Serially Concatenated LT Code with DQPSK Modulation - IEEE Xplore
IEEE WCNC 2011 - PHY
Serially Concatenated LT Code with DQPSK Modulation Iqbal Hussain, Ming Xiao, and Lars K. Rasmussen School of Electrical Engineering, Communication Theory Laboratory KTH - Royal Institute of Technology and the ACCESS Linneaus Center Stockholm, Sweden, Email: {iqbalh,mingx,lkra}@kth.se Abstract—We consider serial concatenation of a Luby Transform (LT) code with a differential quadrature phase-shift-keying (DQPSK) modulator for transmission over an additive white Gaussian noise (AWGN) Channel. Assuming a target average rate for the operation of the rateless LT DQPSK scheme, the degree distribution of the LT code is optimized in terms of convergence threshold using extrinsic information transfer (EXIT) charts. From the EXIT chart analysis, we show that the proposed LT DQPSK scheme has a similar convergence performance, but lower complexity, as compared to a Raptor code with differential modulation, and a LDPC code optimized for DQPSK. The EXIT chart analysis framework is also applied to evaluate the throughput performance for the three schemes in terms of the average code rate as a function of the signal-to-noise ratio. The comparison demonstrates that the proposed structure is wellsuited for adaptive-rate transmission over a wide range of rates.
I. I NTRODUCTION Reliable high data-rate transmission over mobile wireless channels is a challenge due to significant variations in the channel conditions. Optimal strategies have been proposed given that accurate channel state information (CSI) is available at the receiver or at both transmitter and receiver [1]. When CSI is only available at the receiver, adaptive transmission strategies with feedback are required such that either transmit power, code rate, or both can be adapted to instantaneous channel conditions. For variable-rate transmission rateless fountain codes are able to create an unlimited number of parity bits on-the-fly, making such codes attractive for timevarying channels. Code bits are continuously transmitted until the receiver returns an acknowledgement. One of the first practical realizations of a rateless code was the LT code in [2]. However, MacKay pointed out that an LT code is fundamentally “bad” as it suffers an error floor that does not diminish with increasing block length [3]. To address this problem, Raptor codes were proposed in [4], where a high-rate low-density parity-check (LDPC) pre-code cleans up the error floor of the LT code. The LT code belongs to the family of low-density generator matrix (LDGM) codes [3], and it has been shown in [5] that concatenated LDGM codes provide a different approach for eliminating the error floor problem. Rateless codes have also been investigated for noisy channels [6], binary symmetry channels [7], blockfading channels [8], and relay channels [9]. More recently constant-envelope modulation has been considered for LDGM codes in [10], and for LT codes in [11].
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Random phase variations are notoriously difficult to track in fast varying channels. In such cases, differential modulation can be used to eliminate the phase tracking problem. From a coding perspective, differential modulation may be viewed as a memoryless mapper followed by a rate-one recursive convolutional encoder [12], [13]. Following the principles of serially concatenated turbo codes [14], a recursive rate-one inner convolutional code can provide coding gain without additional redundancy. The serial concatenation of a convolutional outer code with an inner differential encoder has been considered in [12], [13]. In this paper we propose a serially concatenated coding scheme with an outer LT code and an inner differential quadrature phase-shift-keying (DQPSK) modulator. As our design tool we apply extrinsic information transfer (EXIT) chart analysis to evaluate the convergence behavior of the proposed concatenated scheme [15]. To enable the design, we define a target operational (average) signal-to-noise ratio (SNR), which can be translated into a corresponding operational (average) rate R of the LT code. Assuming the code is mainly operating at the target rate allows us to use the EXIT curve-fitting approach in [16] to determine a suitable degree distribution for the LT code, specifically designed for the concatenated LT DQPSK structure. Even though EXIT chart analysis is primarily a powerful design tool for fixed-rate codes, using the concept of an operational target rate the framework also provides useful insight into the design of concatenated rateless codes. The remainder of this paper is structured as follows. In Section II the system model is introduced, while the basic principles of EXIT chart analysis is detailed in Section III. The procedure for finding good degree distributions is outline in Section IV together with the improved degree distribution for the concatenated LT DQPSK coding scheme. An extended performance comparison is detailed in Section V, and some numerical results supporting the conclusions are shown in Section VI. Summarizing remarks conclude the paper in Section VII. II. S YSTEM M ODEL Consider the transmitter model in Fig. 1. A sequence of K information bits u = [u0 , u1 , . . . , uK ] is encoded by an LT encoder into a sequence of 2N coded bits c = [c0 , c1 , . . . , cn ], which is subsequently interleaved. The minimum rate for the
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QPSK Mapping u
LT Encoder
c
Interleaver
˜c
Differential Modulator
(01) (00)
(11)
x
s
(10)
D Fig. 1: Transmitter model. 00
7π/4
7π/4 10 11 01
5π/4
5π/4
3π/4
3π/4 10
11
Lch (m) := log
π/4
00
Fig. 2: One trellis section of the DQPSK modulator.
