Space-Time Coding for Rayleigh Fading Channels in CPM system Xiaoxia Zhang

Dept. of Electrical Engineering The Ohio State University Columbus, OH 43210-1272 [email protected]

Michael Fitz

Dept. of Electrical Engineering The Ohio State University Columbus, OH 43210-1272 [email protected]

Abstract

Space-Time codes have shown considerable promise for reliable transmission over wireless fading channels by eciently employing transmit diversity. Past work only considered design of space-time codes with linear modulation. In this paper we give the design criterion of space-time codes for at Rayleigh Fading channel in CPM system. A simple code example is provided to demonstrate its good performance and then applied to an interleaved trellis-coded space-time CPM (TCST-CPM) system. The corresponding adaptive soft-output detection algorithm is also presented.

1 Introduction Diversity techniques are widely employed to combat the detrimental eects of multipath fading in wireless communications. Recently some interesting approaches for transmit diversity have been suggested. In [1] Tarokh et al. introduced space-time trellis coded modulation proposing a joint design of coding, modulation and transmit diversity for fading channels. It can achieve the same diversity advantage as receive diversity by avoiding destructive superposition of the signals transmitted simultaneously from dierent antennas. Meanwhile, for the transmission of digital information over both bandwidthand power-limited channels such as mobile satellite communications and land mobile radio communications, continuous phase modulation (CPM) is an attractive modulation scheme [2]. It is worth investigating the design of codes making use of transmit diversity in CPM system. This paper presents the design criterion for designing space-time codes in CPM system. A code example to achieve maximum diversity level is given. This code is then used in an interleaved TC-CPM scheme to obtain better performance. The corresponding adaptive soft-output detection algorithm without perfect channel state information is also provided.

2 System Model We consider a mobile communication system where there are Lt transmit antennas and Lr receive antennas. In the transmitter shown in Fig. 1, data is encoded by the channel encoder, the encoded data goes through an interleaver and passes the space-time encoder

~ I (k )

~ Channel D (k ) Inter- D (k ) Encoder leaver

D1 ( k )

SpaceTime Encoder

DLt (k )

X 1 (t )

CPM CPM

X Lt (t )

Figure 1: Transmitter Block Diagram to get Lt streams of data. Each stream of data is used as the input to a CPM modulator. The modulated signals are transmitted through Lt transmit antennas at the same time and the signal received at each receive antenna is a noisy superposition of the Lt transmitted signals corrupted by Rayleigh fading and an independent zero-mean complex AWGN. If we limit the CPM scheme to be of single modulation index, the signal received by antenna j , Y j (t), can be given by [2]:

~) = Yj (t; D Xi(t; D~ i) =

Lt X Ci;j (t)Xi(t; D~ i) + Nj (t); j = 1; ; Lr ri=1

Es exp j(t; D~ ) ; i = 1; ; L i t T

X (t; D~ i) = 2hi Di (l)qm (t ? lT ); (k ? 1)T t < kT k

l=0

(1) (2) (3)

where Di(k) is the kth modulation symbol of the ith antenna, hi is the modulation index, qm (t) is the phase smoothing response function, T is the symbol time and Es is the symbol energy. It can be shown that for k Lm where Lm is the modulation memory [2]

(t; D~ i) = i(k) + 2hi

k X

l=k?Lm+1

Di(l)q(t ? lT )

(4)

If the modulation index hi is rational, say hi = 2pm where m and p are relatively prime integers, then i(k) (mod 2) 2 , where [2]

(5)

i = 0; 2p ; 4p 2(p ?p 1) : For rational modulation indices (t; D~ i(k)) is constrained to lie on a trellis. The state of the system S~i(k) at time kT , is de ned as S~i(k) = [ i(k) Di(k ? Lm + 1) Di(k ? 1) ] (6) With these de nitions, [Di (k); S~i(k)] uniquely specify (t; Di(k); S~i(k)) and X i(t; D~ i) in the semi-open interval [kT; (k + 1)T ).

3 Code Design 3.1

Design Criterion

~ = Assume a transmission of length Nc, the maximum likelihood metric of sequence D ~ in the presence of perfect channel state information c(t) is: m(y(t)jc(t); x(t; ~ )) = ?

Lr Z Nc T X j =1 0

jy(t) ?

Lt X i=1

ci;j (t)xi (t; ~i)j2dt:

(7)

~ = The pairwise error probability (PWEP) of confusing D ~ with another sequence ~D = ~, denoted as P ( ~ ~ ! ), is given by:

n

P ( ~ ! ~ ) = Pr m(y(t)jc(t); x(t; ~ )) m(y(t)jc(t); x(t; ~ ))

o

(8)

Following the similar method in [3], it can be shown that the pairwise error probability can be upper bounded by:

2 2H (s) ? 1 H (s) 3Lr 6 H (s) ? 1 N0 77 ~ 6 P ( ~ ! ) 4 QH (s) (s) 5 i=1

i

(9)

where ~(s) are the nonzero eigenvalues of the signal matrix Cs = RCc, H (s) is the number of nonzero eigenvalues, Cc is the covariance matrix of fading coecients and R is a matrix of the form: 0 R R R 1 BB R1121 R1222 R12LLtt CC R=B (10) @ ... ... . . . ... CA RLt 1 RLt2 RLtLt Each element of R is a diagonal submatrices of size Nc by Nc. The diagonal entries of the submatrix Rij are given as

Rij (k) =

Z kT

(k?1)T

i (t; ~ ; ~)j (t; ~ ; ~) for k = 1; ; Nc

(11)

where

i(t; ~ ; ~ ) = xi (t; ~ i) ? xi (t; ~i)

(12)

