Degrees of Freedom of the MIMO Interference Channel with Cooperation and Cognition

1 De grees of Freedom of the MIMO Interference Channel with Cooperation and Cognition Chiachi Huang and Syed A. Jafar Electrical Engineering and Computer Science Uni versity of California Irvine Irvine, CA, USA Email: { chiachih, syed } @uci.edu Abstract In this paper , we explore the benefits, in the sense of total (sum rate) de grees of freedom (DOF), of cooperation and cogniti ve message sharing for a two-user multiple-input-multiple-output (MIMO) Gaussian interference channel with M 1 , M 2 antennas at transmitters and N 1 , N 2 antennas at receiv ers. For the case of cooperation (including cooperation at transmitters only , at receiv ers only , and at transmitters as well as receivers), the DOF is min { M 1 + M 2 , N 1 + N 2 , max( M 1 , N 2 )) , max( M 2 , N 1 ) } , which is the same as the DOF of the channel without cooperation. For the case of cognitive message sharing, the DOF is min { M 1 + M 2 , N 1 + N 2 , (1 − 1 T 2 )((1 − 1 R 2 ) max( M 1 , N 2 ) + 1 R 2 ( M 1 + N 2 )) , (1 − 1 T 1 )((1 − 1 R 1 ) max( M 2 , N 1 ) + 1 R 1 ( M 2 + N 1 )) } where 1 T i = 1 (0) when transmitter i is (is not) a cognitiv e transmitter and 1 Ri is defined in the same fashion. Our results show that while both techniques may increase the sum rate capacity of the MIMO interference channel, only cogniti ve message sharing can increase the DOF . W e also find that it may be more beneficial for a user to have a cogniti ve transmitter than to hav e a cogniti ve receiv er . 2 I . I N T RO D U C T I O N Multiple-input-multiple-output (MIMO) systems ha ve been prov en to be very powerful in point-to-point commu- nication. Follo wing their success in the point-to-point case, MIMO techniques have been widely applied to v arious multiuser communication scenarios. Since the capacity region for most network communication scenarios has been an open question for many years, capacity approximations are needed to pro vide an e valuation of the system performance. The number of degrees of freedom (DOF), which is also known as capacity pre-log or multiplexing gain [1], provides a capacity approximation C Σ ( ρ Σ ) = η log ( ρ Σ ) + o ( log ( ρ Σ )) where η is the number of degrees of freedom, C Σ ( ρ Σ ) is the sum rate capacity , and ρ Σ is the signal-to-noise ratio (the total transmit power of all nodes di vided by the local noise power). The approximation error is within o (log( ρ Σ )) for any ρ Σ and the accuracy of the approximation approaches 100% as ρ Σ increases. The DOF of various multiuser MIMO systems hav e been found. The two-user multiple access channel (MA C) with M 1 , M 2 antennas at the transmitters and N antenna at the receiver has the DOF of min( M 1 + M 2 , N ) [2]. The two-user broadcast channel (BC) with M antennas at the transmitter and N 1 , N 2 antennas at the receiv ers has the DOF of min( M , N 1 + N 2 ) [3]– [5]. The DOF of the two-user MIMO interference channel with M 1 , M 2 antennas at the transmitter and N 1 , N 2 antennas at the recei vers, which will be referred to ( M 1 , M 2 , N 1 , N 2 ) interference channel later in this paper, is min { M 1 + M 2 , N 1 + N 2 , max( M 1 , N 2 ) , max( M 2 , N 1 ) } [6]. Note that in the MIMO MA C and BC, the distributed processing at either the transmitter side or the