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3440 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 2007 On Analysis and Design of Low Density Generator Matrix Codes for Continuous Phase Modulation Ming Xiao, Student Member, IEEE, and Tor M. Aulin, Fellow, IEEE Abstract— We investigate the analysis and design of low density generator matrix (LDGM) codes for continuous phase modulation (CPM). The system uses LDGM codes as an outer code for CPM. For additive white Gaussian noise channels, we derive the union bound to analyze the error ﬂoor performance. Design principles for lowering error ﬂoors are suggested from this analysis. We propose a design approach of jointly considering the LDGM code degree and the CPM modulation index. Then we consider the rate-adaptive system for slowly fading channels. By changing the rate of the LDGM codes, the information rate of the CPM signals is adapted according to channel variations. We use a low-rate LDGM code as the mother code. Higher rates are achieved by puncturing the output of these codes. To exploit the rate-ﬂexible property of punctured LDGM codes, a rate function is proposed to calculate the rate of each transmitted block. Thus, we can have a quasi-continuous information rate. Numerical results show that this approach can improve the energy efﬁciency from a discrete-rate adaptation. Using the rateadaptive approach, up to 11 dB transmitted energy gain can be achieved from the non-adaptive scheme in the low bit-error-rate region (smaller than 10−3 ) for minimum shift keying (MSK). Index Terms— Continuous phase modulation, low density generator matrix codes, rate-adaptive, slowly fading, union bound. I. I NTRODUCTION ONTINUOUS phase modulation (CPM) is an excellent digital modulation scheme which has attracted lots of research [1]. As a constant-envelope modulation scheme, inexpensive nonlinear ampliﬁers can be used for CPM. Due to the built-in memory, CPM can be regarded as a convolutional code over an integer ring [2]. Thus, it is an energy efﬁcient modulator. By avoiding sudden changes in the phase, CPM has a compact power spectrum [1]. Thus, it is widely used in wireless communications. In order to obtain a further improvement in energy efﬁciency, CPM is often combined with an external code. With the same principle as a serially concatenated code (SCC) [3], serially concatenated CPM (SCCPM) with iterative decoding was proposed and showed a signiﬁcant improvement in bit error rate (BER) performance [5]. In this scheme, the built-in coding property of CPM is exploited. CPM is used as a recursive inner code of the concatenated system, and normally a convolutional code (CC) is used as the outer code. They are C Manuscript received February 28, 2006; revised July 3, 2006; accepted August 21, 2006. The associate editor coordinating the review of this paper and approving it for publication was T. Duman. This work was supported by the Swedish VR, under project number 2004-4582. Part of this work was presented at the IEEE Information Theory Workshop (ITW05), Rotorua, New Zealand, Aug. 2005, and IEEE Globecom 2005, St. Louis, USA, Nov. 2005. The authors are with the Telecommunication Theory Group, Department of Computer Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden (e-mail: mxiao@ce.chalmers.se; aulin@ce.chalmers.se). Digital Object Identiﬁer 10.1109/TWC.2007.06020051. separated by a pseudo-random interleaver. In [6], the lowdensity parity-check (LDPC) codes are proposed to encode minimum shift keying (MSK). A further gain is achieved in the convergence threshold. On the other hand, as a special class of low density parity check (LDPC) codes [4] or as a generalization of concatenated single parity check (SPC) codes [7], low-density generator matrix (LDGM) codes have attracted much research interest [8], [9], [10]. It shows the near channel capacity property with simple component codes. One of the main advantages of LDGM codes over standard LDPC codes is that the encoding complexity is linear in code length [8]. Motivated by the merits of both CPM and LDGM codes, LDGM codes are proposed to be combined with CPM [14], [15]. LDGM codes are used as the outer code for CPM. Essentially, irregular repeat CPM of [15] (CPM with the LDGM systems in [14]) can be regarded as a special SCCPM scheme with LDGM codes as the outer code. Since CPM has encoding complexity linear in code length [1], the proposed system has linear encoding complexity. Compared to SCCPM with CC, substantial transmitted energy gains are achieved using degree-optimized LDGM codes. Compared to the results of [6], the systems have comparable convergence thresholds for MSK. The degree optimization approaches of the LDGM codes for CPM are discussed in [14], [15] for early convergence thresholds. Yet in [14], [15], the considered systems are limited to inﬁnite block length and for additive white Gaussian noise (AWGN) channels. In practical communication systems, the codeword length is ﬁnite due to processing delay or the complexity limitations etc. Thus, we will consider analyzing and designing LDGM codes for CPM with a ﬁnite length. The analysis approach is the union bound. Meanwhile, CPM is mainly used for wireless communications, where the channels are often fading. Then it is valuable to investigate the approach to combat fading. Here we will discuss an adaptive code scheme for the system. The paper is organized as follows. First, we give a system description of CPM with LDGM codes. Then we use the union bound to analyze the system error ﬂoor for AWGN channels. Then, design considerations based on the bound analysis are suggested. After that, we discuss CPM with adaptive LDGM codes for slowly fading channels. We change the rate of the LDGM codes by puncturing according to the channel state information (CSI). Two adaptation policies are used to adapt the rate of the LDGM codes depending upon channel conditions. Finally, discussion and conclusions are given. 