System Design for Bit Interleaved Coded Square QAM with Iterative ...
Communication Theory
System Design for Bit Interleaved Coded Square QAM with Iterative Decoding in a Rician Fading Channel H ENRIK S CHULZE University Paderborn, Division Meschede D-59872 Meschede, Germany. [email protected]
Abstract. One-step iterative decoding for bit-interleaved coded QAM with conventional Gray mapping can give a significant improvement of performance for fading channels. Furthermore, iterative decoding with ideally known feed back bits is easier to analyze theoretically than non-iterative decoding. In this paper, we derive analytical expressions for the pairwise error probabilities for bit-interleaved coded QAM with correct feed-back bits. They are used to obtain union bounds for the bit error rate. Numerical simulations show that the performance with only one feedback step comes very close to these ideal theoretical curves. They are therefore a reasonable guideline for system design to choose the right code rate and modulation level for bit interleaved coded QAM in a fading channel.
1
I NTRODUCTION
The need for transmitting higher and higher data rates over band limited fading channels lets system designers venture upon higher level modulation schemes. As an example, 16-QAM and 64-QAM are part of the standards for DVB-T [1] and HIPERLAN/2 [2] at least as possible options - in addition to the the well established and robust standard 4-QAM (QPSK). Mainly because of pragmatic reasons of easy and flexible implementation, standard convolutional codes have been chosen together with conventional Gray mapping. Both systems use OFDM with symbol interleaving in frequency direction. For higher level modulation however, this would not be sufficient, because a deep fade of one QAM symbol would influence several adjacent bits in the coded data stream. To avoid these error burst, an additional bit interleaver has to be introduced. This makes DVB-T and HIPERLAN/2 maybe the first two systems that have implemented the general concept of bit interleaved coded modulation (BICM) [14, 3]. It has been observed and theoretically founded [3] that, for fading channels, the general concept of trellis coded modulation (TCM) with Ungerboeck set partitioning that combines coding and modulation is inferior to the simpler approach that treats both matters separately. Besides these theoretical reasons there are many practical benefits of Submission
BICM: There is a high flexibility in the code rate using only one decoder and punctured convolutional codes, and it offers the possibility to adjust the coding to the transmission rate and the channel. Li and Ritcey [11] have demonstrated for the case of bit-interleaved coded 8-PSK that iterative decoding (ID) improves the performance. Advantage can be taken from the knowledge of correctly decoded bits from preceding decoding steps. However, for their iterative decoding approach they use a hybrid set partitioning instead of the conventional Gray mapping. This provides a higher gain with iterative decoding, but it is inferior without it. For this reason, and because it is implemented in existing transmission sytems, we concentrate ourselves on Gray mapping, even though the gains due to ID may be smaller. In this paper, we investigate bit-interleaved coded QAM (BICQAM) with square constellations and the and the code trade-off between modulation level rate to choose the best combination of both for a given spectral efficiency. For BICQAM with iterative decoding, we derive an exact expression for the probability of an error event of weight that can be used to calculate union bounds for the bit error rate (BER). This expression uses the polar representation of the Gaussian probability integral like described in [7] to average over the fading and all possible combinations of bits to get an expression for that leaves
1
H. Schulze
ENC
CH
!
MCU
DEC
ENC
Figure 1: Block diagram of the system model.
only one simple integral that can be easily computed numerically. Using the same method we can also calculate the so-called expurgated union bounds (EX bounds) for that have been obtained by Caire et al. [3] by using an inverse Laplace transform method. These may be used to obtain tight bounds for the BER without iterative decoding. A comparison of both types of BER curves gives an estimate of the possible gain that can be obtained by iterative decoding. ID improvements turn out to be significant for low code rates, but small for high code rates. However, numerical simulations show that the ID union bounds for the bit error rate is much tighter than the EX union bound, so that the gain by iterative decoding will be overestimated by this method. It can be shown that only one additional iteration step (with hard decision bits) is sufficient to reach practically the ideal ID curves for correctly fed back bits, so the complexity of iterative decoding is quite low. This paper is organized as follows: In section 2 we explain the system model and the notation to be used. Optimum and sub optimum receivers and their metric computations are discussed. In section 3 we derive an expression to calculate bit error rates for BICQAM. We compare the bit error curves with computer simulations and discuss their relevance for system design. In section 4 we draw some conclusions.