code is thus Rmin = K/2N . During the encoding process, the LT encoder randomly selects a certain number of information bits, which are added modulo-2 (XOR) to form a coded bit. The number of information bits involved is determined by sampling a predetermined degree distribution ρ. The coefficients ρj , of the degree distribution must satisfy the following constraints [7]: and
ρX max
ρj = 1,
j=1
where ρmax is the maximum check node degree. The sequence of interleaved coded bits c˜ is then mapped to a sequence of √ QPSK symbols s = [s0 , s1 , . . . , sN ], where si = Es ejθk , using Gray mapping. For notational simplicity the symbol energy is normalized to Es = 1, and θk ∈ [0, π/2, π, 3π/2]. The symbol sequence s is then differentially encoded into x = [x0 , x1 , . . . , xN ] using, xi = xi−1 si and x−1 = 1. The differential modulator is recursive and described by the fourstate trellis section shown in Fig. 2. The resulting encoder represents a serial concatenated code where the DQPSK modulator is the inner code and the LT code is the outer code. The differentially modulated sequence x is transmitted over a complex additive white Gaussian Noise (AWGN) channel with double-sided power spectral density N0 /2. The received sequence is described as y = x + w,
(2)
for m ∈ [1, 2, 3], ∀i = 1, 2, . . . , N + 1, and where p(yi |xi ) is the conditional probability density function of the channel described by
Next state
Present state
0 ≤ ρj ≤ 1
p(yi |xi = ej(mπ/2 ) , p(yi |xi = 1)
(i)
01
π/4
Due to the serial concatenation through an interleaver ideal rateless operation, where the LT encoder in principle can produce an infinite number of code bits, is not possible. The fixed-sized interleaver constrains the rateless transmission to a fixed maximum number of code bits, which in turn corresponds to a fixed minimum rate Rmin = K/2N . The “rateless” transmission rate R of the LT code is therefore constrained by Rmin ≤ R ≤ 1. At the receiver we perform standard iterative decoding for a serially concatenated system [14] as shown in Fig. 3. The inner DQPSK code is first decoded using the BCJR algorithm [17] based on the four-state trellis diagram developed from Fig. 2. The inner decoder receives both channel L-values as well as the a priori L-values from the outer LT decoder. Under the assumption that the receiver is able to track the channel phase, a received observable yi is converted to a corresponding (i) (i) (i) (i) symbol L-value vector, Lch = Lch (1), Lch (2), Lch (3) , as follows:
(1)
where w is a sequence of independent, identically distributed zero mean complex AWGN samples of variance N0 .
p(yi |xi ) = (πN0 )−1/2 e−|yi −xi |
2
/N0
, ∀i = 1, 2, . . . , N + 1. (3)
The symbol L-value vector is in turn converted into two bit L-values for input to the LT decoder. We denoted the iteration between the inner BCJR decoder and the outer LT decoder as a global iteration. The LT code is described by a bipartite factor graph with variable nodes and check nodes [4], where the check node degrees are determined by the degree distribution ρ. Iterative decoding is now performed through message-passing between the variable nodes and the check nodes. We can therefore describe the decoding process as a check node decoder (CND) exchanging messages with a variable node decoder (VND), which is illustrated in Fig. 3. One exchange in each direction is denoted as a local iteration. For each global iteration there are Q > 1 local iterations within the LT decoder. The update rule from check node j to variable node i (l,q) denoted by Lcj →vi is described as [7] (l)
L(l,q) cj →vi
−1
= 2 tanh
tanh
LLT,aprj 2
!
(l,q−1)
×
Y
´i∈N (j)\(i)
tanh
Lv´i →cj 2
!
,
(4)
where l and q are the indices for the global and the local iterations, respectively, N (j) denotes the set of neighboring (l) variable nodes for check node j, and LLT,aprj is the corresponding a priori L-value from the BCJR decoder. Similarly the message passed from variable node i to check node j is
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Hard decision (l)
LBCJR (s)
Lch
Symbol to Bit LLR
(l)
(l)
LBCJR,ext(˜c)
LBCJR (˜c)
+ _
BCJR Decoder
Q−1
(l) LLT,apr (c)
(l,q) Lc→v
CND
VND (l,q) Lv→c
LT Decoder
_ (l−1)
Bit to Symbol LLR
LBCJR,apr (s)
(l)
c) LBCJR,apr(˜
Q
+ (l) LLT,ext(c)
(l)
LLT (c)
Fig. 3: The iterative decoder of the concatenated LT coded DQPSK system. (l,q)
denoted by Lvi →cj , and is determined as [7] X L(l,q) L(l,q) vi →cj = c´j →vi .
(5)
´ j∈N (i)\(j)
After a full cycle of local iterations, the extrinsic L-value from the LT decoder to the inner decoder is determined as ! (l,Q) Y L vi →cj (l) . LLT,extj = 2 tanh−1 tanh (6) 2
where σA is the variance of the a priori L-values. We can then determine the EXIT curve for the DQPSK demapper as follows. A consistent sequence of L-values is generated according to a Gaussian distribution corresponding to a specific a priori MI, and passed through the BCJR decoder along with channel L-values Lch . The output extrinsic L-values are then collected in a histogram, which becomes an estimate of the corresponding output distribution function p(E|x). The output MI is finally determined as
i∈N (j)
The sequence of such L-values are then interleaved and converted to symbol L-values LBCJR,apr(s) to provide new a priori information for the next global iteration. The global iterations are terminated when a suitable stopping criterion is fulfilled, after which the final a-posteriori L-values of the variable nodes, determined as X (7) L(l) (ui ) = L(l,Q) cj →vi , j∈N(i)
are the basis for hard decisions to provide the decoded information sequence u ˆ. III. EXIT C HART A NALYSIS EXIT chart analysis allows us to investigate the convergence behavior of iterative decoding schemes [15], [16] for fixedrate codes. To facilitate the use of EXIT charts, we therefore consider fixed-rate instances of the rateless coding strategy. We model the global iterative process by considering the two constituent decoders, the BCJR DQPSK demapper and the LT decoder, and describing the respective input-output mutual information (MI) transfer functions, referred to as corresponding EXIT curves. The EXIT curve for the BCJR DQPSK demapper is obtained using standard techniques [15]. Under the assumption that the input a priori L-values are Gaussian distributed and consistent, the a priori MI is determined as [15] IA = Z 1−
+∞
−∞
exp
2 ! −1 σA2 log2 (1 + e−y ) √ y− dy, (8) 2 2σA 2 2πσA
IE = Z X 1 2p(e|x) p(e|x) log2 de. (9) 2 e∈E x∈±1 p(e|x = 1) + p(e|x = −1)
To obtain the EXIT curve for the LT decoder, an infinite number of local iterations should in principle be performed within the standard methods detailed above to accurately obtain the input-output MI relationship. However, to get an operational approximation, we iterate until no further improvements in MI are observed. The corresponding EXIT chart then tracks the MI exchange between the two constituent decoders over iterations under the assumption of infinitely long codewords. As long as there is an open tunnel between the two EXIT curves, the global iterative decoding process converges. IV. I MPROVED D EGREE D ISTRIBUTION The error probability performance and the convergence behavior of an LT code are primarily determined by the degree distribution ρ. Suitable degree distributions have been found for the binary erasure channel in [4] and for the AWGN channel in [7]. Here we want to optimize the degree distribution for the serial concatenation of an LT code and a DQPSK mapper, using the EXIT curve-fitting technique originally proposed in [16]. Let the output MI of the DQPSK demapper be denoted by IE,DQPSK and the output MI of the LT decoder by IE,LT . Following the approach in [18] we define the function −1 f (ρ) = min {IE,DQPSK (I) − IE,LT (I)} I∈(0,1)
(10)
to be a metric of whether or not the tunnel between the two EXIT curves is open. Here ρ is the check node degree
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2
3
4
ρ(x) = 0.1403x + 0.4920x + 0.1472x + 0.0905x 5
8
14
+ 0.0739x + 0.0060x + 0.0110x
(11)
+ 0.0346x30.