receiv er side does not cause any loss in the DOF . But in the MIMO interference channel, there may be a significant loss in the DOF due to the distrib uted processing at both transmitter and receiv e sides. For example, while a (1 , n, n, 1) interference channel has only one degrees of freedom, the point- to-point MIMO system with 1 + n antennas at both transmitter and recei ver has 1 + n degrees of freedom. Many techniques are possible candidates to compensate the loss in DOF caused by the distributed processing nature of the MIMO interference channel. In this paper , we consider two of them: user cooperation and cognitiv e message sharing. W e will explore the benefits, in the sense of DOF , of these two techniques for a two-user MIMO Gaussian interference channel. A. Cooperation The basic idea for cooperation is that sev eral nodes cooperate with each other and act as a large virtual antenna array . Nodes can cooperate to form a transmit antenna array or receive antenna array . Cooperation is made possible by allowing noisy links between distributed transmitters or distributed receivers. A two-user interference channel with single antenna at all nodes is considered by Host-Madsen and Nosratinia in [7], [8]. They show that the number of DOF is equal to one when cooperation takes place at transmitters only , at recei vers only , or both at transmitters as well as receiv ers. Howe ver , the DOF for the two-user MIMO interference channel with cooperation remains unkno wn. One of the goals that we pursue in this paper is to answer this question. W e find an upper bound for the DOF of the two-user MIMO Gaussian interference channel with cooperation. The upper bound coincides with the DOF of the channel without cooperation. Thus, we obtain the negati ve result that cooperation can not increase the DOF of the two-user MIMO Gaussian interference channel, a generalized result from the single antenna case. B. Cognitive Messag e Sharing Cogniti ve message sharing refers to genie-aided cooperation in the manner of cognitiv e radio. In the cognitive radio model, some messages are made av ailable to some nodes (other than the intended nodes) non-causally , 3 2 ˆ W 1 ˆ W 1 2 1 M 1 2 2 M 1 W 2 W 1 2 1 N 1 2 2 N 1 3 2 4 MIMO-IC MIMO-IC with coopera tion   2 T 2 ˆ W 1 ˆ W 1 2 1 M 1 2 2 M 1 W 2 W 1 2 1 N 1 2 2 N 1 3 2 4   1 T   1 R   2 R Fig. 1. Channel models for MIMO interference channels with and without cooperation. noiselessly , and for free [9]. The nodes that get the shared messages are called either cognitiv e transmitters or cogniti ve recei vers depending on their roles in the channel. Cooperation among users for the interference channel with single antenna at all nodes has been studied in [10]–[13] in the context of cogniti ve radio channel. The DOF for a ( M , M , M , M ) interference channel with cognitiv e message sharing has been studied in [14]. They find that cogniti ve message sharing can increase the DOF of the channel for some cogniti ve scenarios. They also find that there is no difference, in the sense of DOF , for a user to have a cogniti ve transmitter or to have a cogniti ve receiv er . Ho wev er , the corresponding DOF result and the difference between having a cogniti ve transmitter and a cogniti ve recei ver for a more general ( M 1 , M 2 , N 1 , N 2 ) interference channel remain