CADAMBE AND JAFAR: INTERFERENCE ALIGNMENT AND DEGREES OF FREEDOM OF THE-USER INTERFERENCE CHANNEL3437region of theuser interference channel for anywhereencoded intoindependent streams by transmitter asThe received signal at the th receiver can then be written as This implies thatWe provide an achievable scheme to show that lies in the degrees of freedom region symbol extension of the original of an channel which automatically implies the desired result. In the extended channel, the signal vector at the th user’s receiver can be expressed asAll receivers decode the desired signal by zero-forcing the interferinterference vectors. At receiver 1, to obtain ence free dimensions corresponding to the desired signal from -dimensional received signal vector an , the dimension of the interference should be not more than . This can be ensured by perfectly aligning the interference as follows: from transmitters(49) At the same time, receiver 2 zero-forces the interference from . To extract interference-free dimensions from a -dimensional vector, the dimension of . the interference has to be not more than so that This can be achieved by choosingwhere is an column vector representing the symbol extension of the transmitted symbol , i.e.. . .. . . (50) Notice that the above relations align the interference from transmitters within the interference from transmitter 1 at receiver 2. Similarly, to decode at receiver when we so that the following relations are wish to choose satis ed. (51) so that (49), We now wish to pick vectors have a (50), and (51) are satis ed. Since channel matrices almost surely, (49), (50) and (51) can be equivfull rank of alently expressed asSimilarly and represent symbol extensions of the and , respectively. is a diagonal masymbol extension of the channel as trix representing the shown in the equation at the bottom of the page. Recall that the are drawn independently from a condiagonal elements of tinuous distribution and are therefore distinct with probability . case, message is In a manner similar to the encoded at transmitter 1 into independent streams along vectors so that iswhereis a column vector and -dimensional matrix. Similarlyis a isAt receiver (52). . ..... . .Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 01,2010 at 01:21:44 UTC from IEEE Xplore. Restrictions apply.