conv gets central frequency wrong in voigt profile if you add the frequency offset twice

I can generate a Voigt profile by convolving a Lorentzian and Gaussian line shape using the Matlab conv command. If the Lorentzian and Gaussian line shapes are zero-based, then this works great. But if I have an offset in the x-axis, then that offset gets doubled in the convolution. Here is the code:

    clear; close all;

    lwid = 1;
    gwid = 10;
    fzer = -10;
    fi = linspace(-100,100,10001);
    y1 = (((fi - fzer)./lwid).^2 + 1).^(-1);
    y2 = exp(-(fi-fzer).^2 ./ (2*gwid^2));
    y3 = conv(y1,y2);
    x3 = linspace(2*min(fi),2*max(fi),2*length(fi)-1);
    y3 = y3 ./ max(y3);

    plot(fi,y1,fi,y2,x3,y3);

The convolution y3 has a "center" frequency at twice fzer. Apparently this can be avoided if you only offset ONE of either the Lorentzian or Gaussian profiles.

[Scott]

By the way, this only works if the horizontal scale is symmetric about zero. In the above example, if the array is defined as

    fi=linspace(-120,90,10001);

then the center frequency of the y3 array is off once again! Be careful!

Answer 1

[Matt J]

I don't think there's anything out of the ordinary or unexpected in the result you're seeing. It's well known that signal shifts are additive under convolution.

As a simpler example, it is a basic result that the convolution of impulse(t-T1) with impulse(t-T2) is another impulse centered at t=T1+T2.

[Scott]

Thank you. You are right. In retrospect I should have known this, but somehow I didn't consider it ahead of time. Any ideas about why shifting the min and max of the x-axis array also shifts the convolution?

[Matt J]

It won't shift the convolution. It will shift your plot. Presumably though, you want to be using the syntax

 conv(y1,y2,'same')

so that you don't have to worry about defining a 2nd x-axis.