Concatenated Space-time Codes for Quasi-static Fading Channels: Constrained. Capacity and Code Design. Vivek Gulati and Krishna R. Narayanan. Dept. of ...
Concatenated Space-time Codes for Quasi-static Fading Channels: Constrained Capacity and Code Design Vivek Gulati and Krishna R. Narayanan Dept. of Electrical Engineering Texas A&M University, College Station, TX 77843, USA. Email: vivekgu, krishna @ee.tamu.edu Abstract— The problem of designing codes with close-tocapacity performance on the multiple-input multiple-output (MIMO) quasi-static fading channel (QSFC) is addressed. We consider three different coding schemes – namely, direct transmission of random-like codes, concatenation with linear processing orthogonal space-time block codes (o-STBC) and concatenation with space-time trellis codes (STTC). The constrained modulation outage capacity of each of these schemes is computed. It is shown how Low-density Parity Check (LDPC) codes may be used to approach capacity in these cases. For the STTC, the serial concatenation scheme of [1] turns out to have close to capacity performance.
I. I NTRODUCTION ULTIPLE transmit and receive antenna systems have received considerable attention due to the promise of significant increase in capacity [2]. High data rate packet-data systems motivate the use of the quasi-static fading channel (QSFC) model. In this model, the fade remains constant for a duration of the codeword and changes independently from one codeword to the next. Several schemes address the issue of coding for multipleinput, multiple-output (MIMO) QSFCs. Space-time trellis codes (STTC) [3] and linear processing orthogonal space-time block codes (o-STBC) [4], [5] achieve full spatial diversity. These two families of codes have been used in concatenation schemes to obtain improved performance without sacrificing spatial diversity. The QSFC is a non-ergodic channel and hence its Shannon capacity is zero. Given a transmission rate (no matter how small), there is a non-zero probability that the given channel realization cannot support this rate. Outage capacity is defined as the rate for which the probability that the instantaneous capacity of the channel falls below this rate is less than a constant (called the outage probability, or simply, the outage) :
M
outage
instantaneous
outage
(1)
The outage capacity serves as the measure of performance for the QSFC. The definition for outage capacity assumes infinitely long codewords. However, the comparisons found in most literature do not account for this fact. The performance of many codes, such as the STTCs, depends on the length of the codeword being used. By using sufficiently small codeword length, these codes can be made to appear good whereas when the length is increased the frame error rate approaches one. In this sense, these codes are not “good.” We call a coding scheme “good” if
its performance in terms of frame error rate (FER) improves (or, at least does not deteriorate) as the codeword length approaches infinity. This definition of a “good” coding scheme is motivated by [6]. Further, the known results for the outage capacity of the MIMO-QSFC [2] assume that the modulation is unconstrained. In practice, we are constrained to use a finite alphabet. This introduces a further loss in capacity, which is not accounted for in most published papers. In this paper, we address the issues of long codewords and constrained modulation for three different space-time coding schemes: (a) Direct transmission of random-like codes. (b) A concatenated scheme with o-STBC as the inner code. (c) A concatenated scheme with STTC as the inner code. The direct transmission scheme is a generalization of [7]. This scheme may be thought of as a concatenated system where the inner space-time code is a trivial serial to parallel converter. The transmit sequences for this scheme imitate equiprobable i.i.d. sequences from a finite alphabet. This is to be contrasted with [2] where an i.i.d. Gaussian sequence is shown to achieve capacity on the MIMO QSFC. The symmetric, constrained modulation outage capacity of the channel is the fundamental limit for the performance of such codes. We show how to compute this limit and provide examples of codes that approach this limit. The o-STBCs transform a MIMO channel into an equivalent single-input single-output AWGN channel [4]. This equivalence is used to compute the capacity of a system in which the inner code is fixed to be an o-STBC. We provide examples showing that both the predicted and the simulated performance is very close to capacity. Finally, a concatenation scheme with STTCs as inner codes is considered. We extend the method of [8] to compute the outage capacity of this system. This capacity is identical for both the recursive and the non-recursive realizations of a given space-time trellis code. It is found that the scheme in [1] has a performance close to capacity. We find that at low rates all the three schemes have a constrained capacity close to the unconstrained capacity. For a range of rates the trellis codes offer higher capacity than the other two schemes. For high rates the capacity of the trellis codes and the block codes saturates and direct transmission is the only available choice among these three schemes. A note on the notation is in order. Vectors and matrices are represented in boldface. The superscript represents transpose,
Outer Code
(2)
The channel signal to noise ratio (SNR) at each receive antenna, is independent of the number of transmit antennas. The entries of the channel gain matrix and of the noise vector are independent, zero-mean, unit variance complex Gaussian random variables. The channel gain matrix remains constant for the duration of a codeword and changes independently from one codeword to the next (QSFC). We assume that the transmitter does not know the channel (no CSIT) but the receiver has perfect knowledge of the channel matrix (perfect CSIR). Throughout this paper we also assume that the channel input alphabet is an PSK constellation and the number of receive antennas , though these are not restrictive assumptions.
