ISMICT 2014 1569882609
Iterative Demodulation and Decoding of LDPC-Coded M-ary DCSK Modulation over AWGN Channel Yibo Lyu, Guofa Cai, Lin Wang, Senior Member, IEEE Dept. of Communication Engineering, Xiamen University, Fujian 361005, China Email:
[email protected]
(LLRs) of DCSK demodulator will be derived, where posteriori and extrinsic LLRs are the prior information for LDPC codes. Similarly, those posteriori LLRs produced by LDPC decoder are prior information for DCSK detector. The theoretical derivations and simulation results reveal that the performance of our proposed iterative receiver outperform that of the noniterative ones over AWGN channel due to the mutual information exchanged between demodulator and decoder. The Extrinsic Information Transfer chart also demonstrated this result. The organization of this paper is follows. At first, we will introduce the proposed receiver system model in section II. In section III, we will derive the expression of LLRs that as a prior information for LDPC decoder. There are some simulation results and discussions expressed in section IV. Then we will have a conclusion at the end of this paper.
Abstract—In this article, an iterative demodulation and decoding receiver for M-ary differential chaos shift keying (DSCK) modulation is proposed over Additive White Gaussian Noise (AWGN) channel. The Log-Likelihood Ratios (LLRs) of coherent DCSK demodulator and decoder are derived assuming that its perfect carrier synchronization happened. The simulation results show that the performance of our proposed iterative receiver outperform that of the noniterative receiver over AWGN channel due to extrinsic information exchanged between demodulator and decoder. Furthermore, Extrinsic Information Transfer chart is used to investigate the convergence behavior of our proposed receiver. Index Terms—DCSK; coherent demodulation; LDPC codes; iterative demodulation and decoding; Extrinsic information transfer (EXIT) chart
I. I NTRODUCTION Due to the inherent properties of chaotic signals[1-4], such as low correlations between different signals over sufficient long time interval and the wideband characteristic, those signals are suitable for spread-spectrum modulation. Therefore chaotic modulation techniques, as other spread-spectrum modulation technologies, have the advantages of resisting channel fading, strong anti-inference ability and high communication security. Many chaotic modulation schemes have been studied and proposed in past decades. The differential chaos shift keying (DCSK) modulation scheme[5] , as a kind of chaotic modulation techniques, has presented an excellent performance for multipath fading and time-varying channel. Along with the receiver’s low hardware complexity, this technique can be used in some ultrawideband (UWB) systems, such as wireless personal area networks (WPANs) and wireless local area networks (WLANs)[611]. In order to increase the data rate of DCSK, a general version of M-ary DCSK system in [12][13]is proposed, where Walsh code is used to orthogonalize the transmitted signals. For DCSK demodulation, the coherent receiver is discussed and analyzed in [14][15], which is extended to M-ary DCSK systems in [16][17]. Obviously, we can enhance the performance of 2-ary DCSK for Additive White Gaussian Noise (AWGN) channel by introducing channel coding technology[19][20]. In [19], we can conclude that LDPC codes[18] can dramatically improve the performance of 2-ary DCSK system. However, there are seldom work to discuss the iterative demodulation and decoding scheme[21-23] design for LDPC coded M-ary DCSK modulation for AWGN channel. In this paper, we will propose a joint demodulation and decoding DCSK receiver. And then the Log-Likelihood Ratios
