IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

[10] Q. Yan and R. S. Blum, “Optimum space-time convolutional codes,” in Proc. IEEE Proc. IEEE Wireless Communications and Networking Conf. (WCNC-00), 2000, pp. 1351–1355. [11] R. Wesel, X. Liu, and W. Shi, “Trellis codes for periodic erasures,” IEEE Trans. Commun., vol. 48, pp. 938–947, June 2000. [12] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 55–67, Jan. 1982. [13] C. Köse and R. Wesel, “Code design metrics for space-time systems under arbitrary fading,” in Proc. IEEE Int. Conf. Communications (ICC-01), 2001, pp. 1099–1103. [14] S. G. Wilson and Y. S. Leung, “Trellis coded phase modulation on Rayleigh channels,” in Proc. IEEE Int. Conf. Communications (ICC-87), June 1987, pp. 739–743. [15] D. Divsalar and M. K. Simon, “The design of trellis coded MPSK for fading channels: Performance criteria,” IEEE Trans. Commun., vol. 36, pp. 1004–1012, Sept. 1988. [16] R. D. Wesel, “Trellis code design for correlated fading and achievable rates for Tomlinson-Harashima precoding,” Ph.D. dissertation, Stanford Univ., Stanford, CA, Aug. 1996. [17] J. Shi and R. D. Wesel, “Further error event diagram reduction using algorithmic techniques,” presented at the IEEE International Conference on Communications, Anchorage, AK, 2003. [18] C. Schlegel, Trellis Coding. Piscataway, NJ: IEEE Press, 1997. [19] S. Sandhu, R. Heath, and A. Paulraj, “Space-time block codes versus space-time trellis codes,” in Proc. IEEE Int. Conf. Communications (ICC-01), 2001, pp. 1132–1136. [20] W. Zhu and M. P. Fitz, private communication.

Noncoherent Sequence Detection of Differential Space–Time Modulation Cong Ling, Student Member, IEEE, Kwok H. Li, Senior Member, IEEE, and Alex C. Kot, Senior Member, IEEE

Abstract—Approximate maximum-likelihood noncoherent sequence detection (NSD) for differential space–time modulation (DSTM) in time-selective fading channels is proposed. The starting point is the optimum multiple-symbol differential detection for DSTM that is characterized by exponential complexity. By truncating the memory of the incremental metric, a finite-state trellis is obtained so that a Viterbi algorithm can be implemented to perform sequence detection. Compared to existing linear predictive receivers, a distinguished feature of NSD is that it can accommodate nondiagonal constellations in continuous fading. Error analysis demonstrates that significant improvement in performance is achievable over linear prediction receivers. By incorporating the reduced-state sequence detection techniques, performance and complexity tradeoffs can be controlled by the branch memory and trellis size. Numerical results show that most of the performance gain can be achieved by using an -state trellis, where is the size of the DSTM constellation. Index Terms—Differential space-time modulation (DSTM), multiplesymbol differential detection, noncoherent sequence detection (NSD), time-selective fading, Viterbi decoding.

I. INTRODUCTION Differential space-time modulation (DSTM) is an extension of the standard single-antenna differential modulation scheme to Manuscript received October 30, 2000; revised June 25, 2003. This work was supported in part by the Singapore Millennium Foundation. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]; [email protected]; [email protected]). Communicated by B. Hassibi, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2003.817452

2727

multiple-antenna systems, which allows noncoherent detection and promises significant performance gain in fading channels. The constellations of DSTM can be classified as group versus nongroup. The design [1] based on Alamouti’s two-antenna orthogonal code [2] generally yielded nongroup constellations. Using the tool of the group theory, Hughes [3] and Hochwald and Sweldens [4] designed DSTM for an arbitrary number T of transmit antennas. An appealing feature of the group design is that matrix multiplication may be replaced by addition and table lookup. Besides, the DSTM constellations can be classified as diagonal versus nondiagonal ones. In diagonal DSTM, the design of constellations was simplified considerably [4], however, the diagonal restriction has negative impacts on performance and transmitting amplifiers. It is known that diagonal restriction may result in performance loss [4]. Furthermore, since signal transmission at each antenna is discontinuous in diagonal DSTM, the instantaneous power is T times larger compared to the case in which powers are otherwise distributed on T amplifiers evenly. This requires an amplifier with a large linear region, which is often unwanted. Accordingly, it is sometimes desirable to use nondiagonal constellations. For more on constellation designs of DSTM, see, e.g., [5], [6]. The standard detector for DSTM is the differential detector in which decisions are made over two consecutive observation intervals. If the fading process keeps constant in the duration of two DSTM symbols, the performance penalty of differential detection versus coherent detection is merely 3 dB. However, the mobile radio channels are characterized by time-selective fading due to Doppler spreading. In such channels, differential detection suffers an irreducible error floor, analogous to differential phase-shift keying (DPSK). To mitigate the flooring effect, a number of receiver structures outperforming the differential detector have been developed. These structures applied the reception techniques originally devised for single-antenna systems that basically fall into three categories: block detection, decision-feedback detection (DFD), and sequence detection. Schober and Lampe [7] presented multiple-symbol differential detection (MSDD) for DSTM, of which the computational complexity is exponential in the observation length. To overcome the computation burden, DFD based on linear prediction has been proposed [7]. In addition, Chiavaccini and Vitetta [8] derived sequence detection of DSTM signals by means of the Viterbi algorithm, where the branch metric contains a linear predictor similar to that in DFD. Except the exhaustive MSDD scheme [7], an essential limitation of the afore-mentioned popular linear prediction receivers is that they lack optimality in continuously fading channels for nondiagonal constellations. This is the major motivation of this correspondence. Diagonal constellations were assumed in many works on DFD such as [7] to simplify the receiver design. Specifically, the structure of the linear predictor is the same as that for DPSK, i.e., a time-invariant linear filter subject to an adverse effect that the Doppler frequency shift is multiplied by T . Others, like [8], considered nondiagonal constellations, mainly Alamouti’s two-antenna code, but made an assumption that the fading process is invariant during a DSTM symbol so as to derive the linear predictor. Though the accuracy of this assumption is acceptable in slow fading, the temporal variation is no longer negligible in fast fading. Our recent investigation showed that the ignorance of fading variation might incur severe performance loss in DFD for nondiagonal constellations in fast-fading channels [9]. In this correspondence, we propose approximate maximum-likelihood noncoherent sequence detection (NSD) of DSTM in time-selective fading channels. The philosophy of NSD was originally developed by Colavolpe and Raheli [13, p. 1483] for approximate optimum noncoherent reception of coded linear modulation signals in single-an-

