How to embed a series of Gaussians in a Lorentzian just by adding functions?

Hey,

I want to have the shown function in Matlab and use it in my simulations and codes. It is a series of Gaussians (which has a narrow band of around 200 MHz) embedded in a broadband Lorentzian function (approximately 50 GHz). Could you please help me with how I can write a function just by adding some functions to show this figure? I know how to do that by, for example, removing some elements from the Lorentzian at that point and using concatenation techniques. But I need to show this function in my code as the sum or subtraction of some functions because I want to do other processes on it. I have attached a zoom-out figure which shows what I mean as a whole. Then in other pictures with numbers, I mostly want to say I need to define these series of Gaussians embedded into a broader Lorentzian in a flexible way such that I can obtain figures 1, 2, and 3 by changing some parameters of the added functions.

Your help is so much appreciated!

Answer 1

[Rangesh]

Hi Shaily,

I understand that you would like to embed a series of Gaussian curves into a Lorentzian curve. The provided code accomplishes this task.

x = -100:0.1:100; % Define the x-axis range 
% Define the parameters for the Gaussians and Lorentzian 
gaussians = {[1, 0, 10], [0.5, -40, 1], [0.8, -50, 8],[0.6,-30,5]}; 
lorentzian = [1, 0, 50];  
% Call the function to evaluate the function values 
y = gaussiansInLorentzian(x, gaussians, lorentzian); 

% Plot the resulting function 
plot(x, y); 

In the above code, the range of the x-axis is defined as -100 to 100.The parameters for the Gaussians and the Lorentzian are specified. The function gaussiansInLorentzian is called to evaluate the function values. The resulting function is plotted.

function y = gaussiansInLorentzian(x, gaussians, lorentzian) 
% x: Input vector 
% gaussians: Cell array of Gaussian parameters [amplitude, mean, standard deviation] 
% lorentzian: Lorentzian parameters [amplitude, center, width] 

% Initialize the output vector 
y = zeros(size(x)); 

% Evaluate each Gaussian component 
for i = 1:numel(gaussians) 
    gaussian = gaussians{i}; 
    y = y + gaussian(1) * exp(-(x - gaussian(2)).^2 / (2 * gaussian(3)^2)); 
end 

% Evaluate the Lorentzian component 
y = y + lorentzian(1) ./ ((x - lorentzian(2)).^2 + (lorentzian(3)/2)^2); 
end 

The gaussiansInLorentzian function takes the input vector x, a cell array of Gaussian parameters gaussians, and Lorentzian parameters lorentzian. It initializes the output vector y and then evaluates each Gaussian component through iteration. The Gaussian components are added to y and later, the Lorentzian component is evaluated and added to y.

You can modify the code according to your requirements, such as changing the range of the x-axis or adjusting the parameters for the Gaussians and the Lorentzian.

I hope this resolves your query.

With regards,

Rangesh.