Hello and welcome to JustAnswer. My name is XXXXX XXXXX X am here to help you in any way I can.
Yes, FSI engines are noisier than MPI engines. Allow me to explain. Old MPI engines had their fuel injected at the base of the intake manifold and generally had lower compression ratios. Audi figured out how to inject gasoline directly into the combustion chamber. This required a heavy duty fuel injector, an insane amount of fuel pressure (like 100 bar compared to only 3 bar for MPI systems), and higher compression ratios. This is how diesel engines operate as well. The benefit is a 15 -20% increase in BOTH horsepower and fuel economy. This technology can be found on other cars as well (Mazda, Ford, etc.) and all of them are noisier compared to their MPI counterparts - it is just the nature of the design.
If Audi states that your engine is healthy, then it probably is. You are not the first person to be concerned about the increased noise of a direct-injected gasoline engine. Your vehicle still has the original design 2.0T (the first 4 cylinder engine that had direct-injection). The newer 2.0T launched in 2009 is a tad quieter due to improved acoustics and Audi managed to squeeze even more horsepower and fuel efficiency out of that unit.
I hope this helps. Let me know if you need me to clarify anything.