arXiv:1812.07934v3 [eess.SP] 24 Apr 2019 erd h ult-fsrie(o)o ohsystems. both of (QoS) quality-of-service the degrade challenges radar of the the towards set as CSs new such radar a picture, between brings into sharing systems spectrum o communication used, irrespective and However, being atten- industry. technology the and captured the academia has both [6] of (SAS) tion [3], and Systems [5] (CRs) Access re- [4], Radios Spectrum (LSA/ASA) a Cognitive Access As Shared as systems [2]. Licensed/Authorized such communication [1], and technologies, (CSs) radar through systems between coexistence cellular communicatio sult, as commercial such and systems, (maritim etc.) spectrum radar weather, as of such vision surveillance, incumbents, the federal sp empowered between of has sharing dearth which the in resources, resulted tral have allocation spectrum static efitreec oe tteF SadQSo users. of QoS and BS FD residu the CS’s spectrum at PoD, power performa RS’s proposed power, interference in the transmit self trade-offs RS’s of certain to with utility corresponding its but the framework, maximising para show sharing non-centrality by monotonica its results the and RS technique exploiting Numerical PoD by proposed of of obtained relationship PoD the is increasing which particular, of bound, In interfere performance lower of the CS. the quality mitigate null-spa towards optimises of to CS, the used RS constraints BS at is from to power projection CS’s transmit subject detecti waveform and the of based RS, users probability of at MIMO investigated. (QoS) the service the technique is maximise to of frequency design proposed (PoD) and transceiver is and full time users downlink joint a and same multiple of a the serving comprising While at (BS) (CS), users station system uplink base cellular (FD) MIMO duplex a sys radar and (MIMO) (RS) multiple-output multiple-input a between K(-al{.ig,sdpbsa,t.ratnarajahg}@ed.a sudip.biswas, Edin (e-mail:{k.singh, of UK University Engineering, of School Communications, WHAce nvriy emn emi:[email protected] (e-mail: Germany University, Aachen RWTH QS,tasevrdsg,cne optimization. convex quality-of-ser radar, design, MIMO transceiver sharing, (QoS), spectrum (FD), duplex eerhCucl(PR)udrGatE/047/ n h U the u and Partnerships EP/N014073/1 Thematic DST-UKIERI-2016-17-0060. Initiative Grant Research under and Education (EPSRC) Council Research n prvn tfrpbiainwsPo.SiJin. Shi Prof. revi was the publication coordinating for editor it associate approving The and 2018. 03, August h orsodn uhro hsppri ehvSingh. Keshav is paper this of author corresponding The Informa Theoretical for Institute the with Institute is the Taghizadeh O. with are Ratnarajah T. and Singh, K. Biswas, S. hswr a upre yteUK niern n Physical and Engineering U.K. the by supported was work This aucitrcie aur 3 08 eie ac 4 201 24, March revised 2018; 03, January received Manuscript h rlfrto fwrls eie n evcsaogwi along services and devices wireless of proliferation The Terms Index Abstract oxsec fMM aa n DMIMO FD and Radar MIMO of Coexistence I hswr,tefaiiiyo pcrmsharing spectrum of feasibility the work, this —In ellrSseswt o Considerations QoS with Systems Cellular Mlil-nu utpeotu MM) full- (MIMO), multiple-output —Multiple-input ui iws ehvSnh mdTgiae,adTharmalin and Taghizadeh, Omid Singh, Keshav Biswas, Sudip .I I. n ievra hc a potentially can which vice-versa, and NTRODUCTION amu nefrne eeae by generated interferences harmful c.uk). drgatnumber grant nder ug,Edinburgh, burgh, wo hspaper this of ew inTechnology, tion wth-aachen.de). ;accepted 8; o Digital for Sciences K-India meter. vice tem nce nce ec- lly on ce th al e, n f S h rbe fsetu hrn ewe pulsed, a between sharing and spectrum (primary) an of radar radar problem rotating search a the between CS, sharing spectrum of possibility ytmi emte otasi signals communication transmit the to approach, permitted this between is In sharing CS. spectrum system and for radar [9] in rotating proposed was spectrum opportunistic approach an Similarly, [8]. in investigated eetsuis[9,[2,[3 S n C eeitgae i integrated were CCI and RSI [23] F [22], of [19], potential studies full recent the to harvest literature to Accordingly, user-cluste in techniques. and shown beamforming to been digital the through also (CCI) from suppressed has be interference transmission which the users, co-channel by (UL) the caused uplink users the (DL) downlink is the systems FD predicam mitigat of Another be by techniques. can beamforming caused and digital etc.) through [21] is imbalance, only non-linearit channel chains RSI I/Q (amplifier that receive noise, impairments This and phase extent hardware transmit behind. as the the known left also of to is nature mitigated domain non-ideal (RSI) be the digital SI can and residual SI a analog that the suggest can in FD [20] interference of techniques in performance advancements cellation the antennas recent dominates transmitting However, antennas, systems. the time receiving from its same ( leakage to the self-interference signal the in by However, resources generated systems [19]. spectrum [18], (HD) frequency the duplex and utilising half fully deployed by spectr currently the of double potentially efficiency can that technology emerging direc radar. any a the from minimising for CSs towards effectively by scheme for generated [17] interferences processing radar arbitrary in signal the MIMO proposed a was a Further, radar between CS. MIMO coexistence MIMO the a facilitate and null-space to a propose [12], [14] was in design in waveform considered for was technique (NSP) CS projection of MIMO presence a the and in operating clutter radar [14]–[1 MIMO in a exten while radar was Accordingly, (MIMO) radar multiple-output rotating multiple-input sy atta to the and coexisted probability, diversity the detection waveform at higher increase interference to mitigate null-space Furthermore, etc. to tems. as [11], utilised such beamforming each be techniques, [10], to can state waveforms by accessible of channel where (NSP) be the extent, projection to for an systems allows to both other which of [5], (CSI) [4], the information in LSA/ASA by issue, rad studied this utilised the alleviate was To of being system. operation communication not simultaneous and are prohibiting spectra hence radar, frequency and space nlgto hs hl h uhr n[]adesdthe addressed [7] in authors the while this, of light In eie pcrmsaig uldpe F)i another is (FD) full-duplex sharing, spectrum Besides a Ratnarajah gam 802 . 11 LN(eodr)was (secondary) WLAN fadol if only and if access ,in D, ring tion ded um SI) the ent 7]. ed in ar s- y, n d d 1 - , , 2 the design of transceivers for FD systems. However, most prior works consider perfect channel state information (CSI) at the transmitter, which is improbable to attain due to large training overhead and low signal-to-noise-ratio obtained after beam- forming. Hence, it is important to design robust transceivers, which can provide respectable performance for the co-existed systems even with imperfect or limited CSI estimates. Motivated by the above discussion, in this paper we consider a two-tier spectrum sharing framework, where a multiple- input-multiple-output (MIMO) radar system (RS) is the spec- trum sharing entity and a FD MIMO CS is the beneficiary.