unkno wn. The second goal of this paper is to find the DOF along with the DOF region of a ( M 1 , M 2 , N 1 , N 2 ) interference channel with various cognitive message sharing scenarios. W e find that the total number of DOF of a ( M 1 , M 2 , N 1 , N 2 ) interference channel is gi ven by η 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 = min            M 1 + M 2 N 1 + N 2 (1 − 1 T 2 ) { (1 − 1 R 2 ) max( M 1 , N 2 ) + 1 R 2 ( M 1 + N 2 ) } (1 − 1 T 1 ) { (1 − 1 R 1 ) max( M 2 , N 1 ) + 1 R 1 ( M 2 + N 1 ) }            (1) where 1 T i = 1 if transmitter i is a cogniti ve transmitter and 1 T i = 0 if transmitter i is not a cognitiv e transmitter and 1 Ri is defined in the same fashion. Our results sho w that in general, it may be more beneficial, in the sense of DOF , for a user to ha ve a cognitive transmitter than to hav e cognitiv e receiver . I I . S Y S T E M M O D E L A two-user Gaussian MIMO interference channel (MIMO-IC) is defined by Y [3] = H [31] X [1] + H [32] X [2] + N [3] Y [4] = H [41] X [1] + H [42] X [2] + N [4] (2) where Y [3] is the N 1 × 1 output vector at the node 3 , Y [4] is the N 2 × 1 output vector at node 4 , X [1] is the M 1 × 1 input vector at node 1 , X [2] is the M 2 × 1 input vector at node 2 , N [3] is the N 1 × 1 additive white Gaussian noise 4   2 T 2 ˆ W 1 ˆ W 1 2 1 M 1 2 2 M 1 W 2 W 1 2 1 N 1 2 2 N 1 3 2 4   1 T   1 R   2 R 1 W [ 0, 1, 0, 0 ] [ 0, 1, 1, 0 ]   2 T 2 ˆ W 1 ˆ W 1 2 1 M 1 2 2 M 1 W 2 W 1 2 1 N 1 2 2 N 1 3 2 4   1 T   1 R   2 R 1 W 2 W Fig. 2. Channel models for MIMO interference channels with cognition. T wo scenarios are shown in which [1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 ] = [0 , 1 , 0 , 0] and [1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 ] = [0 , 1 , 1 , 0] separately . (A WGN) v ector at node 3 , N [4] is the N 2 × 1 additi ve white Gaussian noise (A WGN) vector at node 4 , and H [ j i ] is the channel matrix from node i to j . All vectors and matrices are real. W e assume that all channel matrices are fixed and known to all transmitters and recei vers. W e also assume that all channel coef ficients values are drawn from a continuous distribution. This assumption ensures that all channel matrices are full rank with probability one. Furthermore, the transmitters are subject to an average transmit power ρ . There are two independent messages in the channel: W 1 and W 2 where W i is the intended message from node i to node i + 2 , i = 1 , 2 . The message sets are assumed to be functions of ρ , and we indicate the size of the message set by | W i ( ρ ) | . For code words spanning n channel uses, the rate R i ( ρ ) = log | W i ( ρ ) | n is achiev able if the probability of error for W i can be made arbitrarily small. The capacity re gion C ( ρ ) of the channel is defined as the set of all simultaneously achiev able rate tuples R ( ρ ) = ( R 1 ( ρ ) , R 2 ( ρ )) . Similar to the definition of the degrees of freedom region in [14], we define the degrees of freedom region D of the Gaussian MIMO-IC as D , ( ( d 1 , d 2 ) ∈ R 2 + : ∀ ( w 1 , w 2 ) ∈ R 2 + w 1 d 1 + w 2 d 2 ≤ lim sup ρ →∞ sup R ( ρ ) ∈C ( ρ ) w 1 R 1 ( ρ ) + w 2 R 2 ( ρ ) log( ρ ) ! ) The (total) degrees of freedom η of the Gaussian MIMO-IC is defined as η , max D ( d 1 + d 2 ) . W e use the following notational con ventions. The conv ex hull of the set A is denoted by co( A ) . The function max( x, 0) is denoted by ( x ) + . R n + and Z n + represent the sets of n-tuples of non-negati ve real numbers and inte gers respecti vely . 