Degrees of Freedom Regions of Two-User MIMO Z and Full Interference Channels: The Benefit of Reconfigurable Antennas
aa r X i v : . [ c s . I T ] N ov Degrees of Freedom Regions of Two-UserMIMO Z and Full Interference Channels: TheBenefit of Reconfigurable Antennas
Lei Ke,
Student Member, IEEE , andZhengdao Wang (Contact Author),
Senior Member, IEEE
Abstract
We study the degrees of freedom (DoF) regions of two-user multiple-input multiple-output (MIMO)Z and full interference channels in this paper. We assume that the receivers always have perfect channelstate information. We first derive the DoF region of Z interference channel with channel state informationat transmitter (CSIT). For full interference channel without CSIT, the DoF region has been fullycharacterized recently and it is shown that the previously known outer bound is not achievable. Inthis work, we investigate the no-CSIT case further by assuming that the transmitter has the ability ofantenna mode switching. We obtain the DoF region as a function of the number of available antennamodes and reveal the incremental gain in DoF that each extra antenna mode can bring. It is shown thatin certain cases the reconfigurable antennas can bring extra DoF gains. In these cases, the DoF regionis maximized when the number of modes is at least equal to the number of receive antennas at thecorresponding receiver, in which case the previously outer bound is achieved. In all cases, we proposesystematic constructions of the beamforming and nulling matrices for achieving the DoF region. Theconstructions bear an interesting space-frequency interpretation.
Index Terms
Degrees of freedom region, interference channel, multiple-input multiple-output, reconfigurableantenna, antenna mode switching
L. Ke and Z. Wang are with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011USA (e-mail: [email protected]; [email protected]).Part of the work in this paper was accepted by IEEE GLOBECOM 2010.
November 2, 2018 DRAFT
I. I
NTRODUCTION
Characterizing the capacity region of interference channel has been a long open problem. Manyresearchers investigated this important problem, and the capacity regions of certain interferencechannels are known when the interference is strong, e.g. [1]–[3]. However, when the interferenceis not strong, the capacity region is still unknown. Recent progress reveals the capacity regionfor two-user interference channel to within one bit [4], and after that the sum capacity for veryweak interference channel is settled [5]–[7]. Recently, a deterministic channel model has beenproposed and used to explore the capacity of Gaussian interference network [8]–[10] such thatthe gap to capacity region can be bounded up to a constant value.When it comes to multiple-input multiple-output (MIMO) networks, the capacity regions ofcertain MIMO interference channels are known [11], [12]. Instead of trying to characterize thecapacity region completely, the degrees of freedom (DoF) region characterizes how capacityscales with transmit power as the signal-to-noise ratio goes to infinity.It is well-known that in certain cases, the absence of channel state information at transmitter(CSIT) will not affect the DoF for MIMO networks, e.g., in the multiple access channel [13].In other cases, CSIT does play an important role. For example, using interference alignmentscheme, it is shown that the total DoF of a K -user MIMO interference channel is M K/ , where M is the number of antennas of each user [14]. The key idea is to pack interferences frommultiple sources so as to reduce the dimensionality of signal space spanned by interference.The DoF region of two-user MIMO interference channel with CSIT has been obtained in[15], where it is shown that zero forcing is enough to achieve the DoF region. However, it isa different story in two-user MIMO X channel, where each transmitter has a message to everyreceiver. In [16] it is shown that interference alignment is the key to achieving the DoF regionof MIMO X network. The DoF region of two-user MIMO broadcast channel and interferencechannel without CSIT are considered in [17], where there is an uneven trade-off between thetwo users. Except for a special case, the DoF region for the interference channel is known andachievable. Similar, but more general result of isotropic fading channel can be found in [18].The DoF regions of the K -user MIMO broadcast, interference and cognitive radio channels arederived in [19] for some cases. However, the special case in [17] remains unsolved.When only one of the two transmitter-receiver pairs is subject to interference, the interference November 2, 2018 DRAFT channel is termed as
Z interference channel (ZIC). To avoid confusion, we will call the channelwhere both pairs are subject to interference the full interference channel (FIC). The capacityregion of MIMO Gaussian ZIC is established in [20] under very strong interference and alignedstrong interference assumptions. In [21], the authors considered the capacity region of a singleantenna ZIC without CSIT using deterministic approach.Recently, it is shown in [22] that if the channel is staggered block fading, we can explorethe channel correlation structure to do interference alignment, where the upper bound in theconverse can be achieved in some special cases. For example, it is shown that for two-userMIMO staggered block fading FIC with 1 and 3 antennas at transmitters, 2 and 4 antennas attheir corresponding receivers and without CSIT, the DoF pair (1 , . can be achieved. The ideawas further clarified in [23], where a blind interference alignment scheme is also proposed for K -user multiple-input single-output (MISO) broadcast channel to achieve DoF outer bound whenCSIT is absent. Also recently, it is shown in [24] that the previous outer bound is not tight whenthe channels are independent and identical distributed (i.i.d.) over time and isotropic over spatialdomain. So by now the DoF region of two-user MIMO FIC is completely known for both thecase with CSIT and the no CSIT case (receiver-side CSI, or CSIR, is always assumed available),provided that the channel is i.i.d. over time and isotropic over spatial domain. However, whenthe channel is not i.i.d. over time such as in the “staggered” fading channels [22], the DoF couldbe larger.In this paper, we consider the ZIC channel with CSIT, and both ZIC and FIC without CSITbut with reconfigurable antennas. Specifically, we obtain the DoF regions for the cases of:1) ZIC with CSIT. We show that zero forcing is sufficient for achieving the DoF region inthis case (Theorem 1).2) ZIC and FIC when transmitter one has the number K of antennas modes at least equal to N (Theorems 2 and 3). Increasing K beyond N does not bring more gains in DoF.3) ZIC and FIC when M ≤ K < N , in which case each additional antenna mode bringsan incremental gain on the DoF region (Theorem 4).We present joint beamforming and nulling schemes to achieve the DoF region in all cases. Whenreconfigurable antennas are used, our proposed schemes have an interesting space-frequencycoding explanation. November 2, 2018 DRAFT
The rest of the paper is organized as follows. We first present the system model in Section II.Known results on the DoF region of two-user MIMO FIC are also briefly reviewed. The DoFregion of ZIC with CSIT is discussed in Section III. The DoF regions of ZIC and FIC withoutCSIT when there are enough antenna modes are investigated in Section IV. When there are notenough modes, the DoF region is given in Section V. Finally, Section VI concludes this paper.Notation: boldface uppercase (lowercase) letters denote matrices (vectors). R , Z , C are the real,integer and complex numbers sets. CN (0 , denotes a circularly symmetric complex Gaussian(CSCG) distribution with zero mean and unit variance. We use A ⊗ B to denote the Kroneckerproduct between A and B . and denote all one and all zero matrices (vectors), respectively. A T and A † denote the transpose and Hermitian of A , respectively. We also use notation like A m × n to emphasize that A is of size m × n . We use I m to denote a size m × m identity matrixand m to denote an all-one column vector with length m . Denote g n ( a ) := [1 , a, a , . . . , a n − ] T .A size n × m Vandermonde matrix based on a set of element { a , a , . . . , a m } is definedas V n ( a , a , . . . , a m ) = [ g n ( a ) , g n ( a ) , . . . , g n ( a m )] . We use I ( x ; y ) to denote the mutualinformation between x and y . The differential entropy of a continuous random variable x isdenoted as H ( x ) . II. S YSTEM M ODEL AND K NOWN R ESULTS