M

M

M

M

0018-9448/03$17.00 © 2003 IEEE

2728

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

tenna additive white Gaussian noise (AWGN) or slowly fading channels. This approach “moves beyond the crude block detection found in this area, to more natural, logical, and optimal trellis-based detection schemes.” Inspired by Colavolpe and Raheli, we employ the idea of NSD in this correspondence to overcome the limitation of linear-predictive receivers for DSTM, while maintaining practical implementation complexity. The starting point of our proposed scheme is a new version of the optimum MSDD of DSTM given in [7] for flat-fading channels. By truncating the memory of the optimum metric increment, a finite-state trellis for Viterbi decoding is defined in terms of the information symbols. The correspondence is organized as follows. The system model is built up in Section II. NSD of DSTM is introduced in Section III. The bit-error rate (BER) performance is analyzed in Section IV, and compared with the linear prediction Viterbi receiver in [8]. Section V presents reduced-complexity NSD and the associated performance evaluation. A brief comparative study of NSD, MSDD, and DFD is given as well. Finally, conclusions are drawn in Section VI. II. SYSTEM MODEL Consider a multiple-antenna communication system over a flatfading channel, where data are sent from T transmit antennas to R receive antennas. In DSTM, signals fed to the transmission antennas are grouped into an T 2 T matrix S [n] whose row indexes represent different antennas and column indices represent time instants nMT ; . . . ; nMT + MT 0 1. The matrix is properly normalized so that the average power of each column is one. The total transmitted power therefore does not depend on the number of transmit antennas. A rate-R unitary DSTM system contains L = 2RM unitary matrices of size MT 2 MT . Each matrix is drawn from a set

M

M

M

M

G = fG0 ; G1 ; . . . ; GL01 g;

GH l Gl

=

IM

which is also called a constellation [3], [4]. Every RMT bits to be transmitted at time instant nMT are mapped to an L-ary symbol a[n], which then selects a signal matrix from G as G [n] = G a[n] . Before transmission takes place, the matrices G[n] are differentially encoded in a fashion similar to DPSK

S [n] = S [n 0 1]G[n];

S [0] = A

p1

2

x1 x2

0x32 x31

where um 2 f0; 1; . . . ; L 0 1g. Optimal values of um for R = 1, 2, and MT up to 5 are tabulated in [4] through exhaustive search. In this correspondence, we consider a time-selective fading channel in accordance with the Jakes model [11]. The fading processes hij (t) for i = 0; . . . ; MT 0 1; j = 1; . . . ; MR 0 1 are assumed to be complex normal CN (0; 1) and spatially independent. The autocorrelation of a generic fading process h(t) is given by

[k] = E [h(t)h3 (t + kT )] = J0 (2fd kT ) where J0 (:) is the zeroth-order Bessel function of the first kind, fd is the maximum Doppler frequency shift, and T is the duration of each column of G [n] (hence, the duration of an overall DSTM symbol is MT T ). Likewise, the noise samples wj [k] for j = 0; . . . ; MR 0 1 are assumed to be independent across both time and space, and are identically CN (0;  2 ) distributed, where  2 = E [jwj [k]j2 ] is the noise variance. Because of the power normalization, the average bit signal-tonoise ratio (SNR) at each receive antenna is Eb =N0 = 1=(R 2 ). Let si [n] be the ith column of S [n], H i [n] be the MR -by-MT matrix of channel coefficients seen by si [n], and W [n] be an MR -by-MT noise matrix. Then the received data are given by

Y [n] = H[n]S [n] + W [n]

(4)

where the MR -by-MT2 matrix H[n] is obtained by stacking H i [n]

H[n] = [H 0 [n]; H 1 [n]; . . . ; H M 01 [n]] and S [n] is a stretched version of S [n], which is no longer square, but has dimension MT2 -by-MT

S [n] =

s0 [n] 0 0 s1 [n] .. .

0

.. .

0

111 111 ..

0 0 .. .

.