2 2.1
S YSTEM
MODEL
T HE TRANSMISSION
CHAIN
#
2
$&%('*),+
.-0/21436587:96;
5=?@7BAAA*7 $DCFEHG 5% ? 5 %F$IC=E 9 $ % % JLK0M N 7O5 % ?P7 E 7BAA*7 $QCRE
9
/
(1)
determine which ASK-symbol will be transmitted. We regard it as convenient to interpret a QAM symbol as a two-dimensional real symbol instead of a onedimensional complex symbol. Let denote the sequence of ASK symbols. Then is the sequence of inphase symbols and is the sequence of the quadrature component symbols. Each ASK-symbol can take the values
7 TS 7 TU 7VA*AA 7 PW 7 OX 7BAA*A
\_^`7a\_b,^`7BAA*A7a\c3 CRE ;d^ G (2) of the signal constellation Y . Here we have introduced a distance unit ^ which is related to the symbol en maps $e% ')H+ bits ergy. The on a real gfa 7 7VA*symbol AA7 *h mapper symbol . The Gray mapping can be written as
h
j g a f * h - 7 7VA*AA*7 i1 % 3 ClE ; dqsmgnpo r ^`A k8f (3) Figure 2 shows a Eut@C QAM configuration with this mapping. We denote the energy per (two dimensional) QAM-Symbol by and the energy per data bit by . The relation between both is given by
xv w (4) vxwZ%s@')H+ 3 ; vzy r ^ 7 E ?H^ 7p{ r ^ 7VA*AA One can easily show that v w % for 4-QAM, 16-QAM, 64-QAM,... , etc. The sequence of ASK symbols will be written as a (row) vector |}% 3 7 7 S 7VA*AA~; . We consider a discrete channel CH given by %| A (5)
We consider -QAM constellations that are Cartesian products of two -ASK constellations for the (inphase) and the (quadrature) component. Each ASK symbol is labeled by bits. The block diagram for the transmission chain of our system model is shown in figure 1. The encoder (ENC) with code rate produces an encoded bit stream . For practical examples, we will use rate-compatible punctured convolutional (RCPC) codes like described in [13]. The bit interleaver
"
is a pseudo-random permutation of the time index together with a pseudo-random serial-parallel (S/P) conversion. Both are assumed to be statistically independent. This block is given by a random index map that chooses for each time index of the encoded bit a new time position with index and a labeling position for the Gray labeling ( means LSB, means MSB) . For each time index , the bits
vxy
ETT
System Design for Bit Interleaved Coded Square QAM with Iterative Decoding in a Rician Fading Channel
%
c(0) c(1) c(0) c(1) 1 1 2 2
to be fed into The optimum metric for the Viterbi decoder is given by the log-likelihoodratio (LLR)
4 0 1 0 0
3
1 1 0 0
1 0 0 0
0 0 0 0
3 % ?O ; 7 (9) 3 %E ;] see [5]. The probability 3 % ; that the transmitted bit at label 5 has the value under the condition that has been received may depend on hard or soft decision values from other bits * 7T5 l% ¡ 5 that are known from preceding decoding steps. This means that the constellation points may have different apriori probabilities 3 ; . Let Y f and Y be the subset of the the constellation corresponding to % ? and ` %E , resp., and Figure 2: 16-QAM with Gray mapping. V© E E is the vector of received symbols and is the white ¢ 3 ; % £ f¥¤B¦L§¨ C f C % f` r in the probability density for under the condition (10) Gaussian noise vector with variance that each real component. The RF signal-to-noise ratio has been transmitted over a channel with (ideally (SNR) is given by known) fading amplitude . The LLR is then given by F% vxw (6) f %s')H+Zxªs« q¬]® ¯mgnpo ¢¢ 3 ; 3 ; A (11) The fading is described by the diagonal matrix . q¬]® Mmgnpo 3 ; 3 ; « ª The diagonal is the vector of (real) fading amplitudes 3 7: 7 S 7BAA*A~; . The fading amplitudes are normal- In practice, the maxlog- approximation ized to average power one. We have incorporated per ± °³²µ´ ')H+ 3 ¢ 3 ; 3 ;p; fect phase estimation into the model. We can thereq¬]® ¯¦mgnpo « fore work with real quantities. We assume indepenC ²¶q¬]´® M¦mgnpo ')H+ 3 ¢ 3 ; 3 ;p;A (12) dent Rician fading amplitudes. « can be used. If no apriori- information is available, 2.2 R 3 ; %·E for all . If hard decisionr values for the other bits are fed back, 3 ; %¸E for exactly On the receiver side, the counterpart to the symbol f
System Design for Bit Interleaved Coded Square QAM with Iterative Decoding in a Rician Fading Channel H ENRIK S CHULZE University Paderborn, Division Meschede D-59872 Meschede, Germany. [email protected]
Abstract. One-step iterative decoding for bit-interleaved coded QAM with conventional Gray mapping can give a significant improvement of performance for fading channels. Furthermore, iterative decoding with ideally known feed back bits is easier to analyze theoretically than non-iterative decoding. In this paper, we derive analytical expressions for the pairwise error probabilities for bit-interleaved coded QAM with correct feed-back bits. They are used to obtain union bounds for the bit error rate. Numerical simulations show that the performance with only one feedback step comes very close to these ideal theoretical curves. They are therefore a reasonable guideline for system design to choose the right code rate and modulation level for bit interleaved coded QAM in a fading channel.