A. Channel Model
Consider a MIMO interference channel with two transmitters and two receivers, the numberof transmit (receive) antennas at the i th transmitter (receiver) is denoted as M i ( N i ), i ∈ { , } .The system is termed as an ( M , N , M , N ) system, which can be described as y ( t ) = H ( t ) x ( t ) + H ( t ) x ( t ) + z ( t ) (1) y ( t ) = H ( t ) x ( t ) + H ( t ) x ( t ) + z ( t ) (2)where t is the time index, y i ( t ) ∈ C N i , z i ( t ) ∈ C N i are the received signal and additive noise ofreceiver i , respectively. The entries of z i ( t ) are independent and identically CN (0 , distributedin both time and space. The channel between the i th transmitter and the j th receiver is denotedas H ji ( t ) ∈ C N j × M i . We assume the probability of H ji ( t ) belonging to any subset of C N j × M i that has zero Lebesgue measure is zero. For the two-user MIMO ZIC, H ( t ) = 0 . x i ( t ) ∈ C M i November 2, 2018 DRAFT is the input signal at transmitter i and x ( t ) is independent of x ( t ) . The transmitted signalssatisfy the following power constraint:E ( || x i ( t ) || ) ≤ P i = 1 , . (3)Denote the capacity region of the two-user MIMO system as C ( P ) , which contains all therate pairs ( R , R ) such that the corresponding probability of error can approach zero as codinglength increases. The DoF region is defined as follows [17] D := (cid:8) ( d , d ) ∈ R +2 : ∃ ( R ( P ) , R ( P )) ∈ C ( P ) , such that d i = lim P →∞ R i ( P )log( P ) , i = 1 , (cid:27) . B. Reconfigurable antennas
Assume the CSI at receiver (CSIR) is always available. We would like to study the DoFregions of MIMO FIC and ZIC with or without CSIT under an additional assumption that onetransmitter is equipped with reconfigurable antennas. The reconfigurable antennas are differentfrom the conventional antennas as they can be switched to different pre-determined modes sothat the channel fluctuation can be introduced artificially. Similar to [23], we use reconfigurableantennas to explore multiplexing gain other than diversity gain.We assume that only one transmitter is equipped with reconfigurable antennas. We define oneantenna mode as one possible configuration of a single transmit antenna such that by switching toa different mode, the channel between this transmit antenna and all receive antennas is changed.Different antenna modes can be realized via spatially separated physical antennas, or the samephysical antenna excited with different polarizations, and so on. The benefit of antenna modeswitching lies in the fact that channel variation can be artificially created, without the need toincrease the number of RF chains. We let K denote the total number of antenna modes availableat the transmitter with reconfigurable antennas (usually transmitter one).We make the following assumption of the channel in this paper: the channel is block fadingwith coherent length of L symbols. Within each coherent block, the channels between all thetransmitter modes and the receive antennas remain constant. The channels between the K modesof the reconfigurable transmitter and both receivers are isotropic, in the sense of [24]. From blockto block, the channel changes independently.When K > M , the transmitter has the freedom to use different modes at different slots. Fora given antenna mode usage pattern over the length of a whole coherent block, the effective
November 2, 2018 DRAFT channel for the whole block is not isotropic fading and not i.i.d. over the time slots within theblock.One may view our model approximately as a transition from an effective channel where allthe links have exactly the same coherent time as in [24] to an effective channel where thelinks do not have the same coherent time [22]. However, there are two important distinctionsbetween antenna mode switching and variation of channel coherence time: i) Antenna switchingcan be initiated at will at the transmitter, whereas channel coherence structure is in general notcontrollable. ii) The resulting equivalent channel from antenna mode switching is not “staggered”[22], so methods therein do not apply here.
C. Known Results on FIC
We first present some known results on DoF region of MIMO full interference channel whichwill be useful for developing our results.The total degrees of freedom of two-user MIMO full interference channel with CSIT isdeveloped in [15, Theorem 2], which leads to the following DoF regions: d i ≤ min( M i , N i ) , i = 1 , (4) d + d ≤ min(max( N , M ) , max( M , N ) , N + N , M + M ) . (5)An outer bound of degrees of freedom region of two-user MIMO full interference channelwithout CSIT is as follows [18, Theorem 1]: d i ≤ min( M i , N i ) , i = 1 , (6) d + min( N , N , M )min( N , M ) d ≤ min( M + M , N ); (7) min( N , N , M )min( N , M ) d + d ≤ min( M + M , N ) . (8)Note that the same result is also given in [17], though in a less compact form.It is shown in [17] that the outer bound given in (6)–(8) can be achieved by zero forcing ortime sharing except for the case M < N < min( M , N ) , for which it was not known how toachieve ( d , d ) = (cid:18) M , min( M , N )( N − M ) N (cid:19) (9) November 2, 2018 DRAFT in general. The cases when N > N can be converted by switching the user indices. It is shownin [24] that when the channel is isotropic fading and i.i.d. over time, the outer bound given in(6)–(8) is not tight: if N ≤ N , the DoF region of FIC without CSIT can be given as follows: d i ≤ min( M i , N i ) , i = 1 , (10) d + min( N , M ) − α min( N , M ) − α ( d − α ) ≤ min( M , N ) . (11)where α = min( M + M , N ) − min( M , N ) . In other words, (9) is not achievable when M < N < min( M , N ) , as (11) is reduced to d + M min( M , N ) − ( N − M ) d ≤ M + M ( N − M )min( M , N ) − ( N − M ) (12)and the DoF pair ( d , d ) = ( M , N − M ) is on the boundary of the DoF region.III. T WO -U SER
MIMO ZIC
WITH
CSITIn this section, we prove the following theorem.
Theorem 1 (ZIC with CSIT):
The DoF region of a two-user MIMO Z interference channelwith CSIT is described by d i ≤ min( M i , N i ) , i = 1 , (13) d + d ≤ min(max( N , M ) , N + N , M + M ) . (14) Proof:
We split the proof into the achievability and converse parts, as the following twolemmas. The theorem can be proved by showing the regions given by Lemma 1 and Lemma 2are the same for all the cases.
Lemma 1 (Achievability part of Theorem 1):
The following region of two-user MIMO ZICwith CSIT is achievable: d i ≤ min( M i , N i ) , i = 1 , (15) d + d ≤ min( N , M + min( N , M ))1( M < N )+ min( M , N + min( N , M ))1( M ≥ N ) (16)where · ) is indicator function. Proof: If M ≥ N and assume transmitter 1 sends d streams, transmitter 2 can send atmost M − N streams along the null space of H without interfering receiver 1. Transmitter November 2, 2018 DRAFT N − d streams along the row space of H . Therefore user 2 candecode min(( M − N ) + ( N − d ) , N ) streams without interfering receiver 1. If N ≥ M and assume transmitter 2 sends d streams which interfere receiver 1, transmitter 1 can send min( N − d , M ) decodable streams to receiver 1. Combining these two cases, we have theachievable DoF region shown in this lemma. Lemma 2 (Conversepart of Theorem 1):
The region given by (13) and (14) is a valid outerbound for the two-user MIMO ZIC with CSIT.