Unlike existing literature [7]–[17], in this work we consider a Fig. 1: Spectrum coexistence between FD MIMO CS and MIMO RS. framework wherein, by utilising the spectrum of the MIMO RS, the FD CS’s BS serves multiple downlink (DL) and uplink (UL) users simultaneously in same time and frequency over a bandwidth of B Hz. The CS comprises of a FD resources. At this point, we would like to note that though MIMO BS, which consists of M transmit and N receive the consideration of FD mode at the CS provides better QoS 0 0 antennas, and J DL, and K UL users. All DL and UL users for the users, it also induces higher intensity of interference operate in half-duplex (HD) mode and each DL and UL towards the RS, which might affect its probability of detection user is equipped with N receive and M transmit antennas, (PoD). Hence, the requirements for the coexistence of a j k respectively. Furthermore, the MIMO RS is equipped with FD CS, with a MIMO RS requires revisiting. Further, the R receive and R transmit antennas. In the following sub- inclusions of imperfect CSI, hardware impairments and CCI R T sections, we provide details on the architecture of the two in the design of the proposed co-existed network result in a systems. rather arduous problem, which requires rigorous optimisation 1) FD MIMO CS: As shown in Fig. 1, HUL CN0×Mk and analysis and is addressed in this paper. Accordingly, to k ∈ and HDL CNj ×M0 represent the k-th UL and the j-th enable efficient spectrum sharing between a FD MU MIMO j ∈ CS and MIMO RS, we focus on the design of: DL channel, respectively. The SI channel at the FD BS and the CCI channel between the k-th UL and j-th DL users are • Beamforming matrices at the FD CS to mitigate inter- denoted as H CN0×M0 and HDU CNj ×Mk , respectively. ference towards the RS, while also providing data to its 0 jk Finally, the interference∈ channels from∈ the FD BS and kth users. In particular, we formulate an optimisation problem RR×M0 UL user to the RS are denoted as GRB C and UL for maximising the PoD of the RS for a given worst-case CRR×Mk UL∈ Cdk ×1 GRUk , respectively. Now, let sk (l) channel set, subject to the constraints of QoS at the UL ∈ DL ∈ and sDL(l) Cdj ×1 denote the transmitted symbols by the and DL users and powers at the UL users and the FD j the k-th UL∈ user and the FD BS, respectively, at time index l, BS. UL UL H with l = 1,...,L, such that E s (l) s (l) = I UL • Projection matrix at the MIMO RS to mitigate interfer- k k dk DL DL H UL DL ence towards the FD CS. In particular, all the interference and E s (l) s (l) = Ih DL . Here, d iand d j j dj k j channels from the RS to CS are combined into one and denote the data streams from the kth UL user and for the h i a null-space is created. The RS’s signal is then projected jth DL user, respectively, and L is the total number of time onto the null-space of the combined interference channel. UL samples used for CS’s communication. The symbols sk UL The following notations are used throughout this paper. DL UL CMk ×dk and sj are first precoded by matrices Vk DL ∈ Boldface capital and small letters denote matrices and vectors, DL CM0×dj and Vj , respectively, such that the signals respectively. While the transpose and the conjugate transpose transmitted∈ from the -th UL user and the FD BS at time T H k are denoted by ( ) and ( ) , respectively, A F and a 2 UL UL UL · · k k k k index l are given as xk (l) = Vk sk (l) and x0(l) = denote the Frobenius norm of a matrix A and the Euclidean J VDLsDL(l), respectively. Further, similar to [24], we a j=1 j j norm of a vector , respectively. denotes Kronecker product model the imperfections in the transmitter/receiver chains ⊗ I and denotes the statistical independence. The matrices N P(oscillators, analog-to-digital converters (ADCs), digital-to- ⊥0 and M×N denote a N N identity matrix and a M N analog converters (DACs), and power amplifiers) as an additive × E × zero matrix, respectively. The notations ( ) and tr( ) refer to Gaussian noise. In particular, we consider an additive white · A · A expectation and trace, respectively and diag( ) and vec( ) Gaussian term as “transmitter noise” (“receiver distortion”) at generate a diagonal matrix with the same diagonal element each transmit (receive) antenna, whose variance is ψ(υ) times A as and a mn 1 column vector obtained by stacking the the power of the undistorted signal at the corresponding chain. columns of the matrix× A on top of one another, respectively. 2) MIMO RS: Let Wi denote the interference channels shared by the MIMO RS with the CS, where i = 1,..., , II. SYSTEM MODEL DL CN0×RIT with = 1+ J . As shown in Fig. 1, WBR I DL CNj ×RT ∈ We consider the coexistence of a FD multi-user MIMO and Wj denotes the interference channels from CS with a MIMO radar system (RS) as shown in Fig. 1, RS’s transmitter∈ to CS’s BS and j-th DL user, respectively, where the CS operates in the spectrum shared by the RS with WDL, WDL W, j. Let s (l) CRT ×1 rep- { BR j } ⊆ ∀ R ∈ 3
RT ×1 RR×1 resents the transmitted symbol by the RS at time index l, Here, aT C and aR C denote the transmit and E H receive steering∈ vectors of RS’s∈ antenna array. Hereinafter, we with l = 1,...,LR, and sR(l) (sR(l)) = IRT . Here, assume that R = R = R and a (φ) = a (φ) = a (φ). LR is the total number of timeh samples usedi for the RS’s R T R T communication. For the ease of derivation, hereinafter we Accordingly, the i-th element at the r-th column of the matrix assume that the time duration of the RS’s waveform is the A(φ), can be written as same as the communication signals with LR = L [11]. 2π T 3) CSI acquisition: Acquiring CSI at both systems is Air (φ)= exp j [sin (φ); cos (φ)] (zi + zr) , (3) − λ important to ensure an interference free communication. In 1 2 this work, we assume that some amount of CSI, if not full is where zi = zi ; zi is the location of the i-th element of the available at the communication nodes1. For the CS, providing antenna array and λ is the wavelength of the carrier. Since, 3 its CSI to the RS is incentivised by the promise of zero the deterministic parameters αr and φ are unknown , we interference from the RS. On the other hand, it is more adopt the generalised likelihood ratio test (GLRT) [27], which challenging to obtain an accurate estimate of the CSI of the has the advantage of replacing the unknown parameters with RS at the CS, as the RS might not be willing to cooperate their maximum likelihood (ML) estimates for determining the with the CS owing to security concerns. Hence, it might not be probability of detection (PoD). Hereinafter, unless otherwise possible to obtain a full CSI at the CS and only partial CSI may stated, we drop the time index l in this section for notational be obtained through techniques such as blind environmental convenience. The sufficient statistic of the received signal can learning [25], realisation of a band manager with the authority then be found using matched filtering as [28] of exchanging CSI between the RS and CS [26], etc. Hence, to L model the imperfections caused due to imperfect CSI, in this ˆ 1 H √ 1 Y = yRxR = αr LP RA (φ) P + paper we will consider a norm-bounded channel estimation √L l=1 √L XL J error model for the links between the CS and RS. DL DL GRB V s + c0 (4) × l=1 j=1 j j X K X PECTRUM HARING UL UL UL H III. S S MIMO RS + GRUk Vk sk + ck + nR xR . k=1 As discussed in Section I, the considered sharing framework X requires both the RS and CS to work in tandem to mitigate From (4), the vectorization of Yˆ can be written as interference towards each other. Accordingly, in this section, ˆ we formulate the part played by the MIMO RS for efficient ˆy = vec Y spectrum sharing between the two systems. The goal of the = αr√LP Rvec (A (φ) P) RS is to map s (l) onto the null-space of W in order to avoid R L J 1 DL DL interference towards the CS. Accordingly, the symbols sR are + vec GRB V s + c0 √ l=1 j=1 j j first multiplied by a projector matrix P CRT ×RT , such that L ∈ K X X the signal transmitted by the RS is given by x (l)= Ps (l). UL UL UL H R R + GRU V s + c + nR x k=1 k k k k R 1) Target detection: In order to detect a target in the far X field, we apply a binary hypothesis and choose between two , αr√LP Rvec (A (φ) P)+ Ψ, (5) cases: 1) 0: No target but active CS and 2) 1: Both the target andH the CS are active. Further, in this workH we consider where Ψ is zero-mean, complex Gaussian distributed, and has only a single target with no interference from other sources, a non-white covariance matrix of but the CS in order to study the impact of CS’s interference H 2 P (χ˜ + σRIR)P 0 on PoD of RS. . ··· . χ = . .. . . (6) Accordingly, the hypothesis testing problem for an echo . . . 