MATLAB: Fourier transform on conjugate time reversal

fftMATLAB

My first question is if the following transform pair shown by equation 3 is valid:
(1) if F{x(t)} = F(f) 
(2) then F{conj(x(t)} = conj(F(-f)) 
(3) then F{conj(x(-t)} = conj(F(f))
Next, I made a matlab script to test the relation of equation (3). I created a signal x(t) took the conj(fft(x)) (right side of (3)). Then I took the fft(conj(fliplr(x))) (left side of equation 3). I got different answers. The latter has a phase slope difference when compared to the former. Looks like a delay shift in time. Here is part of the MATLAB:
num_samples = 1024;
samplingInterval_sec = 4.0e-09;
gamma = 2.0e+12;
time = ((0:(num_samples-1)) - ((num_samples-1)/2)) * samplingInterval_sec;
phi = 0.5 * gamma * time.^2;
x = exp(1i*2*pi*phi);
spectrum_1 = conj(fft(x)); 
spectrum_2 = fft(conj(fliplr(x)));

Best Answer

My first question is if the following transform pair shown by equation 3 is valid:
It is valid for continuous Fourier transforms. For discrete FTs, it holds for time-reversal that is modulo N. In modulo N time-reversal, x(1) becomes x(1), x(2) becomes x(N), x(3) becomes x(N-1) and so on. This is not equivalent to a fliplr operation, as you have assumed.
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