3766 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
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1 3766 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 6, JUNE 2012 Degrees of Freedom Regions of Two-User MIMO Z and Full Interference Channels: The Benefit of Reconfigurable Antennas Lei Ke and Zhengdao Wang, Senior Member, IEEE Abstract We study the degrees of freedom (DoF) regions of twouser multiple-input multiple-output Z and full interference channels in this paper We assume that the receivers always have perfect channel state information We first derive the DoF region of Z interference channel with channel state information at transmitter (CSIT) For full interference channel without CSIT, the DoF region has been fully characterized recently and it is shown that the previously known outer bound is not achievable In this paper, we investigate the no-csit case further by assuming that one transmitter has the ability of antenna mode switching We obtain the DoF region as a function of the number of available antenna modes and reveal the incremental gain in DoF that each additional antenna mode can bring It is shown that, in certain cases, the reconfigurable antennas can increase the DoF In these cases, the DoF region is maximized when the number of modes is at least equal to the number of receive antennas at the corresponding receiver, in which case the previous outer bound is achieved In all cases, we propose systematic constructions of the beamforming and nulling matrices for achieving the DoF region The constructions bear an interesting space-frequency coding interpretation Index Terms Antenna mode switching, degrees of freedom (DoF) region, interference channel, multiple-input multiple-output (MIMO), reconfigurable antenna I INTRODUCTION CHARACTERIZING the capacity region of interference channel [1], [2] has been a long open problem Many researchers investigated this important problem, and the capacity regions of certain two-user interference channels are known when the interference is strong, eg, [1], [3], and [4] However, when the interference is not strong, the capacity region is still unknown Recent progress reveals the capacity region for two-user interference channel to within one bit [5], and after that the sum capacity for very weak interference channel is settled [6] [8] Recently, a deterministic channel model has been proposed and used to explore the capacity of Gaussian interfer- Manuscript received November 09, 2010; revised October 13, 2011 and November 29, 2011; accepted December 07, 2011 Date of publication January 31, 2012; date of current version May 15, 2012 This work was supported in part by the National Science Foundation under Award ECCS The material in this paper was presented in part at the 2010 IEEE Global Telecommunications Conference L Ke was with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA USA He is now with Qualcomm Inc, San Diego, CA USA ( lke@qualcommcom) Z Wang is with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA USA ( zhengdao@iastateedu) Communicated by S Jafar, Associate Editor for Communications Digital Object Identifier /TIT ence network [9] [11] such that the gap to capacity region can be bounded up to a constant value When only one of the two transmitter receiver pairs is subject to interference, the interference channel is termed as Z interference channel (ZIC), which is also well-known as one-sided interference channel [12] To avoid confusion, we will call the channel where both pairs are subject to interference the full interference channel (FIC) In practice, ZIC may be viewed as an approximation of FIC where one interference link is very weak In theory, ZIC is useful for solving the problem in FIC For example, ZIC has been used in [5] to obtain an outer bound on the sum rate of Gaussian FIC The capacity region of MIMO Gaussian ZIC is established in [13] under very strong interference and aligned strong interference assumptions In [14], the authors considered the capacity region of a single antenna ZIC without channel state information at transmitter (CSIT) using deterministic approach When it comes to multiple-input multiple-output (MIMO) networks, the capacity regions of certain MIMO interference channels are known [15], [16] Instead of trying to characterize the capacity region completely, the degrees of freedom (DoF) region characterizes how capacity scales with transmit power as the signal-to-noise ratio goes to infinity It is well known that in certain cases, the absence of CSIT will not affect the DoF for MIMO networks, eg, in the multiple access channel [17] In other cases, CSIT does play an important role For example, using interference alignment scheme, it is shown that the total DoF of a -user MIMO interference channel is,where is the number of antennas of each user [18] The key idea is to pack interferences from multiple 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 [19], where it is shown that zero forcing is enough to achieve the DoF region However, it is a different story in two-user MIMO X channel, where each transmitter has a message to every receiver In [20], it is shown that interference alignment is the key to achieving the DoF region of MIMO X network The DoF region of two-user MIMO broadcast channel and interference channel without CSIT is considered in [21], where there is an uneven trade-off between the two users Except for a special case, the DoF region for the interference channel is known Similar, but more general result for isotropic fading channels can be found in [22] The DoF regions of the -user MIMO broadcast, interference, and cognitive radio channels are derived in [23] for some cases /$ IEEE
