Instantaneous Amplitude, Phase and Frequency

Here is a link to a good discussion on this with MATLAB code.

Here is another place with a good discussion on doing a phase shift.
The Hilbert transform is fft ---> phase shift ---> ifft.
Doing fft and ifft are easy in numpy, and a phase shift is just multiplying each term in the signal by exp(j*p), where j is the imaginary unit, and p is the desired phase shift.

Variance

Different programs implement variance in different ways. This can lead to confusing results when you move from one set of functions to another.

scipy.stats.cov() normalizes by (N-1), by default, bias=False. By setting bias=True, you will normalize by N.

However, scipy.var() normalizes by N.

In Octave, var( x, 0 ) biases by N-1, and var( x, 1 ) biases by N.

Matlab and R, I think, do things slightly differently.