111

:

(5)

sN 01 [n]

If the fading process is constant in the DSTM symbol duration, i.e., = H [n] for i = 1; . . . ; MT 0 1, then (4) is reduced to the widely used piecewise-constant signal model

H i [n]

(2)

Y [n] = H [n]S [n] + W [n]:

in general the matrix multiplications in (1) need to be done explicitly. If G forms a group under matrix multiplication, then an alternative implementation of (1) is

c[n] = c[n 0 1] 8 a[n]; c[0] = 1 S [n] = AG c[n]

Gl = (G1 )l ; l = 0; 1; . . . ; L 0 1; j 2u =L =L G1 = diag e ; . . . ; ej 2u

(1)

where the initially transmitted matrix A can be any given unitary matrix. For nongroup constellations, such as those based on Alamouti’s twoantenna code [2]

G=

= 1. By restricting G to be Abelian, Hochwald and Sweldens [4] proposed a class of diagonal signals. The cyclic construction of diagonal signals has the form

R

If S [n] is diagonal, in spite of continuously fading, still a formally similar signal model is possible ~ [n]S [n] + W [n] Y [n] = H

(3)

where the addition 8 is defined in accordance with the matrix multiplication. If A is further a member of G , no explicit matrix multiplication is needed at all. Moreover, if G is a cyclic group, 8 becomes the usual addition modulo L. Hughes constructed optimal group constellations for two antennas [3], and recognized that Alamouti’s code forms a cyclic group when

(6)

(7)

~ [n] is obtained by decimating H[n]. where the effective fading matrix H

~ [n] can be expressed as H

~ [n] = [ [n]; [n]; H 0 1

...;

M 01 [n]]

where i [n] is the ith column of H i [n]. In these two cases, the structure of S [n] is preserved.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

III. NONCOHERENT SEQUENCE DETECTION

=0

Suppose that the transmission of DSTM symbols begins at n and ends at n NT . The starting point of the proposed NSD is the optimum block detection for the entire NT DSTM symbols as a whole. Stacking the variables involved in the detection yields the notations

=

+1

S [NT ] =diag[S [0]; S [1]; . . . ; S [NT ]] H[NT ] =[H[0]; H[1]; . . . ; H[NT ]]

[ ] =[Y [0]; Y [1]; . . . ; Y [NT ]] [ ] =[N [0]; N [1]; . . . ; N [NT ]]:

Y NT

(8) [ ] = H[NT ]S [NT ] + N [NT ]: The brute-force MSDD searches S [NT ] that maximizes the condi-

tional probability density function [7]



(

M

1 +1) Det(RS [N ]) 01 Y [N ]R S [N ]Y [N ]

[ ]

T

H

T

[ ]

[ ] = S [N ] (88

where



T

+ 1) [ ]+ [ ]

(

+1 I

N

+ 1)

[ ]=8

R S NT

[ ] =1 [

+1 

N

T

M

) S [N ] and 8 +1 T

H

[ ] [ ]

H S NT S NT

[ ] =1

n

01 T r Y N n R a~; N n Y

[ ] =1 G[1]G[2] 1 1 1 G[n] []

[ ] [ ] = S [0]

H S n1 S n2

n

=1

H

[]

Gi

S

[0]

i n

=

=1

[]

j H n

Gi

i

n

j

= C [n1]C [n2]:

=1

=1

[ ]

[ ]=8

R S NT

N

+1 

[]

Gj

H N

01 [n 0 1] (15)

a; N

[ ] = [Y [n 0 (N 0 1)]; Y [n 0 (N 0 2)]; . . . ; Y [n]] is an observation of length N , R ~ [n] is the N M -by-N M correlation matrix of Y [n] associated with the hypothetical sequence ~, and Y 01 [n 0 1], R ~ 01 [n 0 1] are defined accordingly. From a (11), it is easy to see that R~ [n] only depends on a ~[n 0 (N 0 2)]; a ~[n 0 (N 0 3)]; . . . a~[n]. In this way, the truncation allows the YN n

a; N

T

T

a; N

a; N

[]

[ ] [ ]

a; N

where

N

Gj

H C NT C NT

H N

N

minimization to be realized efficiently by the Viterbi algorithm. Since the truncated incremental metric, or the branch metric in terms of the Viterbi algorithm, depends exclusively on the sequence an0 N 0 ;an0 N 0 ; a n as well, a trellis state may be defined by N 0 information symbols

H

Hence, RS NT can rewritten as

a

a; N

R

Gi

=0

R

[ ] [ ] [n] + M ln (Det(R~ [n])) 0 T r Y 01 [n 0 1]R0~ 1 01 [n 0 1]Y + M ln (Det(R~ 01 [n 0 1])) R

if n > , and C n I if n . We purposely use the notation C n to discriminate from S n that depends on the initial matrix A. Using the differential encoding rule, we have

[]

(13)

Obviously, the minimization of the sequence metric of (12) can be computed recursively via the incremental metrics. As in single-antenna NSD, the implementation difficulty inherent in the optimum metric (14) is the infinite memory that would render the complexity of the detector to increase exponentially with n. In order to limit the complexity, a truncation of the memory is introduced. That is, only the most recent N observations are considered in the incremental metric [10]. With this modification, the incremental metric is approximated by

N

[ ]=

a

a

R

where we define

=1

(12)

a

T

i

T

H

[ ] =1 [I ; C [1]; . . . ; C [N ]] n

a

R

a

(10)

( )

C NT

[ ]=

T

1[n] =1 3[n] 0 3[n 0 1] = T r Y [n]R0~ 1[n]Y [n] + M ln (Det(R~ [n])) 0 T r Y [n 0 1]R0~ 1[n 0 1]Y [n 0 1] + M ln (Det(R~ [n 0 1])) : (14)

[1] . . . [ ]]

C n

H

as well as an incremental metric (9)

I; S ; where S NT ; S NT , and  denotes the Hadamard product. Collect the differentially encoded DSTM symbols in a matrix

0

[ ] [N ] + M ln(Det(R~ [N ]))