0 PH (χ˜ + σ2 I )P wave in a single range-Doppler bin of the RS can be written ··· R R 2 as in (1) as shown on the top of the next page, where 1 J H 2 2 H comprises of the discrete time signal vector received by RS at In the above, χ CR ×R and χ˜ = G VDL VDL ∈ RB j j an angle φ, and 1 comprises of the interference plus noise j=1 K H UL H H P H signals from the CS only. The symbols c0 and ck are defined DL DL UL UL +ψdiag Vj Vj (GRB) + GRUk Vk Vk in Section IV. In the above, PR denotes the transmit power k=1 H UL UL H of RS, αr indicates the complex path loss exponent of the (G ) + ψdiag V V P . However, the × RUk k k radar-target-radar path including the propagation loss and the GLRT in [28] was applied in the presence of white noise 2 coefficient of reflection, nR(l) 0, σRIR , and A (φ) only. Hence, to convert the covariance matrix in (6) into ∼ CN denotes the transmit-receive steering matrix defined as white, we apply a whitening filter ΠH , obtained after the 4 −1 −1 H T Cholesky decomposition of χ as χ = ΠΠ , with Π A (φ) , aR (φ) aT (φ) . (2) 3Due to the availability of CS’s CSI at the RS, we assume that the 1CSI estimation can be performed via the exchange of training sequences covariance matrix of the interference-plus-noise has been accurately estimated and feedback, and the application of usual CSI estimation methods [19]. by the RS. 2For simplicity, we ignore the interference from clutter and false targets. 4Note that χ and χ−1 are both positive-definite Hermitian matrices. 4
Interference from BS
J DL DL 1 : α√P RA (φ) PsR(l)+ GRB Vj sj (l)+ c0 H z j=1 }| { K y (l)= X UL UL UL (1) R + GRU V s (l)+ c +nR(l), 1 l L, k=1 k k k k ≤ ≤ X Interference from UL users : G J VDLsDL(l)+ c + K G VULsUL(l)+ cUL + n (l), 1 l L, 0 RB j=1 j j 0 k=1 RUk k k k R H | {z } ≤ ≤ P P being a lower triangle matrix. Now, the hypothesis testing 2) Projection matrix design at MIMO RS: In order to problem in (1) can be equivalently rewritten as mitigate the interference towards the CS, we need to design the projection matrix P in such as way that it projects onto : α √LP ΠH A (φ)+ ΠH Ψ , ˆy = H1 r R (7) the null space of the interference channels W. Accordingly, : ΠH Ψ , to find the null space of W, we perform its singular value (H0 decomposition (SVD), which can be given as W = RΩXH , where A (φ) = vec (A (φ) P). If p ˆy, αˆ , φ,ˆ and where R and X are unitary matrices and Ω is a diagonal r H1 p (ˆy, 0) denote the probability density function under 1 and matrix whose elements are the singular values of W. Now, , respectively,H and αˆ and φˆ indicate the ML estimationH let H0 r of αr and φ under hypothesis 1, which is expressed as ¯ H Ω , diag(¯ω1 ... ω¯p), (13) αˆ , φˆ = max p ˆy αˆ , φ,ˆ , then the GLRT can be r αr,φ | r H1 givenh byi where p , min(NBS+Users, RT ), ω¯1 > ω¯2 > . . . ω¯q = ω¯ = =ω ¯ =0 and 2 q+1 p H ˆ H ··· A φML ΠΠ ˆy ˜ Ω , diag(˜ω1 ... ω˜R ), (14) ln Lˆy φˆML = ≶ δ¯ , (8) T H 2 ΠH A φˆ ¯ ˜ ML where ω˜r , 0, r q, ω˜r , 1, r > q, with ΩΩ = 0. Using ∀ ≤ ∀ 2 the above definitions, the beamforming matrix for NSP can be H H −1 tr YˆP A φˆ ML χˆ H 1 ¯ defined as [14] = ≶ δ , ˜ H H H −1 P , XΩX . (15) tr A φˆML PP A φˆML χˆ H0
2 where δ¯ denotes the decision threshold and χˆ = χ˜ + σ IR. R Proposition 1: When RT (N0 + JNj ), the beamforming ˆ R ×R ≫ According to [29], the asymptotic statistic of Lˆy φML for matrix P C T T can be projected orthogonally onto the ∈ both the hypothesis is written as null-space of the entire CS involving the full set of interference channels W CNBS+Users×RT . 2 ∈ ˆ 1 : 2 (ρ) , Proof: The proof is given in Appendix A. ln Lˆy φML ∽ H X 2 (9) ( 0 : 2 , H X 2 IV. SPECTRUM UTILISING FD MIMO CS where 2 denotes the central chi-squared distributions with X 2 After establishing the role of the MIMO RS, in this section, two degrees of freedom (DoFs), 2 (ρ) is the non-central chi- squared distributions with two DoFs,X and ρ indicates the non- we formulate the part played by the FD MIMO CS in the central parameter given as proposed spectrum sharing framework. 1) Achievable rate: Using the spectrum of MIMO RS, the 2 H −1 ρ = αr LPRvec (A (φ) P) χ vec (A (φ) P) signals received at the FD BS and the j-th Dl user of the CS | | (10) 2 H H −1 at time index l can be written, respectively as =ΓRσRtr A (φ) PP A (φ) χˆ . K 2 2 UL UL UL where Γ = α LP /σ . The decision threshold δ¯ is set y0(l)= H x (l)+ c (l) + H0 (x0(l)+c0(l)) R | r| R R k k k according to a desired probability of false alarm P as k=1 FA X + e (l)+ P WDLs (l) +n (l), (16) ¯ F−1 0 R BR R 0 δ = X 2 (1 PFA) , (11) 2 − pInterference from RS −1 K where F 2 denotes the inverse central chi-squared distribution X2 DL DL | {z DU } UL UL function with two DoFs. The PoD for the MIMO RS can now yj (l)=Hj (x0(l)+c0(l))+ Hjk xk (l)+ck (l) be given as k=1 DL X DL DL + ej (l)+ PRWj sR(l) +nj (l), (17) −1 F 2 F PD =1 X (ρ) X 2 (1 PFA) , (12) − 2 2 − pInterference from RS
N DL Nj where F 2 is the non-central chi-squared distribution func- where, n (l) C 0 and n (l) C denote the additive X2 (ρ) 0 j| {z } tion with two DoFs. white Gaussian∈ noise (AWGN) vector∈ with zero mean and 5
2 DL 2 UL DL covariance matrix R0 = σ0 IN0 and Rj = σj INj at where Rk,min in (C.1) and Rj,min in (C.2) are the mini- the BS and the j-th DL user, respectively, and cUL(l) mum QoS requirements for the k-th UL and j-th DL users, k ∼ UL UL H UL UL respectively. Further, the constraints (C.3) and (C.4) regu- 0, ψ diag Vk Vk , ck (l) xk (l) and CN ⊥ late the transmit powers at the k-th UL user and the BS, eDL(l) 0,υ diag ΦDL , eDL(l) ˆuDL(l) are j j j j respectively and V = VUL, VDL denotes the set of all the transmit∼ CN and receive distortion at the k-th⊥ UL and j- k j transmit beamforming matrices. Since PD is a monotonically th DL user, respectively with ψ 1 and υ 1. Here, increasing function with respect to the non-central parameter ΦDL = Cov ˆuDL(l) and ˆuDL(≪l) = yDL(l)≪ eDL(l). j { j } j j − j ρ [29], we can equivalently reformulate the problem (P0) as The transmitter/receiver distortion c0(l)/e0(l) can be defined similarly. Like before, we drop the time index l henceforth, H unless otherwise stated. (P1.A) max tr A (φ) PP AH (φ) χˆ−1 (23) V Remark 1: Since the MIMO RS transmits NSP based DL subject to (C.1) (C.4) of (P0), signals, the interference terms in (16), (17) WBRsR(l) and − DL DL DL Wj sR(l) will effectively be WBRxR(l) and Wj xR(l) and hence, will be mitigated. However, they have been retained H H −1 here for tractability of the corresponding expressions, but will Lemma 2: A lower bound for tr A (φ) PP A (φ) χˆ be ignored in calculation of the numerical results. can be given as Remark 2: Since the codeword x0 and the SI channel H0 are known to the BS (its own transmitted signal), the term H x ϕR2 0 0 tr A PPH AH χ−1 (24) (φ) (φ) ˆ RAD 2 , in (16) can be cancelled out and thus, the remaining part H0c0 ≥ I + RσR can be treated as the RSI. However, the term H0x0 has been retained for tractability, but will be ignored in calculation of where ϕ = tr(PPH ) and IRAD is the total interference power the