hilbert transform

The Hilbert transform is a mathematical operation that shifts the phase of each frequency component of a signal by 90 degrees. It is used in signal processing to create an analytic signal, which is a complex-valued signal that has no negative frequency components. The real part of this analytic signal is the original signal, while the imaginary part is the Hilbert transform of the original signal.

Mathematically, for a given real-valued signal ( x(t) ), the Hilbert transform ( \hat{x}(t) ) is defined as:

[ \hat{x}(t) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{x(\tau)}{t - \tau} , d\tau ]

In the frequency domain, the Hilbert transform can be described using the Fourier transform. If ( X(f) ) is the Fourier transform of ( x(t) ), then the Fourier transform of the Hilbert transform ( \hat{x}(t) ) is given by:

[ \mathcal{F}{\hat{x}(t)} = -j \cdot \text{sgn}(f) \cdot X(f) ]

where ( \text{sgn}(f) ) is the sign function:

[ \text{sgn}(f) =
\begin{cases}
1 & \text{if } f > 0 \
0 & \text{if } f = 0 \
-1 & \text{if } f < 0
\end{cases}
]

The analytic signal ( z(t) ) formed by combining the original signal ( x(t) ) and its Hilbert transform ( \hat{x}(t) ) is:

[ z(t) = x(t) + j\hat{x}(t) ]

This analytic signal is useful in various applications, such as amplitude modulation, demodulation, and instantaneous frequency analysis.