Further Results on Interference Cancellation and Space-Time Block Codes Anastasios Stamoulis, Naofal Al-Dhahir, and A. Robert Calderbank AT&T Shannon Laboratory, Florham Park, NJ fas,naofal,
[email protected] Abstract—Space-Time Block Codes (STBC) make use of a rich algebraic structure to provide diversity gains with small decoding complexity. In this work, we show that the rich algebraic structure of STBC reduces the hardware and software complexity of interference cancellation (IC) techniques. Additionally, after the IC stage, transmitted symbols can still be recovered with space-time diversity gains. We present three illustrative examples of IC in wireless networks where co-channel users employ STBC. First, we show that any STBC that is based on an orthogonal design allows IC of two co-channel users with simple linear processing. Second, we show that for the Alamouti STBC, 2 users can be detected with simple linear processing, while still ensuring space-time diversity gains. Third, capitalizing on recent work on single-carrier frequency-domain STBC, we study how the aforementioned IC schemes can be modified for frequency-selective channels.
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I. I NTRODUCTION Since their invention, Space-Time Block Codes (STBC) [10] have sparked a wide interest as they promise to significantly increase transmission rates in wireless communications. STBC provide full diversity gains with simple linear processing at the receiver: with N transmit and M receive antennas, STBC achieve the maximum diversity order NM . An extensive body of work on STBC has illustrated their impact on physical layer (PHY) performance (see [2], [7] and references therein). Interestingly, the significant gains at the PHY translate to a measurable impact at both data link, and TCP layers in 802.11 Wireless LANs [8]. Nevertheless, wireless spectrum is very expensive, a fact which motivates techniques that increase network capacity without requiring additional bandwidth. From this standpoint, interference cancellation (IC) techniques are of high interest, because they can potentially allow mobile users to share the same time slots (in TDMA), or the same codes (in CDMA). In such an environment, it has been shown that K cochannel users, where each user is equipped with N antennas, can be detected with N -order diversity gains, if the receiver is equipped with N (K 1)+1 antennas [11]. We underline that this result assumes that the interfering signals are not correlated. However, the number of receive antennas can be reduced if the rich structure of STBC is exploited: only K receive antennas are needed to provide N -order diversity gains and suppress K 1 co-channel space-time users. For the case where two co-channel users apply the Alamouti [3] code (each user is equipped with two transmit antennas), [6] has developed a simple Interference Cancellation (IC) scheme. Using two receive antennas, [6] shows that, under minor conditions, the diversity order is the same as that of the Alamouti code (i.e., as if only one user was transmitting, and the receiver was equipped with only one antenna). Effectively, [6] showed that it is possible to double the system capacity (in terms of number of users) by applying linear processing at the receiver without sacrificing space-time diversity gains. In this work, we illustrate that, in general, IC of STBC-based transmissions can be carried out with small requirements both in hardware (in terms of number of receive antennas), and in software (only simple linear processing is required). We extend the findings of [6] along three directions. First, we show that simple linear processing can achieve IC of two-cochannel users who employ any STBC based on an orthogonal design. Second, we show that for the Alamouti STBC, K > 2 users can be detected with simple linear processing, while still ensuring space-time diversity gains. Third, capitalizing on recent work on single-carrier frequency-domain STBC, we study how the aforementioned IC schemes can be extended to frequency-selective channels. Throughout the development, illustrative simulations demonstrate the