numerical results. from the CS to the MIMO RS, given as Lemma 1: The approximated aggregate interference-plus- K noise terms at the k-th UL and j-th DL user, can be given RAD UL UL H 5 I = tr GRU V V respectively as in (18) and (19) as shown on the top of the k=1 k k k next page. H X n+ψdiag VUL VUL (G )H Proof: This approximation can be easily proofed k k RUk J from (16) and (17) by considering ψ << 1 and υ << 1 DL DL H o + tr GRB Vj Vj (25) and ignoring the terms ψυ. j=1 X n DL DL H H Hence, from (16), (17) and Lemma 1, the achievable rate +ψdiag Vj Vj (GRB ) , for the -th UL user at the BS and the -th DL user can be k j o given as Proof: The proof is given in Appendix B. UL UL UL H UL UL UL H UL H From Lemma 2, the problem (P1.A) can be equivalently Rk = log2 Ik + Uk Hk Vk Vk Hk transformed into a interference minimisation problem as H −1 UUL UUL ΣUL UUL , (20) × k k k k (P1.B) min IRAD (26) DL H H H V RDL = log I + UDL HDLVDL VDL HDL j 2 j j j j j j subject to (C.1) (C.4) of (P0). H − 1 − UDL UDL ΣDLUDL . (21) × j j j j Now, for analytical simplicity, we write the QoS constraints 2) Beamformer design at FD MIMO CS: The main ob- (C.1) and (C.2) in terms of mean squared errors (MSE). To UL jective of the CS is to provide the cellular users with data by calculate the MSE, we first apply linear receive filters Uk UL N ×dDL ∈ utilising the spectrum of the RS, but without affecting the PoD CN0×dk and UDL C j j to y and yDL to obtain j ∈ 0 j of RS. Hence, we formulate the beamforming design problem the source symbols. Accordingly, the extracted k-th UL user’s as symbols at the BS and j-th DL user’s symbol from the BS can be given by (P0) max PD (22) V K subject to (C.1) RUL RUL , k =1,...,K, UL UL H UL UL UL UL k ≥ k,min ˆsk = Uk Hk Vk sk + ck (27) DL DL k=1 (C.2) Rk Rj,min, j =1, . . . , J, ≥ J X H DL DL DL UL UL +H0 Vj sj + c0 + e0 + WBRsR + n0 , (C.3) tr Vk Vk Pk, k =1,...,K, j=1 ≤ J X J n DL o DL H DL DL H DL DL DL (C.4) tr Vj Vj P0, ˆsj = Uj Hj Vj sj + c0 (28) j=1 ≤ j=1 X n o K X 5 ΣUL ΣDL Note that approximation of k and j is a practical assumption [24]. DU UL UL UL DL DL DL The values of ψ and υ are much lower than 1. However, their values might + Hjk Vk sk + ck + ej + Wj sR + nj . not be negligible under a strong SI channel [24]. k=1 ! X 6
K H H K H H ΣUL HULVUL VUL HUL + ψ HULdiag VUL VUL HUL k ≈ j6=k j j j j j=1 j j j j J X DL DL H XDL DL H H DL DL H 2 + H0 V V + ψdiag V V H0 + PR W W + σ0 IN0 j=1 j j j j BR BR J K X DL DL H H UL UL UL H UL H +υ diag H0V V H0 + υ diag H V V H , (18) j=1 j j j=1 j j j j J J DL X DL DL DL H DL H X DL DL DL H DL H Σj Hj Vi Vi Hj + ψ Hj diag Vi Vi Hj ≈ i6=j i=1 K X DU UL UL H XUL UL H DU H DL DL H 2 + H V V + ψdiag V V H + PR W W + σ IN k=1 jk k k k k jk j j j j K H H J H H +υX diag HDU VUL VUL HDU + υ diag H DLVDL VDL HDL . (19) k=1 jk k k jk i=1 j i i j X X
TABLE I: Simplification of Notations Hence, the MSE for k-th UL and j-th DL users can be H HUL UL UL H UL DL respectively given as ij j ,i ∈S , j ∈S ; 0,i ∈S , j ∈S ; HDU ,i ∈SDL, j ∈SUL; HDL,i ∈SDL, j ∈SDL H ij i UL UL UL UL UL DL Ek ( U , V )= Uk Hk Vk IdUL (29) GR GRU ,j ∈S ; GRB ,j ∈S { } { } − k j j W WDL,i ∈SDL; WDL,i ∈SDL H H H i i BR BR UL UL UL UL UL UL UL DL DL Uk Hk Vk IdUL + Uk Σk Uk , ni n0,i ∈S ; n ,i ∈S × − k i N˜ (M˜ ) N (M ) ,i ∈SUL; N (M ) ,i ∈SDL DL DL H DL DL i i 0 i i 0 E ( U , V )= U H V IdDL (30) j { } { } j j j − j H DL H DL DL DL H DL DL U H V I DL + U Σ U , UL DL j j j dj j j j link, i , and the interference power at the × − ∈ S RADS ∪ S MIMO RS, I can be rewritten, respectively as where U = UUL, UDL denotes the set of all receive beam- k j H H H H forming matrices. Note that, for fixed transmit beamforming Ei = Ui HiiVi Idi Ui HiiVi Idi + Ui ΣiUi, − − (37) matrices, the optimal receive beamforming matrix at the BS for the k-th UL user and at the j-th DL user are MMSE receivers, where which can be expressed as (31) and (32) shown on the top of H H H H the next page. Accordingly, substituting (31) and (32) in (29) Σi = Hij Vj Vj Hij +ψ Hij diag Vj Vj Hij −1 −1 j∈S,j6=i j∈S and (30) and using the property (A + BCD) = A X X −1 −1 −1 −1 −1 − H H H A B DA B + C DA [30], the MSE matrices for +υ diag Hij Vj Vj Hij + PR Wi (Wi) j∈S the k-th UL and j-th DL users can be written as X2I (38) + σi N˜i , UL Ek,MMSE ( V ) (33) { } RAD H H H H H −1 −1 I = tr G V V + ψdiag V V G . (39) = IUL + VUL HUL ΣUL HULVUL , Rj j j j j Rj k k k k k k j∈S X n o Now, using (35), (36), the above simplified notations, and EDL ( V ) (34) epigraph method, [31] the optimisation problem (P1.B) can j,MMSE { } −1 be equivalently reformulated as DL DL H DL H DL −1 DL DL = Ij + Vj Hj Σj Hj Vj . (P1.C) min Γ (40) V,Γ From (33)–(34) and (20)–(21), we obtain the relation between subject to (C.1) IRAD Γ rate and MSE as ≤ −1 UL UL UL UL −1 (C.2) log2 (Ei,MMSE ( Vi )) Ri,min, i , Rk = log2 EMMSE,k ( V ) ; (35) { } ≥ ∈ S { } −1 DL DL DL DL −1 (C.3) log2 (Ei,MMSE ( Vi )) Ri,min, i , Rj = log2 EMMSE,j ( V ) . (36) { } ≥ ∈ S { } H UL (C.4) tr V iVi Pi, i , Before proceeding to the next section, we simplify the ≤ ∈ S H (C.5) tr ViVi P0, notations by combining UL and DL channels similar to [18]. i∈SDL ≤ UL DL Let the symbols and denote the set of K UL and X J DL channels, respectively,S S and the channels from RS to BS V. ROBUST BEAMFORMER DESIGN AT FD MIMO CS and the set of J DL channels from RS to DL users are denoted In order to design a more practical and robust system, in DL DL X X X by BU and BR , respectively. Now, denoting Vi , Ui , di this section, we assume that perfect CSI knowledge of all the andSΣX , X S UL,DL as V , U , d and Σ , respectively, concerned channels is unavailable at the CS6. By considering i ∈{ } i i i i and expressing Hij , GR , ni, and receive (transmit) antenna j 6Note that this assumption doesn’t obtrude with the assumption that RS numbers N˜i M˜ i as shown in Table I, the MSE of the i-th has full CSI knowledge of the CS’s channels and adheres to Section II.3. 7
⋆ H H H H −1 UUL = arg min tr EUL = VUL HUL HULVUL VUL HUL + ΣUL ; (31) k UL k k k k k k k k Uk ⋆ H H H H −1 UDL = arg min tr EDL = VDL HDL HDLVDL VDL HDL + ΣDL . (32) j DL j j j j j j j j Uj
7 the worst-case (norm-bounded error) model [32], the channel where zij and ι are defined as uncertainties can be constructed as H H vec Bi Ui HiiVi Idi Hij ij = H˜ ij + ∆ij : ∆ij F δij , (41) T H H − V B U vec (Hij ) ∈ H k k ≤ j ⊗ i i j∈S,j6=i n o T H H G = G˜ + Λ : Λ θ, j , (42) √ Ξ V B U H (T ) Rj j Rj F ψ ( ℓ j ) i i vec ( ij ) ℓ∈D ∈ G k k ≤ ∈ S ⊗ j j∈S zij = , n o j T H H k √υ Vj Bi Ui Ξℓ vec (Hij ) (R) ℓ∈Di where the nominal values of the CSI are denoted by H˜ ij and ⊗ j∈S T H H ˜ j PR I Bi Ui vec (Wi) k Gj , while ∆ij and Λj represent the channel error matrix and R ⊗ σ vec BH UH δij and θj express the uncertainty bounds. i i i d×d T Lemma 3: Assume that F C is a positive semi-definite V IT vec (Gj ) ∈ j ⊗ j∈S matrix and F 1. Then the maximisation of the function ι= T . √ Ξ V I G (T ) | | ≤ −1 ψ ( ℓ j ) T vec ( j) ℓ∈D f (Q) = tr (QF) + log Q + d is equivalent to log F , ⊗ j j∈S i.e., − | | j k (46)
