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SIMULTANEOUS CHANNEL ESTIMATION AND DECODING FOR DIAGONAL SPACE-TIME CODES A. Lee Swindlehurst Dept. of Electrical & Computer Engineering Brigham Young University Provo, UT 84602 [email protected] ABSTRACT Multiple input, multiple output (MIMO) wireless communications links have been shown to have the potential for significant increases in capacity, provided they are deployed in an environment with rich multipath scattering. To realize these gains, a number of space-time coding strategies have recently been proposed. Most of these algorithms assume that, via training data, the channel is known at least on one end of the link. However, if the channel is time-varying or even just quasi-stationary, the training overhead can offset much of the throughput gain. In this paper, a space-time coding scheme is presented that allows for simultaneous blind or semi-blind channel estimation and decoding of the symbols transmitted by multiple users. The method relies on the use of “diagonal” space-time codes in which the same symbol is successively transmitted from each antenna in turn. This structure leads to a simple subspace-based algorithm that produces closed-form estimates of both the channel and the transmitted symbols. The algorithm is shown to be applicable to cases involving fewer receive than transmit antennas, rank-deficient channels, flat or frequency selective fading, and multiple users. 1. INTRODUCTION The advantages of using multiple antennas at both the transmit and receive ends of a wireless communications link have recently been noted [1, 2]. A number of space-time coding algorithms have been proposed that exploit the potential for dramatically improved throughput and reliability that such systems offer (for example, see [3, 4, 5, 6, 7]). For most algorithms, these gains can only be realized provided that the multiple-input multiple-output (MIMO) channel separating the transmitter and receiver can be identified. While training data can be used to estimate the channel, this approach consumes precious bandwidth and reduces throughput, especially in time-varying scenarios where the channel may be rapidly changing. One approach to overcoming this difficulty is the use of differential space-time codes [8, 9, 10]. In this paper, an alternative method is presented that exploits the temporal structure offered by a particular type of space-time code in order to blindly (or semi-blindly) estimate the channels and decode the signals for multiple users at the same time. Specifically, “diagonal” space-time codes in which the same symbol is successively transmitted from each antenna are exploited. The algorithm is based on the property that a delay in the time domain produces a linear phase shift in the frequency domain. This fact has recently been exploited for both blind and training-signal-based channel This work was supported by the National Science Foundation under Wireless Initiative Grant CCR 99-79452.

estimation [11]. For finite-length data sets, this property holds only approximately due to FFT truncation effects. However, if the transmitted code is not only diagonal but circulant as well, the truncation effects are eliminated. For the application considered here, this requires a few symbols (at least as many as one less than the number of transmit antennas) to be repeated at the end of the data frame. The repeated symbols are referred to as a cyclic prefix, and have been exploited for blind equalization in multicarrier (OFDM) systems [12]. Without a cyclic prefix, the truncation effects will still be negligible provided that the number of transmit antennas is significantly smaller than the number of transmitted symbols. The proposed algorithm is subspace-based, and has been applied to the single-user channel equalization problem in [13]. It estimates the transmitted symbols as those that produce a circulant symbol matrix (due to the circulant space-time code) that is as orthogonal as possible to the row space of the received data matrix. The simple structure of the data model in the frequency domain yields an elegant closed-form solution to this problem. The subspace nature of the algorithm bears some similarity to the recent approach of [14], which modulates the same symbol sequence by a different known code for each transmit antenna. Both blind and semi-blind solutions are presented. For the blind case, only the space spanned by the various users’ signals can be identified if each user has the same number of transmit antennas; separating out the individual user’s signals in such cases requires additional information (such as provided by a finite alphabet [15] or constant modulus [16] assumption). If training symbols are embedded in each user’s signal, then the semi-blind solution presented herein can be used to separate the users provided that they employ linearly independent training data. While the details of the algorithm are presented for the single user flat fading case, extensions to the multiple user case are presented, along with modifications of the algorithm required when there are more transmit than receive antennas, the channel is rank deficient, or the channel is frequency selective. 2. DATA MODEL


     




For the moment, assume a single-user transmit array with elements, a receive array with elements, and a flat-fading channel. If the transmit array broadcasts data vectors, the following model results:

  is an   matrix of received data,  is the   where channel matrix, is additive noise and interference, and is an  matrix containing the transmitted symbols. The subscript (1)



