Symbol mapping diversity in iterative decoding ... - IEEE Xplore
Symbol Mapping Diversity in Iterative Decoding/Demodulation of ARQ Systems Harvind Samra and Zhi Ding University of California, Davis Dept. of Electrical and Computer Engineering Davis, CA 95616 Email: hssamra,[email protected] Abstract— This paper presents a simple, yet effective method of enhancing the inherent diversity among packet retransmissions in digital communication systems that employ high-order modulations such as M -PSK and M -QAM through frequency-selective channels. Diversity is enhanced by uniquely redefining the bitto-symbol mapping for each retransmission. Finding the optimal mappings leads to iterative solutions of the Quadratic Assignment Problem. More importantly, jointly equalizing transmissions with different mappings only requires a marginal increase in complexity, while providing substantial gains in bit error rates (BER) and frame error rates (FER).
I. I NTRODUCTION In many communication systems, if bit errors remain after error correction when processing a transmitted data packet, a frame error is declared and a request for retransmssion is made to the transmitter. In these systems equipped with this Automatic Repeat reQuest (ARQ) mechanism, various approaches have been proposed for packet combining. For example, Chase developed a maximum-likelihood combining scheme for an arbitrary number of coded packets, concatenating M copies of a codeword into a single codeword [1]. Harvey and Wicker also proposed several ARQ strategies, including an approach where soft-decoded codewords from multiple packet transmissions are combined into a single soft codeword [2]. Others, including Hagenauer, Rowitch, and Milstein, have developed schemes involving rate-compatible codes, where retransmitted copies of a packet are each uniquely punctured to improve throughput [3], [4]. Stuber and Narayanan developed an ARQ receiver using error correcting codes where the extrinsic information from the decoding of previous packets is reused [5]. Zhang and Kassam outlined a hybrid ARQ protocol for ratecompatible codes in fading channels that selectively combines a subset of the M received transmissions [6]. Recent works introduced approaches which exploit retransmission diversity at the output of intersymbol interference (ISI) channels [7]– [10]. A general discussion and analysis of ARQ systems from a coding perspective is provided in [11], [12]. In [13], we introduced a simple mapping diversity scheme for systems that employ higher-order modulations such as PSK or QAM. In this paper, we apply mapping diversity in intersymbol interference channels that are stationary or slowly varying in time. By varying the bit-to-symbol mapping for each retransmission, a inherent diversity among the K transmissions is enhanced. We first provide an overview of
Outer Encoder
Interleaver
sn
ψ
ψ
1
M
Fig. 1.
h
+ (1) vn
h
+ v (M) n
yn(1)
y(M) n
Mapping diversity transmitter.
this ARQ method, applying the results in [13] to frequencyselective channels. We also describe a maximum a posteriori (MAP) equalizer whose complexity increases marginally with additional transmissions. Finally, results are presented that verify the performance improvement gained from mapping diversity. II. C REATING S YMBOL M APPING D IVERSITY An overview of a mapping diversity retransmission system is now provided, with K denoting the number of packet transmissions. We begin with a set C of real- or complex-valued points that represents the points of a signal constellation, e.g. 16QAM. Given a packet of bits, consecutive groups of Q = log2 |C| bits are assigned to symbols in C via a symbol mapping function ψ : {0, 1, . . . , |C| − 1} → C. Herein, these groups of Q bits are referred to as labels (consistent with [14]), and label sn ∈ {0, 1, . . . , |C| − 1} represents the decimal equivalent of these bits. With K transmissions of a packet, we define K symbol mapping functions ψ1 , . . . , ψK . Using different mappings enhances the diversity across multiple transmissions. Fig. 1 shows an interleaved, forward error correction coded sequence of labels sn that is ultimately transmitted K times; the transmitter sends symbols ψ1 [sn ], . . . , ψK [sn ]. The receiver obtains samples yn(k) =
L−1
hl ψk [sn ] + vn(k) ,
l=0
where h0 , . . . , hL−1 define the channel’s impulse response, (k) vn is CN (0, σv2 ), and k = 1, . . . , K. We assume that (1) (K) vn , . . . , vn are independent, and that the channel response has unit energy and is identical for all transmissions. By
0-7803-7802-4/03/$17.00 © 2003 IEEE
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the K th mapping from the previous K − 1 mappings. We assume that the first K − 1 mappings have already been determined and we work towards obtaining only the K th mapping, ψK . Our optimization problem then simplifies to
Interleaver yn(1) Joint MAP y(M) n
s^ n
De− Interleaver
Outer MAP Decoder
min
ψK ∈ψ
Equalizer
Fig. 2.