In order to minimize the error probability the rank of Cs should be maximized and so is its determinant. As interleaver cannot be used after CPM otherwise the phase continuity of CPM signals will be disrupted, it is reasonable to assume that the fading coecients remain constant during one error path. Herein we consider the special case where the fading

channels are statistically independent and number of transmit antennas is Lt = 2. Thus the signal matrix can be reduced to 2 R T j (t; ~ ; ~ )j2dt R T (t; ~ ; ~ ) (t; ~ ; ~ )dt 3 1 1 1 1 1 2 2 2 0 0 1 66 77 ... ... 66 R NcT R NcT (t; ~ ; ~ ) (t; ~ ; ~ )dt 77 2 dt ~ j ( t; ~ ; ) j 1 1 1 1 1 2 2 2 6 77 (13) (Nc ?1)TR 1 Cs = 66 R T (Nc?1)T ~ T 2 ~ ~ 77 ~ 2; 2)j dt 0 j2 (t; 66 0 1(t; ~ 1; 1..)2 (t; ~ 2; 2)dt . 75 .. . 4 R Nc T R Nc T ~ 1; ~1 )2(t; ~ 2; ~2 )dt ~ 2; ~2)j2dt (Nc ?1)T 1 (t; (Nc ?1)T j2 (t; 3.2

Code Structure

First we consider the delay diversity scheme (denoted DD) as proposed by Wittneben [4]. This scheme transmits the same information from both antennas simultaneously but with a delay of one symbol interval. It can be easily shown that in DD Cs is guaranteed to have full rank, but it may not achieve maximum coding gain. As we can see from the (13), there are several methods to guarantee Cs full rank. One approach is to make

Z kT 2 j1(t; ~ 1 ; ~1)j dt j2(t; ~ 2 ; ~2)j dt 6= 1 (t; ~ 1 ; ~1)2(t; ~ 2 ; ~2)dt (14) (k?1)T (k?1)T (k?1)T

Z kT

2

Z kT

2

for a given k 2 f1; ; Ncg. According to Scharwtz Inequality this inequality holds as long as 1 (t; ~ 1; ~1) 6= C2(t; ~ 2; ~2 ): (15) Here the dierence between linear modulation and nonlinear modulation comes into play. In linear modulation, for a given k, 1(t; ~ 1 ; ~1) = 1(k) ? 1(k) (16) 2(t; ~ 2 ; ~2) = 2(k) ? 2(k) (17) which means that we always have 1(t; ~ 1 ; ~1) = C2 (t; ~ 2; ~2) (18) Therefore, the above inequality does not hold at all. While in CPM system, as both 1 (t; ~ 1; ~1) and 2 (t; ~ 2; ~2 ) are functions of time, (15) can be easily satis ed. We can use dierent CPM schemes over dierent transmit antennas, which may result in higher complexity or more bandwidth occupation. Another method is to use dierent mapping rules (denoted DM) over dierent antennas to achieve full diversity. Following is just an example to demonstrate this idea. In order to show the performance we need to pick some speci c CPM scheme. Here we consider 4-ary CPFSK with h = 0:5. Consequently, the coding gain provided by delay diversity is 4. If we use the mapping rule as in table 1: D(k) D1 (k) D2 (k) 0 -3 -1 1 -1 +3 3 +1 -3 2 +3 +1 Table 1: Mapping rule of the DM space-time code, Lt = 2, 4-ary

The resulting coding gain is 5:5154. Compared with delay diversity there is a 1:3951dB improvement. Fig. 2 shows the performance of DD and DM. (In all our simulations, each frame consists of 130 transmissions out of each transmit antenna and we choose the normalized Doppler spread to be fD T = 0:01). The improvement of DM is more obvious when receive diversity is available, which can be explained through (9). When there is no receive diversity, although the coding gain for DM is large, the multiplicity for the coding gain is also large and they are of the same signi cance to the error performance; when the receive diversity is available, the coding gain plays a more important role than the multiplicity due to the exponential index Lr . 0

10

−1

Frame Error Probability

10

−2

10

−3

10

2 Tx, 1 Rx, DD 2 Tx, 1 Rx, DM 2 Tx, 2 Rx, DD 2 Tx, 2 Rx, DM 2 Tx, 4 Rx, DD 2 Tx, 4 Rx, DM −4

10

6

8

10

12 SNR in dB

14

16

18

Figure 2: Performance comparison between DD and DM

4 Adaptive Soft-output Demodulation In real application, CPM is very often combined with some outer coding scheme to achieve better performance. Actually the whole system shown in Fig. 1 can be treated as a serially concatenated system which we denote trellis-coded space-time CPM (TC-ST-CPM). It is well known that a soft-output algorithm is necessary in a concatenated system for an iterative detection to be implemented. Here we consider Kd -lag soft-output demodulation of the space-time code in nonlinear modulation without perfect channel state information (CSI). The algorithm only assumes a knowledge of the statistical characteristics of the channel in forming the soft-output a posteriori probability (APP). While the CSI is not explicitly estimated, the algorithm implicitly estimates the CSI and uses this estimate in forming the desired APPs. The algorithms presented here are extensions of the work done in [5, 6, 7, 8]. Modi cations of MLSD are proposed for CPM signals transmitted over frequency at, Rayleigh fading channels in [5]. Seymour [6] gives the MAP algorithms for linearly modulated system in at fading while Balasubramanian [7] extends these results to certain CPM signals. No either transmit diversity or receive diversity is considered in all these works. In [8] soft output receive diversity combining for CPM signals over space-time correlated Rayleigh fading channels is derived. Here we take both the transmit diversity and receive diversity into consideration. The following notations are useful in deriving the algorithm. For any time series ff (0); f (1); ; f (k)g, f~(k) = [f (k) f (0)]T , boldface denotes signals over all receive antennas and Yj (k) = fYj (t) : kT t < (k + 1)T g. And as in general, capital letter indicates random variables while lower case indicates realization.