,
,
merits of our approach. II. M ODEL D ESCRIPTION We provide a brief overview of orthogonal designs and how they are linked to wireless communications through space-time codes. From a high-level viewpoint, an orthogonal design is a matrix whose rows are the symbols to be transmitted at each time slot, and columns are the symbols to be transmitted by a specific antenna. Given a set of real indeterminates x1 ; x2 ; : : : ; xs , a generalized real orthogonal design N is an P N orthogonal matrix (s P ) with entries 0; x1 ; x2 ; : : : ; xs satisfying TN N = ( si=1 x2i ) N . A generalized real orthogonal design can also be written as N = x1 1 + x2 2 + + xs s ; with the i ’s being P N matrices. From the definition of a real orthogonal design, it follows that: Ti j = Tj i , i = j , and Ti i = N , 1 i s. The rate of an orthogonal design is given by R = s=P (see [4], [10] for a detailed treatment of the subject). An example of s = 4, P = 4, N = 3 is the the 4 3 design
f
O
f
A AA
P O O I O A A AA I
A ,A A 6
g
g
0 x O = BB@ ,,xx
1 2
3
3
,x
4
1
x2 x3 x1 ,x4 C C x4 x1 A : ,x3 x2
A
As it will become clear later on, the fact that the entries of i are in 1; 0; 1 greatly simplifies the linear processing involved in IC. A complex orthogonal design is an orthogonal matrix with complex entries (see [4] for details). The simplest complex orthogonal design is
f,
g
x x the 2 2 Alamouti code ,x x . A rate-1/2 generalized comc plex orthogonal design O N can be constructed from a real orthogonal design as O N OcN = conj( (1) ON ) ; 1
2
2
1
where the operator conj( ) replaces each entry of a matrix/vector with its conjugate. The particular construction cN is the focus of this paper, but it is important to note that there are complex designs not of the aforementioned forms [4]. Next we revisit how orthogonal designs form the basis of STBC in a wireless communications set-up.
O
A. Channel Model
The transmitter is equipped with N and the receiver is equipped with M antennas. At each time slot t, the N antennas transmit symbols Ct;i , 1 i N , which arrive at the M receive antennas after going through flat-fading channels. Denoting by hm (n) the channel gain between the n-th transmit and the m-th receive antenna, at time t the received signal at antenna m is
rm (t) =
N X
n=1
hm (n)Ct;n + m (t) ;
where m (t) are samples of AWGN. The channels are assumed to be quasi-static, i.e., constant over a frame of symbols. As in [10], the flat fading coefficients are modeled as i.i.d. complex Gaussian random variables with variance 0.5 per real dimension, the noise is modeled as a zero-mean complex Gaussian process with variance 1/(2SNR) per
real dimension, and the transmitted symbols have average energy 1=N (as a result, the signal-to-noise ratio at the receiver is SNR). With an N design as the basis for the STBC, at time 1, source symbols x1 ; : : : ; xs are available at the transmitter, and populate the entries of N ; at each time slot, the transmitted symbol t;n is set to: t;n = [ N ]t;n , 1 t P , 1 n N . During the time slots 1; : : : ; P , the receiver collects PM symbols which are used for decoding. The orthogonality of the columns of N reduces the decoding complexity, as the maximum likelihood-detection of xi (1 i s) depends only on the received symbols rm (t) (1 m M , 1 t P ), the path gains, and the structure of N [10].