−1 max f (Q) = log F , (43) Here, Ξℓ represents a square matrix with zero as elements, Q∈Cd×d,Q≻0 except for the ℓ-th diagonal element, which is equal to 1, (R) (T ) opt and and denote the set 1 N˜ and 1 M˜ , where the optimum value Q = F−1. Dj Dj { ··· j} { ··· j } H respectively. Now, applying Lemma 3 and decomposing Q = BiBi , the problem (P1.C) under channel uncertainties can be expressed Proof: The proof is provided in Appendix C. as From Lemma 4, the constraint (C.2) of (P2) can be rewritten as (P2) min Γ (44) V,Γ H RAD min max tr Bi EiBi + 2 log Bi + di subject to (C.1) I Γ, GRj j , ∆ {Ui,Bi} − | | ≤ ∀ ∈ G H UL UL (C.2) min max tr Bi EiBi + 2 log Bi + di log(2)Ri,min, i , (47) ∆ {Ui,Bi} − | | ≥ ∈ S UL UL (C.2a) di + 2 log Bi λij log(2)Ri,min, i , j∈S ≥ ∈ S | |−UL UL H log(2)Ri,min, i (C.3) min max tr Bi EiBi + 2 log Bi + di ≥ P ∀ ∈ S (48) ∆ {Ui,Bi} − | | ˆ 2 ⇒ (C.2b) max ˆzij + Zij vec (∆ij ) 2 λij , DL DL ∆ k k ≤ log(2)Ri,min, i , UL ≥ ∈ S ∆ij F δij , i, j , H UL k k ≤ ∀ ∈ S (C.4) tr ViVi Pi, i , ≤ ∈ S H (C.5) tr ViVi P0, ˆ i∈SDL ≤ where ˆzij and Zij are defined as X di×di where ∆ = ∆ij : (i, j) and Bi C , i , is a { ∀ } ∈ ∈ S vec BH UH H˜ V I weight matrix. Due to the constraint (C.1) in (44) and the i i ii i − di inner maximization in (C.2) and (C.3), the problem (P2) is T H H ˜ Vj Bi Ui vec Hij intractable. To make the problem tractable, we consider the ⊗ j∈SUL,j6=i j k following max-min inequality [33] T H H ˜ √ψ (ΞℓVj ) Bi Ui vec Hij T ˆz = ⊗ ℓ∈D( ) ij j j∈S j k min max ( ) max min ( ) log(2)Ri,min,i . T H H ˜ √υ Vj Bi Ui Ξℓ vec Hij R ∆,Λ {Ui,Qi} · ≥ {Ui,Qi} ∆,Λ · ≥ ∈ S ( ) ⊗ ℓ∈Di j∈S (45) j T H H k PR IR Bi Ui vec (Wi) ⊗ H H σivec B U H RAD i i Now, we convert tr Bi EiBi and I into vector forms, (49) given in the following lemma. H RAD Lemma 4: tr Bi EiBi and I can be written in the H 2 RAD 2 7 UL form of vectors as tr B E B = z and I = ι , For sake of simplicity, we consider M˜ = M0 = Mi, i ∈S . i i i k ij k2 k k2 8
VT BH UH i i i where B = R dj + M˜ j , VT B⊗H UH j∈S j ⊗ i i j∈SUL,j6=i T H H P T ˜ √ Ξ V B U (T ) V IT vec Glj ψ ( ℓ j ) i i ℓ∈D j ˆ ⊗ j j∈S ⊗ j∈S Zi = . ˜ι = , j T H H k j T k √υ V B U Ξℓ (R) √ Ξ V I G˜ j i i ℓ∈D ψ ( ℓ j ) T vec lj (T ) ⊗ i j∈S ⊗ ℓ∈D j j∈S j 0 ˜ ˜ ˜ k j k diNi×NiM VT I 0 j R j∈S diN˜i×N˜iM˜ ⊗ ιΛ = T vec (Λ) . (60) (50) √ Ξ V I (T ) ψ ( ℓ j ) R ℓ∈D ⊗ j j∈S Now, using Lemma 5 from Appendix D, we relax the semi- j k EΛ infiniteness of the constraint (C.2b) in (56) and then apply Lemma 6 from Appendix D, to convert the norm constraint of Finally,| the constraint (C.1){z of (P2) is equivalently} rewritten (C.2b) into a LMI form as as H Γ η ˜ι 01×RM˜ λ ˆzH − (C.2b) ij ij ˜ι IB θEΛ 0, (61) H − ˆzij Ididj 0 E I RM˜ ×1 θ Λ η RM˜ H − 0 vec (∆ )H Zˆ η 0. (62) + ij ij 0 . ≥ ˆ " Zij vec (∆ij ) 0didj ×didj # Using the relaxed LMIs in (53), (57) and (61), the prob- (51) lem (P2) can be written as a SDP problem, given as By choosing (P3) min Γ (63) V,Γ,U,B,λ,ǫ≥0,η≥0 λ ˆzH H ˜ιH 0 A ij ij P 0 Zˆ Γ η 1×RM˜ = , = N˜iM˜ ×1, ij , − ˆzij Ididj subject to (C.1) ˜ι IB θEΛ 0, h i H − X = vec (∆ij ) , Q = 1, 0 , (52) 0RM˜ ×1 θEΛ ηIRM˜ − 1×dij − (C.2a) d + 2 logB λ log(2)RUL , i UL, and applying Lemma 5, the constraint (C.2b) of (56) is i | i|− ij ≥ i,min ∀ ∈ S equivalently rewritten as Xj∈S H λij ǫij ˆzij 01×N˜ M˜ H − i λ ǫ ˆz 0 ˜ ˜ ˆ ij ij ij 1×NiM (C.2b) ˆzij Ididj δij Zij 0, − UL H − ˆzij Id d δij Zˆij 0, i, j , (53) i j 0 ˜ ˜ δ Zˆ ǫ I ˜ ˜ H − ∀ ∈ S NiM×1 ij ij ij NiM ˆ − 0 ˜ ˜ δij Zij ǫij I ˜ ˜ UL NiM×1 − NiM i , j , UL ∀ ∈ S ∈ S ǫij 0, i, j , (54) (C.3a) d + 2 log B λ log(2)RDL , i DL, ≥ ∀ ∈ S i | i|− ij ≥ i,min ∀ ∈ S j∈S where ǫ = ǫij : (i, j) . Similar to the transformation of the X { ∀ } ˆzH 0 constraint (C.2) of (P2), the constraint (C.3) of (P2) can be λij ǫij ij 1×N˜iM˜ − ˆ equivalently rewritten as (C.3b) ˆzij Ididj δij Zij 0, H − 0 Zˆ I H N˜iM˜ ×1 δij ij ǫij N˜iM˜ min max tr Bi EiBi + 2 log Bi + di − ∆ {Ui,Bi} − | | i DL, j , DL DL ∀ ∈ S ∈ S log(2)Ri,min, i , (55) (C.4) vec (V ) 2 P , i UL, ≥ ∈ S k i k2 ≤ i ∈ S (C.3a) d + 2 log B λ 2 i i j∈S ij (C.5) vec (Vi) i∈SDL 2 P0, | |−DL DL k⌊ ⌋ k ≤ log(2)Ri,min, i ≥ P ∀ ∈ S (56) where B = Bi, i and λ = λij , i, j . ˆ 2 { ∀ ∈ S} { ∀ ∈ S} ⇒ (C.3b) max ˆzij + Zij vec (∆ij ) 2 λij , Note that the optimisation problem (P3) is not jointly ∆ k k ≤ DL ∆ij F δij , i, j , convex over the optimisation variables V, U and B. However, k k ≤ ∀ ∈ S it is separately convex over each of the variables. Therefore, where(C.3b) is expressed as we adopt an alternating algorithm to solve the problem. This H alternating minimisation process is continued until a stationary λij ǫij ˆz 0 ˜ ˜ ij 1×NiM point is obtained, or a pre-defined number of iterations is − ˆ DL ˆzij Ididj δij Zij 0, i , j , (57) reached. In the following section, we provide details on the H − ∀ ∈ S ∈ S 0 ˜ ˜ δij Zˆ ǫij I ˜ ˜ spectrum sharing algorithm, including the alternating optimi- NiM×1 − ij NiM ǫ 0, i DL, j . (58) sation of the above SDP problem. ij ≥ ∀ ∈ S ∈ S In similar way, we also transform the constraint (C.1) of (P2) VI. SPECTRUM SHARING ALGORITHM as In this section we summarise the roles played in spectrum H H sharing by the RS and CS in the form of algorithms. While Γ ˜ι 0 ιΛ + 0, (59) Algorithm 1 presents the role played by the RS, Algorithm 2 ˜ι IB ιΛ 0B×B 9
Algorithm 1: Spectrum Sharing Phase at RS of diagonal blocks P is SUL ( S + 1) + SDL S + 1. I. Phase 1 [Initial Phase]: | | | | WDL DL WDL W The constraints (C.2b) and (C.3b) create the blocks of size A: Obtain CSI of { i , ∀i ∈S , BR}⊆ through feedback DL UL from CS. aij = 2(NiM˜ + didj + 1) , i , j and II. Phase 2 [Null-space Projection Phase]: a = 2(N˜ M˜ + d d + 1) , i ∈ S UL, j ∈ S DL, re- A: Perform SVD of W, which is obtained from Phase 1. ij i i j ∈ S ∈ S B: Construct Ω¯ and Ω˜ according to (13) and (14) spectively. The size of blocks due to the constraint (C.1) is C: Design the projection matrix P based on (15). a = 2(B + RM˜ + 1), while the constraints for UL power x Ps C: Output: Perform NSP by transmitting waveform R = R. in (C.4) and the BS power in (C.5) make the blocks of size UL UL DL a = Md˜ +1, i and a = M˜ DL d +1, Algorithm 2: Spectrum Sharing Phase at CS i i ∈ S i i∈S i I. Phase 1 [Initial Phase]: respectively. Furthermore, the number required to compute the A: Obtain partial CSI of G , j UL, G . ˜ P { RUj ∀ ∈S RB } unknown variables is n = i∈S 2Mdi +2 +2, where the B: Set minimum QoS requirements for UL and DL users: RUL , ˜ |S| i,min term i∈S 2Mdi correlates with the real and image parts and RDL . P i,min of Vi and the remaining terms are due to the additional V[n] U[n] B[n] C. Initialize , and . slackP variables. Similarly, the number of arithmetic operations D. Set iteration number n = 0, maximum iteration number = nmax. II. Phase 2 [Beamforming Design Phase (Alternating Approach)]: required for Ui,i and Bi,i can be calculated. U[n−1] B[n−1] V[n] ∈ S ∈ S A: n ← n + 1. For fixed i and i , update , ∀i ∈S by solving problem (P3). [n] U[n−1] B: Update Bi , i ∈S by solving the problem (P3) for fixed i V[n−1] VII. NUMERICAL RESULTS and i . U[n] V[n−1] C: Update i , i ∈S by solving the problem (P3) for fixed i B[n−1] and i . In this section, the performance of the proposed two-tier D: Repeat steps II.A – II.C until convergence or n = nmax. coexistence framework between a FD MU MIMO CS and a E: Output: Optimal transceivers: {U⋆, V⋆}. MIMO RS is analysed with the help of computer simulations9 under consideration of QoS of cellular users. The maximum illustrates the role of the CS in the proposed 2 tier spectrum number of iterations is set as 50 with a tolerance value of sharing framework8. In particular, Algorithm 1 performs NSP 10−4. The initialisation points are selected using right singular towards the interference channels of the CS, thereby cancelling matrices initialisation [35] and the results are averaged over the interference from RS to CS. Alternatively, Algorithm 2 100 independent channel realisations. produces the optimal beamforming matrices at the transmitters 1) Simulation Setup: To model the CS, we consider small and receivers of the CS, to minimise the interference from cell deployments under the 3GPP LTE specifications. The the CS to RS (equivalently maximises the PoD of RS), while motivation behind this are: 1) due to low transmit powers, maintaining a particular QoS for the cellular users. Note that short transmission distances and low mobility, small cells are Algorithm 2 is iterative in nature and solves a SDP problem considered suitable for implementation of FD technology [23], in each iteration, which makes it computationally intensive. and 2) FCC has proposed the use of small cells in the 3.5 Below we provide some qualitative analysis