The optimum Kd-lag soft-output algorithm produces the metric p (d(k ? Kd )j~y(k)) which can be written as X ~ p (d(k ? Kd )j~y(k)) = p 1 (k); 2 (k); d(k)j~y(k) (19) f1 (k);2 (k);d~(k)g2?(d(k?Kd ))

where f(k); 2(k); d~(k)g 2 kD and ?(d(k ? Kd)) = f1(k); 2(k); D~ (k) : D(k ? Kd) = d(k ? Kd )g. The set ?(d(k ? Kd )) represents all possible phase trajectories such that D (k ? Kd) = ~d(k ? Kd). p 1 (k); 2 (k); d(k)j~y(k) can be written into p 1 (k); 2(k); d~1(k); d~2(k); j~y(k) by using the structure of space-time encoder and can be computed in an ecient recursive fashion as p (k ); (k ); d~ (k ); d~ (k )j~ y(k ) = (20) f (y(k )j~ y (k ? 1); (k ); (k ); d~ (k ); d~ (k ))p( (k ); (k ); d~ (k ? 1); d~ (k ? 1)j~y(k ? 1))p(d(k )) X f (y(k )j~ y (k ? 1); (k ); d~(k ))p((k ); d~(k ? 1)j~y(k ? 1))p(d(k )) 1

1

2

2

1

1

2

2

1

(k);d~(k)

2

1

2

The key part is to get the innovation PDF f y(k)j~y(k ? 1); 1(k); 2 (k); d~1(k); d~2(k) which can be written as Z f y(k )j~ y (k ? 1); (k ); (k ); d~ (k ); d~ (k ); c(k ) f c(k )j~y (k ? 1); (k ); (k ); d~ (k ); d~ (k ) dc(k ) (21) 1

2

1

2

1

2

1

2

The rst part can be de ned using the Wiener measure [9] while the second part involves an extended Kalman lter [10] to estimate the mean and variance of c(k). Thus the innovation PDF can be computed through a multi-dimensional Gaussian integration. The whole algorithm is derived as follows: f

y(k )j~y (k ? 1); 1 (k ); 2 (k ); d~1 (k ); d~2 (k ); c(k )

Es < qH (k)c(k) ? NEs cH (k)R(k)c(k) (22) = Cf exp 2 N 0

0

where

q(k) = [q1;1 (k); q2;1(k); ; q1;Lr (k); q2;Lr (k)]T c(k) = [c1;1 (k); c2;1(k); ; c1;Lr (k); c2;Lr (k)]T Z (k+1)T 1 q1;j (k) = p yj (t)e?j(t;d (k);~s (k)) dt EsT kT = c1;j (k ? 1) + (k)c2;j (k ? 1) + n1;j (k ? 1) Z (k+1)T 1 yj (t)e?j(t;d (k);~s (k)) dt q2;j (k) = p EsT kT = c2;j (k ? 1) + (k)c1;j (k ? 1) + n2;j (k ? 1) R(k) = diag fA(k); ; A(k)g A(k) =

1 (k)

(k) = T

kT

1

1

(25)

2

2

(26)

1

(27) (28)

ej(t;d1 (k);~s1 (k))?j(t;d2 (k);~s2(k)) dt

(29)

(k)

1 Z (k+1)T

(23) (24)

Actually qi;j (k) is the matched lter output on the j th antenna for [Di(k); S~i(k)] = [di(k);~si(k)]. (Note for simplicity of notation we have dropped the dependence of q( i; j )(k)

on i (k), j (k), d~i(k) and d~j (k)). Notice there exists cross correlation between q1;j (k) and q2;j (k) ((k)) even when the channels are independent due to the signal superposition from dierent transmitter antennas. In order to compute the PDF f c(k)j~y(k ? 1); 1(k); 2 (k); d~1(k); d~2(k) we de ne the following random vectors. Q~ j (k ? 1) = [Q1;j (k ? 1); ; Q1;j (0); Q2;j (k ? 1); ; Q2;1(0)]T (30) i h T Q~ (k ? 1) = Q~ T1 (k ? 1); ; Q~ TLr (k ? 1) (31) It is apparent that the past matched lter outputs are sucient statistics for estimating C(k) and since C(k) and Q~ (k ? 1) are jointly Gaussian, the PDF of C(k) conditioned on the past information is 1 exp n? [c(k) ? ^c(k)]H ? [c(k) ? ^c(k)]o (32) f (c(k )j~ y(k ? 1); (k ); (k ); d~ (k ); d~ (k )) = Lr e j j 1

2

1

2

where ^c(k) and e are given as ^c(k) = gH Q~ (k ? 1)

in which

P = c =

2

1

e

e = E C(k)CH (k) ? gH P

(33)

h~ i H RE C(k ? 1)C (k) h~ i ~ H (k ? 1) R RH E C (k ? 1)C N ?1

(34) (35)

g = c + E0 I P (36) s R = diag (37) I fA;BH ; Ag (38) A = B I B = diagf(k ? 1); ; (0)g (39) Actually (33) can be viewed as a minimum mean square error (MMSE) predictor which is a space-time processor. g serves as a 2Lr dimensional prediction lter of time length k, ^c(k) are the predicted fading coecients at time t = kT and e is the prediction variance. Notice due to the signal cross correlation from dierent transmit antennas, both the lter and the variance are now functions of both channel self correlation and signal cross correlation and their evaluation cannot be carried out o-line, which is unlike the case when only receive diversity is available [8]. Closed form of the innovation PDF can be obtained as f

Lr Y

y(k )j~y (k ? 1); 1 (k ); 2 (k ); d~1 (k ); d~2 (k )

= Nf

(40)

=N0 exp ? 1 + EEs=N jz (k) ? ni (k)j2 + NEs jzi (k)j2 2 i 2 s 0 0 i i=1

where

2 2 21 6 H U 4 0

3

0 0 ... . . . 0 75 U H aea = 0 0 2L2 r n(k) = Ua^c(k) z(k) = Ua?1 q(k)

(41)

aH a = R(k)

(42)

The detector for the whole system is the same as that for standard serially concatenated codes [11] which is shown in Fig. 3. Based on the received signals from all antennas, Y1(t); ; YLr (t), and the a priori information, pe(D(k); i), the inner demodulator calculates the soft information pe(D(k); o). The outer decoder used this soft information to update the a priori information pe(D(k); i). This updated information is then used for the next iteration. After several iterations, the outer decoder calculates the soft information of the information bits, p(I (k~); o) and makes decision. Y1 (t )

~ p e ( D ( k ); i )

Soft- p e ( D (k ); o) Deinteroutput leaver Demod.

