Re: sync function
Raised Cosine Impulse with Roll-Off alfa, have temporal shape similar to Sinc(x) impulse. The substantial difference is that sinc(x) is only a mathematical function (it's in practice impossible to realize an impulse withe infinite duration; an other interpretation is that the TCF of a sinc(x) is a rectangular function and in practice is impossible realize filter with frequency response of that shape).
It is important notice that RRC(alfa) impulse must be limited in the time domain (the transmission impulse must be of finite duration): the great advantage respect sinc(x) is that the quasi-all energy is addensated around the origin. You can control by the parameter alfa, the effective practical duration of the RRC(alfa) impulse, so when truncated, is loss only a trascurable quantity of energy.
You ask the why to use RRC(alfa) in BB communications: it's a limitative ipotesi to work in BB, because You can represent a Transmission System who work at RF, by the Representation in Base Band by use the Complex Envelope of all the signals and the Equivalent Model in Base Band of Your Radio Frequency System.
So, You can use the RRC(alfa) impulses like in PAM (works in BB) like, for example, in QAM (works at RF).