on the complexity GHz band for spectrum sharing [2]. Accordingly, a single of Algorithm 2. hexagonal cell of radius r = 40 m is considered, where the Computational complexity of Algorithm 2: The computa- FD BS is located at the centre of the cell and a MIMO RS tional complexity mainly depends on the number of arithmetic is located 400 m away from the circumference of the cell. operations required to process Phase 2 of Algorithm 2. In The number of UL and DL users is set as K = J = 2 and particular, a SDP problem is solved in Phase 2 in three steps, each user, equipped with N antennas is randomly located in i.e., Step II.A to Step II.C. Hence, for comparison we first the cell. For simplicity, we consider M0 = N0 = N = N˜. consider a standard real-valued SDP problem as Next, to model the path loss in the CS, we consider the close-in (CI) free space reference distance path loss model min cT x (65) x∈Rn as given in [36]. The CI model is a generic model that n describes the large-scale propagation path loss at all relevant subject to A0 + xiAi 0, and x 2 X, i=1 k k ≤ frequencies (> 2 GHz). This model can be easily implemented where Ai is a symmetricX block-diagonal matrix. If P is the in existing 3GPP models by replacing a floating constant diagonal block of matrix A of size a a , l =1,...,P , then with a frequency-dependent constant that represents free space i l × l the number of arithmetic operations required to solve (65) is path loss in the first meter of propagation and is given as upper-bounded by [34] PL(f, d) = PLF (f, d0)+10αc log10 (d/d0)+ σ,d>d0. Here, d is a reference distance at which or closerX to, the path P 1/2 P P 0 2 2 3 loss inherits the characteristics of free-space path loss PLF . (1) 1+ al n n + n al + al . (66) O Further, f is the carrier frequency, αc is the path loss exponent, l=1 ! l=1 l=1 ! X X X d is the distance between the transmitter and receiver and Thus, using (66), we can compute the total number of arith- Xσ is the shadow fading standard deviation. We consider d0 =1 metic operations required to find the optimal Vi, Ui, and Bi in Algorithm 2. For instance, in order to find Vi, the number 9For reference, the numerical results are obtained using MATLAB R2016b 8Note that both algorithms are processed within the same coherence time on a Linux server with Intel Xeon processor (16 cores, each clocked at 2 interval. GHz) having 31.4 GiB of memory. 10
2 4 6 8 10 12 14 16 18 20 6 17 15 10 Antenna number 10 4 16 CS 10 14 RS 10 2 Users
RS [dBm] 15 0 10 1013 -2 1014 -4 1012 1013 -6 1011 -8 1012
-10 10
Number of complex multiplications 10 1011 Number of complex multiplications
Interference power from CS -12
9 -14 1010 10 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 Iteration numbers Number of users in the CS
Fig. 2: Convergence of Algorithm 2. Fig. 3: Complexity of Algorithm 2. m, B = 100 MHz, and carrier frequency =3.6 GHz10. 2) Simulation Results: To enable spectrum sharing, RS The estimated channel gain between the BS and kth UL user deploys Algorithm 1 to null the interference from RS CS, → can be described as H˜ UL = ℘ULHˆ UL, where Hˆ UL denotes i.e., WixR = 0 ,i , which paves the way for the CS to k k k k use the RS’s spectrum.∈ S This is simultaneously followed by the small scale fading followingq a complex Gaussian distribution UL (−A/10) deployment of Algorithm 2 at CS, which maximises the PoD with zero mean and unit variance, and ℘k = 10 , A 11 ∈ of RS by suppressing the interference from CS RS, while LOS, NLOS denotes the large scale fading consisting of → {path loss and shadowing.} LOS and NLOS are computed based also providing QoS to its users. In the following examples, we illustrate the performance of both RS and CS under a spectrum on a street canyon scenario [37]. The parameter αc for LOS and NLOS are set as 2.0 and 3.1, respectively, while the sharing scenario utilising the proposed algorithms. Due to the value of shadow fading standard deviation σ for LOS and iterative nature of Algorithm 2, we begin by showing 1) its NLOS are 2.9 dB and 8.1 dB, respectively. Similarly, we evolution in Fig. 2, i.e., its convergence and 2) its complexity define the channels between UL users and DL users, between analysis in Fig. 3 in terms of complex multiplications required BS and DL users, between BS and RS, and between UL with respect to (w.r.t) increasing number of antennas at CS and users and RS. To model the SI channel, the Rician model RS and users in the CS. It can be seen from Fig. 2 that the cost RAD in [20] is adopted, wherein the SI channel is distributed as function, i.e., I monotonically decreases and converges KR 1 after 25 30 iterations. Further, in Fig. 3, the axes in red (left H˜ 0 Hˆ 0, IN IM , where KR is − ∼ CN 1+KR 1+KR 0 ⊗ 0 and bottom) represent the complexity w.r.t. number of users in ˆ 12 the Rician factorq and H0 is a deterministic matrix . Unless the CS, while the axes in black (right and top) represent the otherwise stated, we consider the following parameters for the complexity w.r.t. number of antennas at RS and the FD BS. It CS and RS. For CS: thermal noise density = 174 dBm/Hz, can be seen that the computational complexity of Algorithm − noise figure at BS (users) 13(9) dB, N˜ = 2, ψ = υ = 70 2 increases as the number of users or antennas are increased. UL DL 5 − dB, δ = θ = 0.1, Ri,min = Rj,min =5.0 10 bps, Pi = 5 Hence, it is imperative that the processing of Algorithm 2 be ×13 dB, P0 = 10 dB and CCI cancellation factor =0.5. For RS: handled centrally at the FD BS, which inadvertently has high −5 R = 8, PFA = 10 , velocity of target = 782 knots, and end computing capabilities. distance of target from RS = 300 m. Next, we analyse the impact of spectrum sharing on the 10The framework presented in this paper is not limited to any particular proposed two-tier model. In particular, we quantify the level frequency band and can also be utilized in other spectrums proposed for of interference towards the RS for various levels of QoS that sharing around the world, such as 2-4 GHz in the UK, 2.3-2.4 GHz in Europe, the CS can support by operating in FD mode. Accordingly, etc., albeit certain changes in frequency dependent path loss, line of sight propagation parameters, etc. in Fig. 4 we show the interference power generated from 11LOS=Line-of-sight and NLOS=Non-line-of-sight. the CS towards the RS as a function of transmitter/receiver 12 ˜ For simplicity, we take KR = 1 and the matrix H0 of all ones for all (ψ/ν) distortion values for two different QoS requirements of simulations [22]. the cellular users. Note that the transmitter/receiver distortion 13It is essential to isolate UL and DL users in a FD system through smart channel assignments at a stage prior to the precoder/decoder design, so that values reflect the amount of RSI left in the FD system. It the CCI is mitigated. This can be done by clustering the users into different can be seen from the figure that, as the RSI cancellation groups through techniques, such as game theory, where the users with very capability of the FD system increases, the interference power strong CCI are not placed in the same group. In this work, the value of CCI cancellation factor represents the following: 0 → 100% cancellation and generated by the CS towards the RS decreases. This can be 1 → 0% cancellation. explained due to the fact that, when RSI is more, the CS 11
10-5 -11.0 4.9 = = -50 dB QoS=5.0 105 bps/user = = -90 dB QoS=5.0 106 bps/user -11.5 4.8
-12.0 4.7
-12.5 4.6 10-5 4.64
-13.0 4.5 4.635
4.63 0.6 dB
-13.5 4.4 4.625 Interference power towards RS (dBm) Interference power towards RS [dBm] 1.98 2 2.02 2.04 2.06 2.08 106 -14.0 4.3 -90 -80 -70 -60 -50 -40 -30 -20 0 0.5 1 1.5 2 2.5 3 3.5 4 Transmitter/Receiver distortion ( = ) [dB] QoS (bps/user) 106
Fig. 4: Interference power towards RS vs. RSI at CS. Fig. 5: Interference power towards RS vs. QoS per user at CS.