H

denotes the Kronecker product, is the N MT NT -by-MT NT autocorrelation matrix of a fading-plus-noise process, whose i; j th entry is given by j0i  2 ij , ij being the Kronecker delta function. R S NT admits a more compact form than (10). It is not hard to show that

(

[ ]

^

[ ]

H

01 T r Y NT Ra~ NT Y

where a is the detected sequence, and  represents a trial sequence. This optimum detection strategy forms the basis of our proposed method in the sequel. Applying the philosophy of [10], let us define a partial sequence metric up to the nth DSTM symbol interval

where RS NT is the autocorrelation matrix of the received signals conditioned on S NT at a particular antenna. RS NT can be expressed as R S NT

[ ]= [ ]

a

T

1 exp 0T r

[]

= [1] [2] . . . [ ]

a

[]

3[n] = T r Y [n]R 0~ 1[n]Y [n] + M ln(Det(R~ [n]))

M

N

[0]

R

Y NT

[ ] S [NT ] =

[ ]

^ = arg min ~

With these we have the signal model

Y NT

An advantage of this new version is considerable computational reduction relative to (10), due to reduced dimension of matrices and element-wise product rather than standard matrix product. It is clear that RS NT is independent of S , i.e., it depends on fG n g exclusively. Since there exist a one-to-one correspondence between the DSTM symbol sequence fG n g and the information symbol sequence a fa ; a ; a NT g, we write RS NT Ra NT to better reflect its sole dependence on a . By doing this and taking the natural logarithm of (9), the MSDD strategy becomes a

N NT

f

2729

(~[ :

(11)

(

2)] ~[

2

(

3)] . . . ~[ ])

[ ] =1 (~a[n 0 (N 0 2)]; a~[n 0 (N 0 3)]; . . . a~[n 0 1]):

n

2730

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

from 0 to N 0 2. Then it is readily seen that the branch metric is reduced to one single term corresponding to k = N 0 1

~ [n] 0 [n] = Y [n]C H

N

01

=1

N i

p

01Y [n 0 i]C~ H [n 0 i]

2

N 01 : (19)

i

Because N 01 is a constant, it may be dropped without affecting the performance. The (N 0 1)th-order linear predictor is determined from Fig. 1.

=

Four-state trellis diagram for two-antenna DSTM with 3.

= 1 and

In the trellis a ~[n] labels the current branch. The resulted number of states of the trellis is S = LN 02 . There are L paths originating from each state, and L paths leading into each state in the trellis, respectively. Therefore, the complexity of the detector becomes only exponential in N 0 1 rather than in NT . Since the fading process can be quite accurately modeled as a finite-order autoregressive (AR) process, the noncoherent sequence detector has the potential to perform close to optimum detection by selecting a value of N no less than the order of the AR process divided by MT . As an example, the trellis of Alamouti’s two-antenna code (2) for BPSK signaling, or equivalently Hughes’ R = 1 constellation

= p1 11 011 2 is shown in Fig. 1, in which L = 4, N = 3, and S = 4. G1

= 01 010

;

A

(16)

The noncoherent sequence detector, in general, does not bring a linear-prediction interpretation of its structure. Two exceptional cases are: a) fading is piecewise constant within the duration of a DSTM symbol, as described in (6); or b) G is diagonal (which does not necessarily imply that the transmitted DSTM symbols S [n] are diagonal), which includes (7) where S [n] is diagonal. In this case, the correlation matrix Ra~; N [n] in (15) can further be simplified to

[ ] = Z ~ [n](880 I

Ra~; N n

H a; N

N

M

)Z ~ [n]

(17)

a; N

where

[ ] =1 diag[C~ [n 0 (N 0 1)]; . . . ; C~ [n]]

Z a~; N n

and 80N is a reduced-dimension correlation matrix of size N -by-N , whose (i; j )th entry is given by [MT (j 0 i)] + ij . By inspection, it is obvious that Det(Ra~; N [n]) is independent of the hypothetical sequence so that it can be dropped from the branch metric. It is also clear that

01 H 01 Ra~; N [n] = Z a~; N [n](8N I M )Z a~; N [n]: Applying the Cholesky decomposition to 801 I M , we have N

01 T r Y N n R a~; N n Y

[ ]

=

N

k

01

=0

0

N

[ ]

H N

[n]

[ 0 k]C~ [n 0 k] H

Y n

010k

=1

N

pi

010kY [n 0 k 0 i]C~ H [n 0 k 0 i]

2 k (18)

i

where pik is the ith coefficient for a k th-order linear predictor, and k is the corresponding variance of the prediction error [12]. The other term 01 T r (Y N 01 [n 0 1]Ra~; N 01 [n 0 1]Y H N 01 [n 0 1]) in the branch metric (15) has a similar expression, but with a smaller range of summation

1 0 01 0(p 01 )3 = (88 ) [  01 0 1 1 1 0 ] N

N

N

T

(20)