II. S YSTEM M ODEL AND N OTIFICATIONS A LDPC codes encoded coherent DCSK modulation scheme is considered. The data to be transmitted are sent to a binary LDPC encoder at the very beginning. Therefore we can get a LDPC codeword C = [c0 , c1 , · · · , cn−1 ] with length n. Then this LDPC codeword is converted to many M-ary symbols which can be defined as ∑M −1 √ Eswm,k p(t − kTc ) Sm (t) = k=0 ∑log ∑ 2 M −1 (1) m = ci 2i = i∈D 2i i=0 D = {k|ck = 1, 0 ≤ k ≤ log2 M − 1, k ∈ Z} whereSm (t), (m = 0, 1, · · · , M − 1) denotes the M-ary symbol, Es is the symbol energy∫ and p(t) presents a chaotic signal which T has the property that 0 c p2 (t)dt = 1/M ,where T = M Tc and Tc present the symbol duration period and chip time respectively. Simultaneously, wm = [wm,0 , wm,1 , · · · , wm,M ]and wj = [wj,0 , wj,1 , · · · , wj,M ] denote two arbitrary chosen Walsh functions that fulfill the equation (2). { M −1 ∑ M, if m = j wm,k wj,k = (2) 0, if m ̸= j k=0
The DCSK modulator structure is shown in fig.1. Assuming those signals are transmitted through AWGN channel, the received signals can be expressed as: y(t) = Sm (t) + n(t);
(3)
where n(t) denotes Gaussian noise. In fig.2, the structure of coherent DCSK demodulator is depicted. For coherent detection,
1
Es p (t )
TC
TC
1) symbol is transmitted. P (m) is a prior probability of the symbol m. At the initial step, we assume that those received symbols are equiprobable, i.e., P (m) = 1/M . p(Z) is the PDF of the received signal vector Z, which is same regardless of the value of m because of p(Z) = ∑M −1 m=0 P (m)p(Z|m) . p(m|Z) is the probability that the mth symbol transmitted given that Z = [z0 , z1 , , zM −1 ]. This probability is used to compute Lz , which is given by ] [ p(ci = 0|Z) (5) Lz (i) = log p(ci = 1|Z)
wm ,1 wm ,2
sm (t )
Walsh function generator
m
wm , M
Fig. 1.
Structure of modulator
y (t ) TC
TC
p (t ) w1,M w1, M
^
m !1
T
$ ".# dt
1
where Lz is a posteriori LLRs which can be used for computing a prior information, i.e., Lv−AP P = Lz − Lz−AP P . P (ci = 0|Z) is the conditional probability of code word ci equal to 0, given the received signal vector Z.
z1
T Tc
Walsh function denerator
w1,1
Es p (t )
III. I TERATIVE D EMODULATION AND D ECODING P ROCESS In this section we will discuss how to derive the LLRs of each bit of the log2 M code bits associated with each received symbol. Assuming that the mth modulated symbol is transmitted through AWGN channel, simultaneously considering that the components of received symbol vector Z are independent, the components z0 , z1 , , zM −1 of the received vector Z are conditionally independent Gaussian random variables, identically distributed except for zm [16,17]: √ zm ∼ N ( Es, σ 2 )(Signal present) (6)
wM , M wM , M
^
m!M
T
zM
$ ".# dt
1
T Tc
Walsh function denerator
wM ,1
Fig. 2.
Channel Signal S(t)
Coherent DCSK demodulator
Structure of coherent demodulator
Soft Decision Result Z =[z0, z1 , ,zM-1]
Lz Symbol probabilities and LLR computation
+
Lv-APP LDPC Decoder
Lz-APP
Fig. 3.
-+
Decoding result
Lv
zi ∼ N (0, σ 2 )i ̸= m(Signal absent)
where N (.) denotes Gaussian distribution and σ is the variance of random Gaussian noise. Therefore the conditional PDF p(Z|m) is given by:
The model for iterative receiver.