III. S YMBOL M APPING A DAPTATION S CHEMES We now address the problem of selecting effective mappings for retransmissions through frequency-selective channels. In [13], we outlined how to select optimal (in an ARQsense) mappings for retransmissions through AWGN channels. Since Viterbi or MAP channel equalizers succesfully mitigate the effects intersymbol interference, the AWGN mappings should be highly proficient for frequency-selective channels. The mapping design found in [13] is summarized below. In AWGN channels, the upper bound expression for the BER is
s=0 u=0 u=s
K 1 Pr[s]B[s, u]Q 2 |ψk [s] − ψk [u]|2 , 2σv k=1
(1) where Pr[s] is the probability of transmitting label s and B[s, u] accounts for the BER produces by a misdetecting label s as label u, B[s, u] =
g [s, ψK [s], u, ψK [u]] ,
number of differing bits between s and u . log2 |C|
(2)
s=0 u=0 u=s
(K)
implicitly assuming that vn , . . . , vn have the same variance and that ψ1 , . . . , ψK map labels to the same set C, the signalto-noise-ratio (SNR) Eb /N0 remains unchanged for all K transmissions. Note that the same interleaver is used for each transmission. This is motivated by our desire to jointly equalize all transmissions, since all transmissions can be viewed as a vector output of a single state machine. Overall, adapting the symbol mapping minimally increases receiver complexity while increasing the Euclidean distances between any two symbols in C. The receiver, shown in Fig. 2, is a turboequalizer that now consists of a joint MAP equalizer/demapper and a MAP decoder. We later describe a joint MAP equalizer whose complexity is marginally higher than a standard MAP equalizer.
|C|−1 |C|−1
with ψ denoting the set of possible mappings, and g[s, a, u, b] is the pairwise BER that results by mapping s to symbol a and u to symbol b in the K th mapping,
Mapping diversity receiver.
(1)
|C|−1 |C|−1
Selecting the K mappings that minimize (1) is a massive combinatorial optimization problem whose solution space contains (|C|!)K possible solutions. Thus, we propose a simpler, and probably sub-optimal, progressive solution by computing
g[s, a, u, b] = Pr[s]B[s, u]Q
h[s, u] =
K−1 k=1
1 (h[s, u] + |a − b|2 ]) , 4σv2
|ψk [s] − ψk [u]|2 .
This solution is optimal for ARQ-type applications which have a secondary objective to minimize the number of transmissions (mappings) needed to achieve a desired BER. In other words, it is desirable to choose the current mapping without relying on future (re)transmissions. Though still computationally difficult, (2) falls into a category of combinatorial optimzation problems commonly referred to as the Quadratic Assignment Problem (QAP). The QAP is one of the most difficult and extensively studied problems in optimization. It was first introduced in 1957 to model the assignment of P economic facilities to P physical locations [15]; a more general version was published in 1963 [16]. Most exact solutions to the QAP involve a branchand-bound search. Typically, lower bounds for the QAP are computationally expensive (O(P 5 )) and generally not very tight. Recent work by Hahn and Grant proposed an efficient lower bounding technique and its inclusion in a branch-andbound scheme [17]. IV. J OINT MAP E QUALIZATION /D EMAPPING Bahl, Cocke, Jelinek, and Raviv (BCJR) developed an optimal method for MAP decoding the output of a channel, producing a log-likelihood ratio (LLR) of the channel’s input [18]. We generalize this method for multiple transmissions under symbol mapping diversity. Multiple transmissions of common data bits can be modeled as a single-input multiple output (SIMO) system. The k th transmission of the sequence of labels sn passes through a FIR (k) channel of length L and adds noise νn to form the received (k) sequence yn . Our equalizer now computes the following MAP probabilities
(K) (K) , Pr sn = (bnQ , . . . , bnQ+Q−1 ) = i Y1 , . . . , YN (3) where i = 0, . . . , |C|−1 are the possible values of the label sn and (bnQ , . . . , bnQ+Q−1 ) are the bits that compose sn . Each
3586
(K)
(1)
(K)
vector Yn = [yn , . . . , yn ]T , with K denoting the number of transmissions. We define the following functions
(K) (4) αn (m) = Pr Sn = m Y1 , . . . , Yn(K) , (K) (K) Pr Yn+1 , . . . , YN Sn = m , (5) βn (m) = (K) (K) (K) (K) Pr Yn+1 , . . . , YN Y1 , . . . , Yn (K) and define the transitional probability γi Yn , m , m as
(6) Pr xn = i, Sn = m, Yn(K) Sn−1 = m ,
where Sn is the state of the channel filter at time n. The MAP probability in (3) is restated using (4), (5), and (6) as γi Yn(K) , m , m αn−1 (m )βn (m) m
m
where m denotes the decimal equivalent of the state Sn = n, and m (sn , . . . , sn−L+1 ) of the channel filter at time (K) denotes the state Sn−1 at time n−1. Using γi Yn , m , m , (4) and (5) are recursive computed,
αn (m) =
|C|−1 γi Yn(K) , m , m αn−1 (m ) m
m
i=0
βn (m) =
m
i=0
|C|−1 (K) γi Yn+1 , m , m αn (m ) m
m
,
.