2 KE AND WANG: DEGREES OF FREEDOM REGIONS OF TWO-USER MIMO Z AND FULL INTERFERENCE CHANNELS 3767 Recently, it is shown in [24] and [25] that if the channel is staggered block fading, we can explore the channel correlation structure to do interference alignment, where the upper bound in the converse can be achieved in some special cases For example, it is shown that for two-user MIMO staggered block fadingficwith1and3antennasattransmitters,2and4antennas at their corresponding receivers and without CSIT, the DoF pair can be achieved The idea was further clarified in [26], where a blind interference alignment scheme is also proposed for -user multiple-input single-output broadcast channel to achieve DoF outer bound when CSIT is absent Also recently, it has been shown in [27] that the previous outer bound is not tight when the channels are independent and identical distributed (iid) over time and isotropic over spatial domainsobynow,thedofregionoftwo-user MIMO FIC is completely known for both the case with CSIT and the no CSIT case [channel state information at receiver (CSIR) is always assumed available], provided that the channel is iid over time and isotropic over spatial domain However, when the channel is not iid over time such as in the staggered fading channels [24], [25], the DoF could be larger The staggered block fading in [24] and [25], albeit theoretically intriguing, is difficult to satisfy for practical channels We investigate whether artificially created channel variations, in the spirit of [26], can be used to increase the system DoF We will introduce reconfigurable antennas at the transmitters, as opposed to at the receivers as in [26] The reconfigurable antennas are capable of switching among a set of preset modes, each corresponding to an independent set of channels to all receivers It is shown that depending on the antenna numbers configuration, in some cases, reconfigurable antennas can enhance the DoF, whereas in other cases they do not We completely characterize the gain offered by reconfigurable antenna in all cases for two-user MIMO ZIC and FIC The main results of this paper are summarized as follows We consider the two-user MIMO interference channel with,,, antennas at the transmitter one, receiver one, transmitter two, and receiver two, respectively We always assume perfect CSIR We consider the ZIC channel with CSIT, and both ZIC and FIC without CSIT but with reconfigurable antennas We characterize the DoF regions for the following cases 1) ZIC with CSIT We show that zero forcing is sufficient for achieving the DoF region in this case (Theorem 1) No reconfigurable antenna is necessary, nor does it increase the DoF region 2) ZIC and FIC with reconfigurable antennas, where the number of antennas modes at transmitter one is at least equal to (Theorems 2 and 3) We show that increasing beyond does not bring more gains in DoF region 3) ZIC and FIC when, in which case each additional antenna mode brings an incremental gain on the DoF region (Theorem 4) We present joint beamforming and nulling schemes to achieve the DoF region in all cases When reconfigurable antennas are used, our proposed schemes have an interesting space-frequency coding explanation The rest of this 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 briefly reviewed The DoF region of ZIC with CSIT is discussed in Section III The DoF regions of ZIC and FIC without CSIT when there are enough antenna modes are investigated in Section IV When there are not enough modes, the DoF region is given in Section V Finally, Section VI concludes this paper Notation: Boldface uppercase (lowercase) letters denote matrices (vectors),, are the real, integer, and complex numbers sets denotes a circularly symmetric complex Gaussian (CSCG) distribution with zero mean and unit variance We use to denote the Kronecker product of and and denote all one and all zero matrices (vectors), respectively and denote the transpose and Hermitian of,respectively We also use notation like to emphasize that is of size Weuse to denote a size identity matrix and to denote an all-one column vector with length Denote Asize Vandermonde matrix based on a set of element is defined as We use to denote the mutual information between and The differential entropy of a continuous random variable is denoted as A Channel Model II SYSTEM MODEL AND KNOWN RESULTS Consider an MIMO interference channel with two transmitters and two receivers For,2,transmitter intends to convey an independent message to receiver The number of transmit antennas at transmitter is denoted as,andthe number of receiver antennas at receiver is denoted as The system is denoted as an system We use to denote the discrete time index Let be the transmitted signal at transmitter, the received signal at receiver,and the additive noise at receiver, respectively The channel between the th transmitter and the th receiver is denoted as Thesystem input output relationship is described by For the two-user MIMO ZIC, The transmitted signals satisfy the following power constraint: We make the following probabilistic assumptions: 1) The entries of are identically distributed and independent across both time and space 2) The probability that belongs to any subset of of zero Lebesgue measure is zero 3) The channel is block fading, and is the length of the coherence interval 4) In the cases where reconfigurable antennas are used, either is sufficiently large, or is a multiple of, being the number of reconfigurable antenna modes (to be defined later), such that the transmitter with reconfigurable antennas can switch times within the coherence interval 5) The CSI is always available at the receivers
