You can extract instantaneous phase information and the envelope of an input signal using Hilbert transforms.
For example, the cosine signal of the input signal x(t) is the following equation:
To find the analytic signal of x(t), calculate the Hilbert transform of x(t) using the following equation:
The analytic signal of x(t) is defined using the equation
where the amplitude of f(t), which is A(t), is the envelope of x(t) and where the phase of f(t), which is ω0t + φ(t), is the instantaneous phase of x(t).
You can obtain the single-sideband spectra using the Fourier transform on the analytic signal f(t), as shown in the following equation:
