Help with fourier transform properties

Hello everyone,

I am trying to remove a phase shift in frequency domain. I simulate a signal (cosine) and I add time delay in it and I try to reverse it in frequency domain. I do it as follows:

Fs = 800;

Time_max = 4; % seconds

t = 0:(1/Fs):Time_max-1/Fs;

delay = pi/3; % One second of delay

f = 5; %Hz

y=exp(1i*2*pi*f*t.^2);

y=real(y);

y1=exp(1i*2*pi*f*(t+pi/3).^2);

y1=real(y1);

SIZE = (length(y));

Y = fft(y,SIZE);

Y2=fft(y1,SIZE);

df = Fs/SIZE;

f1=[(0:1:SIZE/2-1)*df ((SIZE/2:SIZE-1)-SIZE)*df];

for k = 1:SIZE

    L(k) = Y2(k)*exp(-(1i*2*pi*f1(k)*delay));

end

yd = real(ifft(L));

In the case that I use a normal cosine function everythig works out perfectly and I get the original yd==y .

But in this particular case of a qudratic cosine it is not working. What am I doing wrong?

I would like to thank you in advance for the suggestions. They would be very helpful.

Regards,

Ida