A RANK CRITERION FOR QAM SPACE-TIME CODES WITH APPLICATION TO TURBO CODING Youjian Liu, Michael P. Fitz, Oscar Y. Takeshita, and Zhongxin Han
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ABSTRACT
codes to ensure full space diversity. More recently, the design of PSK modulated space-time codes are also addressed Sufficient conditions to ensure QAM space-time codes achieve by Blum [4]. full space diversity in quasi-static fading channel are preIn [5], we provide a theory for the design of space-time sented. The conditions are on code words or generator codes in quasi-static Rayleigh fading channel with higher matrices instead of on every code word pair. This simorder of constellation (22k QAM), such as QPSK, 16 QAM, plifies the check of full space diversity of a code. Based 64 QAM, etc. It includes the BPSK binary rank criterion on these conditions, full space diversity parallel concatein [3] as a special case. For QPSK constellation, it is apnated turbo codes are proposed to use in a multiple eleplicable to GF(4) codes instead of Z4 codes as in [3]. Conment antenna environment. Comparing with space-time sequently, many traditional codes and turbo codes can be trellis codes, the simulations show it has robust performodified to be space-time codes. mance at both quasi-static fading channel and time varySome space–time trellis codes may have low level of diing fading channel. versity in space–time correlated fading channels [2]. Lower performance is caused by the interaction between the channel structure and the code structure. A ”random” coding 1. INTRODUCTION structure would reduce this interaction. Turbo coding is the current best way to build decodable ”random” codes. The For wireless communication, the design goal of so called inherent rich structure of turbo codes will likely to provide “space–time” codes [1] is to take advantage of both the sparobust performance in the variety of channels that will likely tial diversity provided by multiple antennas and the tempobe encountered in wireless practice. Efforts have been made ral diversity available with time–varying fading. to adapt turbo codes to multiple transmit antennas environIn quasi-static Rayleigh fading channel, each possible ment. A discussion of other groups' work can be found code word difference in a coded modulation produces a “sigin [5]. In [6], the BPSK binary rank criterion [3] is used nal” matrix. Increasing the rank of a “signal” matrix into ensure full space diversity of the parallel concatenated creases the amount of diversity in demodulation and reduces turbo codes. Although parallel concatenated QPSK “spacethe pair-wise error probability [1, 2]. If the code length in time turbo codes” has been proposed in [7, 8], full space time, Nc , is larger than the number of transmit antennas, diversity is not guaranteed due to the lack of proper theory. Lt , the maximum possible rank of a “signal” matrix is Lt . The o -rank criterion developed in [5] enables us to design A code is said to achieve full space diversity when the rank full space diversity QPSK turbo codes. of every “signal” matrix corresponding to every code word In Section 2, we give the signal models and explain the pair is equal to Lt . While the ranks of the “signal” matrices space-time code design criteria. Section 3 presents the theare defined over the complex number field, traditional code ory of -rank criterion. As an application of the theory, the o design is usually carried out in finite fields or finite rings. design and performance of QPSK space-time turbo codes This discrepancy causes a serious obstacle in the design. are given in Section 4. Section 5 concludes. The paper by Hammons and El Gamal [3] represents an important first step to bridge this discrepancy. They provided a binary rank criteria for binary BPSK codes and Z4 QPSK 2. MODELS AND PERFORMANCE CRITERIA This work was supported by National Science Foundation under Grant NCR-9706372. Y. Liu, M. P. Fitz, and O. Y. Takeshita are with Department of Electrical Engineering, The Ohio State University, 205 Dreese Lab, 2015 Neil Ave., Columbus OH 43210. Z. Han is with Department of Mathematics.
J
A space-time code word is an Nc by Lt matrix with elements drawn from a finite alphabet. The code symbol ma= f ( ), where f () is an elementtrix is defined as wise constellation mapping from the finite alphabet to points
D
D
J
Definition 1 (Ring Z2k (j )) The ring Z2k (j ) is a finite set and k is a positive integer. Each element V has the form,
C1,1 G D1
G Q1 C2 ,1
V = VI jVQ ;
C1, 2 G Q2
G D2 C 2, 2
Figure 1: A system with 2 transmit antennas and 2 receive antennas. of a constellation on the complex plane. The element at ith column and j th row of , Di (j ), is transmitted from ith antenna at time j . Note that, in this paper, the variable in the subscript is used to indicate the column of a matrix if not explicitly specified. Let Lr be the number of receive antennas. The signals transmitted from each antenna experience spatial independent quasi-static Rayleigh fading. The matched filter output of the received signal at k th receive antenna from time 1 to Nc is Q~ k = C~ k + W~ k ; (1)
D
D
~ k is an Nc by 1 vector, W ~ k is Nc by 1 complex Adwhere Q ditive White Gaussian Noise (AWGN) vector with one side ~ k is an Lt by 1 fading coefficient power spectrum N0 , and C ~ k . Ck;l models vector. Let Ck;l denote the lth element of C a Rayleigh fading channel common in wireless communications by being zero mean complex Gaussian random variable. Ck;l 's are assumed independent for different k or l and known to the receiver. Figure 1 shows an example of such system with 2 transmit antennas and 2 receive antennas. Considering two code symbol matrices, and , is defined as = the code symbol difference matrix ? . With a goal of minimizing the pair-wise error probability, the quasi-static fading channel design criteria [1,2] are to maximize the rank of code symbol difference matrix and maximize the product of nonzero eigenvalues of H for all pairs of code words.