SoftOutput Decoder

~ p( I (k ); i)

YLr (t )

~ p( I (k ); o) used in the final decision

equally likely p e ( D ( k ); i )

~ p e ( D ( k ); o)

Interleaver

Figure 3: Iterative decoder structure 0

0

10

10

−1

10

−1

Frame Error Probability

Frame Error Probability

10

−2

10

−2

10 −3

10

2 Tx, 1 Rx, DD, with csi 2 Tx, 1 Rx, DD, w/o csi 2 Tx, 2 Rx, DD, with csi 2 Tx, 2 Rx, DD, w/o csi 2 Tx, 4 Rx, DD, with csi 2 Tx, 4 Rx, DD, w/o csi

2 Tx, 1 Rx, DM, with csi 2 Tx, 1 Rx, DM, w/o csi 2 Tx, 2 Rx, DM, with csi 2 Tx, 2 Rx, DM, w/o csi 2 Tx, 4 Rx, DM, with csi 2 Tx, 4 Rx, DM, w/o csi −4

10

−3

6

7

8

9 SNR in dB

10

11

Figure 4: Simulation results of DM

12

10

6

7

8

9 SNR in dB

10

11

12

Figure 5: Simulation results of DD

Performance of the derived algorithm is shown in Fig. 4 and Fig. 5. (Note in order to t the algorithm into practical applications we need to cut the channel memory into some nite length Nd). In these simulations, decision lag is chosen to be Kd = 4 and length of channel estimation lter is Nd = 12. There is no outer convolutional code considered here. The space-time codes used here are those given in section 3.2. These gures illustrate that the performance of the derived algorithm is close to that of perfect CSI with degradation being around 1dB when no receive diversity is available while the degradation being less when the receive diversity is available. Again the improvement of DM can be told through these two gures. Performance of TC-ST-CPM is shown in Fig. 6. Here the outer channel encoder is a convolutional code with generator polynomial is g(5; 7) and random interleaver is used.

The simulation results demonstrate several important points. First, by adding outer convolutional code, the whole system performance improves a lot. Second, the iterative detection provides signi cant improvement to the system performance even when CSI needs to be estimated. Third, although the CSI is generated within the inner space-time CPM demodulator, the exchange and update of soft information provides an implicit mechanism for the re-estimation of the unknown CSI. Therefore, the result is closer to that of perfect CSI compared to the case where there is no channel encoder. 0

Frame Error Probability

10

−1

10

w/o cc, w/ csi w/o cc, w/o csi w/ cc, iter.=1, w/ csi w/ cc, iter.=1, w/o csi w/ cc, iter.=5, w/ csi w/ cc, iter.=5, w/o csi −2

10

2

3

4

5

6

7 SNR in dB

8

9

10

11

12

Figure 6: Simulation results of the TC-ST-CPM using DM

5 Conclusion This paper gives the design criterion of space-time codes in CPM system. Its application to serially concatenated CPM system (TC-ST-CPM) is investigated. The adaptive soft-output demodulation without perfect channel state information is also presented. Simulation results show that both the space-time code and the derived adaptive demodulation algorithm give good performance.

References [1] N.Seshadri V.Tarokh and A.R.Calderbank. Space-time codes for high data rate wireless communication: Performance criterion and code construction. IEEE Trans. Inform. Theory, Vol. 44:744{765, Mar 1998. [2] T. Aulin J.B. Anderson and C.W.Sundberg. Digital Phase Modulation. Plenum, New York, 1986. [3] M.P. Fitz, J. Grimm, and S. Siwamogsatham. A new view of performance analysis techniques in correlated Rayleigh fading. IEEE Info. Theory, Submitted 1999. Available at http://eewww.eng.ohio-state.edu/ tz/Papers folder/proberrj.ps. [4] A.Wittneben. A new bandwidth ecient transmit antenna modulation diversity scheme for linear digital modulation. Proc. IEEE`ICC, pages 1630{1634, 1993.

[5] J.H.Lodge and M.L.Mother. Maximum likelihood sequence estimation of CPM signals transmitted over Rayleigh at fading channels. IEEE Trans. Commun., COM38:784{794, June 1990. [6] J.P.Seymour and M.P.Fitz. Near-Optimal Symbol-by-Symbol Detection Schemes for Flat Rayleigh Fading. IEEE Trans. Commun., Vol. COM-41:1525{1533, February/March/April 1995. [7] R. Balasubramanian and M.P. Fitz. Soft Output Detection of CPM Signals in Frequency Flat, Rayleigh Fading. IEEE J. Select. Areas Commun., July 2000. [8] Xiaoxia Zhang and M.P. Fitz. Soft Output Diversity Combining for CPM Signals over Space-time Correlated Rayleigh Fading Channels. submitted to ICC, 2001. [9] H. V. Poor. An Introduction to Signal Detection and Estimation. Springer-Verlag, New York, 1994. [10] H.Meyr A.Aghamohammadi and G.Ascheid. Adaptive synchronization and channel parameter estimation using an extended Kalman lter. IEEE Trans. Commun., Vol. 37:1212{1219, Nov. 1989. [11] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara. A Soft-Input Soft-Output Maximum A Posteriori (MAP) Module to Decode Parallel and Serial Concatenated Codes. TDA Progress Report, Vol. 42-127, November 1996.