6
Outer Encoder
Space−time Code
=< & 6
>@? AB C&
Space−time decoder/ demodulator
MIMO−QSFC
Outer decoder
Fig. 1. Space-time Concatenation Schemes.
Fig. 1 represents the generic concatenation scheme considered in this paper. The three schemes considered in the following are obtained by changing the inner code. III. D IRECT T RANSMISSION
OF
R ANDOM -L IKE C ODES
The codewords of some codes appear like equiprobable i.i.d. sequences. We call such codes random-like codes. Low-density parity-check (LDPC) codes and turbo-codes (with random interleaving before transmission) are examples of such codes especially when the codeword length is large. In this section, we describe the structure of the direct transmission scheme, compute the symmetric constrained modulation outage capacity of the channel and provide an example which approaches this limit. A. Transmitter and Receiver Structures The direct transmission scheme is shown in Fig. 2. The output of the code is converted into parallel streams, the bits are and transmitted over mapped into symbol from the alphabet the various transmit antennas. We can view this scheme as a concatenation in which the inner (space-time) code is simply
D
Loglikelihood Computation Nt
Outer Decoder
Nr
Fig. 2. Direct transmission scheme.
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* ) + ), )-. / 0)-132 ( 4 5 ( 76 98;:
6
S/P
Turbo space−time demodulation and decoding
We consider a system with transmit and receive antennas. In the complex base-band model, one channel use corresponds to the transmission of the vector . The received vec is given by the equation: tor
< :
1
Bits to Sym Bits to Sym
II. S YSTEM M ODEL
5
1
represents conjugation.
a serial-to-parallel converter. The assumption that the code is binary is not restrictive. The receiver performs space-time demodulation and decoding in an iterative fashion similar to the one described in [7]. The log-likelihood generator (space-time demodulator) computes the extrinsic information about all the bits transmitted during a given time instant based on the received signal and the apriors received from the decoder. The decoder processes these likelihoods to compute the decisions on the uncoded bits and the extrinsics corresponding to the transmit symbols. B. Constrained Modulation Outage Capacity Let the transmit vector be drawn equally-likely from the set of all possible transmit vectors . The mutual infor mation between the transmit vector and the receive vector is given by (3) at the top of the next page (obtained from (5) in [9] and replacing the integral with expectation). This mutual information represents the instantaneous symmetric, con strained modulation i.i.d. capacity of the channel . In the absence of CSIT, this mutual information achieved by the equiprobable p.m.f. is the true capacity of the channel. The expectation in (3) is evaluated using Monte Carlo simulations. In order to compute the outage capacity, we calculate the instantaneous capacity for a number of channel realizations (10000 points in the simulation results presented here) and generate a quantile plot. The results for BPSK modulation at 10% outage, computed from (3), are shown in Fig. 3. Note that the loss in capacity is due to two reasons – due to the constrained modulation and due to the assumption of independence among the sequences transmitted from the various transmit antennas.
(
FE .
1.2
1
10% Outage Capacity (b/s/Hz)
denotes transpose conjugate and The operator denotes expectation.