3 3768 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 6, JUNE 2012 Denote the capacity region of the two-user MIMO system as, which contains all the rate pairs such that the corresponding probability of error can approach zero as coding length increases The DoF region is defined as (see, eg, [21]) B Channel With Reconfigurable Antennas The reconfigurable antennas are different from the conventional antennas as they can be switched to different predetermined modes so that artificial channel fluctuation can be introduced [26] We define an antenna mode as one possible configuration of a single antenna such that by switching the RF chain to a different mode, the channels to all the receive antennas are changed Different antenna modes can be realized via spatially separated physical antennas, or the same physical antenna excited with different polarizations, etc 1) Reconfigurable Antennas in the Point-to-Point Channel: To define the reconfigurable antennas, we present the system model of a point-to-point MIMO channel with reconfigurable antennas at both transmitter and receiver Denote the numbers of RF chains at the transmitter and receiver as and,respectively Denote the numbers of antenna modes at transmitter and receiver as and, respectively Notice that we must have and, and the equalities hold when we have only conventional (nonreconfigurable) antennas We assume that the channel is block fading Within each coherent block, the channel,ofsize, between all the transmit modes and all the receive modes remains constant In addition, the channel is isotropic in the sense of [27] From block to block, it changes independently Let be the th column of and the th row of Let and denote the mode indices selected at time by the th transmit and the th receive antennas, respectively We define a transmit mode-switching matrix of size whose th column of is,anda receive modeswitching matrix whose th row is With antenna mode switching, the equivalent channel of size from the transmitter to the receiver is 2) Reconfigurable Antenna in Two-User FIC and ZIC: In two-user MIMO FIC and ZIC, we assume that all the channels have the same coherent time In addition, as we will see, it is sufficient to assume that only one transmitter (say transmitter one) is equipped with reconfigurable antennas that can be switched to different modes within one coherent block The number of modes that are available is denoted as Naturally,themodel (1) in [27] is the special case (no reconfigurability) The artificial channel variation created with antenna mode switching bears resemblance to the structured channel variation in [24], with two important differences: 1) Antenna switching can be initiated at will at the transmitter, whereas channel coherence structure is in general not controllable 2) The resulting equivalent channel from antenna mode switching is not staggered [24], so methods therein do not apply here C Known Results on FIC We first review some known results on DoF region of MIMO FIC which will be useful for developing our results The total DoF of two-user MIMO FIC with CSIT is developed in [19, Th 2], based on which it is simple to show that following is the DoF region: An outer bound of DoF region of two-user MIMO FIC without CSIT is the following [22, Th 1]: Note that the same outer bound is also derived in [21] in a different form It is shown in [21] that the outer bound given in (2) (4) can be achieved by zero forcing or time sharing except for the case, for which it was not known how to achieve The cases when can be converted by switching the user indices It is shown in [27] that when the channel is isotropic fading and iid over time, the outer bound given in (2) (4) is not tight: if, the DoF region of FIC without CSIT is as follows: where Asaresult, (5) is not achievable when,as(7) is reduced to (8), shown at the bottom of the page, and the DoF pair is the corner point of the DoF region (2) (3) (4) (5) (6) (7) (8)
4 KE AND WANG: DEGREES OF FREEDOM REGIONS OF TWO-USER MIMO Z AND FULL INTERFERENCE CHANNELS 3769 III TWO-USER MIMO ZIC WITH CSIT In this section, we prove the following theorem Theorem 1 (ZIC With CSIT): The DoF region of a two-user MIMO ZIC with CSIT is described by (9) (10) Proof: We split the proof into the achievability and converse parts, as the following two lemmas The theorem can be proved by showing that the regions given by Lemma 1 and Lemma 2 are the same for all the cases Lemma 1 (Achievability Part of Theorem 1): The following region of two-user MIMO ZIC with CSIT is achievable: two (and two only) needs to be equipped with reconfigurable antennas can be considered by swapping the transmitter (and receiver) indices Let be the number of available antenna modes at transmitter one We first deal with the case The case will be dealt with in Section V Depending on the antenna numbers configuration, the achievability scheme of ZIC and FIC without CSIT can be divided into two cases In the first case, no reconfigurable antenna is needed to achieve a DoF outer bound reconfigurable antennas are not helpful (see Section IV-B) In the second case, using reconfigurable antennas enlarges the DoF region (see Section IV-C) Our main results in this section are the following two theorems Theorem 2 (ZIC With Enough Reconfigurable Antenna Modes): The DoF region of two-user MIMO ZIC without CSIT is described by the following inequalities: (11) where is indicator function Proof: First consider the case Transmitter 2 can send streams along the null space of without interfering receiver 1 Assume transmitter 1 sends (an integer) streams Transmitter 2 can also send streams along the row space of Therefore user 2 can decode streams without interfering receiver 1 Next, consider the case that Assume transmitter 2 sends (an integer) stream Transmitter 1 can send streams, and receiver 1 will be able to decode Combining these two cases, we have the achievable DoF region showninthislemma Lemma 2 (Converse Part of Theorem 1): The region given by (9) and (10) is a valid outer bound for the two-user MIMO ZIC with CSIT Proof: The inequalities in (9) are single-user MIMO bounds If, we can increase the number of antennas at receiver 1 to Doing so will not increase the DoF region With at least antennas, receiver 1 can decode the message emitted by transmitter 2, after removing the decoded message from transmitter 1 The multiple access channel bound dictates that Based on Lemma 1, zero forcing at receiver is