Dear Mostafa
in current implemtations , usually there is a SAW filter after the LNA , which is off chip this SAW filter handel the conversion and then after it u just add a matching circuit for the input of the mixer
this will not add noise to the LNA .
the meathod ur talking about is great , but it need alot of tweaking , for gain and phase balance between the 2 paths
the third way , use on chip transformer which will add alot of chip area but u need to optimize this tranformer
wish this help u
Khouly