5 I I I . D E G R E E S O F F R E E D O M O F T H E M I M O I N T E R F E R E N C E C H A N N E L W I T H C O G N I T I O N In this section, we find the DOF along with the DOF region of a ( M 1 , M 2 , N 1 , N 2 ) interference channel with v arious cogniti ve message sharing scenarios. W e use the term ”cogniti ve message sharing” to refer to the message sharing in the manner of cognitive radio. W e let 1 T i = 1 (0) to indicate that transmitter i is (is not) a cognitiv e transmitter . 1 Ri is defined in the same fashion. There are total 16 possible combinations of cogniti ve message sharing scenarios. Figure 2 giv es some examples of the possible combinations. Note that in our model, node 1 is transmitter 1 , node 2 is transmitter 2 , node 3 is receiv er 1 , and node 4 is recei ver 2 . A specific cognitiv e message sharing scenario is labeled by [1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 ] . W e use η 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 and D 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 to denote the DOF and the DOF re gion of scenario [1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 ] . W e start from an achie v able scheme. Definition 1: Define A 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 to be the set of all ( d 1 , d 2 ) ∈ Z 2 + satisfying 1 T 1 M 1 + M 2 ≥ 1 T 1 d 1 + d 2 M 1 + 1 T 2 M 2 ≥ d 1 + 1 T 2 d 2 N 1 ≥ (1 − 1 R 1 )( d 2 − (1 T 1 M 1 + M 2 − N 1 ) + ) + + d 1 N 2 ≥ (1 − 1 R 2 )( d 1 − ( M 1 + 1 T 2 M 2 − N 2 ) + ) + + d 2 . The follo wing theorem provides an inner bound for D 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 . Theor em 1: D in 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 , co( A 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 ) ⊆ D 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 . Pr oof: First, we show that any ( d 1 , d 2 ) ∈ A 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 is achie vable. Instead of providing a proof for general scenario [1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 ] , we prove the achiev ability for the scenario [0 , 1 , 0 , 1] to illustrate the ke y ideas and a void the tediousness and complexity of di viding cases in the general scenario. Let r 1 = ( M 1 + 1 T 2 M 2 − N 2 ) + = ( M 1 + M 2 − N 2 ) + r 2 = (1 T 1 M 1 + M 2 − N 1 ) + = ( M 2 − N 1 ) + . Choose v [ 31 ] 1 , . . . , v [ 31 ] r 1 ∈ R M 1 + and v [ 32 ] 1 , . . . , v [ 32 ] r 1 ∈ R M 2 + such that h H [ 41 ] H [ 42 ] i " v [31] 1 . . . v [31] r 1 v [32] 1 . . . v [32] r 1 # = h 0 . . . 0 i When d 1 ≤ r 1 , only v [ 31 ] 1 , . . . , v [ 31 ] d 1 and v [ 32 ] 1 , . . . , v [ 32 ] d 1 are needed. When d 1 > r 1 , choose the remaining v [ 31 ] r 1 + 1 , . . . , v [ 31 ] d 1 and v [ 32 ] r 1 + 1 , . . . , v [ 32 ] d 1 according to an isotropic distrib ution so that the set S 1 = (" v [ 31 ] 1 v [ 32 ] 1 # , . . . , " v [ 31 ] d 1 v [ 32 ] d 1 #) is linearly independent with probability one. Choose v [ 42 ] 1 , . . . , v [ 42 ] r 2 ∈ R M 2 + such that H [ 42 ] h v [ 42 ] 1 . . . v [ 42 ] r 2 i = h 0 . . . 