I am currently working with a sinusoidal signal buried in noise. I filter the signal with a low-pass filter of which I know the transfer function. How can I calculate the SNR after the filtering process assuming white Gaussian noise? How would I calculate the SNR assuming pink noise?
This looks like an exam/homework style question. If it is, it may be helpful if you could provide a bit of background about what are you expected to know, because there are several ways we could solve this.
The most general way would be to get the power-spectral density of the noise. Then we would integrate it over our bandwidth and get the initial power of the noise.
Next we have the noise filtering itself. You convolve the noise with the filter in time domain or you multiply the filter's transfer function with the noise PSD in frequency domain. Then you could integrate the surface under the PSD in frequency domain and get noise power. In the end you get something like in the illustration:
Black rectangle is the PSD of "unfiltered" noise and the red rectangle is the PSD of filtered noise, assuming a filter with square transfer function and AWGN.
In the case of pink noise, the main difference is that the PSD of the noise will look a bit different. Instead of square, you'll get something that looks like a triangle. The procedure is same. You multiply the PSD of the noise with the transfer function of the filter and get a shape. Surface of that shape is your noise power.
What you need is an understanding of Equivalent noise bandwidth. See this.
What this basically says is that if you low-pass filter wide-band white noise with a simple 1st order low pass filter, you are effectively doing the same as a theoretical brick wall filter at 1.57 times the cut-off.
If you have a steeper filter the equivalent bandwidth becomes closer i.e. for a 2nd order filter the equivalent noise bandwidth is 1.22 times the filter's cut-off frequency.
So if you have white noise at x watts per Hz and you have a 1kHz 1st order low pass filter, the total noise at the output is limited to x watts x 1570 Hz