1.0 1.0 P with spectrum sharing, R=8 D P without spectrum sharing, R=8 0.9 D 0.9 P with spectrum sharing, R=12 D P without spectrum sharing, R=12 0.8 D 0.8 P =15 dB R 0.7 0.7 P =5 dB R ) 0.6 ) 0.6 d D 0.95 0.5 0.5 0.9
1.5 dB PoD (P PoD (P 0.4 0.4 0.85 4.5 dB 0.3 0.3 0.8 0.2 P with spectrum sharing 0.2 D 10 15 20 25 P without spectrum sharing D 0.1 0.1
0 0 -30 -20 -10 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 Transmit power (P ) [dB] Probability of false alarm (P ) R FA
Fig. 6: PoD of MIMO RS vs. MIMO RS’s transmit power. Fig. 7: PoD of MIMO RS vs. probability of false alarm at MIMO RS. has to use higher transmission power to overwhelm the RSI determine its PoD. To detect a target in the far-field, the and maintain the QoS of the users, which results in more RS transmits NSP waveforms generated in Algorithm 1 and interference towards the RS. More importantly, it can also be estimates parameters φ and αr from the received signal seen that the minimum guaranteed QoS for each user can be that also includes IRAD (obtained from Algorithm 2). As a increased by a factor of 10 or more if the RS increases its benchmark, we also simulate the scenario without spectrum tolerance threshold for interference× temperature by only 0.6 sharing by generating orthogonal waveforms at the RS and dB. Similarly, in Fig. 5 we show the effect of users’ QoS setting IRAD =0. This scenario relates to the case when the for two different RSI values. It can be seen that as the QoS CS’s BS is unable to provide its users with any connectivity requirements of cellular users increases, the interference from due to lack of spectrum resources. CS towards RS increases linearly. This can be explained due to Accordingly, in Fig. 6, the PoD of the MIMO RS w.r.t. RS’s the fact that, to provide higher QoS to the users, Algorithm 2 transmit power is shown. Here, we consider two scenarios: 1) ensures transmission at higher power at the CS. Nevertheless, R =8 (straight lines) and 2) R = 12 (dashed lines). It can be Algorithm 2 also ensures that for any specific QoS on the seen that for fixed P , in order to achieve a particular P the axis of Fig. 5, the corresponding axis value represents FA D x y RS needs more power (to create NSP waveforms and withstand the− minimum interference that can be− generated from CS interference from CS to enable spectrum coexistence) than towards the RS. the case without spectrum sharing scenario. However, it can After quantifying the interference towards RS, we now be seen that the RS needs more power when R = 8 than 12
R = 12 to achieve similar performance. This is because, which proves that P is an orthogonal projection matrix onto while the number of antennas at the CS (BS and users) the null-space of W. are fixed, increasing the RS’s antennas means that it has more degrees of freedom for reliable target detection and simultaneously nulling out interference towards the CS. This APPENDIX B proves that large antenna arrays can be used at the RS to PROOF OF LEMMA 2 facilitate spectrum sharing without any significant degradation Since A (φ) PPH AH (φ) and χˆ are positive-definite, we in RS’s performance. have Finally, in Fig. 7, we plot PoD for various PFA and PR. Similar to the previous figure, the PoD of the RS is better tr A (φ) PPH AH (φ) χˆ−1χˆ , at high PR and small PFA when the RS is not sharing its H H −1 spectrum. However when both P and P are small, PoD tr A (φ) PP A (φ) χˆ tr (χˆ) . (B.1) R FA ≤ of RS for NSP waveforms is quite comparable to the case without spectrum sharing. Now, a lower bound for tr A (φ) PPH AH (φ) χˆ−1 follows as VIII. CONCLUSION H H A two tier spectrum sharing framework was proposed, tr A (φ) PP A (φ) tr A (φ) PPH AH (φ) χˆ−1 . where 1) transceivers were jointly designed at a hardware ≥ tr (χˆ) impaired FD CS under imperfect CSI considerations 2) null- (B.2) space based waveforms were designed at MIMO RS under per- fect CSI considerations. In particular, the robust optimisation Using the property tr(A1 + B1) = tr(A1) + tr(B1), where H in the CS led to an intractable problem, which was transformed A1 and B1 are square matrices and ϕ = tr(PP ), we can into an equivalent tractable semidefinite programming prob- rewrite (B.2) under RT = RR = R to obtain the desired lem. Next, algorithms were proposed to suppress interference result. at both systems, thus maximising the PoD of RS and also maintaining a specific QoS for each user in CS. Finally, APPENDIX C numerical results were used to demonstrated the effectiveness PROOF OF LEMMA 4 of the proposed algorithms, albeit certain trade-offs in RS’s H transmit power, PoD, and QoS of the users. In particular, it To prove this lemma, we first construct tr Bi EiBi us- was seen that to facilitate spectrum sharing, thereby providing ing (37) as the users of CS with QoS of 5 10−5 bps/user, the MIMO × H H H 2 RS needs to spend an extra power of 1.5 4.5 dB depending tr B EiBi = B U HiiVi Id − i k i i − i kF on the number of antennas it uses. Overall, the designed H H 2 + B U Hij Vj framework provides the essential understanding for successful j∈S,j6=i k i i kF development of future cellular systems in conjunction with X H H 2 + (T ) ψ Bi Ui Hij ΞℓVj F federal incumbents that can operate under same spectrum j∈S ℓ∈Dj k k resources. X X H H 2 + (R) υ Bi Ui ΞℓHij Vj F j∈S ℓ∈Di k k X XH H 2 H H 2 APPENDIX A + PR B U Wi +σ B U , (C.1) k i i k i k i i kF PROOF OF PROPOSITION 1 where (R) and (T ) denote the set ˜ and In order to prove Proposition 1, we first need to show that j j 1 Nj ˜D D { ··· } P is a projector. From (15), we have 1 Mj , respectively, while Ξℓ represents a square matrix with{ ··· zero} elements, except for the ℓ-th diagonal element, H ˜ H H ˜ H H P = (XΩX ) = XΩ X = P , which is equal to 1. Utilising the vec( ) operation, and the 2 H H 2 H · and P = XΩX˜ XΩX˜ = P. (A.1) identity vec (A) 2 = tr AA , (C.1) from above can be reformulatedk as k The above equation holds due to the fact that XXH = I 2 as they are orthogonal matrices and Ω˜ 2 = Ω˜ by construction. tr BH E B = vec BH UH H V I i i i i i ii i − di 2 Now in order to show that P is a projector, we show Px = x, 2 + vec BH UH H V if x range(P). In other words, for some w, x = Pw. Hence, i i ij j 2 from∈ the above and (A.1), we have j∈S,j6=i X H H 2 2 + ψ vec B U H Ξ V Px = P(Pw)= P w = Pw = x, (A.2) i i ij ℓ j 2 j∈S ℓ∈D(T ) P(Px x)= P2x Px = 0. (A.3) X Xj − − 2 + υ vec BH UH Ξ H V (C.2) Hence, Px x null(P), which shows that P is a null-space i i ℓ ij j 2 − ∈ j∈S ℓ∈D(R) projection matrix. Accordingly, X Xi H H 2 2 H H 2 H H H + P vec B U W + σ vec B U . WP = RΩX¯ XΩX˜ = 0. (A.4) R i i i 2 i i i 2
13