D
Z
D
D
D Z
Z Z Z
3. THEORY OF RANK CRITERIA In [5], sufficient conditions are provided to ensure full space diversity of a code using QAM Modulation. The main results are presented here without proof. 3.1. Preliminary Definitions The full space diversity rank criteria developed in [5] are for codes defined on the ring Z2k (j ). In the sequel, n and n are used to denote the modulo n addition and subtraction, and subscript n is dropped if the context is clear. Subscripts I and Q will be used to indicate the real part and imaginary part of a complex number or a matrix.
where the real part VI and imaginary part VQ are integers in Z2k and j 2 = ?1 = ?1 + 2k . In this paper, nonnegative integers in f0; 1; ; 2k ? 1g are used to label elements in Z2k. The addition and multiplication in this ring are the addition and multiplication in complex number field followed by modulo 2k operation on the real part and the imaginary part. Definition 2 (Linear Z2k (j ) Code with Translation Mapping) A linear Z2k (j ) code C is a set of code words which form an additive group. Each code word is an Nc by Lt matrix with elements in the ring Z2k (j ). The linearity implies that if ; 2 C , then 2 C . Each code word matrix is mapped to a complex code symbol matrix by ? = ? (2k ? 1)=2 + j (2k ? 1)=2 . It the translation, results in a 22k QAM constellation.
J
J
J J D J
J J
D
A linear Z2k (j ) code can be represented as a linear transformation from information sequence to code word:
J =
=
J~1 J~2 J~Lt G1 I~ G2I~ GLt I~ ;
(2) (3)
where I~ is an Z2k input information sequence, J~i denotes the ith column of the code word matrix, and i is the Z2k (j ) generator matrix for ith antenna.
G
Definition 3 (o -Coefficients) Coefficients, 1 , 2 , : : :, L , in Z2k (j ) are said to be o -coefficients if there exists i such that ai + bi is odd, where ai jbi = i .
V
Definition 4 (Column o -Rank) A matrix over the ring Z2k(j ) has column o -rank L if L is the maximum number of column vectors of , such that
V
9V = fV~i1 ; : : : ; V~iL g;
L M l=1
l V~il 6= ~0;
for any o -coefficients, 1 ; 2 ; : : : ; L . The row o -rank can be similarly defined. Since column o -rank and row o -rank are equal [5], they are called o -rank.
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over Definition 5 (Full o -Rank) An m by n matrix ring Z2k (j ) is said to be of full o -rank if it has o -rank equal to the minimum of m and n.
3.2.
o -Rank Criterion
Information Size=260, 16 QAM, Rate=4 bps/Hz, NTx=2, NRx=2
0
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The sufficient conditions on code words are given first.
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For linear codes, the conditions can be translated into the conditions on the generator matrices. Theorem 2 Let C be a linear Z2k (j ) code. The ith column of the code word matrix is defined as
J~i = Gi I~;
(4)
G
G
J
-2
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10
fd*T=0, New fd*T=0, AT&T, Delay Diversity -4
10
14
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20 22 Es/N0 per Rx Antenna (dB)
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Figure 2: Performance comparison of 16 QAM space-time trellis codes with rate 4 bits/symbol and 260 information bits/frame. &
I
Recursive Encoder 1
Puncture 1
Channel Interleave 1
Recursive Encoder 2
Puncture 2
Channel Interleave 2
Information Interleaver
J
then 8 nonzero 2 C , is of full o -rank. Thus, the code achieves full space diversity.
18
&
D1 Multiplex
where I~ is the information sequence in Z2k (j ), i is the generator matrix for ith antenna. If for all o -coefficients, 1 ; 2 ; : : : ; Lt , and for all nonzero information sequence, I~, ! Lt M (5) i i I~ 6= ~0; i=1
-1
10
Frame Error Rate
Theorem 1 (o -Rank Criterion) Let C be a linear Z2k (j ) code with translation mapping to 22k QAM constellation. If every nonzero code word 2 C has full o -rank, then C achieves full space diversity.
&
D2
Figure 3: Space-time turbo code encoder for 2 transmit antenna system.