0.8
0.6
0.4
Unconstrained BPSK: Alamouti concatenation BPSK: Equiprobable 0.2
0
1
2
3
4
5 SNR (dB)
6
7
8
9
10
Fig. 3. Outage capacity curves: BPSK modulation, 10% outage.
6
C. Code Design for Random-Like Codes In order to approach the equiprobable outage capacity, we consider the use of an LDPC code as the channel code. The LDPC codes are random codes which may be described with the help of a bipartite graph. The left nodes, or the variable nodes, represent the bits of a codeword. The right nodes, or the check nodes, represent the parity checks. The edges of the graph indicate which bits participate in a given parity check. The coefficients of the polynomials ) are the fraction of edges connected to a left or right node, respectively, of a given degree. A given pair of polynomials is called the degree profile and represents a family of LDPC codes. A particular instance of the code is given by a random graph constructed with the given degree profile. The design of LDPC codes consists of optimizing the degree profile of the code to match a particular channel. Since the channel under consideration changes from one codeword to the next, we must design the code to work for most of the channel realizations. Consider the system with transmit and one receive antenna. The matrix channel reduces to a vector channel *,+-* * /. . * The matrix has only one non-zero 10 eigenvalue ) 32 . The corresponding unit-norm eigenvector is given by:4
& : ?
6
*
!
? &
?
&$'(
(3)
0.15 Probability
0.05 0.0
-5
0
5
10
15
20
LLR
Fig. cases: (a) JLK-M NPO NINIQI4.RIQPS9PDFs TVUWNPof O NIXIReceived NPSZYI[P\ Likelihoods NPO_SZ`PSZXI[IQWina-the UWNPpathological O [IQcb XIYcb J8]^M . The receive SNR is 10 dB.
.
(4) 4 The pathological cases for any code will occur when the num. This ber of independent components 95 than * 65 of*87 becomes* less *87 occurs, for example, when or when or when := A@ < >CB @ * 95;: for some . When and BPSK modulation is used, these pathological cases cause half the bits of 5Dthe : codeword to be nearly erased (log-likelihood ratio, LLR ) while the other half are received with high LLRs. So, the erasure channel is a good approximation of the channel under consideration, especially at high SNRs. This motivates the use of LDPC codes designed for the erasure channel. A rate-1/2 LDPC code that achieves capacity on the erasure channel corrects 50% erasures with an erasure decoder when the other 50%+ of the bits are received perfectly. Asymptotically, when EGFIH approaches infinity, the pathological cases present truly erasure channels and the erasure codes achieve full spatial diversity. We conjecture that the same code will be able to correct most of the erasures with a conventional message-passing decoder when the other 50% bits have high (but not infinite) reliability. , In order to illustrate this point better, consider the BPSK modulation case when the SNR is 10 dB. The pdfs of the received likelihoods for two sets of values for channel gains are
0.15
+ *
shown in Figs. 4,5. These two channel realizations correspond to the pathological cases mentioned above. It should be observed that about half the received bits have LLR close to zero while the remaining bits have high LLRs.
Probability
)& *
636
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0.05
% / D
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0.0
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!5
0.20
0.10
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-5
0
5
10
15
20
LLR
Fig. pathological (b) JLK-M NPO XIR d5.b R dePDFs TfUhg9ofNPO Received dIdiSZXIXPSI\ Likelihoods T6NPO_SZQIinQIR the diS9aj UhgWNPO dc[I[IQIcases: R d . The receive J]eM SNR is 10 dB.
Fig. 6 shows the performance of four different LDPC codes at an overall rate of 1 b/s/Hz – the regular (2048, 3, 6) code, the regular (8192, 3, 6) code, an irregular length 8192 rate-1/2 AWGN code and a length 8192 rate-1/2 erasure code. The performance of these codes does not deteriorate with increase in length. This is to be contrasted with, say the STTCs, whose performance deteriorates with length. The erasure code performs better than the AWGN code by about 0.5 dB and it is only about 2.5 dB away from the constrained outage capacity. , we may not be able to account for all the When lk
0
perform significantly better than the random-like codes. Also, for high rates (more than about 0.8 b/s/Hz), fixing the spacetime scheme to be the Alamouti scheme is not optimal.