(8)
i=0
The transitional probability in (6) is a product of three probabilities
p Yn(K) sn = i, Sn = m, Sn−1 = m × q (sn = i | Sn = m, Sn−1 = m ) × π (Sn = m | Sn−1 = m ) .
The probability q(·|·) is deterministic and is either 0 or 1. Also, the probability π(Sn = m | Sn−1 = m ) is either 0 or 1/|C|, assuming the a priori probabilites are equal (Pr{sn = i} = 1/|C|). (K) Since the elements of the vector Yn are independently Gaussian (given sn , Sn , Sn−1 ), we can restate the probability (K) p(Yn | . . . ) as a product p(Yn(K) | . . . ) = (k) p(yn
i:bnQ+j =0
with the index j = 0, 1, . . . , Q−1. The notation i : bnQ+j = 0 indicates those values of i where the j th bit is zero. Extrinsic information from an independent source regarding the transmitted bits, such as an outer decoder, is incorporated into the equalizer as an additional Gaussian random variable wn . The only substantive change to the equalizer involves the computation of the transitional probability (K) γi wnQ , . . . , wnQ+Q−1 , Yn , m , m , which becomes
(7)
i=0
(K) |C|−1 γi Yn+1 , m, m βn+1 (m )
and recompute the functions αn (m) and βn (m) to update the MAP probabilities. The MAP symbol probabilities are converted into bit log-likelihood ratios (LLR)
Pr sn = i . . . i:b nQ+j =1 (10) Λ(bnQ+j ) = log
,
Pr sn = i . . .
γi
|C|−1 γi Yn(K) , m , m αn−1 (m ) m
is received, we update Overall, when the K th transmission (K) the function γi Yn , m , m using the recursion γi Yn(K) , m , m = γi Yn(K−1) , m , m ×
(9) p ynK sn = i, Sn = m, Sn−1 = m ,
K
p yn(k) sn = i, Sn = m, Sn−1 = m .
k=1
Each | . . . ) can be easily found using the k th transmission channel and the binary precoding polynomial.
Q
Yn(K) , m , m
j=1
Pr wnQ+j sn = i .
Typically, wnQ+j is modelled as xn corrupted by AWGN. Note that the recursion in (9) still applies. Similar to turbodecoding and turbo-equalization, the LLR Λ(bnQ+j ) in (10) is computed with wnQ+j = 0. Note that the complexity increase for equalizing multiple transmissions is slight. The number of states in the equalizer does not increase as the number of transmissions K increases (assuming that the delay spread or length of the transmission channel remains constant). Other than updating the transitional (K) probability γi (Yn , m , m), the algorithm remains identical from transmission to transmission. The majority of the algorithm’s complexity lies in computing the functions αn (m) and βn (m), and the MAP probabilities P r{sn = i| . . .}. The transitional probability update in (9) has a very minor effect on the equalizer’s computational requirements. Thus, as K increases the equalizer’s complexity remains nearly constant. Also, once the transitional probability in (9) is updated, the K th transmission can be discarded. V. R ESULTS We present two sets of results that illustrate the effectiveness of mapping diversity. First, we evaluate the BER improvement in channel equalization provided by mapping diversity retransmissions. We remove coding from the system and send 1000 packets of 1024 bits using both the ML combining ARQ and mapping diversity ARQ systems. We assume that the channel has length L = 3 for all 16QAM and 16PSK, transmissions, and 4 for 8PSK. Each tap is a complex-valued Gaussian random variable (zero mean, unit variance) that varies per
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ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
−1
10
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
−1
10
−2
10 −2
BER
BER
10
−3
10 −3
10
−4
10
−4
10
−5
−8
Fig. 3.
−6
−4
−2
0
2 4 Eb/N0(dB)
6
8
10
12
10
14
BER for 16QAM mapping diversity scheme with no coding.
Fig. 5.
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
−1
10
−5
0
5 Eb/N0(dB)
10
15
BER for 16PSK mapping diversity scheme with no coding.
0
10
−2
FER
BER
10
−3
10
−1
10
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
−4
10
−8
−6
−4
−2
0
2 Eb/N0(dB)
4
6
8
10
12 −8
−6
−4
−2 E /N (dB) b
Fig. 4.