O O O
C
C
O
O
III. STBC AND IC OF TWO CO - CHANNEL USERS We focus on rate-1/2 generalized complex orthogonal designs, and we study IC of two-cochannel users. We prove that only two receive antennas are required, and, after the IC stage, symbols are recovered with space-time diversity gains. For simplicity, we focus on the case where only one receive antenna is available (M = 1). Suppose there is only one user which uses an STBC based on a 2P N generalized complex orthogonal design cN . With = (h(1); : : : ; h(N ))T denoting the path gains, the received signal can be written as
O
h
r rP
1:
P
+1:2
,Ps x A h = ,Psi xi Aii h + ; i i =1
P
(2)
=1
with a 2P 1 vector holding the independent samples of AWGN. After conjugating the last P entries, we arrive at
r:= conj(rrPP 1:
+1:2
X s =
P)
i=1
Aih aixi + ~ ; ai:= Aiconj( h) :
(3) The formulation of (3) naturally leads to the decoding strategy. Capitalizing on the identities
ai aj = 0 ; i 6= j ; ai ai = 2 ,jh(1)j + + jh(N )j = 2jjhjj ; 8i ; 2
2
2
s X
j =1 2 2jjhjj xi + ~i ;
=
~i :=a ~ : i
jjhjj
g
(5)
A
h h
Ah Ps i m r = rr = Psii aa ;i;i xxii + ; am;i:= Aiconj( hm ) ; 1
2
=1
2
1
=1
2
=1
1
=1
2
the y ’s), it is not surprising that we can recover the transmitted symbols. By proper re-arrangement of terms, (7) can be cast in the form
r = F(x : : : xs y : : : ys)T + ; 1
F
1
with a 4P 2s matrix which is a function of the path gains and the structure of cN . What we develop in this section is a simple scheme which i) avoids the inversion of , and ii) ensures diversity order N for the recovery of the transmitted symbols. In our approach, symbol recovery is done in two steps: first, IC is performed using a linear transformation, and, second, symbols are recovered in manner similar to that delineated by (5). For the IC step, we design two (2P ) (2P ) matrices , which decouple the two users, by satisfying the following criterion
O
F
WZ
~r :=r , Wr ~r :=r , Zr 1
1
2
2
does not depend on y1 ; : : : ; ys does not depend on x1 ; : : : ; xs :
2
1
(8) (9)
Given the symmetry of the problem, let us focus on the design of From (7), we easily see that (8) is satisfied if is such that
s X
2
a
=1
=1
2
!
b ;iyi , W 1
s X i=1
W !
b ;i yi = 0 P : 2
2
W. (10)
1
W = 2jjg1 jj , b ; : : : b ;s , b ; : : : b ;s :
aj xj + ai ~ a
1
1
It is easily verified that (10) is satisfied by
It is not surprising that the decoding of symbol xi is accomplished using the matched-filter i . One could view i as a “code sequence” which spreads the transmitted symbol in both time and space. As the i ’s are orthogonal, the matched filter is the optimal receiver1 . Consequently, (5) forms the basis of a maximum-likelihood detecting rule for all transmitted symbols (note that ~i , ~j are independent – a direct 2 consequence of (4)). Furthermore, the factor represents the diversity gains provided by the space-time code. We also note that as i has entries in 1; 0; 1 , all calculations are simplified. Suppose that the receiver has two antennas. With the obvious extension of notation, let us denote by 1 , 2 , the channel vectors corresponding to the path gains from the transmit antenna to the first and the second receive antenna respectively. If only one user is present, (3) becomes:
f,
g g
Ps r = rr = Pisi aa ;i;ixxii (7) Ps + Psi bb ;i;i yyii + ; i Ag i m where bm;i := Aiconj(gm) , m = 1; 2, i = 1; : : : ; s. As r is (4P ) 1, and the total number of unknowns is 2s (the x’s and
i=1
!
a
O
(4)
the decoding of xi is accomplished by forming
r~i := ai r = ai
r r
where 1 , 2 hold the signals received by the two antennas during the time slots 1; : : : ; 2P (after conjugating the last P entries of each vector). Now, let us introduce a co-channel user who transmits symbols y1 ; : : : ; ys using the same cN -based STBC. With 1 , 2 , representing the path gains from the transmit antennas of the second user to the two receive antennas, the received signal is:
