Space-Time Codes Performance Criteria and Design for Frequency Selective Fading Channels Youjian Liu,Michael €? Fitz and Oscar I: Takeshita
[email protected];
[email protected];
[email protected] Department of Electrical Engineering, The Ohio State University be complex Gaussian random variables with equal variance 0.5 and independent. We observe that a 2 transmit antennas by 1 receive antenna system with a 2-tap channel can provide 4 level of diversity in a similar way as a 4 by 1 system with a flat fading channel. At frame error rate 0.01, the 2 taps of IS1 brings about 4 dB benefit compared to the 2 by 1 system in a flat fading channel. This motivates us to address the question of how to design I. INTRODUCTION codes to exploit the extra diversity. Reliable communication in a fading channel can be achieved It has been shown by Tarokh et al. [3] that the achievable diby exploiting the available diversities. The recently developed versity of a space-time code designed for flat fading channel space-time coding, which takes advantage of transmitter (or is not reduced if it is used in frequency selective fading chanspace) diversity, has been studied for fat quasi-static Rayleigh nel. However, a full diversity space-time code designed for f a t fading channel by Tarokh [ 11 and many others. Rayleigh fading channel may not be able to exploit all the available diversity. We propose a design methodology for codes in IS1 channels by introducing the concept of the virtual antenna. This idea permits a formalization of the performance criterion and the use of algebraic rank theory [4,5] to find STC that provide full diversity in an fading IS1 channel. Some example trellis codes which can achieve full space and frequency diversity are designed using the CO-ranktheory [SI. These codes are designed for two situations: without a channel interleaver and with a channel interleaver. If a channel interleaver is not used, the code can be decoded by maximum likelihood joint equalization and decoding. If a channel interleaver is used, the iterative channel equalization and decoding, or turbo equalization, can be employed. 0 2 4 6 8 tb 16 20 Simulation results demonstrate that these codes indeed are able €/No (dB1 to achieve full diversity. Fig. i. The outage probability versus SNR for systems with different number In Section 11, we describe the signal models. The performance of transmit antennas, 1 receive antenna, and different tap vmances at transmission rate 2 bitskymbol. For the case with two taps, each tap coefficient criteria are derived in Section 111. Section IV shows how to use CO-rank theory to design full diversity codes. The simulation is assumed to be equal variance 0.5 and independent results are shown in Section VI. Section VI1 concludes. In this paper, we study the design of space-time codes (STC) 11. SIGNAL MODEL for single carrier transmission over frequency selective quasistatic fading channels with non-negligible intersymbol interferIn a multiple antenna system, let Lt be the number of transence (ISI). These designs take advantage of the frequency di- mit antennas, L, be the number of receive antennas. Let N , be versity and space diversity. We developed the outage probabil- frame size in terms of symbol time. A space-time code word J ity evaluation method [2] to show that the outage probability is is an N , by Lt matrix with elements drawn from a finite alphagreatly improved due to the additional frequency diversity. Fig- bet. The code symbol matrix D is defined as D = f(J), where ure 1 plots the outage probability versus SNR for systems with f ( 0 ) is an element-wise constellation mapping from the finite different number of transmit antennas, 1 receive antenna, and alphabet to points of a constellation on the complex plane. The different number of IS1 taps at transmission rate 2 bitdsymbol. element at ith column and j t h row of D, D i ( j ) ,is transmitted For the case with two taps, the tap coefficients are assumed to from ith antenna at time j .
Absrrucf- This paper studies the space-time code design for single carrier transmission over frequency selective fading channels. The design criteria are derived first and then we apply the algebraic &-rank theory proposed earlier by the authors to show how to design codes to take advantage of space and frequency diversity simultaneously. Finally, example codes are shown to achieve desired level of diversity by simulation results.
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This work was supported by National Science Foundation under Grant NCR9706372.
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We study a simplified symbol spaced tap-delay-line model for frequency selective fading channels. The general channel
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model can be found in [6]. The signals transmitted from each antenna convolute with the spatial independent Rayleigh fading IS1 channel with L, taps. Let Q k ( t ) be the matched filter output of the received signal from kth receive antenna at discrete time t. Let C l , k (t,m ) be zero mean circular symmetric complex Gaussian Rayleigh fading coefficient of tap m at time t , associated with l t h transmit antenna and kth receive antenna. Let Wk(t) be the white complex Gaussian noise of k t h receive antenna at time t with variance No. The matched filter output at time t is L,-1
f i ( t - m)C(t,m) + @(t)
&(t)= m=O
t = 0,1,. .. ,N ,
(1)
+ L , - 1,
where
], 3, $(t) = [ W l ( t ) Wz(t) ... WL,(t) ] , and Q(t> = [ Qi(t) Q 2 ( t ) . f i ( t ) = [ Dl(t) Da(t) . * .