01 N 01 N 01 T where pN 01 = [pN 1 ; p2 ; . . . ; pN 01 ] . This is exactly the linear predictive Viterbi receiver derived by Chiavaccini and Vitetta [8]. The implementation complexity of NSD comprises the algorithmic complexity of the Viterbi algorithm and the computational complexity arising from computing the branch metric. At every level of the trellis, the Viterbi algorithm needs to determine the metrics of LN 01 transitions. This is common to NSD and the linear prediction receiver. Such algorithmic complexity can be circumvented effectively by the reduced-state sequence detection, as will be demonstrated in Section V. The branch metric involves calculation of the inversion and determinant of Ra~; N [n] and Ra~; N 01 [n 0 1]. The computational complexity is on the order of (N MT )3 , which can be large in many situations. Currently we do not know of any efficient method to invert Ra~; N [n] and Ra~; N 01 [n 0 1] for nondiagonal G , because they are not Toeplitz (hence the Levinson–Durbin recursion is not applicable). In comparison, the linear predictive Viterbi receiver has a fixed-coefficient linear filter structure, whose complexity is only linear in N . In case of small values of L and N , a possible solution to reduce the computational complexity of NSD is to predetermine the inversion and determinant of Ra~; N [n] and Ra~; N 01 [n 0 1] for every possible branch, and store the results in a lookup table. The resulting size of this table is proportional to LN 01 . A comment on the noncoherent sequence detector for DSTM ends this section. The structure of the correlation matrix (11) implies that two sequences fS [n]g and fS 0 [n]g will have the same metrics, if one can be obtained from the other after multiplying by any constant unitary matrix. That is, they are indistinguishable to the noncoherent sequence detector. Hence, differential encoding is indeed necessary for the applicability of NSD of the constellation G . IV. ERROR ANALYSIS We proceed to analyze the BER performance of the proposed NSD. The union bound on the BER is given by

NSD 

Pb

1

RMT

fg

P a a

^6=

a

( ^) f ! a^g

d a; a P a

(21)

a

where d(a; a^) is the Hamming distance between the two bit sequences ^, P fag is the a priori probability of a, and associated with a and a ^g is the pairwise error probability (PEP). To evaluate the P fa ! a PEP, consider an erroneous path in the trellis diagram of DSTM that diverges from the correct path at a certain level and remerges with it at a later level. An example is illustrated in Fig. 2 for a 16-state trellis, where the correct path is the all-zero path. Because of the shift property of the states, the last N 0 2 information symbols before the remerge are the same for the correct and erroneous paths. Of course, the two paths must differ at the first position right after the divergence, and do not contain the same N 0 2 symbols in a row at other positions. Accordingly, the length of an error event can be expressed as D +N 02, where D = 1; 2; . . . is the maximum number of information symbols the two paths differ at [13].

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

2731

is a quadratic form. The metric difference can be further rewritten as

= jM=001 j 0 c. The characteristic function of j is determined by [14]

Ra; K [n + D + N 0 3]F ) 8 (s) = Det01 (I + sR

=

KM

01

i=0

(1 + s i )

01

(24)

where i for i = 0; . . . ; KMT 0 1 are the eigenvalues of the matrix Ra; K [n + D + N 0 3]F . Since the fading processes are mutually independent between receive antennas, the characteristic function of M 01 j =0 j is simply given by 8 (s) raised to the MR th power. Consequently, the PEP is given by the integral [15]

P fa ! a^g = P

1 = 2j

Fig. 2. A major error event in a 16-state trellis for two-antenna DSTM with = 1 and = 4.

The PEP P fa ! a^g is the probability that the accumulated metric of an erroneous path is smaller than that of the correct path. Let the error event start at time n and span D + N 0 2 symbol intervals. Then the PEP can be expressed as

P fa ! a^g = P

=1

n+D+N 03 i=n

^[i] 0 [i] < 0 : 

Our approach to evaluate this error probability is to express the metric difference in certain Hermitian quadratic forms and then determine its characteristic function. Note that K = D + 2N 0 3 successive observations (Y [n 0 (N 0 1)]; . . . ; Y [n + D + N 0 3]) are involved in . To express in quadratic forms, we rewrite the branch metric (15) as

[n]

= Tr

Y N [n] Qa^; N [n]Y H N [n]

+ MR ln (Det(Ra^; N [n])) 0 MR ln (Det(Ra^; N 01 [n 0 1])) (22)

where

1

1 Qa^; N [n] = Ra0 ^; N [n] 0

1 Ra0 ^; N 01 [n 0 1] 0M 0M

0M

:

In terms of such matrices, the difference of the path metrics has the expression

with the KMT

1

F=

D+N 03 i=0

= Tr

Y K [n]F Y H K [n]

0c

(23)

2 KMT matrix F defined by

diag [ 0iM

Qa^; N [n + i] 0 Qa; N [n + i] 0(K 0N 0i)M

]

f[ln (Det(Ra; N [n + i])) i=0 0 ln (Det(Ra; N 01 [n + i 0 1]))] 0 [ln (Det(Ra^; N [n + i])) 0 ln (Det(Ra^; N 01 [n + i 0 1]))]g :

c = MR

Let Y jK [n] denote the j th row of Y K [n]. Then

j

=1 Y jK [n]F (Y jK [n])H

01

j < c j =0 "+j 1 esc M  "0j 1

8 (s) ds s

(25)

where the integration path lies within the vertical strip 0 < " < "0 . "0 is the right boundary of the regularity domain of 8 (s). The integration can be calculated efficiently via the Gauss–Chebyshev quadrature [16]. To obtain fast convergence in the numerical integration, the optimum value of " may be identified as the saddle point of the function esc 8M (s) [15]. Nonetheless, the integration is found insensitive to the choice of ". Once the PEP is known, the union bound can be evaluated in principle by listing all error events with different values of D . A tight approximation is obtained by truncating the union bound, i.e., we only consider the error events with D  D0 such that the probability of the error events with D > D0 are negligible [13]. Practically, this can be achieved by increasing D0 step by step till no significant change of the sum is observed. This method is conceptually reasonable because the main error events are characterized by small Hamming distance in Rayleigh fast-fading channels. When G is nondiagonal, the uniform error probability criterion is generally not satisfied. That is, the bit-error probability depends on which code sequence is sent. Therefore, it is necessary to consider all transmitted sequences possible when evaluating the BER. Fortunately, a good approximation of the BER may be obtained by averaging over a moderate number of randomly selected sequences a . For diagonal, cyclic group constellations, it can be checked that the uniform error probability criterion is satisfied. Then, it suffices to consider an arbitrary sequence when evaluating the BER. Therefore, for equally probable signaling, the union bound is simplified to