4. APPLICATIONS The analysis of the existing space-time codes and construction of new space-time codes using the o -rank criterion are detailed in [5]. In this section, we present a 16 QAM example trellis code and QPSK full space diversity turbo code design. 4.1. Space-Time Trellis Code The o -rank criterion can reduce the computation in the search for good space-time trellis codes by identifying full space diversity codes. We give an example of 16 QAM linear Z4(j ) space-time trellis code with rate 4 bits/symbol and 16 states found using the criterion . The generator polynomials with x as dummy variable are
G = G = 1
2
34 2x 1 4 j 34 2x 2x 24 3x 0 4 j 3x 24 x :
The simulation results are plotted in Figure 2 together with the performance of optimal product measure delay diversity 16 state 16 QAM code [1]. In the simulation, each frame corresponds to 260 information bits. With two receive antennas, the performance of our new code, which is the solid line in the plot, is 1 dB better than the delay diversity code, which is the dash line in the plot.
4.2. Space–Time Turbo Codes 4.2.1. Encoder Design The encoder of the proposed space–time turbo code [7] is composed of a turbo code followed by the operation of puncturing, channel interleaving and multiplexing (Figure 3). As an example, we consider a rate 2/4 turbo code [9] with 2 transmit antenna and QPSK modulation. Figure 4 illustrates the structure of the turbo code. Let ij be the matrix corresponding the transfer function from I~(i) to V~ (j ) , where I~(i) and V~ (j ) are the binary input and output of the component ~ (1) ; V~ (2) ) are mapped to QPSK code of the turbo code. If (V symbols using the Gray mapping, then the space-time turbo code can be viewed as a linear Z2(j ) code with ith column of a code word as
H
GiI~ Mi HH P HH P II~~ + ~ H H I j Mi H P H P ; I~ where matrix P corresponds to the information interleaver and matrix Mi corresponds to the operation of puncturing, J~i = =
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11
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(2)
21
12
12
(1)
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channel interleaving and multiplexing.
(1)
(2)
G I (1)
G V (1)
Compare not Full Div., Full Div., no MUX, QPSK, Rate=2 bps/Hz, NTx=2, NRx=1, Size=260, fd*T=0
0
10
G I (2)
G V (2) π1
π2
-1
Frame Error Rate
10
Figure 4: A rate 2/4 turbo code in [9].
-2
Info Prior Component decoder 1
Info LD
Symbol Prior Symbol LD
Component decoder 2
Info Prior
ST-Turbo, #state-CC=4, not Full Div., MUX ST-Turbo, #state-CC=4, Full Div., MUX ST-Turbo, #state-CC=4, Full Div., no MUX ST-TCM, #state=64, AT&T Outage Capacity
Make Decision
Symbol LD
-3
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Decoded Bits
Matched Filter
Symbol Prior
Deinterleaver
Matched Filter
Interleaver
Channel Symbol Decoder
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Info LD
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14 16 18 Es/N0 per Rx Antenna (dB)
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Figure 6: Performance in quasi-static fading channel for 3 different cases.
LD: Likelihood
Figure 5: Iterative Decoder for the case of 2 receive antennas.
It is difficult to find a systematic method to construct the matrix and i so that 1 and 2 satisfy Theorem 2. However, the chance is high that a randomly picked information interleaver and channel interleaver result in a full space diversity code.
P
M
G
G
4.2.2. Decoder The optimal decoding of a space–time turbo code is complicated for the following reasons. First, the received signal is a linear combination of symbols from different transmit antennas and the noise. Secondly, the inputs of the component encoders are interleaved versions of the same information bits sequence as in turbo code. However, we can use a sub– optimal iterative algorithm like that for turbo decoding to decode the code. The decoder diagram is shown in Figure 5 for the case of two receive antennas. The fading coefficients are assumed known to the receiver. The matched filter output is used to calculate the likelihood of the channel symbols. In the first iteration, channel symbols from different transmit antenna are assumed to have equal a priori probability. The turbo decoder uses the likelihood information to do one iteration of standard turbo decoding and produces a priori probability of the channel symbols. The channel symbol decoder uses the “new” a priori probability to refine the likelihood information about the channel symbols which is the start of the second iteration. After several iterations, hard decisions on information bits are made based on the soft information provided by the turbo decoder.