sufficient to achieve the DoF region of ZIC when 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 TWO-USER MIMO ZIC AND FIC WITHOUT CSIT WHEN NUMBER OF MODES In this section, we investigate the DoF region of two-user ZIC and FIC without CSIT but with transmitter side reconfigurable antennas available It will be sufficient to assume that such antenna reconfigurability is available at only one transmitter, chosen depending on the antenna number configuration and assumed to be transmitter one The case when transmitter (12) if either one of the following is true: C1) and transmitter one can switch among antenna modes, or C2) does not hold The DoF region in Theorem 2 is shown in Fig 1 Theorem 3 (FIC With Enough Reconfigurable Antenna Modes): The DoF region of two-user MIMO FIC without CSIT is described by the inequalities (2) (4) if any one of the following is true: C1) and transmitter one can switch among antenna modes, or C2) and transmitter two can switch among antenna modes, or C3) Neither nor holds A Converse Part With the antenna mode switching capability at a transmitter, it is possible to convey information not only using the information symbols, say, but also using the antenna modes However, for the purpose of quantifying the DoF region of the system, we can neglect the information that is conveyed through the antenna mode selection Lemma 3 (Transmit Antenna Mode Selection Conveys Limited Information): For finite, by assuming that the antenna mode selection scheme is available at the receiver, the DoF region of the system is not changed Proof: Assume transmitter one possesses reconfigurable antennas Consider the channel from this transmitter to any particular receiver Let and be the channel input and output, respectively Since there are antenna modes, and RF chains, which is actual number of transmitting antennas at any time, there are at most possible ways of connecting the RF
5 3770 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 6, JUNE 2012 Fig 1 DoF region of two-user MIMO ZIC without CSIT when number of antenna modes Figures (a) (e) are for the case ; figures (f) (h) are for the case chains to the antenna modes Let be the index of transmit antenna selection pattern among these possibilities We can view the transmitter as transmitting both and Wehave Therefore Since this is true for any receiver, it follows that by assuming that the antenna mode selection scheme is available at the receivers the DoF of the system is not affected For this reason, in the following, we assume that the transmit antenna mode switching pattern is known to all the receivers Lemma 4 (Converse Part of Theorem 3): The outer bound of DoF region of two-user MIMO FIC given in (2) (4) is still valid when either or both transmitters are using antenna mode switching Proof: The outer bound (3) has been derived based on the assumption that the rows of and those of are statistically equivalent [21], [22] Similarly, the outer bound (4) has been derived based on the assumption that the rows of and those of are statistically equivalent These assumptions are not affected by antenna mode switching at either or both transmitters Hence, the DoF outer bound is still valid Lemma 5 (Converse Part of Theorem 2): The region specified by (11) and (12) is an outer bound of DoF region of a two-user MIMO ZIC without CSIT and with reconfigurable antennas at transmitter one Proof: This is the direct result of [22, Th 1], as specified in (2) (4), by noticing that there is no interference from transmitter 1 to receiver 2; hence, (4) is not longer needed The antenna switching at transmitter one does not affect the upper bound, for the reason stated in Lemma 4 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) of Theorem 3 Achievability for the remaining cases are left to Section IV-C
6 KE AND WANG: DEGREES OF FREEDOM REGIONS OF TWO-USER MIMO Z AND FULL INTERFERENCE CHANNELS 3771 Lemma 6: For the two-user MIMO ZIC without CSIT, when, the outer bound in Lemma 5 is achievable by zero forcing Proof: When, the corresponding outer regions are shown in Fig 1(f) (h) Noticing that (12) is reduced to, zero forcing is sufficient to achieve the outer bound Lemma 7: When CSIT is absent, the DoF outer region given by Lemma 5 of a two-user MIMO ZIC is the same as that of an ZIC Proof: We give the proof case by case It can be seen that when, reducing the number of antennas at receiver 2 to will not shrink the DoF region When, we can further consider two subcases: and 1) When, corresponding to Fig 1(d) and (e), the DoF bound (12) becomes Hence, the DoF outer region is the same as an ZIC 2) When and, the DoF bound (12) becomes Hence, if, which implies, the DoF outer region is 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 ZIC Hence, the lemma holds We also have the following lemma regarding the relationship between DoF regions of ZIC and FIC Lemma 8: When, the MIMO ZIC and FIC have the same DoF regions Any encoding scheme 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 channel is interference free Conversely, any point achievable in the ZIC region is also achievable in FIC This is based on the fact that the channels are statistically equivalent at both receivers If receiver 1 can decode user 1 s message, then receiver 2, having at least as many antennas, must also 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 8, we can translate all achievability schemes from FIC to ZIC and vice versa when Therefore, the achievability schemes in [21] for FIC when and can be used for ZIC The achievability part for Case C2) of Theorem 2 is complete For the FIC, the achievability for the case, except when, is shown in [21] When, we can swap the indices of the two users, so that except for the Cases C1) and C2) the achievability scheme is known for FIC C Achievability: With Antenna Mode Switching When achievability scheme for the case will be built upon this case Based on Lemma 7 and Lemma 8, we only consider