PbNSD

1  RM T

a^6=a

d(a; a^)P fa ! a^g

( diagonal and cyclic):

(26) If Gray mapping is applied, the BER for NSD of DPSK in a Gaussian channel is dominated by two most probable error events [13]

and c is a constant with respect to the K random observations D+N 03

M

e0 and

= (e[n] = +1; e[n + 1] = 01)

e00

= (e[n] = 01; e[n + 1] = +1) where e[n] = a ^[n] 0 a[n], and e[n + 1] = a^[n + 1] 0 a[n + 1]. Due to the exponential nature of the PEP for Gaussian channels, these two error events bring a BER approximation that asymptotically coincides with the true union bound. This is not the case in fading channels, especially for uncoded systems, because the BER typically decreases linearly with the SNR. Accordingly, the two major error events are not so dominant. The BER obtained by considering the two error events is

2732

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

= 4 of two-antenna DSTM with BPSK = 0 01 and = 1.

Fig. 3. Performance of NSD with signaling (Alamouti’s code) for

no longer an asymptotic upper bound. Instead, it might be viewed as a lower bound, if their overlap is neglected. Nonetheless, it still represents a good approximation of the actual BER of DSTM in many circumstances. This is justified by the following observations: if the fading is slow, then quite often the union bound is loose; if the fading is fast, then short error events dominant the performance at high SNR, while the union bound is loose at low SNR. Since each error event entails two bits in error under Gray mapping, the BER due to these two error events for cyclic constellations is

PbNSD 

P fag(P fa ! a+e0 g+P fa ! a+e00 g): (27)

2

RMT

a

The error event e 0 relative to the all-zero path is plotted in Fig. 2 for = 4, which spans four DSTM symbol intervals. Fig. 3 shows the truncated union bound for Alamouti’s code with the trellis depicted in Fig. 2 over a fading channel with fd T = 0:01 and compares with the approximation which counts in the two dominant error events only. In this figure, the curves corresponding to D0 = 2, 3, and 4 are plotted. It can be observed that the bounds converge at high SNR for different values of D0 . Furthermore, the approximation (26) is quite close to the bounds at high SNR. The minor difference between the approximation and the bounds at high SNR is due to the miss of another major error event (e[n] = +2; e[n + 1] = 02). Computer simulation results are also shown in Fig. 3. Since the union bound diverges at low SNR, it actually overestimates the BER in such circumstances. On the contrary, the approximation fits simulation results quite well. Consequently, we evaluate the BER by the two-error-event approximation henceforth. The performance of coherent detection of a DSTM system with differential encoding and decoding in static fading channels (fd = 0) is included in Fig. 3 as a benchmark for comparison. Its BER is obtained by doubling that of coherent detection of the constellation G without differential encoding, since differential decoding typically doubles the BER. The PEP of coherent detection without differential encoding was given in [17, eq. (C.3)] as

L = 4 and N

P fG i ! Gj g =

1

4

1 01 !

d!

M

=4

2 +1

01

m=0

M

1

1+ m (! 2

2 +1

=4)

Fig. 4. Performance of NSD of two-antenna DSTM with BPSK signaling and = 1. (Alamouti’s code), for various values of

where  = 1= 2 , and m for m = 0; . . . ; MT 0 1 are the singular values of (I + G H j G i )=2. Remarkably, performance close to coherent detection is achieved even with a small value of N = 4 for fd T = 0:01. As fading gets faster, larger values of N are needed. Fig. 4 demonstrates how the performance of coherent detection is approached by increasing N in fading channels with fd T = 0:03 and fd T = 0:1, respectively. By selecting a proper value for N , the BER can decrease linearly with the SNR, which is characteristic of static fading channels. In particular, the performance of NSD with N = 8 is less than 2 dB away from coherent detection for fd T = 0:03. Such performance characteristics might be useful in selection of DSTM constellations for fast fading channels. That is, if a constellation possesses good performance in static fading, then it can also have good performance in time-selective fading when NSD is employed. In Fig. 5, performance of NSD of Alamouti’s code (16) with N = 6 is compared with the linear prediction-based Viterbi receiver in [8]. The linear predictive receiver can be viewed as a noncoherent sequence detector employing a mismatched branch metric for nondiagonal G . Specifically, the correlation matrices Ra~; N [n] and Ra~; N 01 [n 0 1] are (incorrectly) expressed as (17) so that a linear predictor could be derived. The PEP of the linear predictive receiver is simply obtained through the same analytic procedure for NSD, except that (17) is used from (22)–(25). The final BER is calculated by (21). Other methods simpler than this are possible, but this one is easy to manipulate for our purposes. The results shown in Fig. 5 indicate that though the linear predictive receiver has good performance in slow fading with fd T = 0:001, it suffers an error floor greater than 1006 for fading faster than fd T = 0:01. Based on the observations, we conclude that for nondiagonal G , the application of linear prediction-based receivers is limited to slowly fading channels.