In Fig. 4 the EXIT chart for a fixed-rate R = 1/2 LT code concatenated with a DQPSK mapper is shown. EXIT curves are plotted for an LT code using the new degree distribution shown above and the degree distribution optimized for an AWGN channel as detailed in [7], respectively. We note from Fig. 4, that the convergence threshold for the new degree distribution is 1.2 dB as compared to 2.7 dB for the distribution optimized for an AWGN channel. Furthermore, the optimized degree distribution not only exhibits a better threshold but also enjoys lower complexity as the average check node degree is 3.5374 as compared to 11.8437 for the degree distribution in [7]. It is worth noting as well that for LDGM codes, like an LT code, lower check node degrees typically leads to lower threshold but a higher error floor [19]. V. P ERFORMANCE C OMPARISON The rate-adaptive capabilities of rateless codes provide an obvious advantage over fixed-rate codes in transmission scenarios with fast varying channel conditions. It is, however, a challenge to devise a fair comparison between these two classes of coding strategies. Here we have chosen to compare convergence thresholds for fixed-rate instances of the LT coded schemes with similar thresholds for LDPC coded and fixedrate Raptor coded strategies. To conclude we compare the achievable throughput for rateless operation of the LT coded, Raptor coded, and rate-compatible LDPC coded schemes. Systematic LT codes have been shown to perform better than similar nonsystematic codes at lower rates when used with constant-envelope modulation [11]. In Fig. 5 we show the corresponding EXIT curves for a fixed rate R = 1/2 with DQPSK. We observe in this case that the systematic LT code has a better convergence threshold, which turns out to
1 DQPSK, Eb/N0=2.7 dB DQPSK, Eb/N0=1.2 dB LT Decoder, AWGN LT Decoder, DQPSK
0.9 0.8 0.7 I(A,LT),I(E,DQPSK)
−1 distribution of the LT code, and IE,LT (I) is the input MI of the LT decoder as a function of the output MI I. If the tunnel is open, then the EXIT curves only touch in the point of full MI, i.e., for I = 1, in which case f (ρ) = 0. If the tunnel is closed for I < 1, then f (ρ) < 0. We search for a suitable degree distribution as follows. First an initial degree distribution, a target rate R, and an operating Eb /N0 are selected. Then a set of free parameters of the degree distribution is identified for the optimization process, and f (ρ) is evaluated. If f (ρ) = 0, the Eb /N0 is decreased to ensure the tunnel is closed, i.e., f (ρ) < 0. The set of free variables is then randomly perturbed by a zero-mean Gaussian disturbance with variance σd2 to obtain a new degree distribution ρ˜. If f (˜ ρ) ≤ f (ρ), then ρ˜ is discarded and a new perturbation is made. Otherwise, ρ is replaced with ρ˜. If the tunnel opens again, the Eb /N0 is once more decreased. This process continues until a maximum number of steps have been completed, and as the output we have an improved degree distribution ρ for a given operating Eb /N0 value and target code rate R. As an example, we have determined a new degree distribution for the target rate R = 1/2, described as
0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.2
0.4 0.6 I( A,DQPSK),I( E,LT)
0.8
1
Fig. 4: EXIT chart for a fixed rate R = 1/2 LT coded DQPSK scheme optimized for AWGN and DQPSK-AWGN.
be at 1.0 dB as compared to 1.2 dB for the nonsystematic code. Systematic LT codes are found to have a threshold advantage for fixed rates below R = 0.56. Above this rate the nonsystematic counterparts are better. The choice of systematic versus nonsystematic codes therefore depends on the expected operational rate. As discussed in the introduction, LT codes suffer from a high error floor which is independent of the code length [3]. Raptor codes with a high-rate LDPC pre-encoder were then proposed as an efficient solution [4]. Also concatenated LDGM codes have been shown to improve the error floor [5]. Here we compare the convergence performance of a fixed-rate R = 1/2 Raptor code with nonsystematic LT code as inner code based on the degree distribution optimized for DQPSK. For the pre-encoder of the Raptor code, we use a rate R = 0.95 LDPC code with left regular distribution with variable nodes degree 3 and check nodes having Poisson distribution. In Fig. 5 we observe that the two codes have virtually identical EXIT curves, and thus enjoy the same convergence characteristic. By construction, however, the Raptor code will have better error floor performance [6] at the expense of higher complexity. In [18] LDPC codes were optimized for differential modulation. In Fig. 6 we compare our LT coded scheme with a similar LDPC coded scheme optimized for DQPSK and with a fixed-rate R = 1/2. We observe that the two EXIT curves have different shapes, but lead to the same convergence threshold. The advantage of the LT coded scheme is of course that it inherently allows for rateless operation. The relatively high error floor is again a potential drawback. The mapping from bits to symbols in higher-order modulation schemes is important for bit error performance. With iterative decoding, the mapping can be optimized through EXIT chart analysis [20]. For DQPSK only two mappings are relevant; namely Gray mapping and natural mapping. In Fig. 7 we observe that Gray mapping is significantly better in terms of convergence threshold as compared to natural mapping.