1
I NTRODUCTION
The need for transmitting higher and higher data rates over band limited fading channels lets system designers venture upon higher level modulation schemes. As an example, 16-QAM and 64-QAM are part of the standards for DVB-T [1] and HIPERLAN/2 [2] at least as possible options - in addition to the the well established and robust standard 4-QAM (QPSK). Mainly because of pragmatic reasons of easy and flexible implementation, standard convolutional codes have been chosen together with conventional Gray mapping. Both systems use OFDM with symbol interleaving in frequency direction. For higher level modulation however, this would not be sufficient, because a deep fade of one QAM symbol would influence several adjacent bits in the coded data stream. To avoid these error burst, an additional bit interleaver has to be introduced. This makes DVB-T and HIPERLAN/2 maybe the first two systems that have implemented the general concept of bit interleaved coded modulation (BICM) [14, 3]. It has been observed and theoretically founded [3] that, for fading channels, the general concept of trellis coded modulation (TCM) with Ungerboeck set partitioning that combines coding and modulation is inferior to the simpler approach that treats both matters separately. Besides these theoretical reasons there are many practical benefits of Submission
BICM: There is a high flexibility in the code rate using only one decoder and punctured convolutional codes, and it offers the possibility to adjust the coding to the transmission rate and the channel. Li and Ritcey [11] have demonstrated for the case of bit-interleaved coded 8-PSK that iterative decoding (ID) improves the performance. Advantage can be taken from the knowledge of correctly decoded bits from preceding decoding steps. However, for their iterative decoding approach they use a hybrid set partitioning instead of the conventional Gray mapping. This provides a higher gain with iterative decoding, but it is inferior without it. For this reason, and because it is implemented in existing transmission sytems, we concentrate ourselves on Gray mapping, even though the gains due to ID may be smaller. In this paper, we investigate bit-interleaved coded QAM (BICQAM) with square constellations and the and the code trade-off between modulation level rate to choose the best combination of both for a given spectral efficiency. For BICQAM with iterative decoding, we derive an exact expression for the probability of an error event of weight that can be used to calculate union bounds for the bit error rate (BER). This expression uses the polar representation of the Gaussian probability integral like described in [7] to average over the fading and all possible combinations of bits to get an expression for that leaves
1
H. Schulze
ENC
CH
!
MCU
DEC
ENC
Figure 1: Block diagram of the system model.
only one simple integral that can be easily computed numerically. Using the same method we can also calculate the so-called expurgated union bounds (EX bounds) for that have been obtained by Caire et al. [3] by using an inverse Laplace transform method. These may be used to obtain tight bounds for the BER without iterative decoding. A comparison of both types of BER curves gives an estimate of the possible gain that can be obtained by iterative decoding. ID improvements turn out to be significant for low code rates, but small for high code rates. However, numerical simulations show that the ID union bounds for the bit error rate is much tighter than the EX union bound, so that the gain by iterative decoding will be overestimated by this method. It can be shown that only one additional iteration step (with hard decision bits) is sufficient to reach practically the ideal ID curves for correctly fed back bits, so the complexity of iterative decoding is quite low. This paper is organized as follows: In section 2 we explain the system model and the notation to be used. Optimum and sub optimum receivers and their metric computations are discussed. In section 3 we derive an expression to calculate bit error rates for BICQAM. We compare the bit error curves with computer simulations and discuss their relevance for system design. In section 4 we draw some conclusions.