Dept. of Electrical Engineering The Ohio State University Columbus, OH 43210-1272 [email protected]

Michael Fitz

Dept. of Electrical Engineering The Ohio State University Columbus, OH 43210-1272 [email protected]

Abstract

Space-Time codes have shown considerable promise for reliable transmission over wireless fading channels by eciently employing transmit diversity. Past work only considered design of space-time codes with linear modulation. In this paper we give the design criterion of space-time codes for at Rayleigh Fading channel in CPM system. A simple code example is provided to demonstrate its good performance and then applied to an interleaved trellis-coded space-time CPM (TCST-CPM) system. The corresponding adaptive soft-output detection algorithm is also presented.

1 Introduction Diversity techniques are widely employed to combat the detrimental eects of multipath fading in wireless communications. Recently some interesting approaches for transmit diversity have been suggested. In [1] Tarokh et al. introduced space-time trellis coded modulation proposing a joint design of coding, modulation and transmit diversity for fading channels. It can achieve the same diversity advantage as receive diversity by avoiding destructive superposition of the signals transmitted simultaneously from dierent antennas. Meanwhile, for the transmission of digital information over both bandwidthand power-limited channels such as mobile satellite communications and land mobile radio communications, continuous phase modulation (CPM) is an attractive modulation scheme [2]. It is worth investigating the design of codes making use of transmit diversity in CPM system. This paper presents the design criterion for designing space-time codes in CPM system. A code example to achieve maximum diversity level is given. This code is then used in an interleaved TC-CPM scheme to obtain better performance. The corresponding adaptive soft-output detection algorithm without perfect channel state information is also provided.

2 System Model We consider a mobile communication system where there are Lt transmit antennas and Lr receive antennas. In the transmitter shown in Fig. 1, data is encoded by the channel encoder, the encoded data goes through an interleaver and passes the space-time encoder

~ I (k )

~ Channel D (k ) Inter- D (k ) Encoder leaver

D1 ( k )

SpaceTime Encoder

DLt (k )

X 1 (t )

CPM CPM

X Lt (t )

Figure 1: Transmitter Block Diagram to get Lt streams of data. Each stream of data is used as the input to a CPM modulator. The modulated signals are transmitted through Lt transmit antennas at the same time and the signal received at each receive antenna is a noisy superposition of the Lt transmitted signals corrupted by Rayleigh fading and an independent zero-mean complex AWGN. If we limit the CPM scheme to be of single modulation index, the signal received by antenna j , Y j (t), can be given by [2]:

~) = Yj (t; D Xi(t; D~ i) =

Lt X Ci;j (t)Xi(t; D~ i) + Nj (t); j = 1; ; Lr ri=1

Es exp j(t; D~ ) ; i = 1; ; L i t T

X (t; D~ i) = 2hi Di (l)qm (t ? lT ); (k ? 1)T t < kT k

l=0

(1) (2) (3)

where Di(k) is the kth modulation symbol of the ith antenna, hi is the modulation index, qm (t) is the phase smoothing response function, T is the symbol time and Es is the symbol energy. It can be shown that for k Lm where Lm is the modulation memory [2]

(t; D~ i) = i(k) + 2hi

k X

l=k?Lm+1

Di(l)q(t ? lT )

(4)

If the modulation index hi is rational, say hi = 2pm where m and p are relatively prime integers, then i(k) (mod 2) 2 , where [2]

(5)

i = 0; 2p ; 4p 2(p ?p 1) : For rational modulation indices (t; D~ i(k)) is constrained to lie on a trellis. The state of the system S~i(k) at time kT , is de ned as S~i(k) = [ i(k) Di(k ? Lm + 1) Di(k ? 1) ] (6) With these de nitions, [Di (k); S~i(k)] uniquely specify (t; Di(k); S~i(k)) and X i(t; D~ i) in the semi-open interval [kT; (k + 1)T ).

3 Code Design 3.1

Design Criterion

~ = Assume a transmission of length Nc, the maximum likelihood metric of sequence D ~ in the presence of perfect channel state information c(t) is: m(y(t)jc(t); x(t; ~ )) = ?

Lr Z Nc T X j =1 0

jy(t) ?

Lt X i=1

ci;j (t)xi (t; ~i)j2dt:

(7)

~ = The pairwise error probability (PWEP) of confusing D ~ with another sequence ~D = ~, denoted as P ( ~ ~ ! ), is given by:

n

P ( ~ ! ~ ) = Pr m(y(t)jc(t); x(t; ~ )) m(y(t)jc(t); x(t; ~ ))

o

(8)

Following the similar method in [3], it can be shown that the pairwise error probability can be upper bounded by:

2 2H (s) ? 1 H (s) 3Lr 6 H (s) ? 1 N0 77 ~ 6 P ( ~ ! ) 4 QH (s) (s) 5 i=1

i

(9)

where ~(s) are the nonzero eigenvalues of the signal matrix Cs = RCc, H (s) is the number of nonzero eigenvalues, Cc is the covariance matrix of fading coecients and R is a matrix of the form: 0 R R R 1 BB R1121 R1222 R12LLtt CC R=B (10) @ ... ... . . . ... CA RLt 1 RLt2 RLtLt Each element of R is a diagonal submatrices of size Nc by Nc. The diagonal entries of the submatrix Rij are given as

Rij (k) =

Z kT

(k?1)T

i (t; ~ ; ~)j (t; ~ ; ~) for k = 1; ; Nc

(11)

where

i(t; ~ ; ~ ) = xi (t; ~ i) ? xi (t; ~i)

(12)