0 i . When d 2 ≤ r 2 , we only need v [ 42 ] 1 , . . . , v [ 42 ] d 2 . When d 2 > r 2 , choose the remaining v [ 42 ] r 2 + 1 , . . . , v [ 42 ] d 2 according to an isotropic distrib ution so that the set S 2 = n v [ 42 ] 1 , . . . , v [ 42 ] d 2 o 6 is linearly independent with probability one. Note that since all v [ 31 ] i , v [ 32 ] j , and v [ 42 ] k are chosen separately and we require (implicitly or e xplicitly) d 1 + d 2 ≤ M 1 + M 2 , the set S = S 1 [ (" 0 v [ 42 ] 1 # , . . . , " 0 v [ 42 ] d 2 #) is linearly independent with probability one. After choosing all transmit vectors, let X [ 1 ] = d 1 X i =1 v [ 31 ] i x [1] i (3) X [ 2 ] = d 1 X i =1 v [ 32 ] i x [1] i + d 2 X i =1 v [ 42 ] i x [2] i (4) where x [ j ] i represents the i th input used to transmit the code word for message W j . Y [3] = d 1 X i =1 x [1] i  H [31] v [31] i + H [32] v [32] i  | {z } range space dimension = d 1 + r 2 X i =1 x [2] i H [32] v [42] i | {z } =0 + d 2 X i = r 2 +1 x [2] i H [32] v [42] i | {z } range space dimension = ( d 2 − r 2 ) + + N [3] In order to provide enough dimensions to separate the intended signals and the interference, the achie v able scheme requires that N 1 ≥ d 1 + ( d 2 − ( M 2 − N 1 ) + ) + . Note that among all N 1 dimensions at node 3 , there are d 1 dimensions for the intended signals and ( d 2 − r 2 ) + dimensions for the interference. By discarding the dimensions that contain the interference, there are d 1 degrees of freedom for W 1 . W e want to point out that the dimensions of the intersection of the signal space and the interference space is zero with probability one. Since node 4 is a cognitiv e receiv er , it can subtract all the signals that caries W 1 . So we only need N 2 ≥ d 2 to obtain d 2 degrees of freedom for W 2 . Thus, ( d 1 , d 2 ) is achie vable. By time sharing, co( A 0 , 1 , 0 , 1 ) is achie vable. W e need the follo wing lemma for the con verse. Lemma 2: For any ( d 1 , d 2 ) ∈ D 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 , the follo wing statements are true. L 1 : d 1 + d 2 ≤ min( M 1 + M 2 , N 1 + N 2 ) L 2 : d 1 ≤ N 1 L 3 : d 2 ≤ N 2 L 4 : If 1 T 2 = 0 , then d 1 ≤ M 1 L 5 : If 1 T 1 = 0 , then d 2 ≤ M 2 L 6 : If 1 T 2 1 R 2 = 0 , then d 1 + d 2 ≤ max( M 1 , N 2 ) L 7 : If 1 T 1 1 R 1 = 0 , then d 1 + d 2 ≤ max( M 2 , N 1 ) Pr oof: L 1 is tri vial. L 2 and L 4 ( L 3 and L 5 ) are obtained by letting W 2 ( W 1 ) be a dummy message that is kno wn priori for all nodes. W e refer L 6 and L 7 to Theorem 1 and Corollary 1 in [6]. Note that in the proof of Theorem 1 in [6], the message is provided by a genie to a recei ver . But the result is actually stronger in the sense 7 that ev en the message is giv en to both the transmitter and the receiver of the same user , all ar guments in the proof still hold. Cor ollary 3: Define D out 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 as the set of all ( d 1 , d 2 ) ∈ R 2 + that satisfy L 1 to L 7 in Lemma 2. Then D 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 ⊆ D out 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 . Theor em 4: D in 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 = D 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 = D out 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 . Pr oof: Again, we pro vide the proof for scenario [0, 1, 0, 1] to illustrate the key ideas. Using the fact that d 1 ≤ N 1 and d 2 ≤ N 2 ensure that d 1 + d 2 ≤ N 1 + N 