2 2.1
S YSTEM
MODEL
T HE TRANSMISSION
CHAIN
#
2
$&%('*),+
.-0/21436587:96;
5=?@7BAAA*7 $DCFEHG 5% ? 5 %F$IC=E 9 $ % % JLK0M N 7O5 % ?P7 E 7BAA*7 $QCRE
9
/
(1)
determine which ASK-symbol will be transmitted. We regard it as convenient to interpret a QAM symbol as a two-dimensional real symbol instead of a onedimensional complex symbol. Let denote the sequence of ASK symbols. Then is the sequence of inphase symbols and is the sequence of the quadrature component symbols. Each ASK-symbol can take the values
7 TS 7 TU 7VA*AA 7 PW 7 OX 7BAA*A
\_^`7a\_b,^`7BAA*A7a\c3 CRE ;d^ G (2) of the signal constellation Y . Here we have introduced a distance unit ^ which is related to the symbol en maps $e% ')H+ bits ergy. The on a real gfa 7 7VA*symbol AA7 *h mapper symbol . The Gray mapping can be written as
h
j g a f * h - 7 7VA*AA*7 i1 % 3 ClE ; dqsmgnpo r ^`A k8f (3) Figure 2 shows a Eut@C QAM configuration with this mapping. We denote the energy per (two dimensional) QAM-Symbol by and the energy per data bit by . The relation between both is given by
xv w (4) vxwZ%s@')H+ 3 ; vzy r ^ 7 E ?H^ 7p{ r ^ 7VA*AA One can easily show that v w % for 4-QAM, 16-QAM, 64-QAM,... , etc. The sequence of ASK symbols will be written as a (row) vector |}% 3 7 7 S 7VA*AA~; . We consider a discrete channel CH given by %| A (5)
We consider -QAM constellations that are Cartesian products of two -ASK constellations for the (inphase) and the (quadrature) component. Each ASK symbol is labeled by bits. The block diagram for the transmission chain of our system model is shown in figure 1. The encoder (ENC) with code rate produces an encoded bit stream . For practical examples, we will use rate-compatible punctured convolutional (RCPC) codes like described in [13]. The bit interleaver
"
is a pseudo-random permutation of the time index together with a pseudo-random serial-parallel (S/P) conversion. Both are assumed to be statistically independent. This block is given by a random index map that chooses for each time index of the encoded bit a new time position with index and a labeling position for the Gray labeling ( means LSB, means MSB) . For each time index , the bits
vxy
ETT
System Design for Bit Interleaved Coded Square QAM with Iterative Decoding in a Rician Fading Channel
%
c(0) c(1) c(0) c(1) 1 1 2 2
to be fed into The optimum metric for the Viterbi decoder is given by the log-likelihoodratio (LLR)
4 0 1 0 0
3
1 1 0 0
1 0 0 0
0 0 0 0
3 % ?O ; 7 (9) 3 %E ;] see [5]. The probability 3 % ; that the transmitted bit at label 5 has the value under the condition that has been received may depend on hard or soft decision values from other bits * 7T5 l% ¡ 5 that are known from preceding decoding steps. This means that the constellation points may have different apriori probabilities 3 ; . Let Y f and Y be the subset of the the constellation corresponding to % ? and ` %E , resp., and Figure 2: 16-QAM with Gray mapping. V© E E is the vector of received symbols and is the white ¢ 3 ; % £ f¥¤B¦L§¨ C f C % f` r in the probability density for under the condition (10) Gaussian noise vector with variance that each real component. The RF signal-to-noise ratio has been transmitted over a channel with (ideally (SNR) is given by known) fading amplitude . The LLR is then given by F% vxw (6) f %s')H+Zxªs« q¬]® ¯mgnpo ¢¢ 3 ; 3 ; A (11) The fading is described by the diagonal matrix . q¬]® Mmgnpo 3 ; 3 ; « ª The diagonal is the vector of (real) fading amplitudes 3 7: 7 S 7BAA*A~; . The fading amplitudes are normal- In practice, the maxlog- approximation ized to average power one. We have incorporated per ± °³²µ´ ')H+ 3 ¢ 3 ; 3 ;p; fect phase estimation into the model. We can thereq¬]® ¯¦mgnpo « fore work with real quantities. We assume indepenC ²¶q¬]´® M¦mgnpo ')H+ 3 ¢ 3 ; 3 ;p;A (12) dent Rician fading amplitudes. « can be used. If no apriori- information is available, 2.2 R 3 ; %·E for all . If hard decisionr values for the other bits are fed back, 3 ; %¸E for exactly On the receiver side, the counterpart to the symbol f