1 A similar observation on the optimality of the matched filter has also been made in [5].
(6)
11
2
1
21
2
(11) The structure of is not surprising. The “second” part of , ( 2;1 : : : 2;s ) , can be thought of as a bank of matched-filters which recover the y ’s; the “first” part, ( 1;1 : : : 1;s ), scales them properly so that (10) is satisfied (note also the simplicity in the construction of as all i ’s have entries in 1; 0; 1 ). Apart from satisfying (8), possesses another, equally important property: it allows the recovery of the transmitted symbols x1 ; : : : ; xN with simple linear processing. From (8) and (10), we obtain
b
W
b
W
b
A
f,
W
~r = 1
s X
n=1
b
W
g
i xi + ~1 ; i :=(a1;i , Wa2;i ) :
It can be proved (see appendix) that the “spreading sequences” orthogonal, i.e.,
(12)
i are
i j
= 0; i 6= j : (13) As a result, similar to (5), the recovery of xi is made possible by apply-
ing the matched filter:
~r1;i := i ~r1 = i
s X i=1
xi i + ~1
= jj i jj2 xi + noise :
! (14)
−1
importance for our development later on, let us take a closer look at the set of 2 2 matrices
10
One User, 1 Rx One User, 2 Rx Two Users, 2 Rx
h h o H ; := H : H = ,h h ; h ; h 2 C : n
1
(1 1)
−2
10
2
2
1
1
(17)
2
P
b
H
It is straightforward to verify that (1;1) is closed under addition, multiplication, multiplication with a scalar, and inversion (whenever the inverse exists). In next section we will see how these properties of (1;1) will prove to be very useful in interference cancellation using array processing 2 . Let us also define the set (M;N ) of 2M 2N matrices whose every 2 2 submatrix (which contains rows (2m 1; 2m), and columns (2n 1; 2n), 1 m M , 1 n N ) belongs to (1;1) .
H
−3
10
,
−4
10
H
,
H
B. Two Users, Two Receive Antennas −5
10
0
1
2
3
4 SNR (dB)
5
6
7
8
When the receiver is equipped with two antennas, the second antenna can be used to eliminate a co-channel interferer [7]; to see how this is done, let us introduce some notation, and then revisit the framework of [7]. Users 1,2 apply the STBC to the input symbols 1 = x1 (1) x1 (2) T , 2 = x2 (1) x2 (2) T respectively. Let us denote by hk;n (m) the coefficient of the flat fading channel from the m-th antenna of user n to the k-th receive antenna. The received signals on the first receive antenna are can be written as
,
Fig. 1. IC in Complex Orthogonal Designs
jj jj
The factor i 2 denotes the diversity gain (note that the proof for generalized real orthogonal designs follows similar lines). Illustrative Simulation: We simulate the case where 2 co-channel users apply the c3 -based STBC (as given in Equation (37) of [10]); each user is equipped with 3 transmit antennas, QPSK modulation is employed, and the users are equi-powered. The receiver has 2 antennas, and we use Rayleigh fading to model the path gains: each path gain is modeled as a complex Gaussian random variable with variance 0.5 per real dimension – furthermore, we assume that all channels are independent. Fig. 1 depicts the bit-error-rate (BER) performance of this system. We verify that our IC scheme is capable of separating the two users, and then recovering the transmitted symbols with space-time diversity gains.
O
IV. A LAMOUTI CODE : I NTERFERENCE C ANCELLATION FOR N
N RECEIVE ANTENNAS we discuss how N co-channel users (which use the USERS USING
In this section, Alamouti code) can be detected with space-time diversity gains using M = N receive antennas.
x
1
1
2
2
2
1
(15)
r h h n r:= ,r = Hx + n ; H:= ,h h ; n:= n ; 1
2
2
2
1
1
2
(16)
with n1 , n2 independent AWGN processes each with variance No =2 per dimension. Under the assumption of constant channels (over the transmission of two symbols), it is easily seen that is orthogonal, i.e., = ( h1 2 + h2 2 ) 2 . With := r, := , the transmitted symbols can be obtained from: = ( h1 2 + h2 2 ) + : This simple linear processing greatly simplifies the decoding rule, and the x1 , x2 can be recovered independently of each other because has i.i.d. entries [3]. Essentially, the power of Alamouti’s STBC (generalized in [10]) lies in the special form of the resulting channel matrix . Because of its
HH
j j
j j I
H ~r H n~ H n ~r j j j j x n~ n~ H
11
1
11
11
1
2
11
12
12
1
12
11
11
12
12
1
1
12
1
1
n1 (2) denote the AWGN at the two time instants.