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1
1 DQPSK, Eb/N0=1.2 dB Nonsystematic LT Systematic LT Raptor code
0.9 0.8
0.8 0.7 I(A,LT),I(E,DQPSK)
I(A,LT),I(E,DQPSK)
0.7 0.6 0.5 0.4
0.5 0.4 0.3
0.2
0.2
0.1
0.1 0
0.2
0.4 0.6 I( A,DQPSK),I( E,LT)
0.8
0
1
Fig. 5: EXIT chart of systematic and nonsystematic LT codes, and a Raptor code, with DQPSK and a fixed rate R = 1/2.
0.2
0.4 0.6 I( A,DQPSK),I( E,LT)
0.8
1
2 DQPSK, Eb/N0=1.2 dB Nonsystematic LT Optimized LDPC
1.8 1.6
0.7
1.4 Rate in bits/symbol
0.8
0.6 0.5 0.4
1 0.8 0.6
0.2
0.4
0.1
0.2 0
0.2
0.4 0.6 I( A,DQPSK),I( E,LT)
0.8
0 −5
1
LT Coded DQPSK LDPC Coded DQPSK Raptor Coded DQPSK
1.2
0.3
0
0
Fig. 7: EXIT chart of Gray mapping and natural mapping in the DQPSK encoder/decoder.
1 0.9
I(A,LT),I(E,DQPSK)
0.6
0.3
0
Gray, Eb/N0=1.2 dB Natural, Eb/N0=2.8 Natural, Eb/N0=1.2 LT Decoder
0.9
0
5
10
SNR
Fig. 6: EXIT chart of a nonsystematic LT code and a LDPC code with DQPSK and a fixed rate R = 1/2.
For higher-order modulation formats Gray mapping may not necessarily be the best choice [20]. We now consider rateless operation where the target operational rate is R = 1/2 and the minimum rate is Rmin = 1/10; thus we extend the fixed rate coding schemes investigated above to the rateless case, i.e., the LT code, the Raptor code and a rate-compatible punctured version of the LDPC code. Since the LDPC code is a block code, the minimum rate is equal to the target rate R = 1/2, and higher rates are obtained through puncturing. We are in particular interested in the throughput performance in terms of the rate in bits per modulation symbol. It is observed from EXIT charts that for rates above 1.6 bits per symbol, the MI does not converge to 1, and thus the BER will not converge to zero. Therefore, starting from a rate of 1.6 bits per symbol, we determine the convergence threshold
Fig. 8: Rate in bits/symbol of the LT code, Raptor, and ratecompatible LDPC code with DQPSK and a fixed rate R = 1/2.
in terms of SNR using the EXIT chart framework. SNR is defined as SNR = 10 log10 ( σ12 ), where σ is the noise standard deviation. The rate is then subsequently decreased to track the rate as a function of SNR. The throughput performance is illustrated in Fig. 8, where we observe that the LT coded scheme enjoys a higher rate than both the Raptor coded and the LDPC coded alternatives. We especially notice the relatively high gain over the rate-compatible punctured LDPC code; however, the puncturing strategy has not been optimized, and thus some improvements can be expected. The advantage over the Raptor coded scheme is not as significant; however since the LT coded scheme is less complex, it is still a compelling advantage. The inherent drawback of a higher error floor is still a problem that needs further investigation for the LT coded case.
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to a comparable Raptor code and a rate-compatible LDPC code. A drawback of the scheme is the relatively high error floor. The use of concatenated LT codes are currently under investigation to eliminate the error floor while keeping the overall complexity low.
0
10
Nonsystematic LT Systematic LT Raptor −1
10
VIII. ACKNOWLEDGEMENTS
BER
−2
10
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/20072013) / ERC grant agreement n 228044.The work has further been supported in part by the Swedish Research Council under VR grants 621-2008-4249 and 621-2009-4666.
−3
10
−4
10
0
0.5
1
1.5
2 2.5 Eb/N0 in dB
3
3.5
R EFERENCES
4
Fig. 9: BER performance of systematic and nonsystematic LT code with DQPSK for a fixed-rate R = 1/2.
VI. S IMULATION R ESULTS To demonstrate the convergence threshold and error floor for the proposed LT and Raptor coded DQPSK system, we provide some numerical results for the bit error rate (BER) as shown in Fig. 9. We consider both a systematic and nonsystematic LT coded DQPSK scheme for a fixed-rate R = 1/2 over an AWGN channel. The block length is N = 2000 QPSK symbols, corresponding to 2000 information bits. The number of local iterations of the LT decoder was limited to 50, while the number of global iterations was set to 30. We observe that the convergence thresholds are in line with predictions based on the EXIT chart analysis. We also observe an error floor at around 10−4 and 4 · 10−4 , respectively, for the systematic and nonsystematic LT codes. The improved error floor for the systematic code is due to a better average variable node degree as compared to its nonsystematic counterpart. Fig. 9 clearly indicate the better performance of Raptor code in the error floor region. VII. C ONCLUSIONS We have investigated a serial concatenation of an LT code and a DQPSK modulator for transmission over an AWGN channel. With the use of EXIT curve-fitting, and the concept of an expected operational rate of the rateless code, we have proposed a suitable design approach for the degree distribution optimization of the concatenated LT DQPSK scheme. As an example, we considered an operational rate of R = 1/2, for which the obtained degree distribution provided improvements in convergence threshold of 1.5 dB over the degree distribution derived for the AWGN channel. In addition, the proposed distribution leads to comparable convergence performance as for fixed-rate LDPC codes and Raptor codes optimized for DQPSK. Using the EXIT chart analysis strategy, we determined the expected rate of the LT coded scheme when operated as a rateless code as a function of the SNR. Improvements in term of throughput were observed as compared
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Serially Concatenated LT Code with DQPSK Modulation Iqbal Hussain, Ming Xiao, and Lars K. Rasmussen School of Electrical Engineering, Communication Theory Laboratory KTH - Royal Institute of Technology and the ACCESS Linneaus Center Stockholm, Sweden, Email: {iqbalh,mingx,lkra}@kth.se Abstract—We consider serial concatenation of a Luby Transform (LT) code with a differential quadrature phase-shift-keying (DQPSK) modulator for transmission over an additive white Gaussian noise (AWGN) Channel. Assuming a target average rate for the operation of the rateless LT DQPSK scheme, the degree distribution of the LT code is optimized in terms of convergence threshold using extrinsic information transfer (EXIT) charts. From the EXIT chart analysis, we show that the proposed LT DQPSK scheme has a similar convergence performance, but lower complexity, as compared to a Raptor code with differential modulation, and a LDPC code optimized for DQPSK. The EXIT chart analysis framework is also applied to evaluate the throughput performance for the three schemes in terms of the average code rate as a function of the signal-to-noise ratio. The comparison demonstrates that the proposed structure is wellsuited for adaptive-rate transmission over a wide range of rates.