0
2
4
6
0
BER for 8PSK mapping diversity scheme with no coding. Fig. 6.
packet, but remains constant for retransmissions. The channel is normalized to unit energy to maintain the desired Eb /N0 . We use the symbol mapping adaptation schemes that were previously suggested. Figs. 3, 4, and 5 show the results of equalizing K uncoded transmissions for both ARQ systems for 16QAM, 8PSK, and 16PSK, respectively. The gains are consistent with those found in [13], verifying our belief that the MAP equalizer’s ability to mitigate ISI allows for treatment of ISI channels as simple AWGN channels. The gains are quite substantial; for instance using mapping diversity with 16PSK produces approximate gains of 7.5, 9, and 10 dB for 2,3, and 4 transmissions. The second set of results returns to the original ARQ framework, maintaining the same channel lengths as before. Using a (37, 21)8 recursive systematic convolutional (RSC) code, the transmitter encodes a 512 bit data sequence un to produce a 512 code bit sequence. Contained in the original bits prior to RSC encoding, is a 16-bit CRC codeword that is used by the receiver to determine if the received message sequence is correct. The message and code bits are concatenated into a single packet bn . The turbo-equalizer performs a maximum of four iterations for 16QAM and 16PSK, five iterations for
FER for 16QAM mapping diversity scheme with RSC coding.
8PSK. If the CRC check fails after the fourth(fifth) iteration, another copy of the packet is transmitted. This continues until the CRC check is successful or a maximum number (in this case K) of transmissions are processed, then a frame error is declared and the next packet is sent. We compare the frame error rates (FER) of the ARQ system using the mapping diversity approach against a simple ARQ system where the same mapping is used for all K transmissions. Figs. 6, 7, and 8 show the results of both ARQ systems for 16QAM, 8PSK, and 16QAM, respectively, for various values of the maximum number of transmissions K. The gains are still substantial, though not as large as those suggested by the BER plots. In designing the symbol mappings, the outer encoder was altogether ignored. These FER curves indicate that the effects of the encoder should be considered in designing the mappings. Fig. 9 contains the throughputs using 16PSK modulation for K = 3, 4. The throughput is ratio of the number of original data bits (512) versus the number of total transmitted bits (maximum of 1024 × K). At lower Eb /N0 values, mapping diversity provides significantly better throughputs.
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0
0.5
10
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3
0.45
0.4
FER
Throughput
0.35
−1
10
0.3
0.25
0.2
0.15
−6
−5
−4
−3
−2
−1 0 E /N (dB) b
Fig. 7.
1
2
3
4
−6
5
−4
−2
0
2
4 6 E /N (dB) b
0
8
10
12
14
0
Fig. 9. Throughput for 16PSK mapping diversity scheme with RSC coding.
FER for 8PSK mapping diversity scheme with RSC coding.
0
10
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
FER
R EFERENCES
−1
10
−6
Fig. 8.
−4
−2
0
2
4 6 Eb/N0 (dB)
8
10
12
14
FER for 16PSK mapping diversity scheme with RSC coding.
VI. C ONCLUSIONS In this paper, we present a method for improving performance in handling retransmissions of higher-order modulated signals through intersymbol interference channels. The improvement is accomplished by redefining the symbol mapping for each retransmission. This results in a marginal complexity increase for both the transmitter and receiver, while reducing bit error rates and frame error rates significantly. Our simulation results show the effectiveness of this strategy. They also warrant a more rigorous understanding of the effects of various symbol mapping adaption schemes when an outer encoder is included in the system. Also, binary precoding prior to symbol mapping may provide additional gains. Finally, mapping diversity may have other applications such as channel estimation, space-time coding, and packet collision resolution. ACKNOWLEDGEMENTS This work was supported in part by Natioal Science Foundation grants CCR-0196364 and ECS-0121469, and an Accel Partners, Inc. fellowship.