Along the same
lines, we obtain for the second receive antenna
r (1) , x r := ,r(2) = H ; H ; x +n ; h ; (1) h ; (2) where H ; := ,h ; (2) h ; (1) , h (1) (2) , n := n (1) . H ; := ,h ;; (2) hh ;; (1) n (2) 2
2
21
2
21
21
2
2
(19)
21
22
22
1
22
21
21
22
22
2
2
22
2
By grouping together the received signals, we finally obtain:
r H H x n = H ;; H ;; + ; r | {z } | {z } | x{z } | n{z } 1
11
12
1
1
2
21
22
2
2
r(2)
x(2)
H(2)
r = H x +n
(20)
n(2)
H H H
1;1 , 1;2 , 2;1 , (2) (2) (2) (2) . Clearly, and it can be seen that (2) has i.i.d. entries. Though the notation is slightly complicated (as we need to denote user, transmit antenna, receive antenna, and transmitted symbol), the point to be remembered from (20) is that: we have two users, and each user has 2 transmit antennas; at the receiver we have two receive antennas, which collect symbols over two successive time instants (we assume sampling at the symbol rate). Hence, (2) , (2) , (2) are 4 1 vectors, whereas the channel matrix (2) is 4 4. The recovery of the transmitted symbols can be tackled as a multiuser detection problem (i.e., without paying attention to the special structure of (2) ). A clever way of recovering the transmitted symbols
which yields:
space time
The received signals (for two consecutive time slots) can be written as 1
1
1
x
x x x ! x ,! ,x x #
x
r (1) , x r := ,r(2) = H ; H ; x + n ; (18) h ; (1) h ; (2) where H ; := ,h ; (2) h ; (1) , h (1) (2) , n := n (1) , and n (1), H ; := ,h ;; (2) hh ;; (1) n (2)
A. Overview of the Alamouti Code With two transmit antennas, Alamouti’s Space-Time Block Code [3] groups the input symbols into groups of two symbols, = (x1 x2 )T , which are fed to the space-time block encoder:
,
H ; 2H 22
;
(1 1) ,
n
H
x r n
H
2 The underlying algebraic structure of the Alamouti code is exactly what enables IC: quaternions can be viewed as pairs of complex numbers, where the product of quaternions ( 1 2 ), ( 3 4 ) is defined 2 1 4 + 2 ). If we map the quaternion ( 1 2 ) as ( 1 2 )( 3 4 ) = ( 1 3 4 3
x ;x x ;x
to the 2
2 matrix
x ;x x ;x x x , x x! ; x x x x x ;x x1 x2 , then quaternion multiplication coincides with matrix mul , x2 x1
H
tiplication. Quaternions form a normed algebra – consequently, the fact that (N;N ) is closed does not come as a big surprise (see [4] for details on the algebraic structure of STBC).
0
while ensuring space-time diversity gains has been developed in [7]. By applying the block linear filter
10
One User, 1 Rx One User, 2 Rx One User, 3 Rx Two Users, 2 Rx Three Users, 3Rx
, W = ,H I; H,; ,H I; H ; (21) to r, we obtain the interference free y (H ; , H ; H, H ; )x ; = Wr = y (H ; , H ; H,; H ; )x + Wn : As H ; , H ; H,; H ; , H ; , H ; H,; H ; 2 H ; , the transmitted vectors x , x can be recovered with space-time diversity gains. Note that the matrix H plays the role of the correlation matrix in multiuser detection for CDMA so that W plays the role of decorrelat1
(2)
2
11
1 22
12
1
21
2
11
12
22
21
22
1 22 1 11
1 11
21
21
1
12
2
−1
10
(2)
12
b
21
1 22
12
1 11
P
2
(1 1)
2
(2)
ing detector. Next we extend this technique to the case of M
−3
10
= N > 2.