  



on is used to explicitly indicate the number of rows in the matrix, and the symbol is used to differentiate from its noisefree counterpart . To begin, we will assume that and that is full rank, but these assumptions will be relaxed in Section 6, as will the single-user and flat-fading restrictions. The key modeling assumption for the algorithms to be presented is that the following simple “diagonal” space-time code is employed:



  



  23 !! # % $'  # %)(*$& !! # # #  #  +  333

 .. .. ,, )0 ...   (2) ./  .    ." #    1# +  # # )0-.  4 # # 0 is implicitly defined to be which Hankel. Note that 0  makes 5-  , which of transmitted   represents the total number symbols that affect . 0 Assume for the   moment that, out of the total transmitted symbols affecting , the last 6 symbols are repeated values of the first 6 . In other words, the symbol sequence is given by repeated symbols #     # %$&  7,  #     8 #     79;,:   #  6 < >= (3) In OFDM communications systems, this short interval of repeated symbols is referred  to as a cyclic prefix. With a cyclic prefix ? . present, the matrix in (2) is circulant provided that 6 As a result, if the >@ FFT matrix A is defined as !!  C  BB,, CDF EHG 2 33 3 A ! .. .. .. (4) " I 4 . . . C DJEG B, CLK DJEHGNMPO C5 QSR7TU&V D , then the row-FFT of   is given by where 5 WYX, Z)[   A  A ]\_^ /W` (5)   @ W a   where A contains first rows of A , A is the FFT \ ^ is thethediagonal of the noise, matrix defined by \ ^b diagced Wgf (6) W A`d  (7) d     B  ,    N  j i h  W # #  . The vector d is simply the FFT of the and d symbol vector d . It is worth mentioning that, if a cyclic prefix is not used, equation (5) is still approximately true provided that lk . The use of a cyclic prefix is not required for identifiability, but as shown in Sections 4 and 5, it does lead to elegant frequency domain solutions. For blind symbol estimation, is completely unknown. In the semi-blind case, may be decomposed as

d

d

d nd m  dBo  where d m and dBo denote theW known and unknown symbols in d ,  W  W W d d m d d m Apdnm respectively. Thus, as well, where o W A`d o are the FFTs of the known and unknown symbols, and d o respectively. Using the constraint that dnm and d o are non-zero at disjoint time samples, a semi-blind algorithm is developed in the next section that exploits the model of (5) to obtain an estimate of dBo and an unstructured estimate of . The approach uses a leastsquares (row) subspace fitting criterion, and leads to closed form estimates of the desired quantities. Before deriving the semi-blind solution, however, a blind algorithm is presented based on (5) for the case where there is no training data.

 is full rank, then the row space of  and 5W (the noiseIf  and free matrices) will coincide with the row spans of Aq] s\_data ^ r, respectively. state this mathematically, define the SVD tpu5v , whereTov denotes @W is @theWwcomplex xr?t`u?conjugate Wv , where transpose, and note that the SVD of u5W_ y  A v u =  u uz {0 -}h u ^ u@| j , Partition into signal and noise subspaces as |  u ^ u where u and denote the first and remaining columns of , respectively. Then

 u@|p x~ = (8) 3. SUBSPACE RELATIONSHIPS

When noise is present, this relationship does not hold exactly. In Section 4 below, we derive a solution to this problem based on minimizing the norm of

 u@|   A_A v u@|  A H\_^ A v u@|@ (9)   W with respect to d . In Section 5, we extend this approach to the W

  semi-blind case. In both cases, once the estimate d and hence is obtained, the channel matrix may be estimated using €   ‚v     Hv ƒ EHG = (10) Note that if the finite alphabet structure of d is known, the solution

  are in (10) can be preceded by a step in which the elements of projected onto the nearest points in the signal constellation. This provides a simple yet effective way of exploiting the finite alphabet nature of the signal in estimating the channel. Before developing the proposed algorithms, we present the following lemma which will be useful to us later:

h † G ,B †ˆ‡7‰ j is Š‹gŒ and  h Ž G ,, Ž ‡, j >@„ ‹g …. Then  „„ vBˆ‘  ’ vB   „s“• ” 7 „s“• ” –v  where „s“ ” is the q@‹gŒe‹  matrix formed by taking the SchurHadamard product of all possible pairs of columns from „ and  ” , the conjugate of  : „s“@ ” ih † G ‘ Ž” G † G ‘ Ž” T ,, † ‡ ‰ ‘ Ž” ‡  jH= Lemma 1 Suppose is