[1] D. Chase, “Code combining–a maximum-likelihood decoding approach for combining an arbitrary number of noisy packets,” IEEE Trans. Commun., vol. 33, pp. 385–393, May 1985. [2] B. Harvey and S. Wicker, “Packet combining systems based on the Viterbi decoder,” IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1544– 1557, Feb/Mar/Apr 1994. [3] J. Hagenauer, “Rate-compatible punctured convolutional codes (RCPC codes) and their applications,” IEEE Trans. Commun., vol. 36, pp. 389– 400, Apr. 1988. [4] D. Rowitch and L. Milstein, “On the performance of hybrid FEC/ARQ systems using rate compatible punctured turbo (RCPT) codes,” IEEE Trans. Commun., vol. 48, no. 6, pp. 948–959, June 2000. [5] K. Narayanan and G. Stuber, “A novel ARQ technique using the turbo coding principle,” IEEE Commun. Lett., vol. 1, no. 2, pp. 49–51, Mar. 1997. [6] Q. Zhang and S. A. Kassam, “Hybrid ARQ with selective combining for fading channels,” IEEE J. Select. Areas Commun., vol. 17, no. 5, pp. 867–880, May 1999. [7] H. Samra and Z. Ding, “Integrated iterative equalization for ARQ systems.” Orlando, Florida: ICASSP, 2002. [8] ——, “Integrated iterative equalization for ARQ systems.” Lausanne, Switzerland: IEEE Int. Symp. on Info. Theory, 2002. [9] ——, “Precoded integrated equalization for packet retransmissions.” Monterey, California: Thirty-Sixth Asilomar Conf. on Signals, Systems, and Computerss, 2002. [10] D. N. Doan and K. R. Narayanan, “Iterative packet combining schemes for intersymbol interference channels,” IEEE Trans. Commun., vol. 50, no. 4, pp. 560–570, Apr. 2002. [11] S. Lin, J. Daniel J. Costello, and M. J. Miller, “Automatic-repeat-request error-control schemes,” IEEE Commun. Mag., vol. 22, no. 12, pp. 5–18, Dec. 1984. [12] S. B. Wicker, Error Control Systems for Digital Communications and Storage. Englewood Cliffs, NJ: Prentice-Hall, 1994. [13] H. Samra, Z. Ding, and P. M. Hahn, “Optimal symbol mapping diversity for multiple packet transmissions.” Hong Kong, China: ICASSP, 2003. [14] R. D. Wesel, X. Liu, J. M. Cioffi, and C. Komninakis, “Constellation labeling for linear encoders,” IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2417–2431, Sept. 2001. [15] T. C. Koopmans and M. J. Beckmann, “Assignment problems and the location of economic activities,” Econometrica, vol. 25, pp. 53–76, 1957. [16] E. L. Lawler, “The quadratic assignment problem,” Management Science, vol. 9, pp. 586–599, 1963. [17] P. M. Hahn and T. L. Grant, “Lower bounds for the quadratic assignment problem based upon a dual formulation,” Operations Research, vol. 46, no. 6, pp. 912–922, 1998. [18] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, pp. 284–287, Mar. 1974.
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I. I NTRODUCTION In many communication systems, if bit errors remain after error correction when processing a transmitted data packet, a frame error is declared and a request for retransmssion is made to the transmitter. In these systems equipped with this Automatic Repeat reQuest (ARQ) mechanism, various approaches have been proposed for packet combining. For example, Chase developed a maximum-likelihood combining scheme for an arbitrary number of coded packets, concatenating M copies of a codeword into a single codeword [1]. Harvey and Wicker also proposed several ARQ strategies, including an approach where soft-decoded codewords from multiple packet transmissions are combined into a single soft codeword [2]. Others, including Hagenauer, Rowitch, and Milstein, have developed schemes involving rate-compatible codes, where retransmitted copies of a packet are each uniquely punctured to improve throughput [3], [4]. Stuber and Narayanan developed an ARQ receiver using error correcting codes where the extrinsic information from the decoding of previous packets is reused [5]. Zhang and Kassam outlined a hybrid ARQ protocol for ratecompatible codes in fading channels that selectively combines a subset of the M received transmissions [6]. Recent works introduced approaches which exploit retransmission diversity at the output of intersymbol interference (ISI) channels [7]– [10]. A general discussion and analysis of ARQ systems from a coding perspective is provided in [11], [12]. In [13], we introduced a simple mapping diversity scheme for systems that employ higher-order modulations such as PSK or QAM. In this paper, we apply mapping diversity in intersymbol interference channels that are stationary or slowly varying in time. By varying the bit-to-symbol mapping for each retransmission, a inherent diversity among the K transmissions is enhanced. We first provide an overview of
Outer Encoder
Interleaver
sn
ψ
ψ
1
M
Fig. 1.
h
+ (1) vn
h
+ v (M) n
yn(1)
y(M) n
Mapping diversity transmitter.
this ARQ method, applying the results in [13] to frequencyselective channels. We also describe a maximum a posteriori (MAP) equalizer whose complexity increases marginally with additional transmissions. Finally, results are presented that verify the performance improvement gained from mapping diversity. II. C REATING S YMBOL M APPING D IVERSITY An overview of a mapping diversity retransmission system is now provided, with K denoting the number of packet transmissions. We begin with a set C of real- or complex-valued points that represents the points of a signal constellation, e.g. 16QAM. Given a packet of bits, consecutive groups of Q = log2 |C| bits are assigned to symbols in C via a symbol mapping function ψ : {0, 1, . . . , |C| − 1} → C. Herein, these groups of Q bits are referred to as labels (consistent with [14]), and label sn ∈ {0, 1, . . . , |C| − 1} represents the decimal equivalent of these bits. With K transmissions of a packet, we define K symbol mapping functions ψ1 , . . . , ψK . Using different mappings enhances the diversity across multiple transmissions. Fig. 1 shows an interleaved, forward error correction coded sequence of labels sn that is ultimately transmitted K times; the transmitter sends symbols ψ1 [sn ], . . . , ψK [sn ]. The receiver obtains samples yn(k) =
L−1
hl ψk [sn ] + vn(k) ,
l=0
where h0 , . . . , hL−1 define the channel’s impulse response, (k) vn is CN (0, σv2 ), and k = 1, . . . , K. We assume that (1) (K) vn , . . . , vn are independent, and that the channel response has unit energy and is identical for all transmissions. By
0-7803-7802-4/03/$17.00 © 2003 IEEE
3585
the K th mapping from the previous K − 1 mappings. We assume that the first K − 1 mappings have already been determined and we work towards obtaining only the K th mapping, ψK . Our optimization problem then simplifies to
Interleaver yn(1) Joint MAP y(M) n
s^ n
De− Interleaver
Outer MAP Decoder
min
ψK ∈ψ
Equalizer
Fig. 2.
