I have 100,000 samples of a signal $x[n]$ that was sampled at 20kHz. The data is vibration data from a rotating machine, and contains a significant spectral component related to the speed of the machine's rotation.
Because the speed of the machine varies over the duration of the sample, using the peak of the FFT does not yield the result I am looking for.
So I want to use estimators such as Kay's estimator that allow short-term estimates, but assume a signal model of:
$x[n] = A \exp(j \omega n + \theta) + z[n]$
where $n$ = 0... 99,999, $A$ is the amplitude, $\omega$ is the frequency to be estimated, $\theta$ is the initial offset, and $z[n]$ is the complex noise.
However, my signal is real-valued and looks more like:
$x[n] = A \cos(\omega n + \theta) + z_r[n]$
where $z_r$ and $A$ are now real-valued.
How do I transform my real-valued signal into a complex-valued signal, so that I can use Kay's estimator?