@u  | W d d W` i—e˜–™’š^Ÿžs›œ   A  \ ^ A v u  |   (11)   i—e˜–™’š^ ž›œ Tr  \ ^v A  v A ]\_^ A v u@| u |v A ƒ (12) ˆ  ¢   i—e˜–™’š^Ÿž›œ d Wv ¡  A  v A  ˆ‘x A v u | u |v A  d W` (13) ‘ is the Schur-Hadamard product. Thus, a blind estimate where W of d can be obtained by finding the left singular vector of £ |  A  v A  ‘  A v u  | u  |v A   (14) 4. BLIND SYMBOL ESTIMATION

Using the estimate in equation (9), a simple least-squares subspace fitting estimate of can be obtained as follows:

d W d

  d W   …

with smallest singular value. This solution inherently constrains to satisfy , which reflects the fact that in the blind case, is only identifiable to within an arbitrary complex scale factor. When , the procedure for calculating can be considerably simplified by replacing with , and noting and that the diagonal elements of are all that equal to . Using these facts, we may write

¤   | u  |v ¥ - u  d  ^ W u  ^v @ u A v A ¥ A v A  £i|  ¥ - ¡  A  v A  ‘  A v u  ^ u  ^v A   ¢  (15) W which implies that d may also be obtained as the left singular vector of £ ^  A  v A  ˆ‘x A v u  ^ u  ^v A   (16) £ ^ associated with the largest singular value. The use of instead £ | is advantageous if ¦Tp§  , since Lemma 1 may be used of to show that £i^} ¡ A  v “  A v u@ ^  ¢ ¡ A  v “  A v u@ ^  ¢ v = (17) 0  T W Thus, d is also equal to the left singular vector of the matrix ¨ ^ v  v u  ^  A “ A (18) with largest singular value. 5. SEMI-BLIND SYMBOL ESTIMATION

Wv d W o d W d W o d W m  d W o dnm ©A d o © c,ª¬« Y­ ª ­   #  ª ¦ ® ~¯ and #  ª  knownf1 (19)  and let ©±° be its complement on the set of integers from to  . v v Also definev A ² and A ²´³ to be matrices formed by selecting the rows of A whose W indices are in the sets © and ©µ° , respectively. The constraint on d o can then be succinctly written as A ² v d W o ~ = (20) v Since A is orthogonal, we have d W o·¶  A ² v –v¹¸ d W oº col span c  A ²gv ³ –v f = W o as d W o A²Y» ¼ for some   Thus, may parameterize d 0 ^   we vector 0 ^ ¼ , where is the number of training symbols, and for notational simplicity we have defined A ²»  A ²v ³  v = W The constraint on d o can be explicitly incorporated into (13), and the minimization is then over the unstructured elements of the vector ¼ : ¼  —e˜–™µš›œ¾½À¿ d W m  A ² » ¼ÂÁ v ¢ (21)  ¡  A  v A  ‘  A v u@ | u  |v A  ¿ d W m  A ²w» ¼ÂÁ The portion of the signal is then just A ² » ¼  . ofAsthewithunknown the blind algorithm developed above, the d W o estimate

For the semi-blind case, we begin with the minimization problem of (13), we replace with , and then minimize with respect to . However, must be constrained so that its inverse FFT, , is equal to zero at samples where is non-zero. Define to be the set of time indices where the training symbols are non-zero:

@u  ^ ¼  x—n˜–™’š›œ¯½ ¿ d W m  A ²w» ¼ Á v h  ¥ - ¨ ^ ¨ ^v j ¿ d W m  A ²Y» ¼ Á  (22) ¨^  ¥ - ¨ ^ ¨ ^v , the where is defined in (18). Defining à solution for the unknown symbols is easily found to be d W o A ² » ¼  - A ² »  A ² » v ÃpA ² » ƒ EHG A ² » v Õd W m = (23) matrix in the brackets may be reformulated using the signal subspace vectors , as follows:

Ã

The matrix inverse in (23) can be considerably simplified using the structure of and the matrix inversion lemma; in the end, matrix is required. As with the only the inverse of an blind algorithm, matrix products involving all or part of can be implemented via the FFT. Consequently, the computational load required to solve (23) is again dominated by the calculation of , and thus both the blind and semi-blind algorithms have roughly the same level of complexity.