III. S YMBOL M APPING A DAPTATION S CHEMES We now address the problem of selecting effective mappings for retransmissions through frequency-selective channels. In [13], we outlined how to select optimal (in an ARQsense) mappings for retransmissions through AWGN channels. Since Viterbi or MAP channel equalizers succesfully mitigate the effects intersymbol interference, the AWGN mappings should be highly proficient for frequency-selective channels. The mapping design found in [13] is summarized below. In AWGN channels, the upper bound expression for the BER is
s=0 u=0 u=s
K 1 Pr[s]B[s, u]Q 2 |ψk [s] − ψk [u]|2 , 2σv k=1
(1) where Pr[s] is the probability of transmitting label s and B[s, u] accounts for the BER produces by a misdetecting label s as label u, B[s, u] =
g [s, ψK [s], u, ψK [u]] ,
number of differing bits between s and u . log2 |C|
(2)
s=0 u=0 u=s
(K)
implicitly assuming that vn , . . . , vn have the same variance and that ψ1 , . . . , ψK map labels to the same set C, the signalto-noise-ratio (SNR) Eb /N0 remains unchanged for all K transmissions. Note that the same interleaver is used for each transmission. This is motivated by our desire to jointly equalize all transmissions, since all transmissions can be viewed as a vector output of a single state machine. Overall, adapting the symbol mapping minimally increases receiver complexity while increasing the Euclidean distances between any two symbols in C. The receiver, shown in Fig. 2, is a turboequalizer that now consists of a joint MAP equalizer/demapper and a MAP decoder. We later describe a joint MAP equalizer whose complexity is marginally higher than a standard MAP equalizer.
|C|−1 |C|−1
with ψ denoting the set of possible mappings, and g[s, a, u, b] is the pairwise BER that results by mapping s to symbol a and u to symbol b in the K th mapping,
Mapping diversity receiver.
(1)
|C|−1 |C|−1
Selecting the K mappings that minimize (1) is a massive combinatorial optimization problem whose solution space contains (|C|!)K possible solutions. Thus, we propose a simpler, and probably sub-optimal, progressive solution by computing
g[s, a, u, b] = Pr[s]B[s, u]Q
h[s, u] =
K−1 k=1
1 (h[s, u] + |a − b|2 ]) , 4σv2
|ψk [s] − ψk [u]|2 .
This solution is optimal for ARQ-type applications which have a secondary objective to minimize the number of transmissions (mappings) needed to achieve a desired BER. In other words, it is desirable to choose the current mapping without relying on future (re)transmissions. Though still computationally difficult, (2) falls into a category of combinatorial optimzation problems commonly referred to as the Quadratic Assignment Problem (QAP). The QAP is one of the most difficult and extensively studied problems in optimization. It was first introduced in 1957 to model the assignment of P economic facilities to P physical locations [15]; a more general version was published in 1963 [16]. Most exact solutions to the QAP involve a branchand-bound search. Typically, lower bounds for the QAP are computationally expensive (O(P 5 )) and generally not very tight. Recent work by Hahn and Grant proposed an efficient lower bounding technique and its inclusion in a branch-andbound scheme [17]. IV. J OINT MAP E QUALIZATION /D EMAPPING Bahl, Cocke, Jelinek, and Raviv (BCJR) developed an optimal method for MAP decoding the output of a channel, producing a log-likelihood ratio (LLR) of the channel’s input [18]. We generalize this method for multiple transmissions under symbol mapping diversity. Multiple transmissions of common data bits can be modeled as a single-input multiple output (SIMO) system. The k th transmission of the sequence of labels sn passes through a FIR (k) channel of length L and adds noise νn to form the received (k) sequence yn . Our equalizer now computes the following MAP probabilities
(K) (K) , Pr sn = (bnQ , . . . , bnQ+Q−1 ) = i Y1 , . . . , YN (3) where i = 0, . . . , |C|−1 are the possible values of the label sn and (bnQ , . . . , bnQ+Q−1 ) are the bits that compose sn . Each
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(K)
(1)
(K)
vector Yn = [yn , . . . , yn ]T , with K denoting the number of transmissions. We define the following functions
(K) (4) αn (m) = Pr Sn = m Y1 , . . . , Yn(K) , (K) (K) Pr Yn+1 , . . . , YN Sn = m , (5) βn (m) = (K) (K) (K) (K) Pr Yn+1 , . . . , YN Y1 , . . . , Yn (K) and define the transitional probability γi Yn , m , m as
(6) Pr xn = i, Sn = m, Yn(K) Sn−1 = m ,
where Sn is the state of the channel filter at time n. The MAP probability in (3) is restated using (4), (5), and (6) as γi Yn(K) , m , m αn−1 (m )βn (m) m
m
where m denotes the decimal equivalent of the state Sn = n, and m (sn , . . . , sn−L+1 ) of the channel filter at time (K) denotes the state Sn−1 at time n−1. Using γi Yn , m , m , (4) and (5) are recursive computed,
αn (m) =
|C|−1 γi Yn(K) , m , m αn−1 (m ) m
m
i=0
βn (m) =
m
i=0
|C|−1 (K) γi Yn+1 , m , m αn (m ) m
m
,
.