the twouser MIMO ZIC with to prove the Cases C1) for both theorems Case C2) of Theorem 3 is the Case of C1) with user indices swapped Therefore, we want to show that the following DoF pair is achievable for ZIC with modes: We first notice that this point cannot be achieved by zero forcing over one time instant This is because if transmitter 1sends streams, transmitter 2 can only send streams without interfering receiver 1 If transmitter 2 sends more streams, the desired signal and interference are not separable at receiver 1 as transmitter 2 does not know CSI, so it cannot send streams along the null space of Asimpleexample is the case, where the outer bound gives us, which is not achievable via zero forcing over one time slot We make the assumption that the channel stays the same for at least time slots It is sufficient to show that streams can be achieved in time slots We first develop the beamforming andnullingdesignbyassuming that there are antenna modes available at transmitter 1 such that it can use different antenna modes in different slots to create channel variation The transmit mode-switching patterns are [cf, (1)] We will further show that the resultant beamforming and nulling design still work even if there are only modes available Here and after, we use tilde notation to indicate the time extension signals, where the number of slots of time extension signals shall be clear within the contextthetimeextension channel between transmitter 1 and receiver 1 in time slots is and the channel between transmitter 2 and receiver 1 is In this section, we prove a weaker version of the achievability for Case C1) of Theorem 2 and Cases C1) and C2) of Theorem 3 Namely, we assume that the number of antenna modes available is The scheme is simpler in this case, and the as transmitter 2 does not create channel variation We will use precoding at transmitter 2 only and nulling at receiver 1 only Let be the transmit beamforming matrix at transmitter 2 and
7 3772 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 6, JUNE 2012 be the nulling matrix at receiver 1 We propose to use the following structures for them: a specific pairof and,,suchthat is full rank We propose the following: The received signal at receiver 1 can be written as (13) (14) (17) where Let Take the realizations of,,as where is a length vector, and is a length vector After applying nulling matrix,wehave It can be verified that for such choices of and, is a Vandermonde matrix (15) To achieve the DoF for both users, it is sufficient to design our and to satisfy the following conditions simultaneously: 1) 2) 3) The second condition can be easily satisfied Because, we only need to design such that As to the third condition, notice that hence of full rank We also notice that is a leading principal minor of a permuted fast Fourier transform (FFT) matrix with size The permutation is as follows: index the columns of an FFT matrix, and then permute them in an order shown as follows: Based on Lemma 9, if we choose the nulling matrix using as specified in (17), has full rank almost surely One choice of the corresponding matrix with respect to (17) is the following: (18) It is, therefore, sufficient (and also necessary) to have Then, the key is to find a such that the equivalent channel of user 1 after nulling (16) has full rank with probability 1 The matrix is of size and has the following structure: To show that has full rank, we need the following lemma, which is known earlier, and a proof of it can be found in, eg, [28] Lemma 9 [28, Lemma 2]: Consider an analytic function of several variables If is nontrivial in the sense that there exists such that,then the zero set of is of measure (Lebesgue measure in ) zero Because the determinant of is an analytic polynomial function of elements of,, we only need to find which is orthogonal to This completes the achievability part under conditions in Case C1) of Theorem 2 and Cases C1) and C2) of Theorem 3, but with D Achievability: With Antenna Mode Switching When Assuming there are modes available at transmitter 1 and denote these channel vectors between receive antennas of user 1andthe th mode as, and let We choose the following switching patterns: (19) (20) (21) We want to show that under these switching patterns, the equivalent channel in (16) between transmitter one and receiver one after nulling is still full rank To show this, by indexing the columns of in (16) as,wethenpermute and group the columns of in the following way:
8 KE AND WANG: DEGREES OF FREEDOM REGIONS OF TWO-USER MIMO Z AND FULL INTERFERENCE CHANNELS 3773 Denote the permutation result as and it can be expressed as where is a size diagonal matrix and can be expressed as Notice that,where Recall To show is full rank, it is necessary to show is full rank as is full rank with probability 1 It can be verified that, via row and column permutations, can be changed to a block diagonal matrix with the th block being DoF region if or Forallthe other cases, the DoF region is strictly smaller when comparing with the CSIT case This observation can be verified case by case Notice that it is already shown in [21, Th 2] that when, absence of CSIT does not reduce DoF region in two-user MIMO FIC, because MIMO FIC and ZIC have the same DoF region when We only need to consider the subcases when, corresponding to (f) (h) in Fig 1 1) If and, the total DoF of MIMO ZIC is upper bounded by due to (10), so the DoF region remains the same if CSIT is absent 2) If, the DoF region of MIMO ZIC without CSIT is a square only when, in which case loss of CSIT does not reduce the DoF region Otherwise, the maximum total DoF of ZIC with CSIT is,whichis strictly larger than, the maximum total DoF when CSIT is absent 3) Alternative Construction When : When, instead of using the given in (17), we can use the following We need to show that this matrix will lead to a full rank This can be achieved by choosing such that it can be decomposed