C(t,m) =
[
Cl,l(t,m )
QL,.(~)
DLC(t)
. . . C1,L(t,m)
1-
CLt,l(t,m) .... . . CLt,L,(t,m)
Due to the assumption of space independence, Cl,k(t,m)'s are assumed to be independent for different 1 or k . There might be correlation between different taps for the same pair of transmit and receive antenna. In quasi-static fading channel, Cl,k(t,m)remains constant within a frame and is independent from frame to frame. 111. PERFORMANCE CRITERIA
The code word Dim''', k = 1.. .Lt can be viewed as sigspace-time 'Odes for flat fading nals sent through virtual antennas. The same analysis [8] for flat channel has been derived in [ 1,7,8]by minimizing the pairwise fading channel can be applied to get a signal matrix error probability. For fading IS1 channel, similar results can be derived by in= ZCCZH, troducing "virtual transmit antennas". Since the performance criteria will not change with number of receive antennas, in the - =a where code symbol difference matrix = D -D ,covariance following derivation, one receive antenna is assumed. coefficients CC = E [ c C H ] . By collecting the signals from time 0 to N , - 1to a vector or matrix of The pairwise error probability can be upper bounded by a matrix, equation (1) can be expressed as The performance
Of
c,
z
*
where
L
-
D ( L C- 1 )
... D(Lc-l) Lt
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where E, is the total symbol energy of all transmit antennas, Xi's are the nonzero eigenvalues of the signal matrix C,, the diversity gain AH is the number of the nonzero eigenvalues of C,, and K is a constant. The product of nonzero eigenvalues is called product measure or coding gain. The code design problem becomes how to maximize diversity gain and the coding gain. Assuming IS1 taps are independent and with equal power, the C, can be simplified and the following can be proved [8] in the
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case of quasi-static Rayleigh fading and independent Rayleigh fading channels. Quasi-static fading: Xi's are proportional to square of the nonzero singular values of the virtual code symbol difference matrix Z("). AH is the number of nonzero singular values or the rank of Z("); Independent fading: Xi's are proportional to square of the nonzero norm of row vectors of Z("). AH is the number of nonzero rows of z(v); where Z(") is the difference of two virtual code symbol matrices. A virtual code symbol matrix D(")is defined as
We observe that the virtual code symbol matrix D(")contains original code symbol matrix D transmitted through the real antenna and the zero padded versions of D's transmitted through the virtual antennas. In the performance criterion of IS1 channel, D(")plays the same role as D in non-IS1 channel. For quasistatic fading channel, the maximum value of A, is Lt x L,. It is easy to verify that the above statement about diversity gain A, is true even if the independent IS1 taps have unequal power.
IV. CODEDESIGN For space-time code design, if diversity gain is important, one of the possible approaches is to maximize the diversity gain first, then to maximize the coding gain. There are some sufficient conditions, such as zero symmetry [9], binary rank theory [4], and CO-rank theory [SI to help maximize diversity gain. Exhaustive computer search may have to be used to maximize the coding gain. We will apply the algebraic CO-ranktheory to design some full diversity example codes, whose coding gain is not necessarily maximized, to demonstrate the code constructions for fading IS1 channel. CO-ranktheory applies to linear code defined on ring Z2b(j) with translation mapping [5]. A linear ; Z p ( j ) code can be represented as a linear transformation from an information sequence to a code word:
[ Glf
The matrix M corresponds to the channel interleaver. If no channel interleaver is used, M equals to an identity matrix. Once the generator matrices are found, CO-ranktheory [SI can be readily applied to check the diversity level. The details can be found in [2]. Intuitively, CO-ranktheory says that if all of the Colinear combinations of the generator matrices are not singular, the code is guaranteed to achieve full diversity. For trellis codes without channel interleaver, the design is extremely simpleonly the generator polynomials need to be checked [ 5 ] . For the case with channel interleaver, the code and the interleaver need to be jointly designed. One approach is to select a trellis code first, then randomly construct the interleaver. If the code and interleaver pair does not satisfy the &rank criterion, another interleaver is randomly chosen. The process repeats until a full diversity code and interleaver pair is found. For systems using trellis codes, with or without channel interleaver, the trellis codes need to satisfy the following constraints to achieve r level diversity with 1 receive antenna. They are extensions of the corresponding constraints in [ 11. Theorem I : (Without Channel Interleaverj If a trellis code without channel interleaver is used in an L , independent tap IS1 fading channel, to achieve a transmission rate of R bitslsymbol and T level diversity with 1 receive antenna, the code must have at least 2 R ( T - L c )states and the constraint length is at least T L,+l, where the constraint length is defined as minimum length of error path of the original trellis code. Proof: If we combine the trellis code with the channel memory to form a super trellis, then by Lemma 3.3.1 and 3.3.2 in [l], this super trellis must have constraint length T and 2 R ( T - 1 ) states. Since the channel memory can at most increase constraint length by L , - 1, the constraint length of the trellis code is at least r - L, - 1. The same argument in [ 1 , Lemma 3.3.21 implies that the code has at least 2R(T-Lc)states. 0