T T

A

u ^

6. EXTENSIONS OF THE ALGORITHMS

s




6.1. More Transmit Than Receive Antennas



, then is “fat” and the basic subspace relationship (8) If does not hold. However, the Hankel structure of can be exploited to overcome this difficulty by means of data stacking. Define the column cycle matrix as follows:

>@

Ä   Ä ÆÅÇ¥ ÇÈ   -     vector of zeros. The operation   Ä shifts where Ç is a   to the left by one position, and replaces all of the columns of the  last column with the first. By stacking column shifted versions  of on top of one another, we obtain !!   2 33 É X, Z)[ ! . Ä 3 +Ë@ ÍÌ EG ÎÏ (24) Ê " ..  EG 4 ÄHÊ  ? Ð -‚  block Sylvester matrix (identical Ë where is a bÐ+ Î in form to those obtained in blind equalization problems [17]), É ˆ

Í  Ì is and of size  formed    . toThe, new Ê EHG is a HankelË matrix Ð -Ñidentically “channel” matrix is guaranteed

to be full rank as long as columns provided that

is, and will have no fewer rows than

.-. Ði  -  = The algorithms sections can then be directly apÉ of the previous plied to Òinstead- of$ , although prefix must now sat Í  Ì isfy 6s Ð for Ê EHG theto becyclic circulant. 6.2. Rank Deficient Channels   If the channel is rank deficient,  i.e.,   rank Ó  ÑÓ?§+š›Ôœ ce bf , then in the noiseless case, rank

. If we partition the SVD

 of

u@^ so that

Ó contains the first right singular vectors, then Ó  there exists a full rank  matrix Õ that satisfies u ^ Õ  = (25)

This equation is equivalent to a noiseless version of our original model (1) for a case with more transmit than receive antennas (i.e., is fat). Thus, we may use the approach of Section 6.1. Instead of (24), we apply the algorithms to the matrix

Õ

Ð

!! u@ ^ É X7 Z%[ ! .u ^ Ä " ..u  ^ EHG ÄHÊ

4

3 

(26)

is chosen to satisfy

6.3. Frequency Selective Fading An approach similar to those presented above can be used when the multipath is frequency selective. If the channel is assumed to be FIR with a delay spread covering symbol periods, then the data model of (1) becomes

×   ÙÚ ÛÝÜ Ø  ßÞN  ß ÞN /z (27) G  ßÞNß ÞN is the Þ%àâá matrix tap of the channel impulse response, where ˆ

 and isßÞNa Hankel matrix with rows like (2), except its first Ò .  # i 6  × element is . If the cyclic prefix satisfies , then

 ßÞ Ö , ,  7   ×  will all be circulant as well. All of the rows of ß  N Þ 

 except the last are contained in given

 ßÞN for Þ ' ,,  × -Ö , the next term  ßÞ , and   hence only makes  a rank one contribution to the noise-free portion of . Thus, we may re-write (27) as   Õ ÍÌ EHG / (28) Ø ä â à á  where the ã column of Õ , denoted by Õ m , is given by Û ä Õ m ÛPÜ åLæ ç,Ú è mné ØëÍê Ì ä m Ì EG ßÞN (29) G E G é m ä]î ßÞN denotes columnåLìí,ï è of  ßÞð . ê At this point, if   and-+ × , the algorithms§ñ.of Sections and 5 can be applied with no -Ö ,4then modification. If  × the stacking technique of Section 6.1 is used to create more virtual receive channels. 6.4. Multiple Users

