How can I use MATLAB to show that the Doppler effect destroys orthogonality of OFDM subcarriers?

Question

I am trying to get a visual representation of how the Doppler effect destroys the orthogonality of the subcarrier and understand this effect.

I plotted three orthogonal subcarriers in MATLAB:

st = linspace(-10,10,1000);     % sinc, -10 to 10.
y1 = sinc(st);
y2 = sinc(st);
y3 = sinc(st + ((pi/2)/4000));
figure
hold on
plot(st+1,y2,'r',st+2,y2,'b',st,y3,'r--');grid;xlim([-8 11]);
hold off

To observe the effect of the Doppler shift, should I only add a shift for one of the subcarriers or will all subcarriers be shifted?

st = linspace(-10,10,1000);     % sinc, -10 to 10.
y1 = sinc(st+f_doppler);
y2 = sinc(st);
y3 = sinc(st + ((pi/2)/4000));

My goal is to reproduce the following figure:

S. Ahmed and H. Arslan, "Evaluation of frequency offset and Doppler effect in terrestrial RF and in underwater acoustic OFDM systems," MILCOM 2008 - 2008 IEEE Military Communications Conference, San Diego, CA, USA, 2008, pp. 1-7, doi: 10.1109/MILCOM.2008.4753547 Add to Citavi project by DOI.

UPDATES

max_doppler_shift = 5;
% Generate frequency shifts
frequency_shifts = linspace(-max_doppler_shift, max_doppler_shift, 3);
y11 = sinc(st-frequency_shifts(1));
y21 = sinc(st-frequency_shifts(2));
y31 = sinc(((pi/2)/4000)-frequency_shifts(3));

I don't see any effect of adding a Doppler offset or do I read the plots wrong?

Answer 1

The absolute frequency of all the subcarriers is shifted. But the shift can only be approximated by adding a constant. What really happens is that the absolute frequencies are rescaled, ie. multiplied by a constant that’s near to 1. This also means that the frequency spacing between the subcarriers is rescaled by the same constant.

When the constant is very close to 1, or the spacing between the subcarriers is close enough in a narrowband signal, adding a frequency shift is a reasonable approximation. But the general solution multiplies all absolute frequencies by a constant.

Usually the carrier and the subcarrier frequencies are derived from the same frequency reference, so any errors in the reference frequency will also show up as scaling of absolute frequencies by a constant close to 1.