Conjugation Property and Conjugate Symmetry
The conjugation property states that if the $ \mathcal{F} $ of x(t) will be equal to X(jw) then, the $ \mathcal{F} $ of x*(t) will be equal to X*(-jw) This property follows from the evaluation of the complex conjugate of $ X^* (jw)=[\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt]^* $ $ X^* (jw)=\int\limits_{-\infty}^{\infty}x^* (t)e^{(\jmath wt)}dt $ Replacing w with -w, $ X^* (-jw)=\int\limits_{-\infty}^{\infty}x^* (t)e^{(-\jmath wt)}dt $ (The right hand side of this equation is the Fourier transform analysis equation for x*(t)) The conjugation property shows that if x(t) is real, then X(jw) has conjugate symmetry $ X(-\jmath w) = X^* (\jmath w) $ where [x(t) real]