‹

G  7, 2  3 ‡ j   éG .. 5 

.é‡ 4 ‡ Û ñ ÚÛÝÜ = G

Ëô h  ! ó "

where

2 33

Ö- iÐ  Ó_-/ = As before, we require  6   Ð - $ to maintain a circulant code. where

Û Þ 

 domain signal vector associated with user . Letting Þ é represent the Hankel matrix of (2) associated with user , we may also write { +Ëpó‚ (31)

‡ Û @  Ww Ú ÛÝÜ  A nòS\_^ò /?Wp (30) Û Û Û ÛG    d and d W represent the channel, the number of where transmit antennas, the time domain signal vector, and frequency

If (symbol synchronous) users are present, then equation (5) becomes

‹

Note that data obeying (31) could also be generated by a single user whose transmit antennas are divided into groups, with each group employing a different circulant space-time code. Such an approach may be useful in trading off diversity for throughput. To apply the algorithms of the previous sections to (31), we require that be tall (or square). If , data stacking is required as described in Section 6.1, although in this case the stacking factor must be chosen to satisfy

Ð

Ë


‹

Ñ


.Ðs  - ‹‹

¹ Þ àßá Þ Û Û Û Û Û Û Û Û d W o é - A ² » é  A ² » é v à A ² » é ƒ EHG A ² » é v à d W meé  (32) Û Û ¨ Û¨ Û ¨ Û Û    v v u ^ ^ ^ ^ n  ò v , and A ² » é  ¥ A of the “ A training where à , is chosenÞ to coincide with the locations symbols Þ õ' ,B  ‹ will separate for user . Application of (32) for the individual users provided that the matrix h d W meé G ,, d W meé ‡ j and is necessary. In the following discussion, we assume . Application of the semi-blind solufor simplicity that tion in (23) to the multiple user case is straightforward. The unknown portion of the user’s transmitted data can be solved for using (23) customized for user :

is full rank. In general, application of the blind algorithm to the multiple user case can only provide the subspace in which the various users’ signals and their circulantly shifted versions lie. If all of the users share the same number of transmit antennas, then the signals can only be separated from one another in a second step using some type of blind source separation technique (such as [15, 16]) that exploits additional information about the signals. Interestingly, if one of the users has more antennas than any of the others, its signal can be found without a second blind separation step. Assume that user 1 has the largest number of transmit antennas . The signal from this user can be found by calculating the left singular vector of (33)

G

¨ ^ G 'v ö  v u  ^  A “ A with the largest singular value. This because user 1 is the only u| iswith G column shifts (i.e., G  whose G  ,Bsignal G 'isö orthogonal user to   H E G dthis d wayÄ sinced notÄ only).is The other users cannot in u@| withbe resolved n d m orthogonal to m column shifts, so is d G .

0

Mean Squared Symbol Estimation Error

10

tennas”, Wireless Personal Communicaitons, 6(3):311–335, March 1998. [3] G. Foschini, “Layered Space-Time Architecture for Wireless Communication in a Fading Environment when Using MultiElement Antennas”, Bell Labs Tech. J., 1(2):41–59, Autumn 1996.

R=3 −1

10

[4] P. Wolniansky, G. Foschini, G. Golden, and R. Valenzuela, “V-BLAST: An Architecture for Realizing Very High Data Rates Over the Rich Scattering Wireless Channel”, In Proc. IEEE ISSSE Conf., Pisa, Italy, September 1998.

R=8 R=16

−2

10

blind semi−blind

[5] G. Raleigh and J. Cioffi, “Spatio-Temporal Coding for Wireless Communication”, IEEE Trans. on Comm., 46(3):357– 366, March 1998.

−3

10

0

5

10

15 SNR (dB)

20

25

30



Figure 1: Mean square symbol estimation error vs. SNR for various .