as,where whichisfullrankduetovandermonde structure Hence, is full rank It follows that is full rank with probability 1 This completes the achievability part under conditions in Case C1) of Theorem 2 and Cases C1) and C2) of Theorem 3 for For this E Discussion 1) Frequency Domain Interpretation: We note that the matrix is an inverse FFT (IFFT) matrix in our construction (13) and (14), (17) and (18) This observation yields an interesting frequency domain interpretation of our construction The signal of user 2 is transmitted over frequencies corresponding to the last columns of an IFFT matrix, whereas the first user 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 which introduces frequency spread User 1 s signal is spread from one frequency bin to all the frequencies while user 2 s signal remains in the last frequency bins Therefore, the signal in the first bins is interference free, which can be used to decode user 1 s message The nulling matrix applied at receiver 1 has a projection interpretation as well Left multiplying both sides of (15) with yields where is the frequency domain projection matrix We can see that the signal of user 1 is projected from frequencies to the first frequencies 2) The Loss of DoF Due to Lack of CSIT: In two-user MIMO ZIC without CSIT, loss of CSIT does not lead to shrinkage of the which has full rank For this choice of,weonlyuse antenna modes in time slots Therefore, for the two-user MIMO ZIC and FIC when and, -fold time extension is enough to achieve the DoF region We remark that this can be viewed as the generalization of the case we discussed in Section IV-C for and In fact, is the nulling matrix givenin(17)when, 4) Successive Decoding in ZIC: For the two-user MIMO FIC when and CSIT is absent, we need block decoding at both receivers in general, which introduces decoding delay Successive interference cancellation decoder can be used at receiver 2 to reduce decoding delay Taking the case as an example, we can use -fold time extension and choose The corresponding matrix is not necessary to be the last columns of a FFT matrix The following matrix still satisfies the design constraint Here, has a nice structure Every stream of user 2 can be decoded immediately as they are interference free For other cases where cannot divide, we can still find a,
9 3774 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 6, JUNE 2012 pair through numerical simulation such that the upper diagonal parts of are all zeros and contain small number of nonzero entries Such a beamforming matrix can guarantee the immediate decoding of user 2 s signal as the interference only comes from the streams decoded already V TWO-USER MIMO ZIC AND FIC WITHOUT CSIT WHEN In this section, we will present our result for the case Note that the number of antenna modes cannot be smaller than the number of available antennas, that is, The main result of this section is the following theorem Theorem 4: When and the antennas of transmitter 1 can be switched among antenna modes, and, the DoF region of two-user MIMO ZIC and FIC without CSIT is given by the inequalities (22) and (23) shown at the bottom of the page The DoF region of FIC for can be obtained by switching the two user indices The method of proof is heavily based on results in [27] The reader will be referred to [27] for several lemmas and their proofs Some notation is used only in this Section VI We use tilde notation to denote the time-extended signal and is the index of the slot within one block In general, by default, for a vector, and for a matrix, In addition, for a time-extended vector,weuse or to denote a sequence of successive blocks of : Furthermore, is the sequence of which contains all the vector of the th slot of all blocks: Similar notation is defined for matrices as well We use to denote,and for all the channel matrices over blocks In addition, for a random vector, is a corresponding CSCG vector that has the same covariance matrix as slots due to antenna mode switching at transmitter 1 Transmitter 1 has modes with and it can adopt arbitrary switching pattern Let be an full rank random matrix such that,and, the random vector channel between the th antenna mode and receive antennas of user 1 We introduce the fictitious vectors to simplify the proof We assume is isotropic fading and iid over blocks of length each, where satisfies We denote the decomposition of as Recall that is the transmit mode switching pattern matrix, we have,where is of size due to the fictitious vectors At receiver 1, from Fano s inequality, we have where as Denote where Using [27, Th 2], which says that Gaussian input can reduce the mutual information by at most an quantity, and two uses of chain rule we have Using [27, Lemma 2], we have and A Converse Part We prove the converse part of Theorem 4 in the following Recall that for, the proof is equivalent for both FIC and ZIC We will only present the proof for ZIC To make the proof self-contained, we will go through some similar steps as in [27], but without details The converse is developed based on blocking for every slots In each block, the channel, stay the same with the decomposition and, whereas is time varying among Hence, can be further bounded as (24) (22) (23)