In order to minimize the error probability the rank of Cs should be maximized and so is its determinant. As interleaver cannot be used after CPM otherwise the phase continuity of CPM signals will be disrupted, it is reasonable to assume that the fading coecients remain constant during one error path. Herein we consider the special case where the fading

channels are statistically independent and number of transmit antennas is Lt = 2. Thus the signal matrix can be reduced to 2 R T j (t; ~ ; ~ )j2dt R T (t; ~ ; ~ ) (t; ~ ; ~ )dt 3 1 1 1 1 1 2 2 2 0 0 1 66 77 ... ... 66 R NcT R NcT (t; ~ ; ~ ) (t; ~ ; ~ )dt 77 2 dt ~ j ( t; ~ ; ) j 1 1 1 1 1 2 2 2 6 77 (13) (Nc ?1)TR 1 Cs = 66 R T (Nc?1)T ~ T 2 ~ ~ 77 ~ 2; 2)j dt 0 j2 (t; 66 0 1(t; ~ 1; 1..)2 (t; ~ 2; 2)dt . 75 .. . 4 R Nc T R Nc T ~ 1; ~1 )2(t; ~ 2; ~2 )dt ~ 2; ~2)j2dt (Nc ?1)T 1 (t; (Nc ?1)T j2 (t; 3.2

Code Structure

First we consider the delay diversity scheme (denoted DD) as proposed by Wittneben [4]. This scheme transmits the same information from both antennas simultaneously but with a delay of one symbol interval. It can be easily shown that in DD Cs is guaranteed to have full rank, but it may not achieve maximum coding gain. As we can see from the (13), there are several methods to guarantee Cs full rank. One approach is to make

Z kT 2 j1(t; ~ 1 ; ~1)j dt j2(t; ~ 2 ; ~2)j dt 6= 1 (t; ~ 1 ; ~1)2(t; ~ 2 ; ~2)dt (14) (k?1)T (k?1)T (k?1)T

Z kT

2

Z kT

2

for a given k 2 f1; ; Ncg. According to Scharwtz Inequality this inequality holds as long as 1 (t; ~ 1; ~1) 6= C2(t; ~ 2; ~2 ): (15) Here the dierence between linear modulation and nonlinear modulation comes into play. In linear modulation, for a given k, 1(t; ~ 1 ; ~1) = 1(k) ? 1(k) (16) 2(t; ~ 2 ; ~2) = 2(k) ? 2(k) (17) which means that we always have 1(t; ~ 1 ; ~1) = C2 (t; ~ 2; ~2) (18) Therefore, the above inequality does not hold at all. While in CPM system, as both 1 (t; ~ 1; ~1) and 2 (t; ~ 2; ~2 ) are functions of time, (15) can be easily satis ed. We can use dierent CPM schemes over dierent transmit antennas, which may result in higher complexity or more bandwidth occupation. Another method is to use dierent mapping rules (denoted DM) over dierent antennas to achieve full diversity. Following is just an example to demonstrate this idea. In order to show the performance we need to pick some speci c CPM scheme. Here we consider 4-ary CPFSK with h = 0:5. Consequently, the coding gain provided by delay diversity is 4. If we use the mapping rule as in table 1: D(k) D1 (k) D2 (k) 0 -3 -1 1 -1 +3 3 +1 -3 2 +3 +1 Table 1: Mapping rule of the DM space-time code, Lt = 2, 4-ary

The resulting coding gain is 5:5154. Compared with delay diversity there is a 1:3951dB improvement. Fig. 2 shows the performance of DD and DM. (In all our simulations, each frame consists of 130 transmissions out of each transmit antenna and we choose the normalized Doppler spread to be fD T = 0:01). The improvement of DM is more obvious when receive diversity is available, which can be explained through (9). When there is no receive diversity, although the coding gain for DM is large, the multiplicity for the coding gain is also large and they are of the same signi cance to the error performance; when the receive diversity is available, the coding gain plays a more important role than the multiplicity due to the exponential index Lr . 0

10

−1

Frame Error Probability

10

−2

10

−3

10

2 Tx, 1 Rx, DD 2 Tx, 1 Rx, DM 2 Tx, 2 Rx, DD 2 Tx, 2 Rx, DM 2 Tx, 4 Rx, DD 2 Tx, 4 Rx, DM −4

10

6

8

10

12 SNR in dB

14

16

18

Figure 2: Performance comparison between DD and DM

4 Adaptive Soft-output Demodulation In real application, CPM is very often combined with some outer coding scheme to achieve better performance. Actually the whole system shown in Fig. 1 can be treated as a serially concatenated system which we denote trellis-coded space-time CPM (TC-ST-CPM). It is well known that a soft-output algorithm is necessary in a concatenated system for an iterative detection to be implemented. Here we consider Kd -lag soft-output demodulation of the space-time code in nonlinear modulation without perfect channel state information (CSI). The algorithm only assumes a knowledge of the statistical characteristics of the channel in forming the soft-output a posteriori probability (APP). While the CSI is not explicitly estimated, the algorithm implicitly estimates the CSI and uses this estimate in forming the desired APPs. The algorithms presented here are extensions of the work done in [5, 6, 7, 8]. Modi cations of MLSD are proposed for CPM signals transmitted over frequency at, Rayleigh fading channels in [5]. Seymour [6] gives the MAP algorithms for linearly modulated system in at fading while Balasubramanian [7] extends these results to certain CPM signals. No either transmit diversity or receive diversity is considered in all these works. In [8] soft output receive diversity combining for CPM signals over space-time correlated Rayleigh fading channels is derived. Here we take both the transmit diversity and receive diversity into consideration. The following notations are useful in deriving the algorithm. For any time series ff (0); f (1); ; f (k)g, f~(k) = [f (k) f (0)]T , boldface denotes signals over all receive antennas and Yj (k) = fYj (t) : kT t < (k + 1)T g. And as in general, capital letter indicates random variables while lower case indicates realization.