2 , we can remove the constraint d 1 + d 2 ≤ N 1 + N 2 in D out 0 , 1 , 0 , 1 . Reorg anizing the constraints in D out 0 , 1 , 0 , 1 , we ha ve the follo wing D out 0 , 1 , 0 , 1 =        ( d 1 , d 2 ) ∈ R 2 + : d 1 ≤ N 1 d 2 ≤ min( M 2 , N 2 ) d 1 + d 2 ≤ min( M 1 + M 2 , max( M 2 , N 1 ))        Using Lemma 5 bellow , the constraint N 1 ≥ d 1 + ( d 2 − ( M 2 − N 1 ) + ) + in A 0 , 1 , 0 , 1 is equiv alent to d 1 ≤ N 1 and d 1 + d 2 ≤ max( M 2 , N 1 ) . Reorganizing the constraints in A 0 , 1 , 0 , 1 , we find that A 0 , 1 , 0 , 1 = D out 0 , 1 , 0 , 1 ∩ Z 2 + . Observing the constraints in A 0 , 1 , 0 , 1 (or D out 0 , 1 , 0 , 1 ), we can find that all intersections of the boundaries take place at points where x-coordinate and y-coordinate are both nonnegati ve inte gers. Therefore, we hav e D in 0 , 1 , 0 , 1 = co( A 0 , 1 , 0 , 1 ) = D out 0 , 1 , 0 , 1 . Follo wing the similar procedure, one can prov e that the theorem holds for all scenarios. Lemma 5: For all a, b, c, d ∈ Z 2 + ,  ( a, b ) : a + ( b − ( c − d ) + ) + ≤ d  = ( ( a, b ) : a ≤ d a + b ≤ max( c, d ) ) Theor em 6: η 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 is gi ven by (1). Pr oof: The theorem is pro ved by solving the linear programming max D 1 T 1 , 1 T 2 , 1 R 1 , 1 R 2 ( d 1 + d 2 ) for each case. Cor ollary 7: D 0 , 0 , 0 , 1 ⊆ D 0 , 1 , 0 , 0 = D 0 , 1 , 0 , 1 ⊆ D 0 , 1 , 1 , 0 ⊆ D 1 , 1 , 0 , 0 (5) η 0 , 0 , 0 , 1 ≤ η 0 , 1 , 0 , 0 = η 0 , 1 , 0 , 1 ≤ η 0 , 1 , 1 , 0 ≤ η 1 , 1 , 0 , 0 (6) Some interesting observation can be drawn for the corollary . First, it may be more powerful, in the sense of DOF , for a user to have a cognitiv e transmitter than to have a cognitiv e recei ver . Second, for a specific user , after having a cognitiv e transmitter , having a cogniti ve receiver does not increase the DOF . I V . D E G R E E S O F F R E E D O M O F T H E M I M O I N T E R F E R E N C E C H A N N E L W I T H C O O P E R AT I O N In this section, we find the DOF of a ( M 1 , M 2 , N 1 , N 2 ) interference channel with cooperation among users. 8 A. System Model Cooperation among users is made possible by allowing noisy links between users. In order to provide these noisy links, the system model for the ( M 1 , M 2 , N 1 , N 2 ) MIMO-IC defined in (2) is generalized to Y [ i ] ( n ) = 4 X j =1 H [ ij ] X [ j ] ( n ) + N [ i ] ( n ) (7) where n is the index for time slot and the definitions of X [ i ] , Y [ i ] , H [ ij ] , and N [ i ] are similar to those in Section II. Note that in our ne w model, all nodes are allo wed to transmit and recei ve in full duplex mode. But there are still only two messages (as before) - W 1 from node 1 to node 3 and W 2 from node 2 to node 4 . All nodes are subject to an a verage transmit power ρ . W e define X [ i ] n as X [ i ] n , h X [ 1 ] ( 1 ) . . . X [ i ] ( n ) i t . Similar definitions apply to Y [ i ] n and Z [ i ] n . The encoding and decoding functions are X [ i ] ( n ) = f 1 ,n  W i , Y [ i ] ( n − 1)  X [ i +2] ( n ) = f i +2 ,n  Y [ i +2] ( n − 1)  ˆ W i +2 = g i +2  Y [ i +2] N  where N is the codew ords length and for i = 1 , 2 . B. Main Results In order to find the upper bound of the DOF of the ( M 1 , M 2 , N 1 , N 2 ) interference channel with cooperation among users, we define the auxiliary random variables U [1] ( n ) , U [2] ( n ) , U [3] ( n ) , U [4] ( n ) as U [ i ] ( n ) = H [ i 1] X [1] ( n ) + N [ i ] ( n ) , i = 1 , 2 , 3 , 4 The follo wing lemma is needed to pro ve our main theorem. Lemma 8: These statements are true. L 1 : X [1] n ← W 1 , W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1 L 2 : X [2] n , X [3] n , X [4] n ← W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1 L 3 : Y [1] n , Y [2] n , Y [3] n , Y [4] n ← W 2 , U [1] n , U [2] n , U [3] n , U [4] n where A ← B denotes that A can be completely determined by B . Next, we pro vide a genie-based upper bound for the DOF of the ( M 1 , M 2 , N 1 , N 2 ) MIMO-IC with cooperation where N 2 ≥ M 1 . Before providing the theorem, we like to mention that the proof is an extension from the single antenna setting in [8] and [15] to the MIMO setting. While the extension is straightforward for the most part, we include it for the sake of completeness. Theor em 9: When N 2 ≥ M 1 , the DOF of the ( M 1 , M 2 , N 1 , N 2 ) MIMO-IC with cooperation satisfies η ≤ N 2 . Pr oof: Suppose that a genie provides node 3 with side information containing W 2 , U [1] n , U [2] n , U [3] n , and U [4] n . According to lemma 8, node 3 can construct Y [ i ] n , i = 1 , 2 , 3 , 4 using the side information. Using Fano’ s 9 inequality we ha ve the follo wing N R 1 ( ρ ) ≤ I  W 1 ; W 2 , Y [3] N , U [1] N , U [2] N , U [3] N , U [4] N  + N  N (8) = I  W 1 ; W 2 , U [1] N , U [2] N , U [3] N , U [4] N  + N  N (9) = I  W 1 ; U [1] N , U [2] N , U [3] N , U [4] N | W 2  + N  N (10) = H  U [1] N , U [2] N , U [3] N , U [4] N | W 2  | {z } T 1 − H  U [1] N , U [2] N , U [3] N , U [4] N | W 1 , W 2  | {z } T 2 + N  N (11) where T 1 can be expressed and bounded above as follow T 1 = N X n =1 H  U [1] ( n ) , U [2] ( n ) , U [3] ( n ) , U [4] ( n ) | W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1  (12) = N X n =1 H  U [4] ( n ) | W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1  + N X n =1 H  U [1] ( n ) , U [2] ( n ) , U [3] ( n ) | W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1 , U [4] ( n )  (13) ( a ) = N X n =1 H U [4] ( n ) + 4 X i =2 H [4 i ] X [ i ] ( n ) | W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1 ! + N X n =1 H  U [1] ( n ) , U [2] ( n ) , U [3] ( n ) | W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1 , U [4] ( n )  (14) ( b ) = N X n =1 H  Y [4] ( n ) | W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1 , Y [4] n − 1  + N X n =1 H  U [1] ( n ) , U [2] ( n ) , U [3] ( n ) | W 2 , U [1] n − 1 , U [2] n − 1 , U [3] n − 1 , U [4] n − 1 , U [4] ( n )  (15) ( c ) ≤ N X n =1 H  Y [4] ( n ) | W 2 , Y [4] n − 1  + N X n =1 H  U [1] ( n ) , U [2] ( n ) , U [3] ( n ) | U [4] ( n )  (16) ( d ) = H  Y [4] N | W 2 ,  + N X n =1 H  U [1] ( n ) , U [2] ( n ) , U [3] ( n ) | U [4] ( n )  (17) ( e ) ≤ H  Y [4] N | W 2 ,  + N X n =1 3 X i =1 H  U [ i ] ( n ) | U [4] ( n )  . (18) Equality ( a ) is obtained by L 2 in lemma 8. Equality ( b ) is obtained by L 3 in lemma 8. Inequality ( c ) and ( e ) use the fact that conditioning reduces entrop y . The second term in (18) can be bounded above by the follo wing method. 