The Hilbert transform is a linear operator that takes a real-valued signal \( x(t) \) and produces another real-valued signal \( \hat{x}(t) \), often called its quadrature counterpart. In the time domain, it is defined as the convolution of \( x(t) \) with the kernel \( 1/(\pi t) \):
\( \mathcal{H}\{x(t)\} = \hat{x}(t) = \dfrac{1}{\pi} \cdot \text{p.v.} \int_{-\infty}^{\infty} \dfrac{x(\tau)}{t - \tau} \, d\tau \)
where p.v. denotes the Cauchy principal value, which handles the singularity at \( \tau = t \). In practice, the Hilbert transform is most easily understood in the frequency domain.