The optimum Kd-lag soft-output algorithm produces the metric p (d(k ? Kd )j~y(k)) which can be written as X ~ p (d(k ? Kd )j~y(k)) = p 1 (k); 2 (k); d(k)j~y(k) (19) f1 (k);2 (k);d~(k)g2?(d(k?Kd ))

where f(k); 2(k); d~(k)g 2 kD and ?(d(k ? Kd)) = f1(k); 2(k); D~ (k) : D(k ? Kd) = d(k ? Kd )g. The set ?(d(k ? Kd )) represents all possible phase trajectories such that D (k ? Kd) = ~d(k ? Kd). p 1 (k); 2 (k); d(k)j~y(k) can be written into p 1 (k); 2(k); d~1(k); d~2(k); j~y(k) by using the structure of space-time encoder and can be computed in an ecient recursive fashion as p (k ); (k ); d~ (k ); d~ (k )j~ y(k ) = (20) f (y(k )j~ y (k ? 1); (k ); (k ); d~ (k ); d~ (k ))p( (k ); (k ); d~ (k ? 1); d~ (k ? 1)j~y(k ? 1))p(d(k )) X f (y(k )j~ y (k ? 1); (k ); d~(k ))p((k ); d~(k ? 1)j~y(k ? 1))p(d(k )) 1

1

2

2

1

1

2

2

1

(k);d~(k)

2

1

2

The key part is to get the innovation PDF f y(k)j~y(k ? 1); 1(k); 2 (k); d~1(k); d~2(k) which can be written as Z f y(k )j~ y (k ? 1); (k ); (k ); d~ (k ); d~ (k ); c(k ) f c(k )j~y (k ? 1); (k ); (k ); d~ (k ); d~ (k ) dc(k ) (21) 1

2

1

2

1

2

1

2

The rst part can be de ned using the Wiener measure [9] while the second part involves an extended Kalman lter [10] to estimate the mean and variance of c(k). Thus the innovation PDF can be computed through a multi-dimensional Gaussian integration. The whole algorithm is derived as follows: f

y(k )j~y (k ? 1); 1 (k ); 2 (k ); d~1 (k ); d~2 (k ); c(k )

Es < qH (k)c(k) ? NEs cH (k)R(k)c(k) (22) = Cf exp 2 N 0

0

where

q(k) = [q1;1 (k); q2;1(k); ; q1;Lr (k); q2;Lr (k)]T c(k) = [c1;1 (k); c2;1(k); ; c1;Lr (k); c2;Lr (k)]T Z (k+1)T 1 q1;j (k) = p yj (t)e?j(t;d (k);~s (k)) dt EsT kT = c1;j (k ? 1) + (k)c2;j (k ? 1) + n1;j (k ? 1) Z (k+1)T 1 yj (t)e?j(t;d (k);~s (k)) dt q2;j (k) = p EsT kT = c2;j (k ? 1) + (k)c1;j (k ? 1) + n2;j (k ? 1) R(k) = diag fA(k); ; A(k)g A(k) =

1 (k)

(k) = T

kT

1

1

(25)

2

2

(26)

1

(27) (28)

ej(t;d1 (k);~s1 (k))?j(t;d2 (k);~s2(k)) dt

(29)

(k)

1 Z (k+1)T

(23) (24)

Actually qi;j (k) is the matched lter output on the j th antenna for [Di(k); S~i(k)] = [di(k);~si(k)]. (Note for simplicity of notation we have dropped the dependence of q( i; j )(k)

on i (k), j (k), d~i(k) and d~j (k)). Notice there exists cross correlation between q1;j (k) and q2;j (k) ((k)) even when the channels are independent due to the signal superposition from dierent transmitter antennas. In order to compute the PDF f c(k)j~y(k ? 1); 1(k); 2 (k); d~1(k); d~2(k) we de ne the following random vectors. Q~ j (k ? 1) = [Q1;j (k ? 1); ; Q1;j (0); Q2;j (k ? 1); ; Q2;1(0)]T (30) i h T Q~ (k ? 1) = Q~ T1 (k ? 1); ; Q~ TLr (k ? 1) (31) It is apparent that the past matched lter outputs are sucient statistics for estimating C(k) and since C(k) and Q~ (k ? 1) are jointly Gaussian, the PDF of C(k) conditioned on the past information is 1 exp n? [c(k) ? ^c(k)]H ? [c(k) ? ^c(k)]o (32) f (c(k )j~ y(k ? 1); (k ); (k ); d~ (k ); d~ (k )) = Lr e j j 1

2

1

2

where ^c(k) and e are given as ^c(k) = gH Q~ (k ? 1)

in which

P = c =

2

1

e

e = E C(k)CH (k) ? gH P

(33)

h~ i H RE C(k ? 1)C (k) h~ i ~ H (k ? 1) R RH E C (k ? 1)C N ?1

(34) (35)

g = c + E0 I P (36) s R = diag (37) I fA;BH ; Ag (38) A = B I B = diagf(k ? 1); ; (0)g (39) Actually (33) can be viewed as a minimum mean square error (MMSE) predictor which is a space-time processor. g serves as a 2Lr dimensional prediction lter of time length k, ^c(k) are the predicted fading coecients at time t = kT and e is the prediction variance. Notice due to the signal cross correlation from dierent transmit antennas, both the lter and the variance are now functions of both channel self correlation and signal cross correlation and their evaluation cannot be carried out o-line, which is unlike the case when only receive diversity is available [8]. Closed form of the innovation PDF can be obtained as f

Lr Y

y(k )j~y (k ? 1); 1 (k ); 2 (k ); d~1 (k ); d~2 (k )

= Nf

(40)

=N0 exp ? 1 + EEs=N jz (k) ? ni (k)j2 + NEs jzi (k)j2 2 i 2 s 0 0 i i=1

where

2 2 21 6 H U 4 0

3

0 0 ... . . . 0 75 U H aea = 0 0 2L2 r n(k) = Ua^c(k) z(k) = Ua?1 q(k)

(41)

aH a = R(k)

(42)

The detector for the whole system is the same as that for standard serially concatenated codes [11] which is shown in Fig. 3. Based on the received signals from all antennas, Y1(t); ; YLr (t), and the a priori information, pe(D(k); i), the inner demodulator calculates the soft information pe(D(k); o). The outer decoder used this soft information to update the a priori information pe(D(k); i). This updated information is then used for the next iteration. After several iterations, the outer decoder calculates the soft information of the information bits, p(I (k~); o) and makes decision. Y1 (t )

~ p e ( D ( k ); i )

Soft- p e ( D (k ); o) Deinteroutput leaver Demod.