V. RECUDED-STATE DETECTION The algorithmic complexity of NSD is proportional to the number of states S = 2RM (N 02) of the trellis. Even for moderate values of R and MT , the complexity increases drastically with N . For example, the trellis will contain S = 4096 states if R = 1, MT = 3, and N = 6. It might be difficult to implement real-time sequence detection with such a large-size trellis in practice. The reduced-state sequence detection

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

2733

Fig. 5. Performance comparison between NSD and the linear prediction (LP) receiver for two-antenna DSTM with BPSK signaling (Alamouti’s code), for , = 6, and = 1. various values of

Fig. 6. Performance of reduced-state NSD versus DFD for three-antenna = 0 1, = 6, and = 2. nondiagonal DSTM, for

technique can be incorporated to overcome the implementation complexity [18], [19]. For this, a trellis state is defined in terms of most recent 0 information symbols, where 0 < N 0 2

for those in the survivor history. The upper limit of the summation in F and c is reduced to D + 0 0 1. After the characteristic function

0 [n] = (~ a[n 0 0]; a ~[n 0 (0 0 1)]; 1

. . . a~[n 0 1]):

In this way, the number of states is reduced to S = L0 . In reduced-state NSD, the calculation of the branch metric (15) still involves N observations and N 0 1 trial symbols, among which the remaining N 0 2 0 0 symbols unavailable in the state are extracted from the survivor history according to the per-survivor processing (PSP) principle [20]. In the extreme case of 0 = 0, all the N 0 2 symbols are found from history and NSD degenerates into a decision-feedback detector performing symbol-by-symbol detection [21], which is generally different from the linear prediction-based decision-feedback detector. It is also possible to incorporate other reduced-state techniques such as set-partitioning [18] in NSD when the constellation size L is large. Therefore, NSD is very flexible in tradeoffs between performance and complexity. By controlling N and 0, we can range between the optimum MSDD and DFD. Performance analysis of reduced-state NSD is a modification of the procedures for full-state detection described in the previous section. The analysis is complicated to some extent by sporadic erroneous symbols in the survivor history, analogous to DFD. Hence, we follow a widely accepted approach that assumes completely correct feedback symbols. It is well known that there is little error propagation in DFD of DPSK and DSTM. Besides, it is known that reduced-state sequence detection has even less effect of error propagation than DFD [18], [19]. Therefore, we expect the genie-aided analysis will yield the BER close to the actual performance. To be specific, the length of an error event is shortened to D + 0. The difference in cumulative metrics becomes n+D+001 ^[i] 0 [i] 

= i=n which involves K 0 = D + 0 + N 0 1 successive observations (Y [n 0 (N 0 1)]; . . . ; Y [n + D + 0 0 1]). The matrices Ra^; N [n] and Ra^; N 01 [n 0 1] are calculated by imposing correct symbols

a ^[n 0 (N 0 1)] = a[n 0 (N 0 1)];

. . . ; a^[n 0 (0+1)] = a[n 0 (0+1)]

Ra; K [n + D + 0 0 1]F ) 8 (s) = Det01 (I + sR is determined, the PEP is evaluated by (25). Considerable insights into the performance of NSD can be gained by a comparison with DFD and MSDD. We have demonstrated that the performance of the decision-feedback detector can be maintained at the same order as MSDD [21]. Because DFD is the least favorable case of reduced-state NSD from the performance viewpoint, we conclude that reduced-state NSD (0 > 0) has better performance than MSDD for equal values of N . This property was also observed in [10] for single-antenna full-state NSD over AWGN channels by inspecting the metrics. Fig. 6 illustrates the performance of reduced-state NSD as well as two extremes: full-state trellis detection and DFD, where MT = 3, MR = 2, R = 1, N = 6, and fd T = 0:1. The transmitted signal U , where G is the constellation is generated by a transform G0 = U H GU rate-1 diagonal constellation for three antennas [4], and U is selected as a unitary matrix given by the discrete Fourier transform (DFT). After such a transform, the constellation is nondiagonal so that the linear prediction receiver does not work. Fig. 6 shows that DFD exhibits an error floor above 1004 in this fast-fading channel, while reduced-state NSD yield significant gains. A noticeable feature is that a major amount of the performance gain can be achieved by an L-state trellis (0 = 1) in NSD. If 0 = 2, the reduced-state detection performs close to full-state detection. Therefore, we do not need a large trellis in order to achieve satisfactory performance. Performance comparison between the reduced-state noncoherent sequence detector and linear prediction receiver [8] is given in Fig. 7 for Alamouti’s code with BPSK signaling. Both receivers select N = 6, 0 = 1, i.e., the Viterbi decoding is performed on a four-state trellis. Simulation results are extracted from [8], which conform to our theoretic analysis. It is clear from Fig. 7 that our proposed noncoherent sequence detector performs close to coherent detection, while the linear prediction receiver exhibits an error floor (higher than 1004 if fd T = 0:03). In fact, it was observed in [8] that, in comparison with the standard differential detector, this reduced-state linear prediction receiver can only reduce the error floor by one order in magnitude.

2734

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

Fig. 7. Performance comparison between the reduced-state NSD and LP = 6, 0 = 1, and receiver for Alamouti’s code with BPSK signaling, for = 1.