Proof:
It is obvious that adding antennas at the receiver will not shrink the DoF region.Hence, we can add M antennas to receiver 2 resulting an ( M , N , M , M + N ) MIMO FIC,and (14) follows from Corollary 1 in [15]. The outer bound of such a MIMO FIC is a valid outerbound of an ( M , N , M , N ) MIMO ZIC. Combining the trivial upper bound on point-to-pointsystem, we have this lemma.Based on Lemma 1, zero forcing at receiver is sufficient to achieve the DoF region of ZICwhen CSIT is available. The antenna mode switching ability is not needed in this case. However,we shall see later that such an ability is important for the case when CSIT is absent.IV. T WO -U SER
MIMO ZIC
AND
FIC
WITHOUT
CSIT W
HEN N UMBER OF M ODES K ≥ N In this section, we describe the DoF region of two-user ZIC and FIC without CSIT but withtransmitter side reconfigurable antennas. We deal with the case that K , the number of antennamodes is at least equal to the N . The case K < N will be dealt with in Section V.Based on the antenna number configuration, the achievability scheme of ZIC and FIC withoutCSIT can be divided into two cases. In the first case, no reconfigurable antenna is neededto achieve an DoF outer bound — reconfigurable antennas are not helpful (Section IV-B). Inthe second case, the outer bound can be achieved with enough transmit side antenna modes(Section IV-C): reconfigurable antennas enlarges the DoF region. Our main results in this sectionare the following two theorems. Theorem 2 (ZIC with Enough Reconfigurable Antenna Modes):
The DoF region of two-userMIMO Z interference channel without CSIT is described by the following inequalities d i ≤ min( M i , N i ) , i = 1 , (17) d + min( N , N , M )min( N , M ) d ≤ min( M + M , N ) . (18) November 2, 2018 DRAFT if either one of the following is true:C1) M < N < min( M , N ) and transmitter one can switch among N antenna modes, orC2) ( M , N , M , N ) do not satisfy the above condition.The DoF region in Theorem 2 is shown in Fig. 1. Theorem 3 (FIC with Enough Reconfigurable Antenna Modes):
The DoF region of two-userMIMO full interference channel without CSIT is described by the inequalities (6)–(8) if any oneof the following is true:C1) M < N < min( M , N ) and transmitter one can switch among N antenna modes, orC2) M < N < min( M , N ) and transmitter two can switch among N antenna modes, orC3) ( M , N , M , N ) are not one of the two above cases. A. Converse part
We first prove the converse part of the two theorems.
Lemma 3 (Converse part of Theorem 3):
The outer bound of DoF region of two-user MIMOfull interference channel given in (6)–(8) is still valid when either or both transmitters are usingantenna mode switching.
Proof:
The outer bound (7) has been derived based on the assumption that the rows of H and those of H are statistically equivalent [17], [18]. Similarly, the outer bound (8) hasbeen derived based on the assumption that the rows of H and those of H are statisticallyequivalent. These assumptions are not affected by antenna mode switching at either or bothtransmitters. Hence, the DoF outer bound is still valid. Lemma 4 (Converse part of Theorem 2):
The outer bound of degrees of freedom region oftwo-user MIMO Z interference channel without CSIT can be given as when transmitter one hasthe antenna mode switching ability d i ≤ min( M i , N i ) , i = 1 , (19) d + min( N , N , M )min( N , M ) d ≤ min( M + M , N ) . (20) Proof:
This is the direct result of [18, Theorem 1] as in (6)–(8), by noticing that there isno interference from transmitter 1 to receiver 2 hence (8) is not longer needed. The antennaswitching at transmitter one does not affect the upper bound, for the same reason stated inLemma 3.
November 2, 2018 DRAFT0 d d N N ( ) min( , ) N Nf M M N N (cid:116)(cid:14) (cid:100) M min( M , N ) ,( ) min( , ) N N N Mg M M N N (cid:116) (cid:33)(cid:14) (cid:33) ( )
N Nh N M (cid:116)(cid:100) d d N N M min( M , N ) d N N d d N N ( ) M N Na M M N (cid:100) (cid:100)(cid:14) (cid:100) M M ( ) M N Nb M N M M (cid:100) (cid:100)(cid:31) (cid:31) (cid:14) d d N N M M d N N ( ) M N Nc N M (cid:100) (cid:100)(cid:100) M d d min( M , N ) N d d min( M , N ) N M min( , )( ), M N N MM N (cid:167) (cid:183)(cid:16)(cid:168) (cid:184)(cid:169) (cid:185) ( ) ,
N Ne N M N M (cid:100)(cid:31) (cid:31) ( ) min( , ) d M N M N (cid:31) (cid:31) d d min( M , N ) Fig. 1. DoF region of two-user MIMO ZIC without CSIT when number of antenna modes K ≥ N . Figures (a)–(e) are forthe case N ≤ N ; Figures (f)–(h) are for the case N ≥ N . B. Achievability: when antenna mode switching is not needed
In this section, we prove the achievability part for Case C2) of Theorem 2 and Case C3) ofTheorem 3. Achievability for the remaining cases are left to Section IV-C.
Lemma 5:
For the two-user MIMO Z interference channel without CSIT, when N ≥ N ,(20) is achievable by zero forcing. November 2, 2018 DRAFT1
Proof:
When N ≥ N , the corresponding outer regions are shown in Fig. 1 (f)–(h). Noticingthat (20) is reduced to d + d ≤ min( M + M , N ) , zero forcing is sufficient to achieve theouter bound. Lemma 6:
When CSIT is absent, the DoF outer region given by Lemma 4 of a two-userMIMO ( M , N , M , N ) ZIC is the same as that of an ( M , N , min( M , N ) , min( M , N )) ZIC.
Proof:
We give the proof case by case. It is trivial that when M ≤ N reducing the numberof antennas at receiver 2 to M will not shrink the DoF region. When M > N , we can furtherconsider two sub-cases: N ≥ N and N < N .1) When M > N ≥ N , corresponding to Fig. 1 (d) and (e), the DoF bound (20) becomes d N + d N ≤ . Hence the DoF outer region is the same as an ( M , N , N , N ) ZIC.2) When M > N and N < N , the DoF bound (20) becomes d + d ≤ min( M + M , N ) .Hence, if M ≥ N − N , which implies M + min( M , N ) ≥ N , the DoF outer regionis a pentagon or a tetragon; see Fig. 1 (g) and (h). Otherwise, it is a square, see Fig. 1(f). One can show that the region is the same as that of an ( M , N , N , N ) ZIC.Hence, the lemma holds.We also have the following lemma regarding the relationship between DoF regions of ZICand FIC.