By utilising the identity vec(ABC) = [7] M. J. Marcus, “Sharing government with private users: Opportunities and CT A vec (B), (C.2) in the above can be rewritten challenges,” IEEE Wireless Commun. vol. 16, no. 3, pp. 4-5, Jun. 2009. ⊗ 2 [8] F. Hessar and S. Roy, “Spectrum sharing between a surveillance radar and as zij , where zij is defined as IEEE Trans. Aerosp. Electron. Syst. k k2 secondary Wi-Fi networks,” , vol. 52, H H no. 3, pp. 1434-1447, Jul. 2016. vec Bi Ui HiiVi Idi [9] R. Saruthirathanaworakun, J. M. Peha, and L. M. Correia, “Opportunistic T H H − Vj Bi Ui vec (Hij ) sharing between rotating radar and cellular,” IEEE J. Sel. Areas Commun., ⊗ j∈S,j6=i vol. 30, no. 10, pp. 1900-1910, Nov. 2012. T H H √ Ξ V B U H (T ) [10] A. Babaei, W. H. Tranter, and T. Bose, “A nullspace-based precoder ψ ( ℓ j ) i i vec ( ij ) ℓ∈D ⊗ j j∈S zij = with subspace expansion for radar/communications coexistence,” in Proc. j T H H k IEEE Global Commun. Conf. (GLOBECOM), Dec. 2013, pp. 3487-3492. √υ Vj Bi Ui Ξℓ vec (Hij ) (R) ⊗ ℓ∈Di j∈S [11] F. Liu, C. Masouros, A. Li, T. Ratnarajah, and J. Zhou, “MIMO radar and T H H cellular coexistence: A power-efficient approach enabled by interference j PR IR Bi Ui vec (Wi) k ⊗ exploitation,” IEEE Trans. Signal Process., vol. 66, no. 14, pp. 3681-3695, σ vec BH UH i i i Jul. 2018. (C.3) [12] B. Li and A. Petropulu, “MIMO radar and communication spectrum RAD sharing with clutter mitigation,” in Proc. IEEE Radar Conf. (RadarConf), In a similar way, we can also represent I into vector May 2016, pp. 1-6. form as [13] B. Li and A. P. Petropulu, “Joint transmit designs for coexistence of MIMO wireless communications and sparse sensing radars in clutter,” 2 RAD IEEE Trans. Aerosp. Electron. Syst., vol. 53, no. 6, pp. 2846-2864, Dec. I = vec (Glj Vj ) 2 j∈S k k 2017. X 2 [14] A. Khawar, A. Abdelhadi, and T. C. Clancy, “Target detection perfor- + (T ) ψ vec (Glj ΞℓVj ) 2 . (C.4) mance of spectrum sharing MIMO radar,” IEEE Sensors J., vol. 15, no. ℓ∈Dj k k 9, pp. 4928-4940, Sept. 2015. Now, applyingX the same identity vec(ABC) = [15] B. Li, A. Petropulu, and W. Trappe, “Optimum co-design for spectrum T sharing between matrix completion based MIMO radars and a MIMO C A vec (B), (C.4) in above can be constructed communication system,” IEEE Trans. Signal Process., vol. 64, no. 17, ⊗2 as ι 2, where ι is defined as pp. 4562-4575, Sept. 2016. k k [16] J. A. Mahal, A. Khawar, A. Abdelhadi, and T. C. Clancy, “Spectral VT I vec (G ) coexistence of MIMO radar and MIMO cellular system,” IEEE Trans. j T j j∈S ⊗ Aerosp. Electron. Syst., vol. 53, no. 2, pp. 655-668, Apr. 2017. ι= T . (C.5) √ Ξ V I G (T ) ψ ( ℓ j ) T vec ( j) ℓ∈D [17] H. Deng and B. Himed, “Interference mitigation processing for ⊗ j j∈S spectrum-sharing between radar and wireless communication systems,” j k IEEE Trans. Aerosp. Electron. Syst., vol. 49, no. 3, pp. 1911-1919, Jul. APPENDIX D 2013. USEFUL LEMMAS [18] A. C. Cirik, S. Biswas, S. Vuppala and T. Ratnarajah, “Robust transceiver H design for full duplex multiuser MIMO systems,” IEEE Wireless Com- Lemma 5: If P, Q, A are matrices with A = A , mun. Lett., vol. 5, no. 3, pp. 260-263, Jun. 2016. then semi-infinite LMI of the form of A PH XQ + [19] A. C. Cirik, S. Biswas, S. Vuppala and T. Ratnarajah, “Beamforming H H design for full-duplex MIMO interference channels-QoS and energy- Q X P, X : X F ρ, holds iff ǫ 0 such that ∀ k k ≤ ∃ ≥ efficiency considerations,” IEEE Trans. Commun., vol. 64, no. 11, pp. A ǫQH Q ρPH 4635-4651, Nov. 2016. − − 0. (D.1) [20] M. Duarte, C. Dick, and A. Sabharwal, “Experiment-driven characteri- ρP ǫI − zation of full-duplex wireless systems,” IEEE Trans. Wireless Commun., Lemma 6: Schur Complement Lemma [31]: Let Q and R vol. 11, no. 12, pp. 4296-4307, Dec. 2012. [21] W. Li, J. Lilleberg, and K. Rikkinen, “On rate region analysis of half- and are symmetric matrices. Then full-duplex OFDM communication links,” IEEE J. Sel. Areas Commun., Q S vol. 32, no. 9, pp. 1688-1698, Sept. 2014. 0 , R 0, Q SR−1S∗ 0. (D.2) [22] D. Nguyen, L. Tran, P. Pirinen, and M. Latva-aho, “On the spectral effi- S∗ R − ciency of full-duplex small cell wireless systems,” IEEE Trans. Wireless Commun., vol. 13, no. 9, pp. 4896-4910, Sept. 2014. REFERENCES [23] A. C. Cirik, S. Biswas, S. O. Taghizadeh, and T. Ratnarajah, “Robust [1] Federal Communications Commission (FCC), Spec- transceiver design in full-duplex MIMO cognitive radios,” IEEE Trans. trum policy task force, [Online] Available: Veh. Technol., vol. PP, no. 99, pp. 1-1, 2017. https://transition.fcc.gov/sptf/files/IPWGFinalReport.pdf, Nov. 2002. [24] B. P. Day, A. R. Margetts, D. W. Bliss, and P. Schniter, “Full-duplex [2] Federal Communications Commission (FCC), FCC proposes bidirectional MIMO: Achievable rates under limited dynamic range,” innovative small cell use in 3.5 GHz band, [Online] Available: IEEE Trans. Signal Process., vol. 60, no. 7, pp. 3702-3713, Jul. 2012. https://www.fcc.gov/document/fcc-proposes-innovative-small-cell-use- [25] E. A. Gharavol, Y.-C. Liang, and K. Mouthaan, “Robust downlink 35-ghz-band, accessed, Dec. 2012. beamforming in multiuser MISO cognitive radio networks with imperfect [3] S. Haykin, “Cognitive radio: Brain-empowered wireless communica- channel-state information,” IEEE Trans. Veh. Technol., vol. 59, no. 6, pp. tions,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201-220, Feb. 2852-2860, Jul. 2010. 2005. [26] T. W. Ban, W. Choi, B. C. Jung, and D. K. Sung, “Multi-user diversity [4] CEPT, “Licensed Shared Access (LSA),” Electronic in a spectrum sharing system,” IEEE Trans. Wireless Commun., vol. 8, Communications Committees (ECC), [Online] Available: no. 1, pp. 102-106, Jan. 2009. http://www.erodocdb.dk/Docs/doc98/official/pdf/ECCREP205.PDF, [27] L. Xu and J. Li, “Iterative generalized-likelihood ratio test for MIMO Feb. 2014. radar,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2375-2385, Jun. [5] M. Matinmikko, M. Mustonen, D. Roberson, J. Paavola, M. Hoyhtya, 2007. S. Yrjola, and J. Roning, “Overview and comparison of recent spectrum [28] I. Bekkerman and J. Tabrikian, “Target detection and localization using sharing approaches in regulation and research: From opportunistic unli- MIMO radars and sonars,” IEEE Trans. Signal Process., vol. 54, no. 10, censed access towards licensed shared access,” in Proc. IEEE Int. Symp. pp. 3873-3883, Oct. 2006. Dyn. Spectr. Access Netw. (DYSPAN), Apr. 2014, pp. 92-102. [29] S. M. Kay, Fundamentals of statistical signal processing: Detection [6] National Telecommunications and Information Administration theory, vol. 2, Prentice Hall, 1998. (NTIA), An assessment of the near-term viability of accommodating wireless broadband systems in the 1675-1710 MHz, 1755- 1780 MHz, 3500-3650 MHz, 4200-4220 MHz, and 4380- 4400 MHz bands (Fast track report), [Online] Available: https://www.ntia.doc.gov/files/ntia/publications/fasttrackevaluation_111- 52010.pdf, Nov. 2010. 14
[30] A. M. Tulino and S. Verdú, “Random matrix theory and wireless tion: Analysis, Algorithms, Engineering Applications, Philadelphia, PA, communications.” Found. Trends Commun. Inf. Theory., vol. 1, no. 1, USA: SIAM, 2001. pp 1-182, Jun. 2004. [35] H. Shen, B. Li, M. Tao, and X. Wang, “MSE-based transceiver designs [31] S. Boyd and L. Vandenberghe, Convex optimisation, Cambridge, U.K.: for the MIMO interference channel,” IEEE Trans. Wireless Commun., vol. Cambridge University Press, 2004. 9, no. 11, pp. 3480-3489, Nov. 2010. [32] Y. Zhang, E. DallAnese, and G. B. Giannakis, “Distributed optimal [36] J. B. Andersen, T. S. Rappaport and S. Yoshida, “Propagation mea- beamformers for cognitive radios robust to channel uncertainties,” IEEE surements and models for wireless communications channels,” IEEE Trans. Signal Process., vol. 60, no. 12, pp. 6495-6508, Dec. 2012. Commun. Mag., vol. 33, no. 1, pp. 42-49, Jan. 1995. [33] J. Jose, N. Prasad, M. Khojastepour, and S. Rangarajan, “On robust [37] S. Sun et al., “Propagation path loss models for 5G urban micro- weighted-sum rate maximization in MIMO interference networks,” in and macro-cellular scenarios,” in Proc. IEEE 83rd Vehicular Technology Proc. IEEE Int. Conf. Commun. (ICC), Jun. 2011, pp. 1-6. Conference (VTC Spring), May 2016, pp. 1-6. [34] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex optimisa-