10 KE AND WANG: DEGREES OF FREEDOM REGIONS OF TWO-USER MIMO Z AND FULL INTERFERENCE CHANNELS 3775 As to receiver 2, using Fano s inequality and [27, Lemma 2], we have where Hence (25) Notice that by using Gaussian input, the following inequalities hold: (26) (27) Then, letting, multiplying (25) with some positive scalar, adding it to (24), and using (26) and (27), we have the following inequality: We have (31) (32) (33) (34) (35) where (31) and (33) hold as and are full rank square matrices Equation (32) is due to [27, Lemma 2] Equation (34) holds as changing noise variance will not change the DoF Equation (35) is true because has the same distribution as and is independent of Tofind the DoF order of,wefirst notice that for each slot in one block, can be divided into three parts:,,and 1) is of size and equal to 2) is of size and is the same for all It consists of rows of that do not appear in any, 3) is of size and consists of rows of that are neither in nor in Example: Assume,,,,and where s are vectors Assume is the following: where is to be determined and (28) We have (29) Divide (28) by and let,wehavethefollowing inequality on the DoF of two users: where Recall that Wedefine and and Note that remains the same in one block of slots Suppose receiver 1 receives as in (30) and wants to decode the message of that goes through an equivalent channel Then, are the directions of interference from transmitter at time, are those directions that are temporarily interference free at time,and are the directions which are interference free for a whole block The associated noises of the those directions are, similarly, defined as,,and To bound the DoF of of (35), we define (30)
11 3776 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 6, JUNE 2012 and adopt the following notation for simplicity: In addition,,,, are sequences of corresponding vectors of the th slot over blocks The collection of is denoted as We also define,,and similarly Using thechainrule,wehave 2) (38) and (41) are expressing mutual information via entropy; 3) (39) holds as the second term is the entropy of noise when conditioning on ; 4) (40) is based on the fact that conditioning reduces entropy; 5) (43) follows by [27, Lemma 3]; 6) (44) holds due to the fact that the DoF of an point-to-point MIMO channel is at most The third term in (36) can be bounded in a similar fashion We have (45) (46) Now checking the second term in (36), we notice that (36) (47) (37) (48) (49) (38) (50) (39) where 1) (37) and (42) follow by chain rule; (40) (41) (42) (43) (44) (51) (52) where 1) (45) follows by chain rule; 2) (46) and (49) are expressing mutual information via entropy; 3) (47) holds as the second term is the entropy of noise when conditioning on ; 4) (48) is based on the fact that conditioning reduces entropy; 5) (50) and (51) follows by [27, Lemma 4], where the covariance matrix of and are and, respectively In addition, the optimal input of is CSCG with covariance matrix
12 KE AND WANG: DEGREES OF FREEDOM REGIONS OF TWO-USER MIMO Z AND FULL INTERFERENCE CHANNELS 3777 Substituting (44) and (52) into (36), and considering (31) (35), we have (53) Now we go back to (29) Notice that if we choose, has the same distribution as as both and are uniformly distributed and has no fewer columns than Setting,wehavethefollowing Markov chain: (54) Denote Let contain the first rows of,and contain the first elements of We can bound defined in (29) as Fig 2 Space-frequency dimension allocation for the two users when can be achieved over slots with antenna mode switching at transmitter one among modessimilartosectioniv-d,we choose the mode switching patterns as follows: (55) (56) where in (55) we have substituted (53) in (29) and used the Markov chain (54) Notice that the size of is Based on [27, Lemma 3], if we choose the difference of the first two mutual information terms of (56) is at most in the order of and we have We propose to use a generalization of the joint nulling and beamforming design that was investigated in Section IV-C Unlike the frequency nulling that has been used for,this scheme requires that receiver 1 performs nulling in both frequency and spatial domains We hereby use two superscripts and to indicate the matrices that associated with frequency processing and spatial processing The generalized joint nulling and beamforming has the following structure: (58) (59) (60) (61) The received signal at receiver 1 can be written as Recall that We,thus,have the outer bound on the sum DoF as shown in (23) and the proof of the converse part of Theorem 4 is complete B Achievability In order to show the achievability part of Theorem 4, we only need to construct an achievability scheme for the corner point of the DoF region Without loss of generality, we assume that ; otherwise, transmitter 2 can simply use transmit antennas To achieve the corner point it is sufficient to show that the following DoF pair: (57) where is a length vector, and is a length vector After applying nulling matrix,wehave (62) To achieve the DoF pair shown in (57) for both users, it is sufficient to design our and to satisfy the following conditions simultaneously 1) 2) 3)
13 3778 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 6, JUNE 2012 Fig 3 Benefit of antenna mode switching on the DoF region, in the case of 4) We propose to use the following realizations: (63) (64) (65) (66) (67) (68) because This is not surprising as the signal of user 2 transmitted via and is orthogonal in frequency domain The remaining part is used to show that the first condition holds, which is true because here has the same structure as of (22) with replaced by and replaced by This completes the proof of the achievability C Discussion It is not surprising that when, (23) implies where (68) means that Here, we choose tobeasize IFFT matrix, which offers the same frequency domain explanation as discussed in Section IV-E (see also Fig 2) It is trivial to see In other words, receiver 1 will simply ignore the signal in the last frequencies and only use the signal in the first frequencies to decode his own message Therefore, contains the interference directions from all the antennas of transmitter 2 but only in certain frequencies Now, after applying the frequency nulling, there are dimensions remaining, which contain both user 1 s message and the message of user 2 that is transmitted by Among all the dimensions, receiver 1 only requires dimensions to decode his own message, while leaving additional dimensions for user 2 Here, we choose one possible way of decomposing the remaining dimensions Transmitter 2 sends some messages in the first frequencies but only though