10 W e choose i = 1 as an example. H  U [1] ( n ) | U [4] ( n )  = H  H [11] X [1] ( n ) + N [1] ( n ) | H [41] X [1] ( n ) + N [4] ( n )  (19) ( a ) ≤ M 1 X j =1 H  H [11] j X [1] ( n ) + N [1] j ( n ) | H [41] X [1] ( n ) + N [4] ( n )  (20) ( b ) ≤ M 1 X j =1 H  H [11] j X [1] ( n ) + N [1] j ( n ) | H [41] j X [1] ( n ) + N [4] j ( n )  (21) ( c ) ≤ M 1 X j =1 log 1 + k H [11] j k 2 ρ 1 + k H [41] j k 2 ρ ! + log(2 π e ) ! (22) where H [ i 1] j and N [ i ] j denote the channel and noise associated with the j th antenna at node i . Inequality ( a ) and ( b ) use the fact that conditioning reduces entropy . W e refer inequality ( c ) to lemma 1 in [7]. W e would like to mention that inequality ( d ) holds only when H [41] is full rank and N 2 ≥ M 1 which ha ve been assumed. Next, T 2 can be simplified as follo w T 2 = N X n =1 H  U [1] ( n ) , U [2] ( n ) , U [3] ( n ) , U [4] ( n ) | W 1 , W 2 , U [1] n − 1 , U [1] n − 1 , U [1] n − 1 , U [1] n − 1  (23) ( a ) = N X n =1 H  U [1] ( n ) , U [2] ( n ) , U [3] ( n ) , U [4] ( n ) | W 1 , W 2 , U [1] n − 1 , U [1] n − 1 , U [1] n − 1 , U [1] n − 1 , X [1] ( n )  (24) = N X n =1 H  N [1] ( n ) , N [2] ( n ) , N [3] ( n ) , N [4] ( n )  (25) = N X n =1 4 X i =1 H  N [ i ] ( n )  (26) = N ( M 1 + M 2 + N 1 + N 2 ) log (2 π e ) . (27) Equality ( a ) is obtained by L 1 in lemma 8. The simplification of T 2 can be explained as the following. Since W 1 and W 2 are the only messages in the system, after knowing W 1 and W 2 , the uncertainty in U [1] N , U [2] N , U [3] N , and U [4] N is only the uncertain of Gaussian noise. The entropy of Gaussian noise does not increase with ρ . Combining (11), (18), (22), and (27), we have R 1 ( ρ ) ≤ 1 N H  Y [4] N | W 2 ,  + o (log( ρ )) . Using Fano’ s inequality , R 2 ( ρ ) can be bounded abo ve as follo w R 2 ( ρ ) ≤ 1 N I  W 2 ; Y [4] N  +  N (28) = 1 N H  Y [4] N  − 1 N H  Y [4] N | W 2  +  N (29) Adding R 1 ( ρ ) and R 2 ( ρ ) together , we get R 1 ( ρ ) + R 2 ( ρ ) ≤ 1 N H  Y [4] N  + o (log( ρ )) (30) ≤ N 2 log( ρ ) + o (log( ρ )) (31) 11 where the last equality can be obtained from the property of Gaussian random variable. Thus, we prov e that when N 2 ≥ M 1 , the DOF of the system is smaller than equal to N 2 . Cor ollary 10: The DOF of the ( M 1 , M 2 , N 1 , N 2 ) MIMO-IC with cooperation satisfies η ≤ min { max( M 1 , N 2 ) , max( M 2 , N 1 ) } . Pr oof: If N 2 ≥ M 1 , Theorem 9 can be applied directly to obtain η ≤ N 2 . If M 1 > N 2 , let us add more antennas to node 4 so that node 4 has M 1 antennas too. Adding antennas does not hurt, so the con verse argument remains. W e then apply Theorem 9 to the new MIMO-IC to obtain η ≤ M 1 . Thus, we ha ve η ≤ max( M 1 , N 2 ) for all possible cases. η ≤ max( M 2 , N 1 ) can be obtained by switching indices 1 to 2 and 2 to 1 . Corollary 10 along with Theorem 2 in [6], which gi ves the DOF of the MIMO-IC without cooperation, lead to the follo wing corollary . Cor ollary 11: The DOF of the ( M 1 , M 2 , N 1 , N 2 ) MIMO-IC with cooperation satisfies η = min { M 1 + M 2 , N 1 + N 2 , max( M 1 , N 2 ) , max( M 2 , N 1 ) } . Our result sho ws that cooperation can’t increase the DOF of the MIMO-IC. This result can be thought of as a generalization of the single antenna case in [7]. V . C O N C L U S I O N S W e in vestigate the de grees of freedom of the Gaussian MIMO interference channel with cooperation and cognition. W e find the general forms of the DOF and the DOF region of the Gaussian MIMO-IC with all possible cogniti ve message sharing scenarios. 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