C. IC for N
> 2 co-channel users With N > 2 users and M = N receive antennas, (20) is extended in a
0 BB B@ |
1 0 H ; H ;N C B .. .. .. C B . . . = C B A @ } | HN; {z HN;N
straightforward manner to:
r r
1
11
2
.. .
2
r{zN
1
1
N 1
N 2N
2
10 CC BB CA B@ }|
x x
1 2
.. .
x{zN
N 1
2
1 0 CC BB CA + B@ } |
n n
1 2
.. .
n{zN
N 1
2
−4
10
H x
0
1
2
3
4 SNR (dB)
5
6
7
8
Fig. 2. IC of Three Users
1 C C C A; }
V. IC IN F REQUENCY-S ELECTIVE C HANNELS Without imposing an unbearable burden on the computational complexity, the aforementioned IC schemes can be modified for frequencyselective channels building on recent work on space-time single-carrier transmissions [1]. Let us start from the single-user case, and denote the nth symbol of the kth transmitted block from antenna i by x(ik) . At (k ) (k) times k = 0; 2; 4; : : : pairs of length-N blocks 1 (n) and 2 (n) (for 0 n N 1) are generated by the mobile user. Inspired by Alamouti STBC, we encode the information symbols as follows [1]
(22) or (N ) = (N ) (N ) + (N ) . Our idea is to iteratively recover the symbols of each user starting from the N -th user, proceeding to the (N 1)-th user, and eventually to the first user. To recover the symbols of the N -th user we partition the matrix (N ) as
r
−2
10
n
,
x
,
x
H A B x k (n) = ,x k ((,n)N ) and xxk (n) = x k ((,n)N ) H N = , NN NN ; for n = 0; 1; : : : ; N , 1 and k = 0; 2; 4; : : : with A N , B N , , N , N denoting respectively the 2(N , 1) (24) 2(N , 1) upper-left, 2(N , 1) 2 upper-right, 2 2(N , 1) lower-left, where ()N denotes the modulo-N operation. In addition, a cyclic preand 2 2 lower-right submatrices of H N . fix (CP) of length (the maximum order of the FIR wireless channel) In a manner similar to (21), let us define the block linear filter is added to each transmitted block to eliminate inter-block interference , ! (IBI) and make all channel matrices circulant. At the receiver, the CP I , B N , N N W N = ,,N A, ; part of each received block is discarded to eliminate IBI. The resultI N ing length-N blocks are then processed in pairs where they are first transformed to the frequency domain using the Fast Fourier Transform which is applied to r N to yield: ! (FFT), resulting in the two blocks r (A N , B N ,N , N )x N , N, yN = ( N , , N A,N B N )xN +W N n N : Y k X k ! = , +noise ; (25) k (23) ,Y k X | {z } | {z } It readily follows that the 2(N , 1) 1 vector r N , will be used | {z } Y X for the recovery of the symbols of users (N , 1); : : : ; 1, whereas the symbols of the N -th user will be recovered from yN . k and X k are the FFTs of the information blocks x k In the appendix, we prove that N , , N A,N B N 2 H ; , where X k , and A N , B N N , N 2 H N , ;N , . Consequently, we can and x , respectively. Since circulant matrices are diagonalized by recover xN from yN with space-time diversity gains. Furthermore, the FFT, inter-carrier interference is eliminated and , are diagowe can iteratively apply the same procedure to r N , to recover the nal matrices containing the FFT coefficients of the underlying wireless channels. To eliminate inter-antenna interference, the linear combiner symbols of the remaining (N , 1) users. is applied to Y. Due to the orthogonal structure of , a second(
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Illustrative Simulation: We simulate the case where 3 co-channel users apply the Alamouti code. Each user is equipped with 2 transmit antennas, QPSK modulation is employed, and the users are equi-powered. The receiver has 3 antennas, and we use Rayleigh fading to model the path gains: each path gain is modeled as a complex Gaussian random variable with variance 0.5 per real dimension – furthermore, we assume that all channels are independent. Fig. 2 depicts the bit-error-rate (BER) performance of this system. Similar to Fig. 1, we verify that our IC scheme is capable of separating the three users, and then recovering the transmitted symbols with space-time diversity gains.