I. I NTRODUCTION Reliable high data-rate transmission over mobile wireless channels is a challenge due to significant variations in the channel conditions. Optimal strategies have been proposed given that accurate channel state information (CSI) is available at the receiver or at both transmitter and receiver [1]. When CSI is only available at the receiver, adaptive transmission strategies with feedback are required such that either transmit power, code rate, or both can be adapted to instantaneous channel conditions. For variable-rate transmission rateless fountain codes are able to create an unlimited number of parity bits on-the-fly, making such codes attractive for timevarying channels. Code bits are continuously transmitted until the receiver returns an acknowledgement. One of the first practical realizations of a rateless code was the LT code in [2]. However, MacKay pointed out that an LT code is fundamentally “bad” as it suffers an error floor that does not diminish with increasing block length [3]. To address this problem, Raptor codes were proposed in [4], where a high-rate low-density parity-check (LDPC) pre-code cleans up the error floor of the LT code. The LT code belongs to the family of low-density generator matrix (LDGM) codes [3], and it has been shown in [5] that concatenated LDGM codes provide a different approach for eliminating the error floor problem. Rateless codes have also been investigated for noisy channels [6], binary symmetry channels [7], blockfading channels [8], and relay channels [9]. More recently constant-envelope modulation has been considered for LDGM codes in [10], and for LT codes in [11].
978-1-61284-254-7/11/$26.00 ©2011 IEEE
Random phase variations are notoriously difficult to track in fast varying channels. In such cases, differential modulation can be used to eliminate the phase tracking problem. From a coding perspective, differential modulation may be viewed as a memoryless mapper followed by a rate-one recursive convolutional encoder [12], [13]. Following the principles of serially concatenated turbo codes [14], a recursive rate-one inner convolutional code can provide coding gain without additional redundancy. The serial concatenation of a convolutional outer code with an inner differential encoder has been considered in [12], [13]. In this paper we propose a serially concatenated coding scheme with an outer LT code and an inner differential quadrature phase-shift-keying (DQPSK) modulator. As our design tool we apply extrinsic information transfer (EXIT) chart analysis to evaluate the convergence behavior of the proposed concatenated scheme [15]. To enable the design, we define a target operational (average) signal-to-noise ratio (SNR), which can be translated into a corresponding operational (average) rate R of the LT code. Assuming the code is mainly operating at the target rate allows us to use the EXIT curve-fitting approach in [16] to determine a suitable degree distribution for the LT code, specifically designed for the concatenated LT DQPSK structure. Even though EXIT chart analysis is primarily a powerful design tool for fixed-rate codes, using the concept of an operational target rate the framework also provides useful insight into the design of concatenated rateless codes. The remainder of this paper is structured as follows. In Section II the system model is introduced, while the basic principles of EXIT chart analysis is detailed in Section III. The procedure for finding good degree distributions is outline in Section IV together with the improved degree distribution for the concatenated LT DQPSK coding scheme. An extended performance comparison is detailed in Section V, and some numerical results supporting the conclusions are shown in Section VI. Summarizing remarks conclude the paper in Section VII. II. S YSTEM M ODEL Consider the transmitter model in Fig. 1. A sequence of K information bits u = [u0 , u1 , . . . , uK ] is encoded by an LT encoder into a sequence of 2N coded bits c = [c0 , c1 , . . . , cn ], which is subsequently interleaved. The minimum rate for the
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QPSK Mapping u
LT Encoder
c
Interleaver
˜c
Differential Modulator
(01) (00)
(11)
x
s
(10)
D Fig. 1: Transmitter model. 00
7π/4
7π/4 10 11 01
5π/4
5π/4
3π/4
3π/4 10
11
Lch (m) := log
π/4
00
Fig. 2: One trellis section of the DQPSK modulator.