10
2.2
−1
10
2
1% Outage Capacity (b/s/Hz)
Frame Error Rate at 1 b/s/Hz
irregular AWGN code, length 8192 erasure code, length 8192 unconstrained outage equiprobable i.i.d. bpsk outage
−2
10
6
8
10
12 SNR (dB)
14
16
18
Fig. 6. Performance of direct transmission of LDPC codes with BPSK modulation.
1.8
1.6
1.4
1.2
Unconstrained Alamouti AT&T 4−state AT&T 8−state AT&T 16−state
1 12
pathological cases with a single code. Also, the characterization k of the pathological cases for seems difficult. These are subjects of our current research.
&
13
14
15
16 SNR (dB)
17
18
19
20
Fig. 7. Outage capacity curves: QPSK modulation, 1% outage.
B. Achieving Capacity
IV. O RTHOGONAL S PACE -T IME B LOCK C ODES 1
The orthogonal space-time block codes are designed in such a way that they orthogonalize the paths from the various transmit antennas at the expense of rate. It is well known that this orthogonalization also incurs a penalty in the unconstrained capacity except when the channel is of rank one . In this section, we show that even for the rank one channel these codes suffer from a loss in capacity when a constrained modulation is used. We then show how LDPC codes may be used to achieve this constrained capacity. A. Capacity Computation It can be easily shown [4] that the linear processing o-STBCs convert the multiple-input multiple-output (MIMO) channel into an equivalent single-input single-output (SISO) AWGN channel with an effective SNR that equals the MIMO channel SNR scaled by the sum of the channel gains. In order to compute the fundamental limits of this scheme at a given rate , we need to find the channel SNR such that the probability of the effective SNR at the output of the o-STBC receiver falling below the constrained capacity of the AWGN channel with PSK modulation is less than the outage probability . That is,
5
>
5
5
eff
5
AWGN
E E
(5)
where eff tr . For all linear processing o-STBCs H 0 in [4], [5], the effective SNR has a closed form distribution. For the one receive antenna (rank one channel) case and the Alamouti: scheme as the o-STBC, we have eff ) H where ) random variable with 4 degrees of freedom is a and a variance of 0.5. Thus, the capacity computation in (5) is pretty straightforward. The results of this exercise are shown for BPSK in Fig. 3 and for QPSK in Fig. 7. It is evident that for a range of rates the concatenation scheme with the Alamouti inner code should
636
5
5
Outer Code
ML Receiver for Alamouti scheme
Alamouti Scheme
Outer Decoder
Nr
Channel as seen here is an instantaneously Gaussian channel
Fig. 8. Concatenation of LDPC codes with o-STBC.
In order to achieve the capacity of this scheme, we propose the concatenation of an outer LDPC codes with inner o-STBCs (Fig. 8). Since the equivalent channel at the output of the oSTBC receiver is an AWGN channel, the LDPC codes that are optimal for the AWGN channel are also optimal for our case. When higher order modulation is used, LDPC codes matched to the constellation being used (matched-BICM) should be employed. A significant advantage of using LDPC codes is the ease with which the performance of this concatenation scheme can be predicted. Since LDPC codes exhibit a threshold phenomenon, we can assume that if the effective SNR eff is below the threshold of the code code , the decoder will always make an error. Otherwise, the decoder can be assumed to decode the codeword correctly. Since the closed form distribution of the effective SNR eff is known, the codeword error rate of the concatenation is simply the integral of this distribution from the threshold of the code to infinity:
5
5
5
eff
(6)
code
Fig. 9 shows the agreement between the prediction and the simulated performance when an irregular, rate-1/2 code of length 10000 bits is concatenated with the Alamouti scheme and BPSK modulation is used. It should also be noted that the predicted performance is less than 0.4 dB away from the constrained capacity of this scheme.
It is to be noted that the transmit sequences are no longer independent. The o-STBC introduces correlation between the transmit sequences and it performs better than the constrained modulation i.i.d. case for low rates. Thus, we have shown a method to partially recover the loss in capacity due to the assumption of independence between the transmit sequences,: at least for a range of rates (less than about 0.75 b/s/Hz for a outage).