order diversity gain is achieved. Then, the two decoupled blocks at the output of the linear combiner are equalized separately using the MMSE Frequency-Domain-Equalizer which consists of N complex taps that mitigate inter-symbol interference. Finally, the MMSE-FDE output is transformed back to the time domain using the inverse FFT where decisions are made. Equation (25) constitutes the basis for IC of STBC in frequencyselective channels. The frequency-selective channels are transformed to flat-fading taps on the FFT grid. Consequently, the IC techniques of Section IV-C can be readily applied in the frequency domain (a similar
WT W = (scalar)IN , (26) amounts to showing tr((WAi h )(Aj h )T ) + tr((Ai h )(WAj h )T ) = 0 :
0
Using (4), and that
10
One User, 1Rx One User, 2Rx Two Users, 2Rx
2
−1
1
1
2
10
The latter equation holds because
tr((WAi h2 )(Aj h1 )T ) + tr((Ai h1 )(WAj h2 )T ) = = tr(Ai ATj Wh2 hT1 ) + tr(Ai ATj h1 hT2 WT ) = tr(ATj Ai (Wh2 hT1 )) + tr((h1 hT2 WT )T Aj ATi ) = tr(ATj Ai (Wh2 hT1 )) + tr(ATi Aj (Wh2 hT1 )) =0;
−2
P
b
10
−3
10
AA
,ATi Aj .
as Tj i = straightforward.
−4
10
The extension to complex channel vectors is
−5
10
0
1
2
3
4 SNR (dB)
5
6
7
III. IC WITH S PACE -T IME G AINS
8
Section IV-C asserts that
A(N ) , B(N ) ,1 ,(N )
Fig. 3. IC of Two Users in HiperLan/2
(
technique, but for only two co-channel users has appeared in [9] for OFDM transmissions). Illustrative Simulation: We simulate the case where 2 co-channel users apply the Alamouti code. Each user is equipped with 2 transmit antennas, QPSK modulation is employed, and the users are equi-powered. The receiver has 2 antennas, and we use the HiperLan/2 type-A channels to model the path gains – furthermore, we assume that all channels are independent. Fig. 3 depicts the bit-error-rate (BER) performance of this system; as in the previous cases, we observe that our IC cancellation technique is capable of separating the co-channel users without sacrificing performance. VI. C ONCLUSIONS In this paper we have showed how the rich structure of Space-Time Block Codes (STBC) enables interference cancellation (IC) with simple linear processing. In particular, we have showed that STBC based on orthogonal designs allow IC of two co-channel users. Second, we have showed that for the Alamouti STBC, K > 2 users can be detected with simple linear processing, while still ensuring space-time diversity gains. Third, exploiting recent work on single-carrier frequencydomain STBC, we have illustrated how the aforementioned IC schemes can be modified for frequency-selective channels. A PPENDIX I. O RTHOGONALITY OF
ai
Proving that (4) holds involves simple algebraic manipulations, and the use of the following property of the trace tr( ) operator: tr( )= tr( ) = tr(( )T ), where , are square matrices with appropriate dimensions. Suppose that the entries of are real. Then, T i , we obtain tr( T j T ) = starting from Ti j = j i
AB AB AB h A A ,A A A A hh ,tr(ATj Ai hhT ) ) tr(ATi Aj hhT ) = ,tr((ATj Ai hhT )T ) ) tr((ATi Aj )(hhT )) = ,tr((hhT )(ATj Ai )) ) tr(hhT ATi Aj ) = 0 ) tr(Aj hhT ATi ) = 0 ) (Aj h)T Ai h = 0. The orthogonality of the ai ’s follows after some algebra if we write h = Re(h) + p ,1Im(h).