we assume that the demodulator can obtain perfect carrier synchronization. It is known that M-ary DCSK modulation signals are equivalent to the set of orthogonal signals. The received signals thus can be expressed by[16] Z
= [z0 , z1 , · · · , z√ m , · · · , zM −1 ] = [n0 , n1 , · · · , Es + nm , · · · , nM −1 ]
(7)
2
p(Z|m)
=
(2πσ[2 )− 2 ×√ ( ]∏ ) 2 zi2 M −1 exp − exp − (zm −2σ2Es) i=0,i̸=m 2σ 2 M
(8) Due to Bayes’ rule, the conditional PDF p(m|Z) can be written by p(Z|m)P (m) p(m|Z) = (9) p(Z) Computation of LLRs Lz : According to the bits-to-symbols mapping relationship described in (1), the probability P (ci = 0|Z) can be computed from (9), which is given by ∑ p(ci = 0|Z) = p(m|Z) (10)
(4)
where n0 , n1 , , nM −1 denote statistically independent Gaussian random variable with zero mean and variance σ 2 = N o/2. The receiver structure we consider in this paper is presented in fig.3. Before LDPC decoding procedure, we use the soft decision variable Z = [z0 , z1 , · · · , zM −1 ] to compute the corresponding bit probabilities and its LLRs as the prior information for LDPC decoder. Then LDPC decoder utilizes those LLRs to produce new extrinsic information Lv that are fed back to detector as a prior information. The whole procedure is depicted in fig.3. Contrast to other iterative receiver, our proposed system does not need interleaver and deinterleaver, due to LDPC codes that we used here is a kind of random codes. Therefore, this design reduces the implementation complexity. There are some notifications used in our discussions: p(Z|m) is the conditional Probability Density Function (PDF) of the received signal vector Z, given that mth, (m = 0, 1, , M −
m:ci (m)=0
where function ci (m) returns the ith bit associated with the mth modulated symbol. Then at the lth(l = 1, 2, · · · ) iteration, the LLRs for the ith codeword ci can be derived as below: ] [ p(ci = 0|Z) l Lz (i) = log p(ci = 1|Z) [∑ ] (11) m:ci (m)=0 p(m|Z) = log ∑ m:ci (m)=1 p(m|Z)
2
Inserting (8), (9) and (10) into (11) and ignoring the common terms p(Z), the expression (11) can be derive by [∑ ] m:ci (m)=0 p(Z|m)P (m) l Lz (i) = log ∑ (12) m:ci (m)=1 p(Z|m)P (m)
0
10
−1
10
−2
Bit error probility
10
where P (m) = 1/M at initial step, then it will update in each iteration. Since the value of P (m) iteratively updated, it can converge toward real probability of the symbol m from initial value. Therefore, our proposed iterative demodulation and decoding scheme can improve the error correction performance when M > 2. When M = 2, i.e., binary DCSK, P (m = 1) = P (ci = 1) and P (m = 0) = P (ci = 0). So the equation (11) can be simplified in following form: [ ] [ ] p(Z|0) p(0) + log Llz (i) = log p(Z|1) p(1) [ ] (13) p(Z|0) l−1 = log + Ll−1 (i) − L (i) v z p(Z|1) l−1 Llz−AP P (i) = Llz (i) − Ll−1 v (i) + Lz (i) [ ] √ p(Z|0) Es(z0 − z1 ) = log = p(Z|1) σ2
−3
10
BER of Uncode PEG irregular 1008 Joint PEG irregular 1008 Separate PEG regular 1008 Joint PEG regular 1008 Separate PEG irregular 504 Joint PEG irregular 504 Separate PEG regular 504 Joint PEG regular 504 Separate
−4
10
−5
10
−6
10
−7
10
2
Fig. 4.
3
4
5
6 Es/No
7
8
9
10
The simulation results for 4-ary DCSK modulation, R = 0.5.
0
10
(14)
−1
10
−2
10 Bit error probility
Obviously, the value of equation (14) only reflects channel condition. In another word, it is irrelative with iterative operation. So we can infer that when M = 2, our proposed system yield no improvement. However, as expressed above, our proposed design can bring iteration gain when M > 2. Simulation results demonstrate this conclusion in following section.
−3
10
BER of Uncode PEG irregular 1008 Joint PEG irregular 1008 Separate PEG regular 1008 Joint PEG regular 1008 Separate PEG irregular 504 Joint PEG irregular 504 Separate PEG regular 504 Joint PEG regular 504 Separate
−4
10
−5
10
−6
IV. S IMULATION R ESULTS AND EXIT C HART A NALYSIS
10
To illustrate the effectiveness of the proposed iterative demodulation and decoding scheme for M-ary DCSK, we conducted an extensive set of simulations. The modulation scheme is M = 4 and M = 16. Both regular and irregular LDPC codes with short frame length[24] (n = 504 and n = 1008) for this system over AWGN channel are chosen. The code rate R = 1/2. The maximum iteration number between demodulation and decoder is equal to 5 and the maximum local iteration number for LDPC decoder is equal to 50. Simulation results depicted in fig.4 and fig.5 show that our proposed scheme enhances the performance of M-ary DCSK communication system. In fig.4 the Bit Error Rate (BER) versus Es/N o performance for 4-ary coherent DCSK modulation is depicted. From these results we can find that iteration system yield error correction improvement. Here we use solid lines present decoding results of regular LDPC codes and dash lines denote that of irregular ones. When frame length equal to 504, we can observe that proposed scheme has obtained about 0.3 dB iterative gain for irregular LDPC code at BER = 10−6 and more than 1.2 dB gain for regular one. While for n = 1008, we can find that there are a gap of 1.3 dB between iterative and noniterative receiver for regular LDPC code at BER = 10−6 . Also for irregular LDPC code, our system can acquire 0.6 dB gain at BER = 10−6 .