code is thus Rmin = K/2N . During the encoding process, the LT encoder randomly selects a certain number of information bits, which are added modulo-2 (XOR) to form a coded bit. The number of information bits involved is determined by sampling a predetermined degree distribution ρ. The coefficients ρj , of the degree distribution must satisfy the following constraints [7]: and
ρX max
ρj = 1,
j=1
where ρmax is the maximum check node degree. The sequence of interleaved coded bits c˜ is then mapped to a sequence of √ QPSK symbols s = [s0 , s1 , . . . , sN ], where si = Es ejθk , using Gray mapping. For notational simplicity the symbol energy is normalized to Es = 1, and θk ∈ [0, π/2, π, 3π/2]. The symbol sequence s is then differentially encoded into x = [x0 , x1 , . . . , xN ] using, xi = xi−1 si and x−1 = 1. The differential modulator is recursive and described by the fourstate trellis section shown in Fig. 2. The resulting encoder represents a serial concatenated code where the DQPSK modulator is the inner code and the LT code is the outer code. The differentially modulated sequence x is transmitted over a complex additive white Gaussian Noise (AWGN) channel with double-sided power spectral density N0 /2. The received sequence is described as y = x + w,
(2)
for m ∈ [1, 2, 3], ∀i = 1, 2, . . . , N + 1, and where p(yi |xi ) is the conditional probability density function of the channel described by
Next state
Present state
0 ≤ ρj ≤ 1
p(yi |xi = ej(mπ/2 ) , p(yi |xi = 1)
(i)
01
π/4
Due to the serial concatenation through an interleaver ideal rateless operation, where the LT encoder in principle can produce an infinite number of code bits, is not possible. The fixed-sized interleaver constrains the rateless transmission to a fixed maximum number of code bits, which in turn corresponds to a fixed minimum rate Rmin = K/2N . The “rateless” transmission rate R of the LT code is therefore constrained by Rmin ≤ R ≤ 1. At the receiver we perform standard iterative decoding for a serially concatenated system [14] as shown in Fig. 3. The inner DQPSK code is first decoded using the BCJR algorithm [17] based on the four-state trellis diagram developed from Fig. 2. The inner decoder receives both channel L-values as well as the a priori L-values from the outer LT decoder. Under the assumption that the receiver is able to track the channel phase, a received observable yi is converted to a corresponding (i) (i) (i) (i) symbol L-value vector, Lch = Lch (1), Lch (2), Lch (3) , as follows:
(1)
where w is a sequence of independent, identically distributed zero mean complex AWGN samples of variance N0 .
p(yi |xi ) = (πN0 )−1/2 e−|yi −xi |
2
/N0
, ∀i = 1, 2, . . . , N + 1. (3)
The symbol L-value vector is in turn converted into two bit L-values for input to the LT decoder. We denoted the iteration between the inner BCJR decoder and the outer LT decoder as a global iteration. The LT code is described by a bipartite factor graph with variable nodes and check nodes [4], where the check node degrees are determined by the degree distribution ρ. Iterative decoding is now performed through message-passing between the variable nodes and the check nodes. We can therefore describe the decoding process as a check node decoder (CND) exchanging messages with a variable node decoder (VND), which is illustrated in Fig. 3. One exchange in each direction is denoted as a local iteration. For each global iteration there are Q > 1 local iterations within the LT decoder. The update rule from check node j to variable node i (l,q) denoted by Lcj →vi is described as [7] (l)
L(l,q) cj →vi
−1
= 2 tanh
tanh
LLT,aprj 2
!
(l,q−1)
×
Y
´i∈N (j)\(i)
tanh
Lv´i →cj 2
!
,
(4)
where l and q are the indices for the global and the local iterations, respectively, N (j) denotes the set of neighboring (l) variable nodes for check node j, and LLT,aprj is the corresponding a priori L-value from the BCJR decoder. Similarly the message passed from variable node i to check node j is
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Hard decision (l)
LBCJR (s)
Lch
Symbol to Bit LLR
(l)
(l)
LBCJR,ext(˜c)
LBCJR (˜c)
+ _
BCJR Decoder
Q−1
(l) LLT,apr (c)
(l,q) Lc→v
CND
VND (l,q) Lv→c
LT Decoder
_ (l−1)
Bit to Symbol LLR
LBCJR,apr (s)
(l)
c) LBCJR,apr(˜
Q
+ (l) LLT,ext(c)
(l)
LLT (c)
Fig. 3: The iterative decoder of the concatenated LT coded DQPSK system. (l,q)
denoted by Lvi →cj , and is determined as [7] X L(l,q) L(l,q) vi →cj = c´j →vi .
(5)
´ j∈N (i)\(j)
After a full cycle of local iterations, the extrinsic L-value from the LT decoder to the inner decoder is determined as ! (l,Q) Y L vi →cj (l) . LLT,extj = 2 tanh−1 tanh (6) 2
where σA is the variance of the a priori L-values. We can then determine the EXIT curve for the DQPSK demapper as follows. A consistent sequence of L-values is generated according to a Gaussian distribution corresponding to a specific a priori MI, and passed through the BCJR decoder along with channel L-values Lch . The output extrinsic L-values are then collected in a histogram, which becomes an estimate of the corresponding output distribution function p(E|x). The output MI is finally determined as
i∈N (j)
The sequence of such L-values are then interleaved and converted to symbol L-values LBCJR,apr(s) to provide new a priori information for the next global iteration. The global iterations are terminated when a suitable stopping criterion is fulfilled, after which the final a-posteriori L-values of the variable nodes, determined as X (7) L(l) (ui ) = L(l,Q) cj →vi , j∈N(i)
are the basis for hard decisions to provide the decoded information sequence u ˆ. III. EXIT C HART A NALYSIS EXIT chart analysis allows us to investigate the convergence behavior of iterative decoding schemes [15], [16] for fixedrate codes. To facilitate the use of EXIT charts, we therefore consider fixed-rate instances of the rateless coding strategy. We model the global iterative process by considering the two constituent decoders, the BCJR DQPSK demapper and the LT decoder, and describing the respective input-output mutual information (MI) transfer functions, referred to as corresponding EXIT curves. The EXIT curve for the BCJR DQPSK demapper is obtained using standard techniques [15]. Under the assumption that the input a priori L-values are Gaussian distributed and consistent, the a priori MI is determined as [15] IA = Z 1−
+∞
−∞
exp
2 ! −1 σA2 log2 (1 + e−y ) √ y− dy, (8) 2 2σA 2 2πσA
IE = Z X 1 2p(e|x) p(e|x) log2 de. (9) 2 e∈E x∈±1 p(e|x = 1) + p(e|x = −1)
To obtain the EXIT curve for the LT decoder, an infinite number of local iterations should in principle be performed within the standard methods detailed above to accurately obtain the input-output MI relationship. However, to get an operational approximation, we iterate until no further improvements in MI are observed. The corresponding EXIT chart then tracks the MI exchange between the two constituent decoders over iterations under the assumption of infinitely long codewords. As long as there is an open tunnel between the two EXIT curves, the global iterative decoding process converges. IV. I MPROVED D EGREE D ISTRIBUTION The error probability performance and the convergence behavior of an LT code are primarily determined by the degree distribution ρ. Suitable degree distributions have been found for the binary erasure channel in [4] and for the AWGN channel in [7]. Here we want to optimize the degree distribution for the serial concatenation of an LT code and a DQPSK mapper, using the EXIT curve-fitting technique originally proposed in [16]. Let the output MI of the DQPSK demapper be denoted by IE,DQPSK and the output MI of the LT decoder by IE,LT . Following the approach in [18] we define the function −1 f (ρ) = min {IE,DQPSK (I) − IE,LT (I)} I∈(0,1)
(10)
to be a metric of whether or not the tunnel between the two EXIT curves is open. Here ρ is the check node degree
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2
3
4
ρ(x) = 0.1403x + 0.4920x + 0.1472x + 0.0905x 5
8
14
+ 0.0739x + 0.0060x + 0.0110x
(11)
+ 0.0346x30.
