Chapitre 5

If the Fourier transform of a signal is causal, it is considered analytic.

Consider a real signal \(x\left(t\right)\) with a Fourier transform \(X\left(f\right)\). Let the signal \(z_x(t)\), whose transform \(Z_x\left(f\right)\) equals \(X\left(f\right)\) over the domain of positive frequencies and is zero over the domain of negative frequencies, be defined as :

\(Z_x\left(f\right)=\left\{\begin{matrix}2\ X\left(f\right)\ \ \ \ \ \ f>0\\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f<0\\\end{matrix}\right.\) \(\enspace\) \(\enspace\) (46)

We introduce coefficient 2 solely for energy or power conservation purposes.

The signal \(z_x(t)\) is called the analytic signal associated with the real signal \(x\left(t\right)\). We can already note that its Fourier transform does not possess, by construction itself, the property of Hermitian symmetry. Therefore, it is a purely complex signal. Let's calculate its temporal expression \(z_x(t)\) :

\(Z_x\left(f\right)=\left\{\begin{matrix}\ X\left(f\right)+X\left(f\right)\ \ \ \ \ f>0\\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f<0\\\end{matrix}\right.\) (47)

We can write :

\(Z_x\left(f\right)=X\left(f\right)+sgn\left(f\right)X\left(f\right)=X\left(f\right)-j^2sgn(f)X\left(f\right)\) (48)

Using another notation :

\(Z_x\left(f\right)=X\left(f\right)-j.\left[\ j.sgn(f)X\left(f\right)\right]\) (49)

According to equation (36), we find :

\(Z_x\left(f\right)=X\left(f\right)+j\hat{X}\left(f\right)\) (50)

Since a signal \(x(t)\) is real by assumption, its Hilbert transform is also real. This immediately implies a reciprocal relationship :

\(x\left(t\right)=\mathfrak{R}\left\{z_x\left(t\right)\right\}\) (51)