J', ]
J = [ =
where
G2f
G L ~ ~ ] ,
where+fis an N I by 1 input information sequence defined on Z2b, Ji denotes the ith column of the code word matrix, and Gi is the Z 2 s ( j ) generator matrix for ith antenna. By the concept of virtual antenna, the code design for frequency selective fading channels is equivalent to the design of generator matrices with the following structure for flat fading channel:
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Theorem 2: (With Channel Interleaverj Assuming ML detection, if a trellis code with channel interleaver is used in an L, independent tap IS1 fading channel, to achieve a transmission rate of R bits/symbol and T level diversity with 1 receive antenna, the code must have at least 2 R ( T / L c - 1 )states and the constraint length is at least r/L,, where the constraint length is defined as minimum length of error path of the original trellis code. Proof: Suppose the trellis code has constraint length K . When a constraint length error event happens, the code symbol difference can be non-zero at most in K time interval. In the ideal case, an interleaver can spread the code symbol differences over time so that the resulting virtual code symbol matrix Z(")
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has K L , non-zero rows. If K is less than r / L c , the code can not achieve r-level diversity. The same argument in [ 1, Lemma 3.3.21 implies that the code has at least 2R(‘ILc-1) states. 0
Some example QPSK trellis codes achieving full diversity
LtLc for systems with and without channel interleaver are designed as follows. All the lower bounds in Theorem l and 2 are achieved in some of the example codes. Hence, those lower bounds are tight.
Fig. 2 Block diagram of a multiple antenna system utilizing trellis code and channel
TABLE I EXAMIUC O ~ FOR S SYSTEMS W I T H 2 T R A N S M I T ANTENNAS A N I ) 1 R E C E l V t A N T k N N A IN A 1st C H A N N E L WITH 2 INIIEPkNDENTTAI’S.
Exa.
#Diversity
#State
1
4
16
2 4
4 4 4
4 4 16
5
3
4
3
Channel Source Interleaver Random Random Search Blockln,o Grimm et.al r91 Random Grimm et al. [9] Without Random Search Without Tarokh et al. [ 11
TABLE I t GENERATOR 1’OI.YNOMIALS
OF THE E X A M P L E CODES. X IS A D U M M Y VARIABLE.
Fig. 3. Block diagram of multiple antenna system utilizing trellis code without channel interleaver.
v.
EQUALIZATION A N D DECODER
For multiple antenna systems utilizing trellis codes, two equalization and decoding scheme are constructed according to whether channel interleavers are present. If there is channel interleaver, iterative BCJR [lo] Soft Input Soft Output (SISO) equalizer and decoder are used as shown in Figure 2. The soft information P(Q,priorlD) and P(D,evidence) have exact probability meaning in the first iteration. In the expression, “prior” refers to a priori probability of D and “evidence” refers to likelihood of D. The iterative method has been shown to have near ML performance [ 1 I]. If there is no channel interleaver, the Maximum Likelihood detector combines the IS1 channel and trellis code to form a super trellis as shown in Figure 3. VI. SIMULATION RESULTS
In order to verify whether the designed codes can achieve the desired diversity levels, computer simulation is conducted to evaluate the performance. Figure 4-6 show the frame error rate versus SNR. The frame size is 130 symbol time, corresponding to 260 information bits. The transmission rate is 2 bits/symbol. The system has 2 transmit antennas, 1 receive antenna. The fading IS1 channel has 2 equal power independent taps. For examThe example codes together with some non-full-diversity ples with channel interleavers, the SISO iterative equalization codes to compare with are summarized in Table I and Table and decoding takes as low as 3 iterations to converge. To verify the diversity level of codes with channel inter11. Example 1-3 are for system with channel interleaver. The generator polynomial of the trellis code of example is found by leavers, Figure 4 shows the performance of Example 1, a 16random trials with a large product measure for the shortest er- state code, and Example 2 and 3 , 4-state code. At frame error events. Example 2 and 3 use the same generator polynomi- ror rate 0.01, the performances of the 16-state code and 4-state als corresponds to the 4-state code in [9]. The interleavers are code are less than 2.5 dB and 4dB away from the outage probjointly designed for them to achieve full diversity. Example 4 ability. For the 4-state code, both a rectangular and a random and 5 are for system without channel interleaver. Example 4 is interleavers are found to facilitate full diversity and the correfound by random trials. Example 5 is the delay diversity 4-state sponding performance is very close. Although iterative equalcode in [ 11 for the purpose of comparison. Example 4 achieves ization and decoding is sub-optimal, the example codes appear full diversity but example 5 does not. No attempt is made to op- to achieve full diversity as indicated by their slopes. In an independent 2-tap IS1 channel without interleaving at timize the coding gain. We only want to illustrate the design of the transmitter, full rank codes for flat fading might not achieve full diversity codes.