7. SIMULATION EXAMPLE A series of simulations was conducted assuming two users with 2 and 3 transmit antennas, respectively. The user signals were random uncorrelated symbol-rate sampled QPSK waveforms, and the elements of the additive noise matrix were unit variance circular complex zero-mean Gaussian random variables, independent across the receive antennas and from sample to sample. The and channel matrices and were also generated with unit-variance, zero-mean, independent Gaussian elements. The signal-to-noise ratio (SNR) at the receive array for user is defined to be SNR



i (

 G

T

. $

ã

m  me÷ mT  m . Theofsignal where ÷ m is the amplitude of the elements ampli SNRT .ofAd block tudes were chosen so that SNR G 50 samples

was collected for each trial, and each experiment involved 1000 independent trials. Of the 50 transmitted symbols, 10 were assumed to be training data to be used by the semi-blind algorithm. The training data were located at symbols 3-12 for user 1, and symbols 4-13 for user 2. Different random channels, user signals, and noise realizations were generated for each trial. The Mean-Squared Error (MSE) of the estimated signals is plotted in Figure 1 as a function of SNR for three different re. For , a stacking factor of ceive array sizes: was employed to build up the dimension of the row signal subspace. No stacking was used for . Only the MSE performance for user 2 is plotted, since both a blind and semiblind estimate were possible for this signal ( ). The solid and dashed lines correspond to the semi-blind and blind estimates for each . The training symbols provide a slight improvement in performance, particularly for , but only at low SNR for . The main benefit of the training data is that it allows the semi-blind algorithm to recover the signal from each user.

Ð ñú

  ûøgBBù

 ( –ø¯B,ù

 (  õøg,Bù T
Ñ G  (

8. REFERENCES [1] I. Telatar, “Capacity of Multiple Antenna Gaussian Channels”, AT&T Technical Memorandum, June 1995. [2] G. Foschini and M. Gans, “On Limits of Wireless Communications in a Fading Environment when Using Multiple An-

[6] A. Naguib, V. Tarokh, M. Seshadri, and A. R. Claderbank, “A Space-Time Coding Modem for High-Data-Rate Wireless Communications”, IEEE J. Sel. Areas in Commun., 16(8):1459–1478, October 1998. [7] V. Tarokh, A. Naguib, M. Seshadri, and A. R. Calderbank, “Space-Time Codes for High Data Rate Wireless Communication: Performance Criteria in the Presence of Channel Estimation Errors, Mobility, and Mulitple Paths”, IEEE Trans. Commun., 47(2):199–207, February 1999. [8] B. L. Hughes, “Differential Space-Time Modulation”, In IEEE Wireless Commun. and Networking Conf., New Orleans, LA, Sept. 1999. [9] V. Tarokh and H. Jafarkhani, “A Differential Detection Scheme for Transmit Diversity”, In IEEE Wireless Commun. and Networking Conf., New Orleans, LA, Sept. 1999. [10] B. Hochwald and W. Sweldens, “Differential Unitary SpaceTime Modulation”, Technical report, Bell Laboratories, Lucent Technologies, March 1999. [11] A. Swindlehurst and J. Gunther, “Methods for Blind Equalization and Resolution of Overlapping Echoes of Unknown Shape”, IEEE Trans. SP, 47(5):1245–1254, May 1999. [12] G. Giannakis and Jr. R. Heath, “Multirate Pre-coding for Blind Channel Equalization in OFDM”, In Proc. 31st Conf. on Info. Sciences & Systems (CISS ’97), pages 769–774, Baltimore, MD, March 1997. [13] A. Swindlehurst and J. Gunther, “Direct Semi-Blind Symbol Estimation for Multipath Channels”, In Proc. 32nd Asilomar Conf. on Signals, Systems, and Computers, 1998. [14] P. Vandaele, G. Leus, and M. Moonen, “A Non-Iterative Blind Signal Separation Algorithm Based on Transmit Diversity and Coding”, In Proc. 33rd Asilomar Conf. Signals, Systems, and Computers, Oct. 1999. [15] S. Talwar, M. Viberg, and A. Paulraj, “Blind Separation of Synchronous Co-Channel Digital Signals Using an Antenna Array. Part I. Algorithms”, IEEE Trans. Sig. Proc., 44(5):1184–1197, May 1996. [16] A. van der Veen and A. Paulraj, “An Analytical Constant Modulus Algorithm”, IEEE Trans. Sig. Proc., 44(5):1136– 1157, May 1996. [17] E. Moulines, P. Duhamel, J.F. Cardoso, and S. Mayrargue, “Subspace Methods for the Blind Identification of Multichannel FIR Filters”, IEEE Trans. Sig. Proc., 43(2):516–525, Feb. 1995.