SoftOutput Decoder

~ p( I (k ); i)

YLr (t )

~ p( I (k ); o) used in the final decision

equally likely p e ( D ( k ); i )

~ p e ( D ( k ); o)

Interleaver

Figure 3: Iterative decoder structure 0

0

10

10

−1

10

−1

Frame Error Probability

Frame Error Probability

10

−2

10

−2

10 −3

10

2 Tx, 1 Rx, DD, with csi 2 Tx, 1 Rx, DD, w/o csi 2 Tx, 2 Rx, DD, with csi 2 Tx, 2 Rx, DD, w/o csi 2 Tx, 4 Rx, DD, with csi 2 Tx, 4 Rx, DD, w/o csi

2 Tx, 1 Rx, DM, with csi 2 Tx, 1 Rx, DM, w/o csi 2 Tx, 2 Rx, DM, with csi 2 Tx, 2 Rx, DM, w/o csi 2 Tx, 4 Rx, DM, with csi 2 Tx, 4 Rx, DM, w/o csi −4

10

−3

6

7

8

9 SNR in dB

10

11

Figure 4: Simulation results of DM

12

10

6

7

8

9 SNR in dB

10

11

12

Figure 5: Simulation results of DD

Performance of the derived algorithm is shown in Fig. 4 and Fig. 5. (Note in order to t the algorithm into practical applications we need to cut the channel memory into some nite length Nd). In these simulations, decision lag is chosen to be Kd = 4 and length of channel estimation lter is Nd = 12. There is no outer convolutional code considered here. The space-time codes used here are those given in section 3.2. These gures illustrate that the performance of the derived algorithm is close to that of perfect CSI with degradation being around 1dB when no receive diversity is available while the degradation being less when the receive diversity is available. Again the improvement of DM can be told through these two gures. Performance of TC-ST-CPM is shown in Fig. 6. Here the outer channel encoder is a convolutional code with generator polynomial is g(5; 7) and random interleaver is used.

The simulation results demonstrate several important points. First, by adding outer convolutional code, the whole system performance improves a lot. Second, the iterative detection provides signi cant improvement to the system performance even when CSI needs to be estimated. Third, although the CSI is generated within the inner space-time CPM demodulator, the exchange and update of soft information provides an implicit mechanism for the re-estimation of the unknown CSI. Therefore, the result is closer to that of perfect CSI compared to the case where there is no channel encoder. 0

Frame Error Probability

10

−1

10

w/o cc, w/ csi w/o cc, w/o csi w/ cc, iter.=1, w/ csi w/ cc, iter.=1, w/o csi w/ cc, iter.=5, w/ csi w/ cc, iter.=5, w/o csi −2

10

2

3

4

5

6

7 SNR in dB

8

9

10

11

12

Figure 6: Simulation results of the TC-ST-CPM using DM

5 Conclusion This paper gives the design criterion of space-time codes in CPM system. Its application to serially concatenated CPM system (TC-ST-CPM) is investigated. The adaptive soft-output demodulation without perfect channel state information is also presented. Simulation results show that both the space-time code and the derived adaptive demodulation algorithm give good performance.

References [1] N.Seshadri V.Tarokh and A.R.Calderbank. Space-time codes for high data rate wireless communication: Performance criterion and code construction. IEEE Trans. Inform. Theory, Vol. 44:744{765, Mar 1998. [2] T. Aulin J.B. Anderson and C.W.Sundberg. Digital Phase Modulation. Plenum, New York, 1986. [3] M.P. Fitz, J. Grimm, and S. Siwamogsatham. A new view of performance analysis techniques in correlated Rayleigh fading. IEEE Info. Theory, Submitted 1999. Available at http://eewww.eng.ohio-state.edu/ tz/Papers folder/proberrj.ps. [4] A.Wittneben. A new bandwidth ecient transmit antenna modulation diversity scheme for linear digital modulation. Proc. IEEE`ICC, pages 1630{1634, 1993.

[5] J.H.Lodge and M.L.Mother. Maximum likelihood sequence estimation of CPM signals transmitted over Rayleigh at fading channels. IEEE Trans. Commun., COM38:784{794, June 1990. [6] J.P.Seymour and M.P.Fitz. Near-Optimal Symbol-by-Symbol Detection Schemes for Flat Rayleigh Fading. IEEE Trans. Commun., Vol. COM-41:1525{1533, February/March/April 1995. [7] R. Balasubramanian and M.P. Fitz. Soft Output Detection of CPM Signals in Frequency Flat, Rayleigh Fading. IEEE J. Select. Areas Commun., July 2000. [8] Xiaoxia Zhang and M.P. Fitz. Soft Output Diversity Combining for CPM Signals over Space-time Correlated Rayleigh Fading Channels. submitted to ICC, 2001. [9] H. V. Poor. An Introduction to Signal Detection and Estimation. Springer-Verlag, New York, 1994. [10] H.Meyr A.Aghamohammadi and G.Ascheid. Adaptive synchronization and channel parameter estimation using an extended Kalman lter. IEEE Trans. Commun., Vol. 37:1212{1219, Nov. 1989. [11] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara. A Soft-Input Soft-Output Maximum A Posteriori (MAP) Module to Decode Parallel and Serial Concatenated Codes. TDA Progress Report, Vol. 42-127, November 1996.