If we have a 12V DC input voltage and we drop that voltage using a switching regulator (LM2596) to 9V and then use a linear regulator (7805) to drop 9V to 5V, what is the output voltage going to look like?
Is the output going to be like a switching regulator output "noisy/non-linear" or linear?
voltages and part numbers are just for this example...
The output from the linear regulator will be typical linear - smooth and flat.
That's not to say that your circuit as a whole will be clean. The switching frequency from the switching regulator can get into your circuit other ways. There can be ground bounce or ground loops where the switching noise gets into your analog signals. There can also be radiated interference from the switching regulator that your analog sections pickup and amplify.
Using a linear regulator to clean up after a switching regulator helps, but it doesn't get every thing.
It's a common thing to do for noise-sensitive circuits. The linear regulator attenuates the noise on its input to a considerable degree over a wide bandwidth. This is part of what is called "line regulation" and "ripple rejection". The specific numbers or graphs are usually readily available in the datasheet.
For example, this TI datasheet covers the LM7805. Look at Figure 5 on page 9, which shows a ripple rejection figure of at least 50 dB all the way up to 100 kHz, and as high as 80 dB at lower frequencies.