(8)
i=0
The transitional probability in (6) is a product of three probabilities
p Yn(K) sn = i, Sn = m, Sn−1 = m × q (sn = i | Sn = m, Sn−1 = m ) × π (Sn = m | Sn−1 = m ) .
The probability q(·|·) is deterministic and is either 0 or 1. Also, the probability π(Sn = m | Sn−1 = m ) is either 0 or 1/|C|, assuming the a priori probabilites are equal (Pr{sn = i} = 1/|C|). (K) Since the elements of the vector Yn are independently Gaussian (given sn , Sn , Sn−1 ), we can restate the probability (K) p(Yn | . . . ) as a product p(Yn(K) | . . . ) = (k) p(yn
i:bnQ+j =0
with the index j = 0, 1, . . . , Q−1. The notation i : bnQ+j = 0 indicates those values of i where the j th bit is zero. Extrinsic information from an independent source regarding the transmitted bits, such as an outer decoder, is incorporated into the equalizer as an additional Gaussian random variable wn . The only substantive change to the equalizer involves the computation of the transitional probability (K) γi wnQ , . . . , wnQ+Q−1 , Yn , m , m , which becomes
(7)
i=0
(K) |C|−1 γi Yn+1 , m, m βn+1 (m )
and recompute the functions αn (m) and βn (m) to update the MAP probabilities. The MAP symbol probabilities are converted into bit log-likelihood ratios (LLR)
Pr sn = i . . . i:b nQ+j =1 (10) Λ(bnQ+j ) = log
,
Pr sn = i . . .
γi
|C|−1 γi Yn(K) , m , m αn−1 (m ) m
is received, we update Overall, when the K th transmission (K) the function γi Yn , m , m using the recursion γi Yn(K) , m , m = γi Yn(K−1) , m , m ×
(9) p ynK sn = i, Sn = m, Sn−1 = m ,
K
p yn(k) sn = i, Sn = m, Sn−1 = m .
k=1
Each | . . . ) can be easily found using the k th transmission channel and the binary precoding polynomial.
Q
Yn(K) , m , m
j=1
Pr wnQ+j sn = i .
Typically, wnQ+j is modelled as xn corrupted by AWGN. Note that the recursion in (9) still applies. Similar to turbodecoding and turbo-equalization, the LLR Λ(bnQ+j ) in (10) is computed with wnQ+j = 0. Note that the complexity increase for equalizing multiple transmissions is slight. The number of states in the equalizer does not increase as the number of transmissions K increases (assuming that the delay spread or length of the transmission channel remains constant). Other than updating the transitional (K) probability γi (Yn , m , m), the algorithm remains identical from transmission to transmission. The majority of the algorithm’s complexity lies in computing the functions αn (m) and βn (m), and the MAP probabilities P r{sn = i| . . .}. The transitional probability update in (9) has a very minor effect on the equalizer’s computational requirements. Thus, as K increases the equalizer’s complexity remains nearly constant. Also, once the transitional probability in (9) is updated, the K th transmission can be discarded. V. R ESULTS We present two sets of results that illustrate the effectiveness of mapping diversity. First, we evaluate the BER improvement in channel equalization provided by mapping diversity retransmissions. We remove coding from the system and send 1000 packets of 1024 bits using both the ML combining ARQ and mapping diversity ARQ systems. We assume that the channel has length L = 3 for all 16QAM and 16PSK, transmissions, and 4 for 8PSK. Each tap is a complex-valued Gaussian random variable (zero mean, unit variance) that varies per
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ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
−1
10
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
−1
10
−2
10 −2
BER
BER
10
−3
10 −3
10
−4
10
−4
10
−5
−8
Fig. 3.