antennas, as shownin(67)noticethat which means that the choice of as given in (68) is sufficient to set It is clear that for the interference signal sent via, receiver 1 only need to do spatial zero-forcing in our scheme, which can be seen from the fact due to (65) To satisfy the second condition, notice that and,itissufficient to show that, which is obvious as which is the same as (3) and that in [21, Th 3] when For the scheme that we discussed earlier, disappears and it is the DoF achievable scheme that we developed in Section IV-C In addition, when, (23) becomes (8) and disappears, the general scheme reduces to the DoF-optimal spatial zero-forcing as shown in [27] Hence, for one extra mode at transmitter 1, we can further align streams of interference over slots The incremental gain per slot is reduced when increases (see Fig 3) Our result reveals the fundamental benefit that can be obtained from reconfigurable antenna modes when there is no CSIT and 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 in frequency and spatial domains suffice regardless of the CSIT assumption VI CONCLUSION We derived the DoF region for the MIMO Z and FICs when perfect CSI is available at receivers, including 1) the ZIC with CSIT; 2) the ZIC and FIC without CSIT, but with reconfigurable antennas at the transmitters For both FIC and ZIC, when the number of antenna modes at the transmitter with the reconfigurable antennas is no less than the number of receive antennas at the corresponding receiver, the DoF region is maximized and no longer depends on the number of antenna modes Otherwise, each additional antenna mode can bring extra gain in the DoF region when for both FIC and ZIC, and when for FIC The incremental gain diminishes as increases The achievability schemes we designed for the reconfigurable antenna cases rely on time extension and joint beamforming and
14 KE AND WANG: DEGREES OF FREEDOM REGIONS OF TWO-USER MIMO Z AND FULL INTERFERENCE CHANNELS 3779 nulling over the time-extended channel Interestingly, they also bear a space-frequency coding interpretation We completely characterized the DoF regions for both ZIC and FIC when transmitter antenna mode switching is allowed Our result can specialize to previously known cases when there is no antenna mode switching by simply setting the number of antenna modes equal to the number of transmit antennas Our study reveals how the channel variation introduced by the extra antenna mode switching brings benefits in the sense of the DoF region ACKNOWLEDGMENT The authors would like to thank one anonymous reviewer of an early draft of our work for suggesting that we consider antenna mode associated with each antenna They would also like to thank the associate editor Syed Ali Jafar for several useful comments, which have been incorporated in the paper REFERENCES [1] A Carleial, A case where interference does not reduce capacity, IEEE Trans Inf Theory, vol21,no5,pp , May 1975 [2] A Carleial, Interference channels, IEEE Trans Inf Theory, vol 24, no 1, pp 60 70, Jan 1978 [3] H Sato, The capacity of the Gaussian interference channel under strong interference, IEEE Trans Inf Theory, vol 27, no 6, pp , Jun 1981 [4] AEGamalandMCosta, Thecapacityregionof a class of deterministic interference channels, IEEE Trans Inf Theory, vol 28, no 2, pp , Feb 1982 [5] RHEtkin,DNCTse,andHWang, Gaussian interference channel capacity to within one bit, IEEE Trans Inf Theory, vol 54, no 12, pp , Dec 2008 [6]XShang,GKramer,andBChen, Anewouter bound and the noisy-interference sum-rate capacity for Gaussian interference channels, IEEE Trans Inf Theory, vol 55, no 2, pp , Feb 2009 [7] V S Annapureddy and V Veeravalli, Sum capacity of MIMO interference channel in the low interference regime, IEEE Trans Inf Theory, vol 57, no 5, pp , May 2011 [8] A Motahari and A Khandani, Capacity bounds for the 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state information at transmitters, IEEE Trans Inf Theory, vol 58, no 2, pp , Feb 2012 [22] Y Zhu and D Guo, Isotropic MIMO interference channels without CSIT: The loss of degrees of freedom, in Proc Allerton Conf Commun, Control, Comput, 2009, pp [23] C S Vaze and M K Varanasi, The degrees of freedom regions of MIMO broadcast, interference, and cognitive radio channels with no CSIT 2009 [Online] Available: [24] S A Jafar, Exploiting channel correlations simple interference alignment schemes with no CSIT 2009 [Online] Available: abs/ [25] S A Jafar, Exploiting heterogeneous channel coherence intervals for blind interference alignment, Tech Rep, Center for Pervasive Communications and Computing, Univ California Irvine, CA, Oct 2011 [Online] Available: [26] T Gou, C Wang, and S Jafar, Aiming perfectly in the dark-blind interference alignment through staggered antenna switching, IEEE Trans Signal Process, vol 59, no 6, pp , Jun 2011 [27] Y Zhu and D Guo, The degrees of freedom of isotropic MIMO interference channels without state information at the transmitters, IEEE Trans Inf Theory, vol 58, no 1, pp , Jan 2012 [28] T Jiang, N Sidiropoulos, and J ten Berge, Almost-sure identifiability of multidimensional harmonic retrieval, IEEE Trans Signal Process, vol 49, no 9, pp , Sep 2001 Lei Ke received the BE degree and the MSc degree in Electronic Engineering and Information Science from the University of Science and Technology of China (USTC), Hefei, China in 2003 and 2006, and PhD in Electrical Engineering from Iowa State University Currently, he is a senior system engineer at Qualcomm His general interests include multiuser information theory, wireless communication and signal processing Zhengdao Wang (S 00 M 02 SM 08) received his BS degree in Electronic Engineering and Information Science from the University of Science and Technology of China (USTC), 1996, the MSc degree in Electrical and Computer Engineering from the University of Virginia, 1999, and PhD in Electrical and 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