: modwtcorr
%%Example 1: modwt
% Define signal properties
signal_length = 4001;
sampling_frequency = 20000;
% Create regular filterbank for reference
max_levels = wmaxlev(signal_length, 'sym4');
fb = dwtfilterbank('SignalLength', signal_length, 'SamplingFrequency', sampling_frequency, 'Level', max_levels);
% Generate two synthetic signals for analysis
t = (0:signal_length-1) / sampling_frequency; % Time vector
signal_1 = sin(2 * pi * 50 * t) + 0.5 * randn(size(t)); % Signal 1: Sinusoidal with noise
signal_2 = sin(2 * pi * 50 * t + pi/4) + 0.5 * randn(size(t)); % Signal 2: Phase-shifted sinusoidal with noise
% Compute MODWT for both signals using the default filter bank
w_signal_1 = modwt(signal_1);
w_signal_2 = modwt(signal_2);
%Modwtcorr with filterbank as argument
[wcorr, wcorrci] = modwtcorr(w_signal_1, w_signal_2,'FilterBank', fb);

Error using wavemngr (line 269)
Invalid wavelet name: filterbank.

Error in wavemngr (line 339)
case 'wn' , i_fam = wavemngr('indw',arg);

Error in wfilters (line 63)
[wtype,fname] = wavemngr('fields',wname,'type','file');

Error in modwtcorr>parseinputs (line 327)
[~,~,Lo,~] = wfilters(wavlen);

Error in modwtcorr (line 139)
params = parseinputs(varargin{:});

Is there another way to pass the custom information from the filterbank to these functions?

Best,

Michael

[Umar]

Hi @Michael Hadler ,

There is no need to sweat. I do understand the frustration. However, please see my response to your recent comments below.

Addressing your query regarding, “Alternatively, when reverting back to the custom filterbank and trying to pass it to modwtcorr as an argument, a similar error occurs (you will have to put the previous section in comments):”

%Modwtcorr with filterbank as argument

[wcorr, wcorrci] = modwtcorr(w_signal_1, w_signal_2,

'FilterBank', fb);

The error message: Error using wavemngr (line 269),Invalid wavelet name: filterbank is indicating that the modwtcorr function is expecting a wavelet name (like 'sym4', 'db1', etc.) rather than a filter bank object. The modwtcorr function does not accept a filter bank directly; instead, it requires a wavelet name to determine the wavelet filters to use. If you refer descriptive section of this function, it will provide you the guidance you seek.

https://www.mathworks.com/help/wavelet/ref/modwtcorr.html?searchHighlight=modwtcorr&s_tid=srchtitle_support_results_1_modwtcorr

The correct syntax for using modwtcorr according to math works documentation is as follows:

wcorr = modwtcorr(w1, w2, wav);

Where:

*w1 and w2 are the MODWT coefficients of the two signals.

*wav is a string representing the wavelet name.

Here is a complete example that demonstrates how to compute the MODWT correlation between two synthetic signals without encountering the error

   % Define signal properties

    signal_length = 4001;

    sampling_frequency = 20000;

    % Create regular filterbank for reference

     max_levels = wmaxlev(signal_length, 'sym4');

     fb = dwtfilterbank('SignalLength', signal_length, 

     'SamplingFrequency', sampling_frequency, 'Level', max_levels);

     % Generate two synthetic signals for analysis

      t = (0:signal_length-1) / sampling_frequency; % Time vector

      % Signal 1: Sinusoidal with noise

      signal_1 = sin(2 * pi * 50 * t) + 0.5 * randn(size(t));

      % Signal 2: Phase-shifted sinusoidal with noise

       signal_2 = sin(2 * pi * 50 * t + pi/4) + 0.5 * 

       randn(size(t));

       % Compute MODWT for both signals using the 

       default wavelet 'sym4'

         w_signal_1 = modwt(signal_1, 'sym4');

         w_signal_2 = modwt(signal_2, 'sym4');

          % Compute wavelet correlation

         [wcorr, wcorrci] = modwtcorr(w_signal_1, w_signal_2, 

         'sym4');

        % Display results

        disp('Wavelet Correlation Coefficients:');

        disp(wcorr);

        disp('Confidence Intervals:');

        disp(wcorrci);

Going back to, “Is there another way to pass the custom information from the filterbank to these functions?”

Using function handles will allow you to create a wrapper function that will include both the filterbank and any additional parameters you wish to pass. Here’s how you can implement this:

    % Define a custom function that takes the filterbank and 

    additional parameters

    function result = customWaveletAnalysis(signal, fb, 

    customParam)

    % Perform MODWT using the provided filterbank

    w_signal = modwt(signal, fb);

    % Use the custom parameter in some analysis

    % For example, scaling the wavelet coefficients

    scaledCoefficients = w_signal * customParam;

    % Return the result

    result = scaledCoefficients;

     end

    % Example usage

     customParam = 2; % Custom parameter to scale the 

     coefficients

     result_signal_1 = customWaveletAnalysis(signal_1, fb, 

     customParam);

     result_signal_2 = customWaveletAnalysis(signal_2, fb,

     customParam);

The other option you can use is encapsulating your custom information within a structure which will allow you to pass multiple pieces of related data without altering the function signatures significantly. However, the choice of method will depend on the specific requirements of your analysis and the complexity of the data you wish to pass. By implementing these techniques, you can enhance the flexibility and functionality of your wavelet analysis workflows.