3 Fig. 3. Three transmitters wish to align their signals at two receivers. x$_{1}$ x$_{2}$ x$_{3}$ y$_{1}$ y$_{2}$ 1 1 1 a b c T T T a T b T c T T T T Since n is an arbitrary integer, any alignment efficiency close to 1 is possible. Hence, the partial alignment approaches the perfect alignment. available at both receivers are used for data, which in turn gives us the total DOF of the channel. In the two user X channel, we have observed that interfering signals from two different sources can be easily aligned at a single receiver. Moreover, two interfering streams are received with the same direction occupying only 1 3 of the available dimensions of the receivers. This is in fact the best efficien cy that one can hope for in reducing the number of waste dimen- sions. This perfect alignment is not possible in general. In the following example, we show that partial alignment can be used instead to provide the same performance asymptotically. Let us consider a communication scenario in which three transmitters try to align their signals at two different receivers. The channel is depicted in Figure 3. In order to shed light on the alignment part of the signaling, the intended receivers are removed from the picture. Alignment can be done at the first receiver by sending a single data stream with direction 1 from each of the trans- mitters; whereas alignment at the second receiver requires bc , ac , and ab as chosen transmit directions for first, second, and third transmitters, respectively. In general, it is not possible to simultaneously align three single data streams at two different receivers. Therefore perfect alignment is not feasible by transmitting single data streams from each transmitter. The solution to this problem is partial alignment, which is first introduced in [3]. In this technique, instead of send ing just one data stream, several data streams are transmitted from each transmitter. The idea is to choose the transmit directions based on channel coefficients in such a way that the number of received directions is minimum. For the sake of simplicity, we choose the same directions at all transmitters. Let T denote the set of transmit directions. A direction T ∈ T is chosen as a transmit direction if it can be represented as For the multiple transmitter and receiver, the above approach can be easily extended. In fact, it can be shown that asymp- totically perfect alignment is possible for any finite numbe r of transmitters and receivers. In the following section, we need the following theorem which in fact summarizes the conditions needed to achieve the multiplexing gain of 1 m per data stream in a system. For the proof please see [13]. Theorem 3: Consider there are K transmitters and K ′ re- ceivers in a system parameterized by the channel coefficient vector h . Transmitter i sends M data stream along directions T$_{i}$ = { T$_{i}$$_{0}$, T$_{i}$$_{2}$, . . . , T$_{i}$$_{(}$$_{M}$$_{−}$$_{1)}$ } for all i ∈ { 1 , 2 , . . . , K } . The data streams intended for the j ’th receiver arrive at L$_{j}$ directions, which are T$_{j}$ = { ¯ T$_{j}$$_{0}$, ¯ T$_{j}$$_{2}$, . . . , ¯ T$_{j}$$_{(}$$_{L}$$_{j}$$_{−}$$_{1)}$ } . More- over, the interference part of the received signal at the j ’th receiver has L ′ j effective data streams with received directions T ′ j = { ¯ T ′ j $_{0}$, ¯ T ′ j $_{2}$, . . . , ¯ T ′ j ( L ′ $_{j}$− $_{1)}$} for all j ∈ { 1 , 2 , . . . , K $^{′}$} . Let the following conditions for all j ∈ { 1 , 2 , . . . , K $^{′}$} hold: where 0 ≤ s$_{i}$ ≤ n − 1 for all i ∈ { 1 , 2 , 3 } . In this way, the total number of transmit directions is L$_{1}$ = n $^{3}$. To compute the efficiency of the alignment, one needs to find the set of received directions in the first and second receivers, which are denoted by T$_{1}$ and T$_{2}$ , respectively. Since all transmit directions arrive at the first receiver intact, T$_{1}$ = T . To compute the set of received directions at the second receiver, we look at the received directions due to the first, second, and third transmitters separately. Since all of them are multiplied by a , the received directions due to the first transmitter are of the form a s$_{l}$ $^{+1}$b $^{s$_{2}$}$c$^{s$_{3}$}$, where 0 ≤ s$_{i}$ ≤ n − 1 for all i ∈ { 1 , 2 , 3 } . Similarly, a $^{s$_{l}$}$bs$_{2}$ $^{+1}$c s$_{3}$ and a $^{s$_{l}$}$b$^{s$_{2}$}$cs$_{3}$ +1 are the types of received directions due to the second and third transmitter, respectively. Taking the union of all these directions, one can compute T$_{2}$ . However, we can easily see that the set of directions formed by a $^{s$_{l}$}$b$^{s$_{2}$}$c$^{s$_{3}$}$, where 0 ≤ s$_{i}$ ≤ n for all i ∈ { 1 , 2 , 3 } includes T$_{2}$ and can be used as an upper bound on the number of received directions. This set has ( n +1) $^{3}$, which is an upper bound for L$_{2}$ . Hence, we conclude that The K -user GIC models a network in which K transmitter- receiver pairs (users) sharing a common bandwidth wish to Remark 2: If C3 does not hold, then by adding a virtual data stream in the direction 1 at the receiver, one can conclude that 1 m +1 is achievable for all data streams. where m is the maximum received directions among all receivers, i.e., m = max$_{i}$ L$_{i}$ + L ′ $_{i}$. Then, by encoding each data stream using the constellation U = ( − Q, Q )$_{Z}$ where Q = γP 1 $_{−}$ϵ 2( m + ϵ ) and γ is a constant, the following DOF is achievable for almost all realizations of the system: T = a $^{s$_{1}$}$b$^{s$_{2}$}$c$^{s$_{3}$}$, (3) r$_{sum}$ = L$_{1}$ + L$_{2}$ + · · · + L$_{K}$ ′ m , (4) III. PROOF OF THE MAIN THEOREM A. System Model η = L$_{1}$ L$_{2}$ > ( n n + 1 ) 3 . C1 Components of T$_{i}$ are distinct member of G ( h ) and linearly independent over the field of rational numbers. C2 Components of T$_{i}$ and T i are all distinct. C3 One of the elements of either T$_{i}$ or T i is 1.