"to" followed by "-ing" form of a verb

Hi

I was told that "to" is always followed by a noun phrase. I have also read that "to" is followed by base form of a verb. For instance, I went to the kitchen to drink water. I believe "drink water" is a noun phrase. So, no violation of the rule.

But many a time I have also seen "to" being followed by "-ing" form of a verb (which I think is called gerund and gerund is a noun). So, there is no violation of the rule because gerund is a noun and "to" must be followed by a noun phrase. But I still get confused when a gerund is used. My confusion is as follows. What difference would it make if instead of "to doing that" the phrase "to do that" had been used? Would there be any difference in meaning? Please help me with it. By the way, I just came across the following sentence and I thought it's the time to clear the confusion; you can use any other sentence for the purpose of explanation. Thank you.

The essential goal in designing systems is that they respond in the right way. Therefore, we must be able to calculate the response of a system to any arbitrary input signal. As we will see throughout this text, there are multiple approaches to doing that.

Regards
Jackson