Lemma 7:
When N ≤ N , the MIMO ZIC and FIC have the same DoF regions. Any encodingscheme that is DoF optimal for one channel is also DoF optimal for the other. Proof:
Any point in the FIC is also trivially achievable in the ZIC because user 2’s channelis interference free. Conversely, any point achievable in the ZIC region, is also achievable inFIC. This is based on the fact that the channels are statistically equivalent at both receivers. Ifreceiver 1 can decode user 1’s message, then receiver 2, having at least as many antennas, mustalso be able to decode the same message. Receiver 2 can then subtract the decoded message,which renders the resulting channel the same as in the ZIC.Due to Lemma 7, we can translate all achievability schemes from FIC to ZIC and vice versawhen N ≤ N . Therefore the achievability schemes in [17] for FIC when N ≤ N and M ≥ N can be used for ZIC. Therefore, the achievability part for Case C2) of Theorem 2 iscomplete. November 2, 2018 DRAFT2
For the FIC, the achievability for the case N ≤ N , except when M < N < min( M , N ) ,is shown in [17]. When N ≥ N , we can swap the indices of the two users, so that except forthe Cases C1) and C2) the achievability scheme is known for FIC. C. Achievability: with antenna mode switching when K ≥ M N In this subsection, we prove a weaker version of the achievability for Case C1) of Theorem 2and Cases C1) and C2) of Theorem 3. Namely, we assume that the number of antenna modesavailable is K ≥ M N . The scheme is simpler in this case, and the achievability scheme forthe case K = N will be built upon this case.Based on Lemma 6 and Lemma 7, we only consider the two-user MIMO ZIC with M It is therefore sufficient (and also necessary) to have QP = . Then the key is to find a Q suchthat the equivalent channel of user 1 after nulling ˜ A = ( Q ⊗ I N ) ˜ H (27)has full rank M N with probability 1. The matrix ˜ A is of size M N × M N and has thefollowing structure ˜ A = q H (1) q H (2) · · · q N H ( N ) q H (1) q H (2) · · · q N H ( N ) ... ... . . . ... q M H (1) q M H (2) · · · q M N H ( N ) . To show that ˜ A has full rank, we need the following lemma, which is known before, and aproof of it can be found in e.g., [25]. Lemma 8: [25, Lemma 2] Consider an analytic function h ( x ) of several variables x =[ x , . . . , x n ] T ∈ C n . If h is nontrivial in the sense that there exists x ∈ C n such that h ( x ) = 0 ,then the zero set of f ( x ) Z := { x ∈ C n | h ( x ) = 0 } is of measure (Lebesgue measure in C n )zero.Because the determinant of ˜ A is an analytic polynomial function of elements of H ( t ) , t =1 , . . . , N , we only need to find a specific pair of Q and H ( t ) , t = 1 , . . . , N , such that ˜ A isfull rank. We propose the following: Q = [ V N (1 , ω N , . . . , ω M − N )] T , (28)where ω N := exp( − j π/N ) .Let ω := exp( − j π/N ) . Take the realizations of H ( t ) , t = 1 , . . . , N , as H ( t ) = V N ( ω t − , ω N + t − , . . . , ω ( M − N + t − ) . (29)It can be verified that for such choices of Q and H ( t ) , ˜ A is a Vandermonde matrix: ˜ A = V M N (1 , ω N , . . . , ω ( M − N , ω , ω N +1 , . . . ,ω ( M − N +1 ,. . . , ω N − , ω N − , . . . , ω M N − ) , November 2, 2018 DRAFT5 hence of full rank. We also notice that ˜ A is a leading principal minor of a permuted fast Fouriertransform (FFT) matrix with size N × N . The permutation is as follows: Index the columnsof an FFT matrix , , . . . , N − , and then permute them in an order shown below: (0 , N , N , . . . , ( M − N ) , (1 , N + 1 , N + 1 , . . . , ( M − N + 1) , ... Based on Lemma 8, if we choose the nulling matrix using Q as specified in (28), ˜ A hasfull rank almost surely. One choice of the corresponding P matrix with respect to (28) is thefollowing P = V N ( ω − M N , ω − ( M +1) N , . . . , ω − ( N − N ) , (30)which is orthogonal to Q . This completes the achievability part under conditions in Case C1)of Theorem 2 and Cases C1) and C2) of Theorem 3, but with K ≥ M N . D. Achievability: with antenna mode switching when K = N Assuming there are N modes available at transmitter 1 and denote these channel vectorsbetween receive antennas of user 1 and the i th mode as h i , ≤ i ≤ N and let ˆ H N × N =[ h , h , . . . , h N ] . We choose the antenna modes to be switched cyclically: H (1) = [ h , h , . . . , h M ] , (31) H (2) = [ h , h , . . . , h M +1 ] , (32)... H ( N ) = [ h N , h , . . . , h M − ] . (33)We want to show that under this switching pattern, the equivalent channel ˜ A in (27) betweentransmitter one and receiver one after nulling, is still full rank. To show this, indexing thecolumns of ˜ A in (27) as , , . . . , N M − , we then permute and group the columns of ˜ A inthe following way: (0 , M , M , . . . , ( M − N ) , (1 , M + 1 , M + 1 , . . . , ( M − N + 1) , ... November 2, 2018 DRAFT6 Denote the permutation result as ˜ A ′ and it can be expressed as ˜ A ′ = ˆ H ˆ H · · · ˆ HG ˆ H ω − N G ˆ H · · · ω − M N G ˆ H ... ... . . . ... G M ˆ H ( ω − N G ) M ˆ H · · · ( ω − M N G ) M ˆ H , where G is a size N × N diagonal matrix and can be expressed as G = diag (1 , ω N , ω N , . . . , ω N − N ) . (34)Notice that ˜ A ′ = R ( I M ⊗ ˆ H ) , where R = I N I N · · · I N G ω − N G · · · ω − M N G ... ... . . . ... G M ( ω − N G ) M · · · ( ω − M N G ) M . Recall ω N = exp( − j π/N ) . To show ˜ A is full rank, it is necessary to show R is full rank as I M ⊗ ˆ H is full rank with probability 1. It can be verified that via row and column permutations R can be changed to a block diagonal matrix with the i th block being V M ( ω iN , ω i − N , · · · , ω i − M +1 N ) , (35)which is full rank due to Vandermonde structure. Hence R is full rank. It follows that A is fullrank with probability 1. This completes the achievability part under conditions in Case C1) ofTheorem 2 and Cases C1) and C2) of Theorem 3 for K = N . E. Discussion1) Frequency domain interpretation: We note that the matrix [ Q † , P ] is an inverse FFT(IFFT) matrix in our construction (23), (24), (28) and (30). This observation yields an interestingfrequency domain interpretation of our construction. The signal of user 2 is transmitted overfrequencies corresponding to the last N − M columns of an IFFT matrix, whereas the firstuser’s signal is transmitted on all frequencies. Due to the antenna mode switching at transmitter 1,the channel between transmitter 1 and receiver 1 is now time-varying and we manually introducefrequency spread. User 1’s signal is spread from one frequency bin to all the frequencies while November 2, 2018 DRAFT7 user 2’s signal remains in the last N − M frequency bins. Therefore the signal in the first M bins is interference free, which can be used to decode user 1’s message. The nulling matrixapplied at receiver 1 has a projection explanation as well. Left multiplying the left and righthand sides of (26) with ˜ Q † yields ˜ Q † ˜ Q ˜ y = ˜ Q † ˜ Q ˜ H ˜ x + ˜ Q † Q ˜ z = (( Q † Q ) ⊗ I N )( ˜ H ˜ x + ˜ z ) , where Q † Q is the frequency domain projection matrix. We can see that the signal of user 1 isprojected from N frequencies to the first M frequencies. 