Gaussian wavelet generation with a given frequency
If I understand the question, I don't think you can do what you want to do. The spectrum of a Gaussian time series must contain frequencies down to DC, i.e. very low frequencies (Python code from scipy.org):



A wavelet whose spectrum is a Gaussian is called a Ricker wavelet, or sometimes Mexican Hat wavelet. I often use this wavelet to model seismic reflection data. It has a central frequency, and is bandlimited. As such, the wavelet oscillates around zero amplitude — it does not have a DC component: