On Differential Modulation in Downlink Multiuser MIMO Systems
OOn Differential Modulation in Downlink MultiuserMIMO Systems
Fahad Alsifiany ∗ , Aissa Ikhlef † , and Jonathon Chambers ∗∗ ComS IP Group, School of Electrical and Electronic Engineering, Newcastle University, NE1 7RU, UK{f.a.n.alsifiany2, jonathon.chambers}@ncl.ac.uk † School of Engineering and Computing Sciences, Durham University, DH1 3LE, [email protected]
Abstract —In this paper, we consider a space time block codedmultiuser multiple-input multiple-output (MU-MIMO) systemwith downlink transmission. Specifically, we propose to usedownlink precoding combined with differential modulation (DM)to shift the complexity from the receivers to the transmitter.The block diagonalization (BD) precoding scheme is used tocancel co-channel interference (CCI) in addition to exploitingits advantage of enhancing diversity. Since the BD schemerequires channel knowledge at the transmitter, we propose touse downlink spreading along with DM, which does not requirechannel knowledge neither at the transmitter nor at the receivers.The orthogonal spreading (OS) scheme is employed in order toseparate the data streams of different users. As a space time blockcode, we use the Alamouti code that can be encoded/decodedusing DM thereby eliminating the need for channel knowledgeat the receiver. The proposed schemes yield low complexitytransceivers while providing good performance. Monte Carlosimulation results demonstrate the effectiveness of the proposedschemes.
Index Terms —Differential modulation, Alamouti STBC, mul-tiuser MIMO, block diagonalization, orthogonal spreading code.
I. I
NTRODUCTION
Future wireless systems require effective transmission tech-niques to support high data rate and reliable communications.As such, a potential technique to utilize as part of multiple an-tenna systems to enhance system diversity is space-time blockcode (STBC) [1]. In the multiuser multiple-input multiple-output MU-MIMO downlink, transmit diversity gain can bemaximized by using downlink transmission techniques suchas transmit precoding, e.g., block diagonalisation (BD), andtransmit spreading, such as the orthogonal spreading (OS)scheme. These techniques allow the MU-MIMO channels to bedecomposed into parallel single user non-interfering channels,and hence eliminate co-channel interference (CCI) [2], [3].For the MU-MIMO downlink, the availability of channelstate information (CSI) at the transmitter makes it possible forthe precoder to precancel the CCI at each user. The authorsin [2] proposed a framework that uses BD to cancel theCCI and assumed full CSI knowledge at the transmitter. TheCSI between the transmitter and the receivers is estimatedat the receivers then fed back to the transmitter. This leadsto increased complexity of the receivers. In [3], the authorsproposed a method that combines the precoding technique in[2] and the Alamouti STBC. The proposed method providesa substantial gain in terms of spatial diversity with a low decoding complexity. However, for the decoding process, eachreceiver still needs to know the composite channel formed bythe precoder and the channel in order to coherently decodethe Alamouti STBC. In practice, each receiver acquires thecomposite channel by direct estimation.The prior focus of STBC MU-MIMO downlink transmissiontechniques has been on cases where CSI is available at thereceivers and transmitter. However, for some systems, due tohigh mobility and the cost of channel training and estimation,CSI acquisition is impossible [4]. One alternative method forsuch systems is differential modulation (DM). In this work, theuse of DM for downlink transmission in a MU-MIMO systemis considered. Specifically, we show how to use DM combinedwith the BD and OS schemes. Furthermore, DM is consideredfor both schemes based on the Alamouti STBC in orderto eliminate the need for estimating the composite channelsformed by the precoders and the channels at the receivers. Inthe BD scheme, the use of DM is to simplify the complexityof the receivers by eliminating the need for CSI as well asto cancel CCI. In particular, in order to have low complexityreceivers, it is assumed that the channels are estimated at thetransmitter, since it can tolerate more complexity comparedto the receivers. Once the channels are estimated at the BS,the transmitter computes the precoder as in [2], [3]. However,since the BD scheme still requires CSI at the transmitter, adownlink OS scheme combined with DM is proposed. In theOS scheme, unlike the BD scheme, the transmitter does notrequire any knowledge of the CSI to separate the data streamsof multiple users [5], [6]. Therefore, implementing the OSscheme with the DM will result in a system that does not needCSI at either ends. The proposed schemes facilitate the pre-cancelling of CCI, enhance diversity, as well as achieve a lowcomplexity transmitter and receivers. Moreover, transmissionoverhead is significantly reduced using the proposed scheme,since neither feedback nor the estimation of the compositechannels are required.The rest of the paper is organized as follows. Section IIintroduces the system model of STBC MU-MIMO. Section IIIdescribes downlink transmission for interference cancellation.Section IV presents DM-STBC in a MU-MIMO system withdownlink transmission. In Section V, the simulation results areshown. Finally, conclusions are drawn in Section VI. a r X i v : . [ c s . I T ] J u l I. S
YSTEM M ODEL
Consider a MU-MIMO downlink broadcast channel wherethe base station (BS) transmits multiple streams to K users(e.g., mobile stations), as shown in Fig. 1. The BS has N t transmit antennas and each user has N k , k = 1 , · · · , K ,receive antennas. The total number of receive antennas forall users is N r , i.e., N r = P Kk =1 N k . Z k STBCSTBCSTBC
Base Station (BS)s K s k s X X k X K
12 H H k H K MS KMS kMS 1
Mobile Station (MS)Z K N K N k N Y Y k Y K N t F K or V K F k or V k F or V Z Fig. 1. STBC MU-MIMO downlink transmission system.
A. Channel Model
The channel matrix H k ∈ C N k × N t for each user k is aRayleigh flat fading matrix given by H k = h ( k )1 , · · · h ( k )1 ,N t ... . . . ... h ( k ) N k , · · · h ( k ) N k ,N t = h ( k )1 ... h ( k ) N k , (1)where the element h ( k ) i,j is the channel coefficient between the j th transmit antenna and the i th receive antenna of user k ,and C denotes the set of complex numbers. It is assumed thatthe channel coefficients are quasi-static over T transmissionslots. The elements of H k are independent and identicallydistributed (i.i.d.) complex Gaussian random variables withzero mean and unit variance, i.e., CN (0 , . B. Space-Time Block Coding - Alamouti Code
The multiple data streams s k for each user are encodedby the Alamouti encoder to generate the STBC codeword.Let X k ∈ C × , k = 1 , · · · , K , be the transmitted AlamoutiSTBC signal, satisfying the following condition [3], [7]: X Hk X k = X k X Hk = I . (2)The generator matrix for the Alamouti code is given as X k = 1 √ • s ,k − s ∗ ,k s ,k s ∗ ,k ‚ , (3)where s ,k and s ,k ∈ Z are the two input symbols to theAlamouti STBC encoder for user k . Z and ( . ) H denote theconstellation set and the Hermitian operator, respectively. III. D OWNLINK T RANSMISSION FOR I NTERFERENCE C ANCELLATION
In this section, two different methods are used to cancelCCI in downlink transmission. The first scheme, referred toas the BD scheme, is suitable for the case where the CSI isavailable at the transmitter and the second scheme, referred toas the OS scheme, is suitable for the case where the CSI isnot available at the transmitter.
A. BD Scheme
The received signal Y ( BD ) k ∈ C N k × at the k th user can beexpressed as Y ( BD ) k = H k F k X k + H k K X j =1 ,j = k F j X j + Z k = H k F k X k + P k + Z k , (4)where F k ∈ C N t × is the precoding matrix, Z k ∈ C N k × isan AWGN noise matrix. P k ∈ C N k × is the CCI componentat the k th user. Note that, at the BS, the precoding matrix F k for the k th user is multiplied by the symbol vector and addedto the precoded signals from the other users to produce thecomposite transmitted matrix, i.e., P Kk =1 F k X k .The BD method employs precoding matrices F k , k =1 , · · · , K , to completely suppress the CCI at the receivers. Tocancel the CCI, the following constraint should be satisfied[2], [3] H j F k = 0 , j, k = 1 , ..., K, j = k. (5)Let ¯ H k ∈ C ¯ N k × N t , where ¯ N k = N r − N k , denote the channelmatrix for all K users excluding the k th user’s channel, whichis defined as ¯ H k = £ H H · · · H Hk − H Hk +1 · · · H HK ⁄ H . (6)Therefore, the zero-interference constraint in (5) is re-expressed as ¯ H k F k = 0 , k = 1 , ..., K. (7)According to [3], to satisfy (7), one solution is to construct F k as F k = ( I − ¯ H † k ¯ H k ) Φ k , (8)where Φ k ∈ C N t × is an eigenmode selection matrix, and ( . ) † denotes the pseudo-inverse. The magnitude, i.e, the vectornorm of the precoding matrix F k has to be unity to ensure aconstant transmission power for the k th user, i.e., F Hk F k = I , k = 1 , · · · , K. (9)Therefore, to satisfy (9), the unitary F k matrix can beconstructed as a linear combination of the column spacespanning vectors of ( I − ¯ H † k ¯ H k ) , which can be obtained bythe Gram-Schmidt orthogonalization (GSO), or the standardQR decomposition. In this paper, QR decomposition is usedfor its simplicity.To compute Φ k , a singular value decomposition (SVD) of H k ( I − ¯ H † k ¯ H k ) is performed. This is done by selecting thewo singular vectors corresponding to the two largest singularvalues of H k ( I − ¯ H † k ¯ H k ) . The resulting received signal forthe k th user after cancelling out the CCI is given by Y ( BD ) k = H k F k X k + Z k = ˘ H k X k + Z k , (10)where ˘ H k ∈ C N k × is the effective channel for user k . B. OS Scheme
In the OS case, the received signal matrix Y ( OS ) k ∈ C N k × KN t for the k th user is given by [5] Y ( OS ) k = H k X k V k + H k K X j =1 ,j = k X j V j + Z k , (11)where V k ∈ C N t × KN t is the orthogonal spreading matrixfor user k , Z k ∈ C N k × KN t is an AWGN noise matrix. Thecomposite transmitted matrix is P Kk =1 X k V k . Note that, inorder to apply Alamouti STBC along with the orthogonalspreading code, the number of transmit antennas at the BShas to be limited to two, i.e, N t = T = 2 .In the OS scheme, each user is assigned a unique orthogonalspreading code to separate the data of the users at the receivers.The STBC codeword for each user is multiplexed by its ownspecific spreading code and then transmitted. As in the BDmethod case, to eliminate CCI, the spreading code matrix hasto obey the following conditions V k V Hk = I N t , k = 1 , ..., K. (12) V j V Hk = 0 , k, j = 1 , ..., K, and j = k. (13)The OS code for each user can be constructed as a submatrixof the Hadamard matrix, or from a discrete Fourier transform(DFT) matrix. Hadamard matrices are of interest because oftheir simplicity. Hadamard codes are a set of orthogonal codeswhich are built repeatedly from the basic building block A = 1 √ • +1 +1+1 − ‚ (14)according to A n +1 = 1 √ n +1 • A n A n A n − A n ‚ , (15)where the dimension of the Hadamard matrix in (15) is n +1 × n +1 . Note that in our case n +1 = KN t .Due to the orthogonality of the spreading matrices usedat the transmitter, at each receiver, the original informationsignal is retrieved by despreading the received signal withthe synchronized duplicate of the spreading code. Therefore,the received signal matrix Y ( OS ) k in (11) for the k th user isdespread by multiplying it with V Hk , which yields ˆ Y ( OS ) k = Y ( OS ) k V Hk = H k X k + ˆ Z k , (16)where ˆ Y ( OS ) k ∈ C N k × N t is the despread received signal, and ˆ Z k ∈ C N k × N t is the despread AWGN noise. C. Complexity Analysis
In this section, the computational complexity with the notionof flops is introduced here, where flops denotes the floatingpoint operation. At the transmitter, the BD scheme uses thespatial dimension to cancel CCI, whereas the OS scheme usesthe time dimension. In the BD scheme, in order to cancel CCIcompletely, the system must satisfy [2], [3] N t ≥ K X j =1 ,j = k N j + 2 . (17)The complexity of the BD scheme is mainly based on pseudo-inverse ¯ H † k = ¯ H Hk ¡ ¯ H k ¯ H Hk ¢ − , and the QR decompositionof ( I − ¯ H † k ¯ H k ) . The complexity of both the pseudo-inverseoperation and the QR decomposition follows [8], [9] O KN t K X j =1 ,j = k N j . (18)In the OS scheme, the precoder is independent from thenumber of receive antennas. Thus, the complexity of theOS scheme is only based on Hadamard matrix constructionwhich is already given. Hence, it does not incur any com-putational complexity. Obviously, the OS scheme has lowercomputational complexity than the BD scheme, but in termsof throughput, the OS scheme throughput is K times smallerthan that of the BD scheme. Note that, the computationalcomplexity at the receiver side for both schemes is the same,and we will explore more about the DM decoder in thefollowing section.IV. D IFFERENTIAL
STBC
FOR
MU-MIMO
WITH D OWNLINK T RANSMISSION
In this section, the differential encoding and decodingprocess for downlink transmission in a MU-MIMO systemis discussed. In particular, this section demonstrates how touse the BD and OS schemes in differential STBC MU-MIMOsystems.
A. Differential Encoding
The particular encoding algorithm utilized for DM buildsupon the works in [7], [10]. The algorithm requires that unitarySTBCs such as the Alamouti code are used. In the encodingprocess, the X matrix is used as a reference code, in whichthe transmitted matrix for the initial block of each user k isset to be identity as X ,k = I T , k = 1 , · · · , K. (19)Then, for the BD scheme, the unitary Alamouti STBC matricesare encoded differentially for the subsequent blocks as follows B ( BD ) n = K X k =1 F k ˆ n Y i =0 X i,k ! , n = 0 , ..., N. (20)or the OS scheme, the encoding process is as follows B ( OS ) n = K X k =1 ˆ n Y i =0 X i,k V k ! , n = 0 , ..., N, (21)where B ( q ) n , q ∈ { BD , OS } , is the n th encoded block, N + 1 is the total number of encoded signal blocks, and F k and V k represent the precoding matrix and spreading matrix for user k , respectively.The performance of the differential modulation system de-pends on the length of time over which the channel coefficientsremain constant. Ordinarily, the reference (known) symbol X ,k must be sent periodically, based on the channel coherencetime. Accordingly, generating the downlink precoding matrix F k or the downlink spreading matrix V k for the new channelcoefficient matrix only needs to be done when there are newchannel coefficients. B. Differential Decoding
For the MU-MIMO downlink system, the differential trans-missions are implemented in blocks, in which each user k receives the sum of all the transmit waveforms of other users;then the received signal blocks for each user must be detectedindependently. Thus, if G k denotes the matrix having all N +1 received signal blocks for the k th user, i.e., G k = [ Y ,k Y ,k · · · Y N,k ] , (22)then the received signal block at the k th user during the n thiteration block, i.e., Y n,k can be expressed as Y n,k = H k B ( q ) n + Z n,k , n = 0 , ..., N, (23)where q ∈ { BD , OS } , and Z n,k is the k th user AWGN noiseduring the n th block. For DM encoding, it is assumed that thechannel matrix H k changes slowly (channel coherence timeis large enough) and extends over several matrix transmissionperiods. In such a case, the BS transmission starts with areference matrix, followed by several information matrices.When encoding using (20) or (21), the decoding process for X n,k would be according to the last two blocks of G k as inthe following notation [7], [10] G k = • Y ,k Y ,k | {z } · · · Y n − ,k Y n,k | {z } · · · Y N − ,k Y N,k | {z }‚ . (24)For the BD method, to make this more explicit, define Y n,k ∆ = • Y n − ,k Y n,k ‚ ∆ = " H k B ( q ) n − + Z n − ,k H k B ( q ) n + Z n,k , (25)and recall from (5) that the interference of other users issuppressed, thus the two blocks in (25) become a single userblock matrix as Y n,k ∆ = • H k F k X n − ,k + Z n − ,k H k F k X n − ,k X n,k + Z n,k ‚ . (26)The code matrices that affect Y n,k are D X n,k = • X n − ,k X n − ,k X n,k ‚ . (27) Assuming that N t = T , and using these results, as well as (2)and (9), the matrices in (27) can be expressed as D HX n,k D X n,k = 2 I N t , (28)therefore, these matrices represent unitary block codes. When X n − ,k is known to the receiver, the optimal decoder for thisblock is the quadratic receiver as [10] ˆ X n,k = arg max X n,k trace n Y n,k D X n,k D HX n,k Y Hn,k o . (29)Since we have D X n,k D HX n,k = • I T X Hn,k X n,k I T ‚ , (30)the decoder in (29) can be re-written as follows [10], [7] ˆ X n,k = arg max X n,k trace (• Y n − ,k Y n,k ‚ • I T X Hn,k X n,k I T ‚ • Y n − ,k Y n,k ‚ H ) = arg max X n,k ℜ ' trace ' X n,k Y Hn,k Y ( n − ,k ““ , (31)where ℜ ( . ) denotes the real part, and trace( . ) denotes the traceof a matrix. Similarly, the equivalent differential decoder forthe OS scheme can be constructed. Note that when the CSIis available at the receiver, the standard Alamouti decoderis used before the maximum likelihood (ML) detection isimplemented upon the combined signals.V. S IMULATIONS R ESULTS AND D ISCUSSION
In this section, the performance of the differential and co-herent Alamouti STBC for MU-MIMO downlink transmissionis examined. Alamouti codes with QPSK are used throughoutthe simulation.Fig. 2 plots the symbol error rate (SER) for coherentmodulation (CM) and DM with one receive antenna per user.For BD scheme, the performance curve is plotted for a singleuser system with 2 transmit antennas at the BS and a four-usersystem with 5 transmit antennas at the BS. For OS scheme,the number of transmission antenna has been set to be alwaystwo against 1 and 4 users. We observe that CM and DM forboth BD and OS schemes achieve the same performance asa single-user STBC-MISO link; that is, CCI is completelyeliminated and full diversity is achieved with the Alamouticode. Ordinarily, the differential detection underperforms thecoherent detection by about 3 dB.Fig. 3 illustrates the results of repeating the experimentwith two receive antennas per user. Similarly, the MU-MIMOsystem of CM and DM for both schemes behave as a singleuser STBC-MIMO link, but with better performance than theone receive antenna per user system. For BD scheme, CCIelimination requires that the number of transmit antennas issufficient to achieve full diversity with the given number ofreceive antennas, so N t = 8 is chosen. For OS scheme, wehave got the same performance but with fixed number oftransmit antennas, e.g., N t = 2 . Consequently, unlike BDscheme, the number of receive antenna per user is independentfrom the number of transmission antenna.ig. 4 shows the performance of exploiting DM combinedwith BD and OS schemes with three receive antennas peruser. The high mobility and multipath propagation may resultin multiple access interference (MAI) in OS scheme andimperfect channel estimation in BD scheme, which destroy theorthogonality of the precoders. Hence, Fig. 4 also shows theimpact of possible errors in both schemes. For OS scheme, theerror spreading matrix for user 1 is ¯ V = V + α V , where α is the error coefficient [5]. The values of α are chosen tobe . , . , respectively. For BD scheme, imperfect channelmatrix at the BS for user 1 is ¨ H = H + E , where H is the perfect channel estimate for user 1 and E is the errormatrix [3]. Entries of E are i.i.d. Gaussian variables withdistribution zero mean and covariance of σ . The values of σ are chosen to be . and . . From Fig. 4, it is clear that theOS is more robust against errors compared to the BD scheme. SNR (dB) S y m b o l E rr o r R a t e ( S E R ) -5 -4 -3 -2 -1 DM-BD : 1 user, 2 TxDM-BD : 4 users, 5 TxCM-BD : 1 user, 2 TxCM-BD : 4 users, 5 TxDM-OS : 1 user, 2 TxDM-OS : 4 users, 2 TxCM-OS : 1 user, 2 TxCM-OS : 4 users, 2 Tx
Fig. 2. SER performance of MU-MIMO STBC downlink precoding withcoherent and differential detection using BD and OS schemes for N k = 1 . SNR (dB) S y m b o l e E rr o r R a t e ( S E R ) -5 -4 -3 -2 -1 DM-BD : 1 user, 2 TxDM-BD : 4 users, 8 TxCM-BD : 1 user, 2 TxCM-BD : 4 users, 8 TxDM-OS: 1 user, 2 TxDM-OS: 4 users, 2 TxCM-OS: 1 user, 2 TxCM-OS: 4 users, 2 Tx
Fig. 3. SER performance of MU-MIMO STBC downlink precoding withcoherent and differential detection using BD and OS schemes for N k = 2 . VI. C
ONCLUSION
In this paper, a low complexity differential STBC schemefor MU-MIMO with downlink transmission has been pro-posed. In particular, DM combined with either the BD schemeor the OS scheme overcame the need for CSI at the receivers
SNR (dB) S y m b o l E rr o r R a t e ( S E R ) -6 -5 -4 -3 -2 -1 DM-OS, α = 0 : with 2 user, 2 Tx ants.DM-OS, α = 0 . α = 0 . σ = 0 : with 2 user, 5 Tx antsDM-BD, σ = 0 . σ = 0 . Fig. 4. SER performance of differential detection system using BD and OSschemes for N k = 3 with the impact of precoding errors on user 1. as well as cancelled CCI. On the other hand the use ofSTBC can achieve full diversity without needing CSI atthe transmitter. It has been demonstrated that implementingthe BD scheme with DM will establish a system that doesnot need CSI at the receivers to decode the signals, whilecombining the OS scheme with DM will establish a systemthat requires CSI at neither the transmitter nor at the receivers.The differential modulation for both systems loses typically3dB in performance relative to the coherent detection method,but this is offset by the reduction in complexity of the receiversand the transmitter. The BD scheme is more complex than theOS scheme; however, the BD scheme has a higher throughput.Moreover, it was shown that the OS is more robust againstprecoding errors compared to the BD scheme.R EFERENCES[1] S. M. Alamouti, “A simple transmit diversity technique for wirelesscommunications,”
IEEE J. Sel. Areas Commun. , vol. 16, no. 8, pp. 1451–1458, Oct. 1998.[2] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing meth-ods for downlink spatial multiplexing in multiuser MIMO channels,”
IEEE Trans. Signal Process. , vol. 52, no. 2, pp. 461–471, Feb. 2004.[3] R. Chen, J. Andrews, and R. Heath, “Multiuser space-time block codedMIMO with downlink precoding,” in
Proc. IEEE Int. Conf. Commu ,vol. 5. IEEE, Jun. 2004, pp. 2689–2693.[4] V. Tarokh and H. Jafarkhani, “A differential detection scheme fortransmit diversity,”
IEEE J. on Selec. Areas in Commun. , vol. 18, no. 7,pp. 1169–1174, Jul. 2000.[5] Y. Hong, E. Viterbo, and J.-C. Belfiore, “A space-time block coded mul-tiuser MIMO downlink transmission scheme,” in
Information Theory,2006 IEEE International Symposium on . IEEE, 2006, pp. 257–261.[6] H. El Gamal and M. O. Damen, “Universal space-time coding,”
IEEETrans. Inf. Theory , vol. 49, no. 5, pp. 1097–1119, May 2003.[7] B. M. Hochwald and W. Sweldens, “Differential unitary space-timemodulation,”
IEEE Trans. Commun. , vol. 48, no. 12, pp. 2041–2052,Dec. 2000.[8] G. H. Golub and C. F. Van Loan,
Matrix computations . JHU Press,2012, vol. 3.[9] K. Zu, R. C. de Lamare, and M. Haardt, “Generalized Design ofLow-Complexity Block Diagonalization Type Precoding Algorithms forMultiuser MIMO Systems.”
IEEE Trans. Commun. , vol. 61, no. 10, pp.4232–4242, Oct. 2013.[10] B. L. Hughes, “Differential space-time modulation,”