2) The Loss of DoF due to lack of CSIT: In two-user MIMO Z interference channel withoutCSIT, losing CSIT will not shrink degrees of freedom region if M ≤ N or M > N ≥ N + M .For all the other cases, the degrees of freedom region is strictly smaller when comparing withthe CSIT case.This observation can be verified case by case. Notice that it is already shown in [17, Theorem2] that when M ≤ N ≤ N absence of CSIT does not reduce DoF region in two-user MIMOFIC. Because MIMO FIC and ZIC has the same DoF region when N ≤ N . We only need toconsider the sub cases when N > N , corresponding to (f)–(h) in Fig. 1.1) If M < N and N > N , the total DoF of MIMO ZIC is upper bounded by N due to(14), so the DoF region remains the same if CSIT is absent.2) If M > N > N , the DoF region of MIMO ZIC without CSIT is a square only when M + N ≤ N , same as that of ZIC with CSIT. Otherwise, the maximum total DoF ofZIC with CSIT is min( M , N + N , min( M , N ) + N ) , strictly larger than N which isthe maximum total DoF when CSIT is absent, hence loss of CSIT reduces the DoF region. 3) Alternative construction when N /M = β ∈ Z : When N /M = β ∈ Z , instead of usingthe Q given in (28) we can use the following Q M × N = I M ⊗ Tβ . We need to show that this Q matrix will lead to a full rank ˜ A . This can be achieved by choosing ˜ H such that it can be November 2, 2018 DRAFT8 decomposed as ˜ H = I M ⊗ ˜ H ′ , where ˜ H ′ = H (1) . . . H (2) . . . ... ... . . . ... . . . H ( β ) N β × N . For this ˜ H ˜ A = ( I M ⊗ Tβ ⊗ I N )( I M ⊗ ˜ H ′ )= I M ⊗ (( Tβ ⊗ I N ) ˜ H ′ ) , which has full rank. For this choice of Q , we only use βM = N antenna modes in N timeslots.Therefore, for the two-user MIMO ZIC and FIC when M < N < min( M , N ) and N /M = β ∈ Z , β fold time expansion is enough to achieve the DoF region. We remarkthat this can be viewed as the generalization of the case we discussed in Section IV-C for N = β and M = 1 . In fact Tβ is the nulling matrix Q given in (28) when N = β, M = 1 . 4) Successive Decoding in ZIC: For the two-user MIMO FIC when M < N < min( M , N ) and CSIT is absent, we need block decoding at both receivers in general, which introducesdecoding delay. Successive interference cancellation decoder can be used at receiver 2 to reducedecoding delay. Taking the case N /M = β ∈ Z as an example, we can use β fold timeexpansion and choose Q = Tβ . The corresponding P matrix is not necessary to be the last β − columns of an β × β FFT matrix. The following P matrix still satisfies the designconstraint P β × ( β − = I β − Tβ − . (36)Here, P has a nice structure. Every stream of user 2 can be decoded immediately as they areinterference free. For other cases where M cannot divide N , we can still find a Q , P pairthrough numerical simulation such that the upper diagonal parts of P are all zeros and containsmall number of nonzero entries. Such a beamforming matrix can guarantee the immediatedecoding of user 2’s signal the interference only comes from the streams already decoded . November 2, 2018 DRAFT9 V. T WO -U SER MIMO ZIC AND FIC WITHOUT CSIT W HEN N UMBER OF M ODES K < N In this section, we will present our result for the K < N case. The main result of this sectionis the following theorem. Theorem 4: When M < N < min( M , N ) and the antennas of transmitter 1 can be switchedamong K antenna modes, where K < N , the DoF region of two-user MIMO ZIC and FICwithout CSIT is given by the following inequalities d i ≤ min( M i , N i ) , i = 1 , (37) d + K min( M , N ) − ( N − K ) d ≤ M + K ( N − M ) + (min( M , N ) − N )( K − M )min( M , N ) − ( N − K ) (38)The DoF region of FIC for M < N < min( M , N ) can be obtained by switching the twouser indices.The method of proof is heavily based on that in [24], to which the reader is referred forseveral lemmas that will be used and their proofs. Some notation that is used in this section arethe following. We use tilde notation to denote the time expanded signal over L time slots and t ∈ [1 , L ] is the index of the slot within one block. In general, by default, for a vector x , ˜ x = vec ( x (1) , x (2) , · · · , x ( L )) and for a matrix V , ˜ V = diag ( V (1) , V (2) , · · · , V ( L )) . In addition,for a time expanded vector ˜ x , we use ˜ x n or { ˜ x } n to denote a sequence of n successive blocks of ˜ x : ˜ x n = vec ( x (1) , x (2) , . . . , x ( nL )) . Furthermore, x ( t ) n is the sequence of x ( t ) which containsall the vector x of the t th slot of all n blocks: x ( t ) n = vec ( x ( t ) , x ( t + L ) , . . . , x ( t + ( n − L )) .Similar notation is defined for matrices as well. We use H denotes ( H , H , H , H ) ,hence ˜ H n denotes all the channel matrices over n blocks. In addition, for a random vector x , x G is a corresponding CSCG vector that has the same covariance matrix as x . A. The Converse Part We prove the converse part of Theorem 4 in the following. Recall that for M < N < min( M , N ) , the proof is equivalent for both FIC and ZIC. We will only show the proof forZIC. To make the proof self-contained, we will go through some similar steps as in [24], butavoiding details. November 2, 2018 DRAFT0 The converse is developed based on blocking for every L slots. In each block, the channel H , H stay the same with the decomposition H = W Λ V † and H = W Λ V † ,whereas H is time-varying among L slots due to antenna mode switching at transmitter 1.Transmitter 1 has K modes with K < N and it can adopt arbitrary switching pattern. Let H be an N × N full rank random matrix such that H = [ h , h , · · · , h N ] and h i , ≤ i ≤ K is the random vector channel between the i th antenna mode and receive antennas of user 1. Weintroduce the fictitious vectors { h i , K + 1 ≤ i ≤ N } to simplify the proof. We assume H isisotropic fading and i.i.d. over blocks of length L each, where L naturally satisfy L ≥ ⌈ K/M ⌉ .We denote the decomposition of H as ˜ W ˜ Λ ˜ V † .Furthermore, let E ( t ) of size N × M denote the antenna mode selection matrix for time t .Let e m , ≤ m ≤ N be the m th column of I N . Let i ( t ) denote the mode index selected byantenna i at time t . Then the i th column of E ( t ) is e i ( t ) . We have H ( t ) = H E ( t ) .At receiver 1, from Fano’s inequality, we have nLR − δ nL ≤ I (˜ y n ; ˜ x n | ˜ H n ) . (39)where δ nL → as n → ∞ . Denote ˜ r = ˜ H ˜ x G + ˜ H † ˜ x + ˜ z (40) ˜ r = ˜ W † ˜ H ˜ x G + ˜ V † ˜ x + ˜ n (41)where ˜ n = ˜ W † ˜ z . Using [24, Theorem 3], which says that Gaussian input can reduce themutual information by at most an o (log( P )) quantity, and two uses of chain rule we have nLR − n o (log( P )) ≤ I (˜ r n ; x n G | ˜ H n ) (42) = I (cid:16) ˜ r n ; x n G | ˜ x n , ˜ H n (cid:17) + I (˜ r n ; ˜ x n | ˜ H n ) − I (cid:16) { ˜ H † ˜ x + ˜ z } n ; ˜ x n | ˜ H n (cid:17) . (43)Using [24, Lemma 2], we have I (cid:16) { ˜ H † ˜ x + ˜ z } n ; ˜ x n | ˜ H n (cid:17) = I (cid:16) { ˜ W ˜ Λ ˜ V † ˜ x + ˜ z } n ; ˜ x n | ˜ H n (cid:17) (44) = I (cid:16) { ˜ Λ ˜ V † ˜ x + ˜ n } n ; ˜ x n ˜ H n (cid:17) (45) ≥ I (cid:16) { ˜ V † ˜ x + ˜ n } n ; ˜ x n | ˜ H n (cid:17) − n o (log( P )) , (46) November 2, 2018 DRAFT1 and I (˜ r n ; ˜ x n | ˜ H n ) = I (cid:16) { ˜ W † ˜ H ˜ x G + ˜ Λ ˜ V † ˜ x + ˜ n } n ; ˜ x n | ˜ H n (cid:17) (47) ≤ I (cid:16) ˜ r n ; ˜ x n | ˜ H n (cid:17) + n o (log( P )) . (48)Hence R can be further bounded as nLR − n o (log( P )) ≤ I (cid:16) ˜ r n ; ˜ x n G | ˜ x n , ˜ H n (cid:17) + I (˜ r n ; ˜ x n | ˜ H n ) − I (cid:16) { ˜ V † ˜ x + ˜ n } n ; ˜ x n | ˜ H n (cid:17) . (49)As to receiver 2, using Fano’s inequality and [24, Lemma 2], we have nLR − δ nL ≤ I (˜ y n ; ˜ x n | ˜ H n ) (50) = I (cid:16) { ˜ W ˜ Λ ˜ V † ˜ x + ˜ z } n ; ˜ x n | ˜ H n (cid:17) (51) ≤ I (cid:16) { ˜ V † ˜ x + ˜ n } n ; ˜ x n | ˜ H n (cid:17) + n o (log( P )) , (52)where ˜ n = ˜ W † ˜ z . Hence nLR − n o (log( P )) ≤ I (cid:16) ˜ r n ; ˜ x n | ˜ H n (cid:17) − I (cid:16) ˜ r n ; ˜ x n | ˜ H n (cid:17) + I (cid:16) { ˜ V † ˜ x + ˜ n } n ; ˜ x n | ˜ r n , ˜ H n (cid:17) . (53)Notice that by using Gaussian input, the following inequalities hold I (cid:16) ˜ r ; ˜ x n G | ˜ x n , ˜ H n (cid:17) ≤ E log (cid:18) det( I LN + PM ˜ H ˜ H † ) (cid:19) (54) = nLM log( P ) + nLo (log( P )) , (55) I (cid:16) ˜ r n ; ˜ x n | ˜ H n (cid:17) ≤ n E log det( I LN + PM ˜ W ˜ W † + PM ˜ H ˜ H † )det( I LN + PM ˜ H ˜ H † ) (56) = nL ( N − M ) log( P ) + nLo (log( P )) . (57)Then let n → ∞ , multiply (53) with some positive scalar µ , add it with (49) and use (55),(57), we have the following inequality nL [ R + uR − o (log( P ))] ≤ nLM log( P ) + µnL ( N − M ) log( P ) + η, (58)where µ is to be determined and η = µ I (cid:16) { ˜ V † ˜ x + ˜ n } n ; ˜ x n | ˜ H n (cid:17) −I (cid:16) { ˜ V † ˜ x + ˜ n } n ; ˜ x n | ˜ H n (cid:17) +(1 − µ ) I (˜ r n ; ˜ x n | ˜ H n ) . (59) November 2, 2018 DRAFT2 Divide (58) by nL log( P ) and let P → ∞ , we have the following inequality on the DoF of twousers d + µd ≤ M + µ ( N − M ) + λ, (60)where λ = 1 nL lim P →∞ η log( P ) . Recall that ˜ r = ˜ W † ˜ H ˜ x G + ˜ V † ˜ x + ˜ n and H ( t ) = H E ( t ) . We define ˜ r = ˜ Λ − ˜ W † ˜ W ˜ V † ˜ x + ˜ V † ˜ E ˜ x G + ˜ Λ − ˜ W † ˜ W ˜ n (61) ˜ r = ˜ W † ˜ W ˜ V † ˜ x + ˜ V † ˜ E ˜ x G + ˜ Λ − ˜ W † ˜ W ˜ n (62) ˜ r = ˜ V ˜ W † ˜ W ˜ V † ˜ x + ˜ E ˜ x G + ˜ V ˜ Λ − ˜ W † ˜ W ˜ n (63) ˜ r = ˜ V ˜ W † ˜ W ˜ V † ˜ x + ˜ E ˜ x G + ˜ n (64) ˜ r = ˜ V † ˜ x + ˜ E ˜ x G + ˜ n (65)We have I (˜ r n ; ˜ x n | ˜ H n ) = I (˜ r n ; ˜ x n | ˜ H n ) (66) = I (˜ r n ; ˜ x n | ˜ H n ) + o (log( P )) (67) = I (˜ r n ; ˜ x n | ˜ H n ) + o (log( P )) (68) = I (˜ r n ; ˜ x n | ˜ H n ) + o (log( P )) (69) = I (˜ r n ; ˜ x n | ˜ H n ) + o (log( P )) , (70)where (67) due to [24, Lemma 2]; (66) and (68) hold as ˜ W ˜ Λ and ˜ V are full rank squarematrices. (69) holds as changing noise variance will not change the DoF. (70) is true because ˜ V ˜ W † ˜ W ˜ V † has the same distribution as ˜ V † and ˜ V ˜ W † ˜ W is independent of ˜ V † .To find the DoF order of I (˜ r n ; ˜ x n | ˜ H n ) , we first notice that for each slot t in one block, V † can be divided into three parts: V † ,a ( t ) , V † ,b ( t ) and V † ,c .1) V † ,a ( t ) is of size M × M and consists of M non-zero rows of E ( t ) V † .2) V † ,c is of size ( N − K ) × M and is the same for all ≤ t ≤ L . It consists of N − K rows of V † that do not appear in any V ,a ( t ) † , ≤ t ≤ L . November 2, 2018 DRAFT3 V † ,b ( t ) is of size ( K − M ) × M and consists of K − M rows of V † that neither in E ( t ) V ,a ( t ) † nor in V † ,c . Example: Assume N = 5 , M = 2 , L = 6 , K = 4 and V = [ v , v , . . . , v N ] where v i ’s are M × vectors. Assume E ( t ) is the following E (1) = [ e , e ] , E (2) = [ e , e ] , E (3) = [ e , e ] , E (4) = [ e , e ] , E (5) = [ e , e ] , E (6) = [ e , e ] . We have V † ,a (1) = [ v , v ] † , V † ,a (2) = [ v , v ] † , V † ,a (3) = [ v , v ] † , V † ,a (4) = [ v , v ] † , V † ,a (5) = [ v , v ] † , V † ,a (6) = [ v , v ] † , V † ,b (1) = [ v , v ] † , V † ,b (2) = [ v , v ] † , V † ,b (3) = [ v , v ] † , V † ,b (4) = [ v , v ] † , V † ,b (5) = [ v , v ] † , V † ,b (6) = [ v , v ] † , and V † ,c = v † . Note that V † ,c remains the same in one block of L slots.Suppose receiver 1 receives r as in (65) and wants to decode the message of x thatgoes through an equivalent channel V † . Then V † ,a ( t ) are the directions of interference fromtransmitter at time t , V † ,b ( t ) are those directions that are temporarily interference-free at time t , and V † ,c are the directions which are interference free for a whole block. The associatednoises of the those directions are similarly defined as n ,a ( t ) , n ,b ( t ) and n ,c ( t ) .To bound the DoF of I (˜ r n ; ˜ x n | ˜ H n ) of (71), we define V † ,ab ( t ) = V † ,a ( t ) V † ,b ( t ) , n ,ab ( t ) = n ,a ( t ) n ,b ( t ) , (71) V † ,bc ( t ) = V † ,b ( t ) V † ,c , n ,bc ( t ) = n ,b ( t ) n ,c ( t ) , (72)and adopt the following notation for simplicity y a ( t ) = V † ,a ( t ) x ( t ) + x G ( t ) + n ,a ( t ) (73) y b ( t ) = V † ,b ( t ) x ( t ) + n ,b ( t ) (74) y c ( t ) = V † ,c x ( t ) + n ,c ( t ) (75) y bc ( t ) = V † ,bc ( t ) x ( t ) + n ,bc ( t ) (76) November 2, 2018 DRAFT4 In addition, y a ( t ) n , y b ( t ) n , y c ( t ) n , y bc ( t ) n are sequences of corresponding vectors of the t th slotover n blocks. The collection of y a (1) n , y a (2) n , . . . y a ( t ) n is denoted as { y (1: t ) a } n . We also define { y (1: t ) b } n , { y (1: t ) c } n and { y (1: t ) bc } n similarly. Using the chain rule, we have I (˜ r n ; ˜ x n | ˜ H n ) = I (cid:16) { ˜ V † ,c ˜ x + ˜ n ,c } n ; ˜ x n | ˜ H n (cid:17) + I (cid:16) { ˜ V † ,b ˜ x + ˜ n ,b } n ; ˜ x n |{ ˜ V † ,c ˜ x + ˜ n ,c } n , ˜ H n (cid:17) + I (cid:16) { ˜ V † ,a ˜ x + ˜ x G + ˜ n ,a } n ; ˜ x n |{ ˜ V † ,bc ˜ x + ˜ n ,bc } n , ˜ H n (cid:17) (77)Now checking the second term in (77), we notice that I (cid:16) { ˜ V † ,b ˜ x + ˜ n ,b } n ; ˜ x n |{ ˜ V † ,c ˜ x + ˜ n ,c } n , ˜ H n (cid:17) = L X t =1 I (cid:16) y b ( t ) n ; ˜ x n |{ y (1: t − b } n , ˜ y nc , ˜ H n (cid:17) (78) = L X t =1 H (cid:16) y b ( t ) n |{ y (1: t − b } n , ˜ y nc , ˜ H n (cid:17) − H (cid:16) y b ( t ) n | ˜ x n , { y (1: t − b } n , ˜ y nc , ˜ H n (cid:17) (79) = L X t =1 H (cid:16) y b ( t ) n |{ y (1: t − b } n , ˜ y nc , ˜ H n (cid:17) − H (cid:16) y b ( t ) n | x ( t ) n , y b ( t ) n , ˜ H ( t ) n (cid:17) (80) ≤ L X t =1 H ( y b ( t ) n | y c ( t ) n , H ( t ) n ) − H ( y b ( t ) n | x ( t ) n , y c ( t ) n , H ( t ) n ) (81) = L X t =1 I ( y b ( t ) n ; x ( t ) n | y c ( t ) n , H ( t ) n ) (82) = L X t =1 [ I ( y b ( t ) n , y c ( t ) n ; x ( t ) n | H ( t ) n ) − I ( y c ( t ) n ; x ( t ) n | H ( t ) n )] (83) ≤ L X t =1 (cid:18) N − M N − K − (cid:19) I ( y c ( t ) n ; x ( t ) n | H ( t ) n ) (84) ≤ nL ( K − M ) log( P ) + o (log( P )) (85)where: • (78) and (83) follow by chain rule. • (79) and (82) are expressing mutual information via entropy. • (80) holds as the second term is the entropy of noise when conditioning on x ( t ) n . • (81) is based on the fact that conditioning reduces entropy. November 2, 2018 DRAFT5 • (84) follows by [24, Lemma 3]. • (85) holds due to the fact that the DoF of an ( N − K ) × M point-to-point MIMO channelis at most min( N − K, M ) = N − K .The third term in (77) can be bounded in a similar fashion. We have I (cid:16) { ˜ V † ,a ˜ x + ˜ x G + ˜ n ,a } n ; ˜ x n |{ ˜ V † ,bc ˜ x + ˜ n ,bc } n , ˜ H n (cid:17) = L X t =1 I (cid:16) y a ( t ) n ; ˜ x n |{ y (1: t − a } n , ˜ y nbc , ˜ H n (cid:17) (86) = L X t =1 H (cid:16) y a ( t ) n |{ y (1: t − a } n , ˜ y nbc , ˜ H n (cid:17) − H (cid:16) y a ( t ) n | ˜ x n , { y (1: t − a } n , ˜ y nbc , ˜ H n (cid:17) (87) = L X t =1 H (cid:16) y a ( t ) n |{ y (1: t − a } n , ˜ y nbc , ˜ H n (cid:17) − H (cid:16) y a ( t ) n | x ( t ) n , y bc ( t ) n , ˜ H n (cid:17) (88) ≤ L X t =1 H ( y a ( t ) n | y bc ( t ) n , H ( t ) n ) − H ( y a ( t ) n | x ( t ) n , y bc ( t ) n , H ( t ) n ) (89) = L X t =1 I ( y a ( t ) n ; x ( t ) n | y bc ( t ) n , H ( t ) n ) (90) ≤ n L X t =1 I ( y aG ( t ); x G ( t ) | y bcG ( t ) , H ( t )) (91) ≤ nL log (cid:18) det (cid:18) PM I M + I M + PM I M (cid:19) det (cid:18) PM I ( N − M ) + I ( N − M ) (cid:19)(cid:19) − nL log (cid:18) det (cid:18) PM I ( N − M ) + I ( N − M ) (cid:19) det (cid:18) PM I M + I M (cid:19)(cid:19) (92) = o (log( P )) (93)where: • (86) follows by chain rule. • (87) and (90) are expressing mutual information via entropy. • (88) holds as the second term is the entropy of noise when conditioning on x ( t ) n . • (89) is based on the fact that conditioning reduces entropy. • (91) and (92) follows by [24, Lemma 3], where the covariance matrix of x G ( t ) + n ,a ( t ) and n ,bc ( t ) are PM I M + I M and I ( N − M ) , respectively. In addition, the optimal input of x ( t ) is CSCG with covariance matrix PM I ( N − M ) . November 2, 2018 DRAFT6 Substitute (85) and (93) in to (77), we have I (˜ r n ; ˜ x n | ˜ H n ) ≤ I (cid:16) { ˜ V † ,c ˜ x + ˜ n ,c } n ; ˜ x n | ˜ H n (cid:17) + nL ( K − M ) log( P ) + o (log( P )) (94)Now we go back to (59). Notice that if we choose D = [ N × (min( M ,N ) − N ) , I N ] , ( DV † , Dn ) has the same distribution as ( V † , n ) as both V and V are uniformlydistributed and V has no fewer columns than V . (Please refer to [24, Sec. IV-C2] formore details). We have the following Markov chain: ˜ x — ˜ V † ˜ x + ˜ n — ˜ V † ˜ x + ˜ n — ˜ V † ,c ˜ x + ˜ n ,c . (95)Denote J = min( M , N ) − ( N − K ) . Let V ,a contain the first J rows of V , and n ,a contain the first J elements of n . We can bound η as η ≤ µ I (cid:16) { ˜ V † ˜ x + ˜ n } n ; ˜ x n |{ ˜ V † ,c ˜ x + ˜ n ,c } n , ˜ H n (cid:17) − I (cid:16) { ˜ V † ˜ x + ˜ n } n ; ˜ x n |{ ˜ V † ,c ˜ x + ˜ n ,c } n , ˜ H n (cid:17) + (1 − µ ) nL ( K − M ) log( P ) + o (log( P )) (96) = µ I (cid:16) { ˜ V † ,a ˜ x + ˜ n } n ; ˜ x n |{ ˜ V † ,c ˜ x + ˜ n ,c } n , ˜ H n (cid:17) − I (cid:16) { ˜ V † ,ab ˜ x + ˜ n } n ; ˜ x n |{ ˜ V † ,c ˜ x + ˜ n ,c } n , ˜ H n (cid:17) + (1 − µ ) nL ( K − M ) log( P ) + o (log( P )) (97)Notice that the size of V † ,ab is K × M . Based on [24, Lemma 3], if we choose µ = KJ (98)the difference of the first two mutual information terms of (97) is at most in the order of o (log( P )) and we have λ ≤ (cid:18) − KJ (cid:19) ( K − M ) = (min( M , N ) − N )( K − M )min( M , N ) − ( N − K ) (99)Recall that d + µd ≤ M + µ ( N − M ) + λ . We thus have the outer bound on the sum DoFas shown in (38) and the proof of the converse part of Theorem 4 is complete. November 2, 2018 DRAFT7 B. Achievability In order to show the achievability part of Theorem 4, we only need to construct an achievablescheme for the corner point of the DoF region. Without loss of generality, we assume that M = min( M , N ) ; otherwise, transmitter 2 can simply use N transmit antennas. Since K ( N − M ) + ( M − N )( K − M ) = M ( K − M ) + M ( N − K ) , it is sufficient to show that thefollowing DoF pair ( d , d ) = ( KM , M ( K − M ) + M ( N − K )) (100)can be achieved over K slots with antenna mode switching at transmitter one among K modes.Similar to Section IV-D, we choose the mode switching pattern as follows: E (1) = [ e , e , . . . , e M ] , E (2) = [ e , e , . . . , e M +1 ] , ... E ( K ) = [ e K , e , . . . , e M − ] . We propose to use a generalization of the joint nulling and beamforming design that isinvestigated in Section IV-C. Unlike the frequency nulling that has been used for K = N ,this scheme requires that receiver 1 performs nulling in both frequency and spatial domains.We hereby use two superscripts F and S to indicate the matrices that associated with frequencyprocessing and spatial processing.The generalized joint nulling and beamforming has the following structure: ˜ Q = Q F M × K ⊗ Q S K × N , (101) ˜ P = [ ˜ P a , ˜ P b ] , where (102) ˜ P a = [ P F a ] K × ( K − M ) ⊗ [ P S a ] M × M , (103) ˜ P b = [ P F b ] K × M ⊗ [ P S b ] M × ( N − K ) . (104)The received signal at receiver 1 can be written as ˜ y = ˜ H ˜ x + ˜ H [ ˜ P a , ˜ P b ]˜ x + ˜ z (105)where ˜ x is a length M K vector, and ˜ x is a length M ( K − M ) + M ( N − K ) vector. November 2, 2018 DRAFT8 (cid:13)(cid:11)(cid:11)(cid:11)(cid:11)(cid:14)(cid:11)(cid:11)(cid:11)(cid:11)(cid:15) (cid:13)(cid:11)(cid:11)(cid:14)(cid:11)(cid:11)(cid:15) Interference space of user 2 via Sigal space of user 1 Interferencespace of user 2 via (cid:105) a P (cid:105) b P M N K (cid:16) K M (cid:16) K S p a ti a l do m a i n Frequency domain (cid:13)(cid:11)(cid:11) (cid:14) (cid:11)(cid:11) (cid:15) (cid:13) (cid:14) (cid:15) Fig. 2. Space-Frequency dimension allocation for the two users when K < N . After applying nulling matrix ˜ Q , we have ˜ Q ˜ y = ˜ Q ˜ H | {z } ˜ A ˜ x + (cid:20) ˜ Q ˜ H ˜ P a | {z } ˜ B , ˜ Q ˜ H ˜ P b | {z } ˜ C (cid:21) ˜ x + ˜ Q ˜ z . (106)To achieve the degrees of freedom pair shown in (100) for both users, it is sufficient to designour ˜ P and ˜ Q to satisfy the following conditions simultaneously1) rank ( ˜ A ) = M K ,2) rank ([ ˜ P a , ˜ P b ]) = M ( K − M ) + M ( N − K ) ,3) ˜ B = ,4) ˜ C = .We propose to use the following realizations: Q F = [ V K (1 , ω K , . . . , ω M − K )] T , (107) P F a = V K ( ω − M K , ω − ( M +1) K , . . . , ω − ( K − K ) . (108) P F b = ( Q F ) † , (109) P S a = I M , (110) November 2, 2018 DRAFT9 P S b = [ I N − K ; ] , (111) Q S = null ( H P S b ) T , (112)where (112) means that Q S H P S b = . Here, we choose (( Q F ) † , P F a ) to be a size K × K IFFTmatrix, which offers the same frequency domain explanation as discussed in Section IV-E; seealso Fig. 2. It is trivial to see ˜ B = . In other words, receiver 1 will simply ignore the signalin the last K − M frequencies and only using the signal in the first K frequencies to decodehis own message. Therefore, ˜ P a contains the interference directions from all the antennas oftransmitter 2 but only in certain frequencies. Now, after applying the frequency nulling, thereare N K dimensions remaining, which contain both user 1’s message and the message of user2 that is transmitted by ˜ P b . Among all the N K dimensions, receiver 1 only requires M K dimensions to decode his own message, while leaving additional K ( N − M ) dimensions foruser 2. Here we choose one possible way of decomposing the remaining dimensions. Transmitter2 sends some messages in the first M frequencies but only though N − K antennas, as shownin (111). Notice that ˜ C = ˜ Q ˜ H ˜ P b = ( Q F ⊗ Q S )( I K ⊗ H )( P F b ⊗ P S b ) (113) = ( Q F P F b ) ⊗ ( Q S H P S b ) (114)which means that the choice of Q S as given in (112) is sufficient to set ˜ C = 0 . It is clear thatfor the interference signal sent via ˜ P b , receiver 1 only need to do spatial zero-forcing in ourscheme, which can be seen from the fact Q F P F b = I K due to (109).To satisfy the second condition, notice that rank ( ˜ P a ) = M ( K − M ) and rank ( ˜ P a ) = M ( N − K ) , it is sufficient to show that ˜ P a ⊥ ˜ P b , which is obvious as ˜ P † b ˜ P a = ( Q F P F a ) ⊗ (( P S b ) † I M ) = (115)because Q F P F a = . This is not surprising as the signal of user 2 transmitted via ˜ P a and ˜ P b are orthogonal in frequency domain. The remaining part is to show the first condition holds,which is true because here ˜ A has the same structure as ˜ A ′ of (34) with N replaced by K and h i replaced by Q S K × N h i . November 2, 2018 DRAFT0 M N min( M , N ) ( N − M 1) min( M , N N K varies from M N N − M N − M 1) + (min( M , N − N (cid:16) − M K (cid:17) d d Fig. 3. The benefit of antenna mode switching on the DoF region, in the case of M < N < min( M , N ) . C. Discussion It is not surprising that when K = N , (38) implies d + N min( M , N ) d ≤ N (116)which is the same as (7) and that in [17, Theorem 3] when M < N < min( M , N ) . For thescheme that we discussed above, ˜ P b disappears and it is the DoF achievable scheme that wedeveloped in Section IV-C. In addition, when K = M , (38) becomes (12) and ˜ P a disappears,the general scheme reduces to the DoF-optimal spatial zero-forcing as shown in [24]. Hence, forone extra mode at transmitter 1, we can further align min( M , N ) − N streams of interferenceover K slots. The incremental gain per slot is reduced when K increases; see Fig. 3. Our resultreveals the fundamental benefit that can be obtained from reconfigurable antenna modes whenthere is no CSIT and M < N < min( M , N ) . In addition, combining with the known results,we know that in order to achieve the DoF region of two-user FIC and ZIC, zero-forcing infrequency and spatial domains suffice regardless of the CSIT assumption. November 2, 2018 DRAFT1 VI. C ONCLUSIONS We derived the exact DoF region for the MIMO Z and full interference channels when perfectchannel state information is available at receivers, including i) the Z interference channel withchannel state information at the transmitter; ii) the Z and full interference channel without channelstate information at the transmitter, but with reconfigurable antennas at the transmitters. For bothFIC and ZIC, when the number of antenna modes K at the transmitter with the reconfigurableantennas is not less than the number of receive antennas at the corresponding receiver, the DoFregion is maximized and no longer depends on the number of antenna modes. Otherwise, eachadditional antenna mode can bring extra gain in the DoF region when M < N < min( M , N ) for both FIC and ZIC, and when M < N < min( M , N ) for FIC. 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