&
−1
10
FER
Constrained capacity Predicted performance Simulated performance
−2
10
−3
10
6
7
8
9
10 SNR (dB)
11
12
13
14
Fig. 9. Concatenation of an LDPC code with Alamouti scheme. The code threshold is 0.3894 dB, the modulation is BPSK.
V. S PACE -T IME T RELLIS C ODES The space-time trellis codes (STTCs) are essentially Markov structures. Given a channel realization, it is easy to determine the Markov state transition diagram for a given STTC. Consider, the 4-state, 4-PSK code of* [3]. : for example, + * Let > denote the 4-PSK symbols. Let de note the two fade coefficients ( ). Define *,+ + * + * + + * + , etc. In all, there are > : , each of which 16 possible signals represents a valid pair of transmit symbols. Given the channel realization , there is a one-to-one mapping from the set to the set . The channel dependent trellis of this code is shown in Fig. 10.
!
!&! !
8
! & B 8 ! !& ! !&
6
!
g g g g 0 1 2 3
g g g g 4 5 6 7
g g g g 8 9 10 11
g g g g 12 13 14 15
Fig. 10. Channel dependent 4-state STTC trellis.
In [8] a modified BCJR algorithm is used to determine the capacity of binary-input channels with memory. Once the Markov structure of the STTC code for a given channel realization is determined, what we have is a non-binary channel with memory. We extend the method of [8] to determine the capacity of the concatenation scheme with STTC as the inner code as follows. The mutual information between the transmit sequence
and the received signal
)
may be written as: *
*
(7) , we have * Since the received signal* * *
+ . The estimate of is obtained by a single long simulation of A sin gle forward BCJR recursion yields which ap
*
proaches for large [8]. In our simulations, we choose : ::: The outage probability of the code is again determined by simulating a number of channel realizations and generating a quantile plot. Fig. 7 shows the computed capacity at 1% outage for a number of different QPSK based STTCs taken from [3]. The advantage of using STTCs as inner codes in a concatenation scheme has, thus, a higher capacity as compared to the scheme with the Alamouti scheme as an inner code. Inspite of the better capacity of STTCs, it is not straightforward to achieve this capacity. As noted earlier, the performance of STTCs degrades with length and a clever way of using these codes must be devised. Once such method is to use the recursive realizations of these codes in serial and parallel concatenation schemes. This method is discussed in details in [1]. The simulated performance of the example presented therein is about 1 dB away from the constrained modulation capacity of the 4-state STTC at 1 b/s/Hz. A possible reason for this difference may be the absence of the maximum likelihood decoder. VI. C ONCLUSIONS
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8 ) ) !F) ! ! ) ) ! ?
We have analyzed three different concatenated space-time coding schemes for use over quasi-static channels from a capacity perspective. The constrained modulation outage capacities for the these schemes were computed. It was shown how to approach the outage capacity for these cases. R EFERENCES [1] V. Gulati and K. R. Narayanan, “Concatenated codes for fading channels based on recursive space-time codes,” To appear in IEEE Trans. Wireless Commun. Available for download at http://ee.tamu.edu/ ˜krishna, Mar. 2001. [2] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communications: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, no. 2, pp. 744–765, Mar. 1998. [4] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE JSAC, vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [5] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [6] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, no. 2, pp. 399–431, Mar. 1999. [7] A. Stefanov and T. M. Duman, “Turbo-coded modulation for systems with transmit and receive antenna diversity over block fading channels: System model, decoding approaches, and practical considerations,” IEEE JSAC, vol. 19, no. 5, pp. 958–968, May 2001. [8] D. Arnold and H. A. Loeliger, “On the information rate of binary-input channels with memory,” in Proc. ICC 2001, Jun. 2001, vol. 9, pp. 2692– 2695. [9] E. Baccarelli, “Evaluation of the reliable data rates supported by multipleantenna coded wireless links for QAM transmissions,” IEEE JSAC, vol. 19, no. 2, pp. 295–304, Feb. 2001.
