When we design a discrete Gabor filter for a 1D signal, how do we determine the variance of the filter, depending on the time step of the signal? Is there any empirical theorem/result that we should know about the Discrete Gabor Transform?
What I look for in such parameter choice problems is empirical experiments. Whatever works best for my application is the correct value to use.
You can develop a parameter search algorithm looking for the optimal values in some training data set. Keep in mind that what works for one application may not perform well in others.
I'm not aware of mathematical arguments estimating gabor's $\theta$. These estimates are typically conservative anyway.