In Fig. 4 the EXIT chart for a fixed-rate R = 1/2 LT code concatenated with a DQPSK mapper is shown. EXIT curves are plotted for an LT code using the new degree distribution shown above and the degree distribution optimized for an AWGN channel as detailed in [7], respectively. We note from Fig. 4, that the convergence threshold for the new degree distribution is 1.2 dB as compared to 2.7 dB for the distribution optimized for an AWGN channel. Furthermore, the optimized degree distribution not only exhibits a better threshold but also enjoys lower complexity as the average check node degree is 3.5374 as compared to 11.8437 for the degree distribution in [7]. It is worth noting as well that for LDGM codes, like an LT code, lower check node degrees typically leads to lower threshold but a higher error floor [19]. V. P ERFORMANCE C OMPARISON The rate-adaptive capabilities of rateless codes provide an obvious advantage over fixed-rate codes in transmission scenarios with fast varying channel conditions. It is, however, a challenge to devise a fair comparison between these two classes of coding strategies. Here we have chosen to compare convergence thresholds for fixed-rate instances of the LT coded schemes with similar thresholds for LDPC coded and fixedrate Raptor coded strategies. To conclude we compare the achievable throughput for rateless operation of the LT coded, Raptor coded, and rate-compatible LDPC coded schemes. Systematic LT codes have been shown to perform better than similar nonsystematic codes at lower rates when used with constant-envelope modulation [11]. In Fig. 5 we show the corresponding EXIT curves for a fixed rate R = 1/2 with DQPSK. We observe in this case that the systematic LT code has a better convergence threshold, which turns out to
1 DQPSK, Eb/N0=2.7 dB DQPSK, Eb/N0=1.2 dB LT Decoder, AWGN LT Decoder, DQPSK
0.9 0.8 0.7 I(A,LT),I(E,DQPSK)
−1 distribution of the LT code, and IE,LT (I) is the input MI of the LT decoder as a function of the output MI I. If the tunnel is open, then the EXIT curves only touch in the point of full MI, i.e., for I = 1, in which case f (ρ) = 0. If the tunnel is closed for I < 1, then f (ρ) < 0. We search for a suitable degree distribution as follows. First an initial degree distribution, a target rate R, and an operating Eb /N0 are selected. Then a set of free parameters of the degree distribution is identified for the optimization process, and f (ρ) is evaluated. If f (ρ) = 0, the Eb /N0 is decreased to ensure the tunnel is closed, i.e., f (ρ) < 0. The set of free variables is then randomly perturbed by a zero-mean Gaussian disturbance with variance σd2 to obtain a new degree distribution ρ˜. If f (˜ ρ) ≤ f (ρ), then ρ˜ is discarded and a new perturbation is made. Otherwise, ρ is replaced with ρ˜. If the tunnel opens again, the Eb /N0 is once more decreased. This process continues until a maximum number of steps have been completed, and as the output we have an improved degree distribution ρ for a given operating Eb /N0 value and target code rate R. As an example, we have determined a new degree distribution for the target rate R = 1/2, described as
0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.2
0.4 0.6 I( A,DQPSK),I( E,LT)
0.8
1
Fig. 4: EXIT chart for a fixed rate R = 1/2 LT coded DQPSK scheme optimized for AWGN and DQPSK-AWGN.
be at 1.0 dB as compared to 1.2 dB for the nonsystematic code. Systematic LT codes are found to have a threshold advantage for fixed rates below R = 0.56. Above this rate the nonsystematic counterparts are better. The choice of systematic versus nonsystematic codes therefore depends on the expected operational rate. As discussed in the introduction, LT codes suffer from a high error floor which is independent of the code length [3]. Raptor codes with a high-rate LDPC pre-encoder were then proposed as an efficient solution [4]. Also concatenated LDGM codes have been shown to improve the error floor [5]. Here we compare the convergence performance of a fixed-rate R = 1/2 Raptor code with nonsystematic LT code as inner code based on the degree distribution optimized for DQPSK. For the pre-encoder of the Raptor code, we use a rate R = 0.95 LDPC code with left regular distribution with variable nodes degree 3 and check nodes having Poisson distribution. In Fig. 5 we observe that the two codes have virtually identical EXIT curves, and thus enjoy the same convergence characteristic. By construction, however, the Raptor code will have better error floor performance [6] at the expense of higher complexity. In [18] LDPC codes were optimized for differential modulation. In Fig. 6 we compare our LT coded scheme with a similar LDPC coded scheme optimized for DQPSK and with a fixed-rate R = 1/2. We observe that the two EXIT curves have different shapes, but lead to the same convergence threshold. The advantage of the LT coded scheme is of course that it inherently allows for rateless operation. The relatively high error floor is again a potential drawback. The mapping from bits to symbols in higher-order modulation schemes is important for bit error performance. With iterative decoding, the mapping can be optimized through EXIT chart analysis [20]. For DQPSK only two mappings are relevant; namely Gray mapping and natural mapping. In Fig. 7 we observe that Gray mapping is significantly better in terms of convergence threshold as compared to natural mapping.
