Study of the properties of X(f), Xp(f) and Xi(f)
\(X(f)=TF[x_p\ (t)]+TF[x_i (t)]=Re[X(f)]+j.Im[X(f)]\) \(\enspace\) \(\enspace\) \(\enspace\) \(\enspace\)(40)
Using the equation (39) :
\(Re\left[X\left(f\right)\right]=TF\left[x_p\left(t\right)\right]=TF\left[x_i\left(t\right).sign\left(t\right)\right]=TF\left[x_i\left(t\right)\right]\ast TF\left[sign\left(t\right)\right]\)
\(Re\left[X\left(f\right)\right]=TF\left[x_p\left(t\right)\right]=TF\left[x_i\left(t\right).sign\left(t\right)\right]=TF\left[x_i\left(t\right)\right]\ast TF\left[sign\left(t\right)\right]=\frac{1}{2\pi}X_i\left(f\right)\ast\frac{2}{j2\pi f}=\frac{1}{\pi}\left(X_i\left(f\right)\ast\frac{1}{j\omega}\right)\)
\(\quad\) \(\quad\) \(\quad\) \(=\frac{1}{2\pi}\left(j.Im\left[X\left(f\right)\right]\ast\frac{1}{j\pi f}\right)\) \(\quad\) \(\quad\) \(\quad\) \(\quad\) \(\quad\) (41)
\(Re\left[X(f)\right]=\frac{1}{2\pi}\left(Im\left[X\left(f\right)\right]\ast\frac{1}{\pi f}\right)\) \(\enspace\) \(\enspace\) \(\enspace\) \(\enspace\)(42)
Equation (42) is referred to as the Hilbert transform of \(X(f)\) in the frequency domain.
\(j.Im\left[X\left(f\right)\right]=TF\left[x_i\left(t\right)\right]=TF\left[x_p\left(t\right).sign\left(t\right)\ \right]=TF\left[x_p\left(t\right)\right]\ast TF\left[sign\left(t\right)\right]=\frac{1}{2\pi}X_p\left(f\right)\ast\frac{2}{j2\pi f}=\frac{1}{2\pi}\left(X_p\left(f\right)\ast\frac{1}{j\pi f}\right)=\frac{1}{2\pi}\left(Re\left[X(f)\right]\ast\frac{1}{j\pi f}\right)\) (43)
\(Im\left[X\left(f\right)\right]=-\frac{1}{2\pi}\left(Re\left[X(f)\right]\ast\frac{1}{\pi f}\right)\) \(\enspace\) \(\enspace\) \(\enspace\) \(\enspace\)(44)
In the frequency domain, we refer to equation (44) as the Hilbert transform. It can be observed that :
\(Re\left[X(f)\right]=TH\left\{Im\left[X\left(f\right)\right]\right\} et Im\left[X\left(f\right)\right]=-TH\left\{Re\left[X(f)\right]\right\}\) \(\enspace\) \(\enspace\) \(\enspace\) \(\enspace\) (45)
A causal signal \(x\left(t\right)\) has a spectrum \(X\left(f\right)\), where the real and imaginary parts are related by the Hilbert transform.