10
−7
0
Fig. 5.
2
4
6
8
10 Es/No
12
14
16
18
The simulation results for 16-ary DCSK modulation, R = 0.5.
A similar trend of iterative gains can be found when M = 16, which results are depicted in fig 5. When M = 16, we can get more throughput with the drawback of more power consumption. According the simulation results depicted on fig.4 and fig.5, we find that our scheme improve the performance of regular LDPC codes more than the one of irregular LDPC codes. This phenomenon is because those LDPC codes which are tested here are with short frame length. As the frame length increasing, we infer that irregular LDPC codes will perform better than regular LDPC codes in this system. By means of Monte Carlo simulation approach, we can find out that those LLRs exchanged between decoder and demodulator (i.e., Lv and Lz ) can be modeled by Gaussian distribution[25][26]. Therefore, we can compute that mutual information
3
V. C ONCLUSION
1
In this work, we have proposed an iterative demodulation and decoding receiver for M-ary DSCK modulation. The expressions of LLRs have been derived for coherent DCSK demodulator over AWGN channel. By means of the iterative receiver scheme, the LDPC coded M-ary DCSK modulation system can obtain better BER performance. Through EXIT chart, we also can find that our proposed scheme has lower convergence threshold. Therefore, we can reduce the power consumption of the DCSK communication system by using iterative receiver. Although the iterative receiver structure is more complicated and will cause longer processing delay, however, for power-limited applications, our proposed scheme will still be an excellent candidate.
0.9 0.8 0.7
IE/IZ
0.6 0.5 0.4 Es/N0 = 4dB Iterative Es/N0 = 4dB Noniterative Es/N0 = 6dB Iterative Es/N0 = 6dB Noniterative Es/N0 = 8dB Iterative Es/N0 = 8dB Noniterative Outer LDPC Decoder
0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 IA/IV
0.6
0.7
0.8
0.9
1
ACKNOWLEDGMENT Authors thank the support from National Science Foundation of China (No. 61271241) and National Science Foundation of China (No. 61001073).
Fig. 6. EXIT char of our proposed receiver. The modulation schem is 4-ary DCSK modulation. The LDPC code test is regular LDPC code with length = 1008, code rate = 0.5.
R EFERENCES by using follow equation:
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Iz\v = 1− ( ) ∫ ∞ (ξ − σ 2 /2)2 1 √ exp − log2 (1 + exp(−ξ))dξ 2σ 2 2πσ 2 −∞ = J(σ) (15) where σ 2 /2 represents the corresponding mean of the LLRs (i.e., Lv and Lz ) values that obtained by Monte Carlo simulations. So we can investigate the convergence behavior of this receive system with Extrinsic Information Transfer (EXIT) chart. The simulation results for M = 4 are shown on fig.6. Here we use solid line with circle presents mutual information Iv send from outer LDPC decoder, and other solid lines denote Iz which is transmitted from soft detector in iterative system. By means of iterative operation, the value of Iz and Iv are increased at the end of each iteration. However, for noniterative receiver, Iz is constant, so it is presented by horizontal dash lines in fig.6. From fig.6 we can infer that an increasing in Es/N o relates to a vertical shift (raise) of demodulator transfer characteristic. We define the region between the two curves (i.e., Iz and Iv ), as the decoding tunnel [26]. If the curve of Iz above the one of Iv , we call that the tunnel is open, otherwise, is close. When the tunnel is open, this phenomenon means that the whole system convergence toward low BER. Obviously, when Es/N o = 8 dB, the decoding tunnel for proposed system is open, as well as for noniterative system, on the contrary, when Es/N o = 4 dB, neither of them are convergent. We also can infer from fig.6 that when Es/N o equal to 6 dB, for proposed system, the tunnel is open, however, for noniterative receiver, the tunnel is closed. The EXIT chart demonstrated that the convergence threshold of iterative system is lower than that of noniterative one, which suggests that our system can start working at lower energy consumption circumstance.
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