BA
II. O RTHOGONALITY AFTER IC Assuming that the entries of all channel vectors are real, by expanding (13) we obtain:
(Ai h1 , WAi h2 )T (Aj h1 , WAj h2 ) = 0 :
(26)
N)
N , , N A,N B N 2 H (
)
(
2 H N , ;N , (
1
1 )
( ) (1;1) , and . This is a direct result of 1)
)
(
the following (n;n) is closed under addition, multiplication, and inversion; (m;n) is closed under addition; if H1 (m;n) and H2 (n;k) , then H1 H2 (m;k) . Closeness under addition and multiplication are straightforward to verify – herein we sketch how closeness under inversion can be proved. Consider (n;n) , and the formula on the inverse of block matrices
H H
2H
2H
2H
H2H A D , = C B , D(B , CA, D), (A , DB, C), , A ; ,(B , CA, D), CA, (B , CA, D), with A, D, C, B denoting respectively the 2(N , 1) 2(N , 1) upper-left, 2(N , 1) 2 upper-right, 2 2(N , 1) lower-left, and 2 2 lower-right submatrices of H. The proof that H, 2 H n;n 1
1
1
1
1
1
1
1
1
1
follows by induction on n.
1
1
(
)
R EFERENCES [1]
N. Al-Dhahir. Single-carrier frequency-domain equalization for space-time block-coded transmissions over frequency-selective fading channels. IEEE Communications Letters, 5(7):304–306, July 2001. [2] N. Al-Dhahir, C. Fragouli, A. Stamoulis, W. Younis, and A.R. Calderbank. Space-time coding for broadband wireless transmission. IEEE Communications Magazine, (to appear). [3] S.M. Alamouti. A simple transmit diversity technique for wireless communications. IEEE Journal on Selected Areas in Communications, 16(8):1451–1458, October 1998. [4] A.R. Calderbank and A.F. Naguib. Orthogonal designs and third generation wireless communication. In J.W.P Hirschfeld, editor, Surveys in Combinatorics. London Mathematical Soc. Lecture Notes, 288, Cambridge Univ. Press, Cambridge, UK, 2001. [5] G. Ganesan and P. Stoica. Space-Time block codes: A maximum SNR approach. IEEE Trans. on Information Theory, 47(4):1650–1656, May 2001. [6] A.F. Naguib, N. Seshadri, and A.R. Calderbank. Applications of space-time block codes and interference suppression for high capacity and high data rate wireless systems. In Proc. 32nd Annual Asilomar Conference on Signals, Systems and Computers, pages 1803–1810, Pacific Grove, California, November 1998. [7] A.F. Naguib, N. Seshadri, and A.R. Calderbank. Increasing Data Rate over Wireless Channels: Space-Time Coding and Signal Processing for High Data Rate Wireless Communications. IEEE Signal Processing Magazine, 17(3):76–92, May 2000. [8] A. Stamoulis and N. Al-Dhahir. 802.11 network throughput gains due to space-time block codes. October 2002. AT&T Technical Document no. 53ENAD. [9] A. Stamoulis, Z. Liu, and G.B. Giannakis. Space-time coded generalized multicarrier CDMA with block-spreading for multirate services. In Allerton Conference on Communication, Control, and Computing, volume 2, pages 1076–1085, Monticello, Illinois, 2000. [10] V. Tarokh, H. Jafarkhani, and A.R. Calderbank. Space-time block codes from orthogonal designs. IEEE Trans. on Information Theory, 45(5):1456–1467, July 1999. [11] J.H. Winters, J. Salz, and R.D. Gitlin. The impact of antenna diversity on the capacity of wireless communication systems. IEEE Trans. on Communications, 42(2):1740 –1751, FEBRUARY/MARCH/APRIL 1994.