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Fig. 4. Performance of Example I , 2, and 3 in an independent 2-tap IS1 channel compared with the outage probability.
Fig. 6. Performance comparison of Example 1 (with channel interleaver) and Example 4 (without channel interleaver).
code design problem is translated to that of flat fading channel. Example codes, which can take advantage of the extra diversity available in IS1 channel, are designed using the CO-ranktheory. Both systems with and without channel interleaver are considered. The simulation results show the example codes achieve desired level of diversity.
LL
P
1
REFERENCES V. Tarokh. N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trurzs. on Info. Th., vol. 44, no. 2, pp. 744-765, Mar. 1998. H. El Gamal, A. R. Hammons Jr.. Y. Liu, M. P. Fitz, and 0. Y. Takeshita, “On the design of space-time and space-frequency codes for MIMO frequency selective fading channels,” IEEE Info. Theory, srrbmitted, Feb. 2001. V. Tarokh, A. Naguib, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility and multiple paths,” IEEE Trans. on Cornin., vol. 47, no. 2, pp. 199-206, Feb. 1999. A. R. Hammons Jr. and H. El Gamal, “On the theory of space-time codes for PSK modulation,” IEEE Itzfo. Theory, vol. 46, no. 2, pp. 524-542, Mar. 2000. Y. Liu, M. P. Fitz, and 0. Y. Takeshita, “A rank criterion for QAM spacetime codes,” submitted to IEEE Trans. on Info. Theory, Mar.2000. W. van Etten, “Maximum likelihood receiver for multiple channel transmission systems,” IEEE Truns. Cornmun., vol. 24, pp. 276-283, Feb. 1976. J. C. Guey, M. P. Fitz, M. R. Bell, and W. Y. Kuo, “Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels,” in Proc. of IEEE Vehicular Technology Conference.Atlanta, CA, 1997, vol. 1, pp. 136-140. M. P. Fitz, J. Grimm, and S. Siwamogsatham, “A new view of performance analysis techniques in correlated Rayleigh fading,” in IEEE WCNC, New Orleans, LA, Sep. 1999. J. Grimm, M. P. Fitz, and J. V. Krogmeier, “Further results in spacetime coding for Rayleigh fading,” in Pmc. 36th Annual Allertorz Conf; on Cornmunicution. Control, und Comnputing, Monticello, Illinois, USA, September 1998, pp, 391400. L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Ir@bnn. Theory, vol. 20, pp. 248-287, Mar. 1974. S. Benedetto, D. Divsalar, G . Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: performance analysis, design, and iterative decoding,” IEEE Inji). Theory, vol. 44, pp. 909-926, May 1998.
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idB)
Fig. 5. Performance of Example 4 and 5 and relevant outage probabilities.
full diversity. For example, consider the 4-state QPSK delay diversity code of [I]. It is of full rank in flat fading channels, but it does not have enough states to achieve full diversity via Theorem 1. Figure 5 shows the performance of the delay diversity code (AH = 3) and Example code 4 that achieve full diversity (AH = 4) along some related outage probabilities. The outage probability of 3 transmit antenna is to indicate the 3 levels of diversity. The purpose of Figure 5 is not to insinuate this is a fair comparison (4 states vs 16 states) but to compare the attained diversity levels of the two codes. Figure 6 compares the performance of Example 1 and Example 4. Both codes have 16 states. With channel interleaver, the performance improves about 1 dB. The simulations results show that the example codes all achieve desired level of diversity. This demonstrates the effectiveness of the design procedure. VII. CONCLUSIONS We studied the space-time code design for fading IS1 channel. By introducing the concept of virtual transmit antenna, the
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