−6
−4
−2
0
2 4 Eb/N0(dB)
6
8
10
12
10
14
BER for 16QAM mapping diversity scheme with no coding.
Fig. 5.
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
−1
10
−5
0
5 Eb/N0(dB)
10
15
BER for 16PSK mapping diversity scheme with no coding.
0
10
−2
FER
BER
10
−3
10
−1
10
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
−4
10
−8
−6
−4
−2
0
2 Eb/N0(dB)
4
6
8
10
12 −8
−6
−4
−2 E /N (dB) b
Fig. 4.
0
2
4
6
0
BER for 8PSK mapping diversity scheme with no coding. Fig. 6.
packet, but remains constant for retransmissions. The channel is normalized to unit energy to maintain the desired Eb /N0 . We use the symbol mapping adaptation schemes that were previously suggested. Figs. 3, 4, and 5 show the results of equalizing K uncoded transmissions for both ARQ systems for 16QAM, 8PSK, and 16PSK, respectively. The gains are consistent with those found in [13], verifying our belief that the MAP equalizer’s ability to mitigate ISI allows for treatment of ISI channels as simple AWGN channels. The gains are quite substantial; for instance using mapping diversity with 16PSK produces approximate gains of 7.5, 9, and 10 dB for 2,3, and 4 transmissions. The second set of results returns to the original ARQ framework, maintaining the same channel lengths as before. Using a (37, 21)8 recursive systematic convolutional (RSC) code, the transmitter encodes a 512 bit data sequence un to produce a 512 code bit sequence. Contained in the original bits prior to RSC encoding, is a 16-bit CRC codeword that is used by the receiver to determine if the received message sequence is correct. The message and code bits are concatenated into a single packet bn . The turbo-equalizer performs a maximum of four iterations for 16QAM and 16PSK, five iterations for
FER for 16QAM mapping diversity scheme with RSC coding.
8PSK. If the CRC check fails after the fourth(fifth) iteration, another copy of the packet is transmitted. This continues until the CRC check is successful or a maximum number (in this case K) of transmissions are processed, then a frame error is declared and the next packet is sent. We compare the frame error rates (FER) of the ARQ system using the mapping diversity approach against a simple ARQ system where the same mapping is used for all K transmissions. Figs. 6, 7, and 8 show the results of both ARQ systems for 16QAM, 8PSK, and 16QAM, respectively, for various values of the maximum number of transmissions K. The gains are still substantial, though not as large as those suggested by the BER plots. In designing the symbol mappings, the outer encoder was altogether ignored. These FER curves indicate that the effects of the encoder should be considered in designing the mappings. Fig. 9 contains the throughputs using 16PSK modulation for K = 3, 4. The throughput is ratio of the number of original data bits (512) versus the number of total transmitted bits (maximum of 1024 × K). At lower Eb /N0 values, mapping diversity provides significantly better throughputs.
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0
0.5
10
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3
0.45
0.4
FER
Throughput
0.35
−1
10
0.3
0.25
0.2
0.15
−6
−5
−4
−3
−2
−1 0 E /N (dB) b
Fig. 7.
1
2
3
4
−6
5
−4
−2
0
2
4 6 E /N (dB) b
0
8
10
12
14
0
Fig. 9. Throughput for 16PSK mapping diversity scheme with RSC coding.
FER for 8PSK mapping diversity scheme with RSC coding.
0
10
ML Combining, K = 4 Mapping Diversity, K = 4 ML Combining, K = 3 Mapping Diversity, K = 3 ML Combining, K = 2 Mapping Diversity, K = 2
FER
R EFERENCES
−1
10
−6
Fig. 8.
−4
−2
0
2
4 6 Eb/N0 (dB)
8
10
12
14
FER for 16PSK mapping diversity scheme with RSC coding.
VI. C ONCLUSIONS In this paper, we present a method for improving performance in handling retransmissions of higher-order modulated signals through intersymbol interference channels. The improvement is accomplished by redefining the symbol mapping for each retransmission. This results in a marginal complexity increase for both the transmitter and receiver, while reducing bit error rates and frame error rates significantly. Our simulation results show the effectiveness of this strategy. They also warrant a more rigorous understanding of the effects of various symbol mapping adaption schemes when an outer encoder is included in the system. Also, binary precoding prior to symbol mapping may provide additional gains. Finally, mapping diversity may have other applications such as channel estimation, space-time coding, and packet collision resolution. ACKNOWLEDGEMENTS This work was supported in part by Natioal Science Foundation grants CCR-0196364 and ECS-0121469, and an Accel Partners, Inc. fellowship.
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