For dispersion compensation techniques we sometimes use dispersion compensation fiber(DCF) and sometimes fiber grad grating(FBG) which technique is better and why?
Overall DSF is a more mature technology, while FBG is newer and more compact. Which one is better, depends on your specific application. Here is a comparison for WDM transmission:
The fibers with opposite signs of second order GVD can also be combined to produce an overall small or zero dispersion. (See nonlinear Fiber Optics by G.P. Agrawaal)
In fact it depends on your specific application: DCF (not DSF as Mr. Solnsev metioned) has higher losses, in reality you need about 1/4 of SMF length to compensate dispersion - on the other hand you can buy it! FBG seems to be a compact version, but for preparation of an about 242 nm grating period one needs a high tec method (maybe laser interference pattern?); furthermore the "bandwidth" of dispersion compensation is mostly not known. Thus you have an application specific preparation, which is more expensive...
As mentioned by other researchers, the choice is application sensitive. Another difference is the device length. DCF is order of few km at least, but FBG is very short one. Few cm in FBG(small dispersion) and few m in CFBG(large disperison) is required, which can be changed depending on the degree of dispersion compensation. The reference addressed by Alexander S. Solntsev may become good starting point in your decision.