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1
1 DQPSK, Eb/N0=1.2 dB Nonsystematic LT Systematic LT Raptor code
0.9 0.8
0.8 0.7 I(A,LT),I(E,DQPSK)
I(A,LT),I(E,DQPSK)
0.7 0.6 0.5 0.4
0.5 0.4 0.3
0.2
0.2
0.1
0.1 0
0.2
0.4 0.6 I( A,DQPSK),I( E,LT)
0.8
0
1
Fig. 5: EXIT chart of systematic and nonsystematic LT codes, and a Raptor code, with DQPSK and a fixed rate R = 1/2.
0.2
0.4 0.6 I( A,DQPSK),I( E,LT)
0.8
1
2 DQPSK, Eb/N0=1.2 dB Nonsystematic LT Optimized LDPC
1.8 1.6
0.7
1.4 Rate in bits/symbol
0.8
0.6 0.5 0.4
1 0.8 0.6
0.2
0.4
0.1
0.2 0
0.2
0.4 0.6 I( A,DQPSK),I( E,LT)
0.8
0 −5
1
LT Coded DQPSK LDPC Coded DQPSK Raptor Coded DQPSK
1.2
0.3
0
0
Fig. 7: EXIT chart of Gray mapping and natural mapping in the DQPSK encoder/decoder.
1 0.9
I(A,LT),I(E,DQPSK)
0.6
0.3
0
Gray, Eb/N0=1.2 dB Natural, Eb/N0=2.8 Natural, Eb/N0=1.2 LT Decoder
0.9
0
5
10
SNR
Fig. 6: EXIT chart of a nonsystematic LT code and a LDPC code with DQPSK and a fixed rate R = 1/2.
For higher-order modulation formats Gray mapping may not necessarily be the best choice [20]. We now consider rateless operation where the target operational rate is R = 1/2 and the minimum rate is Rmin = 1/10; thus we extend the fixed rate coding schemes investigated above to the rateless case, i.e., the LT code, the Raptor code and a rate-compatible punctured version of the LDPC code. Since the LDPC code is a block code, the minimum rate is equal to the target rate R = 1/2, and higher rates are obtained through puncturing. We are in particular interested in the throughput performance in terms of the rate in bits per modulation symbol. It is observed from EXIT charts that for rates above 1.6 bits per symbol, the MI does not converge to 1, and thus the BER will not converge to zero. Therefore, starting from a rate of 1.6 bits per symbol, we determine the convergence threshold
Fig. 8: Rate in bits/symbol of the LT code, Raptor, and ratecompatible LDPC code with DQPSK and a fixed rate R = 1/2.
in terms of SNR using the EXIT chart framework. SNR is defined as SNR = 10 log10 ( σ12 ), where σ is the noise standard deviation. The rate is then subsequently decreased to track the rate as a function of SNR. The throughput performance is illustrated in Fig. 8, where we observe that the LT coded scheme enjoys a higher rate than both the Raptor coded and the LDPC coded alternatives. We especially notice the relatively high gain over the rate-compatible punctured LDPC code; however, the puncturing strategy has not been optimized, and thus some improvements can be expected. The advantage over the Raptor coded scheme is not as significant; however since the LT coded scheme is less complex, it is still a compelling advantage. The inherent drawback of a higher error floor is still a problem that needs further investigation for the LT coded case.
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to a comparable Raptor code and a rate-compatible LDPC code. A drawback of the scheme is the relatively high error floor. The use of concatenated LT codes are currently under investigation to eliminate the error floor while keeping the overall complexity low.
0
10
Nonsystematic LT Systematic LT Raptor −1
10
VIII. ACKNOWLEDGEMENTS
BER
−2
10
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/20072013) / ERC grant agreement n 228044.The work has further been supported in part by the Swedish Research Council under VR grants 621-2008-4249 and 621-2009-4666.
−3
10
−4
10
0
0.5
1
1.5
2 2.5 Eb/N0 in dB
3
3.5
R EFERENCES
4
Fig. 9: BER performance of systematic and nonsystematic LT code with DQPSK for a fixed-rate R = 1/2.
VI. S IMULATION R ESULTS To demonstrate the convergence threshold and error floor for the proposed LT and Raptor coded DQPSK system, we provide some numerical results for the bit error rate (BER) as shown in Fig. 9. We consider both a systematic and nonsystematic LT coded DQPSK scheme for a fixed-rate R = 1/2 over an AWGN channel. The block length is N = 2000 QPSK symbols, corresponding to 2000 information bits. The number of local iterations of the LT decoder was limited to 50, while the number of global iterations was set to 30. We observe that the convergence thresholds are in line with predictions based on the EXIT chart analysis. We also observe an error floor at around 10−4 and 4 · 10−4 , respectively, for the systematic and nonsystematic LT codes. The improved error floor for the systematic code is due to a better average variable node degree as compared to its nonsystematic counterpart. Fig. 9 clearly indicate the better performance of Raptor code in the error floor region. VII. C ONCLUSIONS We have investigated a serial concatenation of an LT code and a DQPSK modulator for transmission over an AWGN channel. With the use of EXIT curve-fitting, and the concept of an expected operational rate of the rateless code, we have proposed a suitable design approach for the degree distribution optimization of the concatenated LT DQPSK scheme. As an example, we considered an operational rate of R = 1/2, for which the obtained degree distribution provided improvements in convergence threshold of 1.5 dB over the degree distribution derived for the AWGN channel. In addition, the proposed distribution leads to comparable convergence performance as for fixed-rate LDPC codes and Raptor codes optimized for DQPSK. Using the EXIT chart analysis strategy, we determined the expected rate of the LT coded scheme when operated as a rateless code as a function of the SNR. Improvements in term of throughput were observed as compared
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