4.2.3. Simulation Results The performance of the example code without puncture mentioned in Section 4.2.1, whose turbo encoder is shown in Figure 4, is evaluated by simulation. In the simulation, each frame corresponds to 260 information bits. The rate of the code is 2 bits/symbol. The performance with one receive antenna in quasi-static fading channel is shown in Figure 6. There are three cases. In the first case, the information interleaver and the channel interleaver are randomly chosen. The outputs of each component code of the turbo code are multiplexed to two transmit antennas. The code does not satisfy the o -rank criterion and rank deficient error event did happen in the simulation. In the second case, the generator matrices satisfy the o -rank criterion after several trials of random information interleaver and channel interleaver. Therefore, it is a full space diversity code. The third case is also a full space diversity code. But different from the first and the second case, there is no multiplexing, which means that all the outputs from one component code are transmitted only on one antenna. For comparison, the performance of a 64 state space-time trellis code in [1] and the outage capacity [10] are also shown in the plot. At low SNR, all the three cases have performances close to the 64 state trellis code. When the SNR increases, the frame error rates of the first two cases do not decrease as much as that of the 64 state code. The performance of the second case showed better slope than the first case since it is a full diversity code. The performance curve of the third case has the same slope as the 64 state code and is 2.5 dB away from the outage capacity at frame error rate 0.01. The third case performs better than the second case. This is counter intuitive since in the second case, with multiplexing, the outputs of one component code experience spatial in-
Time Varying, Turbo, Full Div., No MUX, QPSK Rate=2 bps/Hz, NTx=2, NRx=1, Size=260, #state=4
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Frame Error Rate
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fd*T=0 fd*T=0.001 fd*T=0.01 fd*T=independent -3
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Es/N0 per Rx Antenna (dB)
Figure 7: Robustness of performance in slow to fast fading channel. Turbo-FullDiv.-NoMux Compare with Grimm-32-State-TCM Frame Error Rate=0.01, Size=260, QPSK, Rate=2 bps/Hz, NTx=2, NRx=1 7
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[1] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. on Info. Th., vol. 44, no. 2, pp. 744–765, Mar. 1998.
[3] A. R. Hammons Jr. and H. El Gamal, “On the theory of space-time codes for PSK modulation,” to appear in IEEE Trans. On Information Theory, 1999.
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6. REFERENCES
[2] 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.
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0
Gain of E /N per Rx Antenna (dB)
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sity [5] are presented. The conditions are on code words or generator matrices instead of on every code word pair. As an application example, a full space diversity 16 QAM trellis code is found, which has 1 dB gain over the one with delay diversity [1] when two receive antennas are used. Methods are given to design full space diversity parallel concatenated turbo codes. The simulation results of an example code demonstrate that we can use very simple component codes to obtain equal performance as a space–time trellis codes with 64 states [1]. In fast fading channels, the performance improves by 8 dB with the additional temporal diversity.
0
0.001 0.01 Doppler Spread, fd*T
independent
Figure 8: Performance gain over 32 state space-time trellis code at different Doppler spread. dependent fading and should results in better performance. Further investigation is needed to understand it. Figure 7 shows the performance of case 3 with different Doppler spread. The performance improved significantly when the temporal diversity is available. At frame error rate 0.01, the performance corresponds to the independent fading has a gain of 8 dB over the quasi-static fading case. The performance of case 2 is also compared with the performance of a 32 state space-time trellis code [11] at different Doppler spreads. At frame error rate 0.01, the performance gain in terms of Es =N0 per receive antenna is shown in Figure 8. The gain increases with the Doppler spread. In independent fading channel, the gain is as much as 6.4 dB. These figures show that the space-time turbo code has a significantly rich structure to take advantage of both the temporal and spatial diversity. 5. CONCLUSIONS For QAM error correcting codes in multiple transmit antenna environment, sufficient conditions of full space diver-
[4] R. S. Blum, “Analytical tools for the design of space-time convolutional codes,” submitted to IEEE Trans. On Information Theory, Feb. 2000. [5] Y. Liu, M. P. Fitz, and O. Y. Takeshita, “A rank criterion for QAM space-time codes,” submitted to IEEE Trans. on Info. Theory, Mar. 2000. [6] Hsuan-Jung Su and Evaggelos Geraniotis, “Spectrally efficient turbo codes with full antenna diversity,” in Proc. Multiaccess Mobility and Teletraffic for Wireless Communications (MMT' 99), Scuola Grande di San Giovanni Evangelista Venice, Italy, October 6-8 1999. [7] Y. Liu and M. P. Fitz, “Space–time turbo code,” in Proc. 37th Annual Allerton Conf. on Communication, Control, and Computing, Monticello, Illinois, USA, September 1999. [8] Andrej Stefanov and Tolga M. Duman, “Turbo coded modulation for wireless communications with antenna diversity,” in Proc. IEEE Vehicular Technology Conference (VTC), Amsterdam, The Netherlands, September 1999, Fall. [9] D. Divsalar and F. Pollara, “Turbo trellis coded modulation with iterative decoding for mobile satellite communications,” in IMSC 97, June 1997. [10] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, pp. 311–335, March 1998. [11] J. Grimm, Transmitter Diversity Code Design for Achieving Full Diversity on Rayleigh Fading Channels, Ph.D. thesis, Purdue University, Dec. 1998.