VI. CONCLUSION AND DISCUSSION We have proposed NSD of DSTM in continuously fading channels. The idea was to truncate the memory of the MSDD metric so that a trellis for Viterbi decoding could be defined. NSD overcomes the limitation of the existing linear prediction-based Viterbi receiver as it can accommodate nondiagonal constellations. For diagonal constellations and 0, the two techniques are equivalent to each other. By tuning reduced-state NSD can range between optimum block detection and DFD. Performance analysis for full-state and reduced-state NSD was carried out, although admittedly, it brought few insights into how the performance changes with system parameters like diversity gain. The approximation considering two major error events might be used as a lower bound on the BER, and is fairly accurate in many cases. The analysis showed that reduced-state NSD with 0 > 0 has better performance than MSDD for equal values of N , and a major amount of performance gain in NSD could be obtained by an L-state trellis. Since the number of trellis states can be controlled by 0, the major implementation difficulty in NSD will arise from computing the branch metrics. The table lookup technique suggested in Section III requires a table of size LN 01 , even if reduced-state sequence detection is employed. So its application is limited to small values of L and N . When L and/or N are large, we need to compute the inverse and determinant of Ra~; N [n] and Ra~; N 01 [n 0 1] for each branch transition in real time. In numerical evaluation of the performance, we found that the BER is nearly unchanged if Det(Ra~; N [n]) and Det(Ra~; N 01 [n 0 1]) are ignored in the branch metric (15). But the matrix inversion still involves considerable computational burden. Moreover, numerical instability was sometimes observed as a side effect of matrix inversion in performance evaluation at high SNR. The NSD requires knowledge of the fading autocorrelation function and noise variance. So we need to estimate the second-order statistics of the channel. When such knowledge is unavailable at the receiver, the linear prediction receiver is ready to be adapted by the classical least mean-square (LMS) or recursive least-square (RLS) algorithm. The noncoherent sequence detector, however, lacks a linear filter structure, thereby seemingly not admitting adaptation. Such implementation-related issues warrant future research.

N

REFERENCES [1] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit diversity,” IEEE J. Select. Areas Commun., vol. 18, pp. 1169–1174, July 2000.

[2] S. M. Alamouti, “A simple transmit diversity techniques for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [3] B. L. Hughes, “Differential space-time modulation,” IEEE Trans. Inform. Theory, vol. 46, pp. 2567–2578, Nov. 2000. [4] B. M. Hochwald and W. Sweldens, “Differential unitary space-time modulation,” IEEE Trans. Commun., vol. 48, pp. 2041–2052, Dec. 2000. [5] A. Shokrollahi, B. Hassibi, B. M. Hochwald, and W. Sweldens, “Representation theory for high-rate multiple-antenna code design,” IEEE Trans. Inform. Theory, vol. 47, pp. 2335–2367, Sept. 2001. [6] B. Hassibi and B. M. Hochwald, “Cayley differential unitary space-time codes,” IEEE Trans. Inform. Theory, vol. 48, pp. 1458–1503, June 2002. [7] R. Schober and L. H. J. Lampe, “Noncoherent receivers for differential space–time modulation,” IEEE Trans. Commun., vol. 50, pp. 768–777, May 2002. [8] E. Chiavaccini and G. Vitetta, “Further results on Tarokh’s space-time differential technique,” Proc. Int. Conf. Communication (ICC’02), pp. 1778–1782, Apr./May 2002. [9] C. Ling, K. H. Li, and A. C. Kot, “On decision-feedback detection of nondiagonal differential space-time modulation in temporally correlated fading channels,” in Proc. Int. Conf. Communications (ICC’03), Anchorage, AK, May 2003, pp. 2648–2652. [10] G. Colavolpe and R. Raheli, “Noncoherent sequence detection,” IEEE Trans. Commun., vol. 47, pp. 1376–1385, Sept. 1999. [11] W. C. Jakes, Microwave Mobile Communications. Piscataway, NJ: IEEE Press, 1993. [12] J. H. Lodge and M. L. Moher, “Maximum likelihood sequence estimation of CPM signals transmitted over Rayleigh flat-fading channels,” IEEE Trans. Commun., vol. 38, pp. 787–794, June 1990. [13] G. Colavolpe and R. Raheli, “Theoretical analysis and performance limits of noncoherent sequence detection of coded PSK,” IEEE Trans. Inform. Theory, vol. 46, pp. 1483–1494, July 2000. [14] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. [15] C. W. Helstrom, Elements of Signal Detection and Estimation. Englewood Cliffs, NJ: Prentice-Hall, 1995. [16] E. Biglieri, G. Caire, G. Tarcco, and J. Ventura-Traveset, “Simple method for evaluating error probabilities,” Electron. Lett., vol. 32, pp. 191–192, Feb. 1996. [17] B. M. Hochwald and T. L. Marzetta, “Unitary space–time modulation for multiple-antenna communication in Rayleigh flat-fading,” IEEE Trans. Inform. Theory, vol. 46, pp. 543–564, Mar. 2000. [18] M. V. Eyuboglu and S. U. H. Qureshi, “Reduced-state sequence estimation with set partitioning and decision feedback,” IEEE Trans. Commun., vol. 36, pp. 13–20, Jan. 1988. [19] A. Duel-Hallen and C. Heegard, “Delayed decision-feedback sequence estimation,” IEEE Trans. Commun., vol. 37, pp. 428–436, May 1989. [20] R. Raheli, A. Polydoros, and C. K. Tzou, “Per-survivor processing: A general approach to MLSE in uncertain environment,” IEEE Trans. Commun., vol. 43, pp. 354–364, Feb./Mar./Apr. 1995. [21] C. Ling, K. H. Li, and A. C. Kot, “Decision-feedback multiple-symbol differential detection of differential space-time modulation in continuously fading channels,” in Proc. Int. Conf. Acoustics, Speech, and Signal Processing (ICASSP’03), Hong Kong, China, May 2003, pp. IV_45–IV_48.