1536-1276/07$25.00 c 2007 IEEE XIAO AND AULIN: ON ANALYSIS AND DESIGN OF LOW DENSITY GENERATOR MATRIX CODES FOR CONTINUOUS PHASE MODULATION 3441 II. S YSTEM D ESCRIPTION OF CPM WITH AN LDGM S YSTEM The schematic of the LDGM system with CPM is shown in Fig. 1. On the transmitter side, the outer LDGM code consists of repetition encoders with different degrees (the number of input edges) as the variable node (VN), a random edge interleaver and single parity check (SPC) encoders as the check node (CN). The SPC encoder only outputs the parity bits. The output codewords from the SPC codes are serially input to CPM. If the alphabet size of CPM is larger than 2, a mapper is used to convert bits into symbols. The interleaver between the CN and the VN is produced randomly. The positions of the CNs are randomly set. Due to the built-in encoding property of CPM, the system performs well in the low bit error rate (BER) region, though an LDGM code alone is not asymptotically good [8]. We note that there is some similarity between the structures of the proposed scheme and the irregular concatenated Zigzag codes [16] if we see the differential phase modulation process of CPM [1] as a Zigzag encoding process. The LDGM code used here can be systematic or nonsystematic. If we see CPM as a recursive CC [1], all output bits from the LDGM codes (including systematic bits) are encoded by CPM before transmission. Thus, the meaning of systematic codes here is slightly different from the original deﬁnition [8], [9], where systematic bits denote uncoded information bits. To trigger the decoding process, part of the CNs of nonsystematic codes must have degree-1 (explained below), and each of them connects to one VN. This approach is called doping [17]. Hence, we can regard the systematic codes as a special kind of a nonsystematic code with all VNs being doped. We do not have to separately consider the nonsystematic or systematic codes in the analysis process. The transmitted CPM signals are distorted by both fading and additive noise. Then the received continuous-time signals are given by (1) y(t) = A(t)s(t, U ) + n(t) where n(t) is a complex baseband representation of AWGN, having a double-sided power spectral density No /2. A(t) is the complex baseband channel gain due to the multipath fading process. For AWGN channels, A(t) = 1. s(t, U ) is the the complex baseband representation of CPM signals [1]. It can be described as s(t, U ) = 2Es /Ts exp{j(ψ(t, U ))}, t ≥ 0 (2) function (CPDF) of the CPM signal. Here the CPDF is often called the transfer probability or the channel observation in the literature. To reduce complexity, the principle component analysis (PCA) receiver [11] is used. With the CPDF, an iterative detection approach is used: the BCJR [12] processor is used for CPM and the sum-product algorithm [13] for the repetition and SPC code. The extrinsic information is ﬁrst passed from the BCJR processor to the CN for decoding, then to the VN after edge deinterleaving. The extrinsic information of the VN is sent back to the CN after interleaving, ﬁnally back to the BCJR processor. This concludes one iteration of the decoding process. In the decoding process, we use the log-likelihood ratio value (Lvalue) of the channel observation. It can be calculated as L = log2 P (y|x=0) where x and y denote the transmitted bit P (y|x=1) and its observation at the receiver, respectively. The L-values for the VN and CN [13] are, respectively, dv,i LV i,out = j=1,j=i LV j,in (3) and LC = log2 i,out 1− 1+ L dc,i 1−2 j,in j=1,j=i LC 1+2 j,in L dc,i 1−2 j,in j=1,j=i LC 1+2 j,in C C (4) V where LV i,out and Lj,in are the output and input values for the VN, respectively. LC and LC are the output and input Li,out j,in values for the CN, respectively. The subscript i is the mark of the edge of the VN or the CN. During the decoding process, we initialize the L-values for all edges to zero before decoding (assuming the bit values to have equal probabilities for being 0 and 1 ). dc,i and dv,i are the degrees (the number of edges in the interleaver size) of the CN and VN [8], respectively. Hence, the edges connecting to the CNs that have degrees larger than 1 cannot be decoded during the ﬁrst iteration by (4). Decoding has to rely on the doped CNs (with degree-1) until nonzero L-values propagate to other CNs from the VNs connecting to the degree-1 CNs. This is the reason why doping is necessary. III. E RROR F LOOR A NALYSIS FOR AWGN C HANNELS To measure the performance, different criteria are used for different systems. For an inﬁnite block length (in practice very large) system, the convergence threshold is of most concern. The degree optimization approach for the low threshold is discussed in [14] for the proposed system. Yet for most practical systems, interleaver sizes are ﬁnite. Thus, it is essential to know the system error ﬂoor. The error ﬂoor is the part of the curve in the BER-to-SNR (signal-to-noise ratio) plot, which is ﬂat at low BER (say lower than 10−4 ). Error ﬂoors can be evaluated by simulations or analysis bounds. The latter is more reliable and can reach a low BER region which is hard to use for simulations. As in [3], [5], the traditional analysis bound is the union bound that works well in error ﬂoor regions. Hence, we use it below. where ψ(t, U ) is the information-carrying phase. Let t = t + nTs (n = 1, 2, · · · , ∞), then in the nth time slot, ψ(t, U ) = n−L L−1 ψ(t + nTs , U ) = 2πh i=0 Ui + 4πh i=P Un−i f (t + iTs ) mod .2π, 0 ≤ t ≤ Ts [1], [2]. Es , Ts and f (t) denote symbol energy, symbol time and phase response, respectively. The parameter h = K/P is referred to as the modulation index. U = {U1 , U2 , · · · , Ui , · · · , } is an M -ary symbol sequence with Ui ∈ {0, 1, · · · , M − 1}. If h = 1/2, M = 2 and the phase response type is rectangular, the CPM type is MSK [1]. At the receiver side, a CPM receiver (a ﬁlter bank [5]) is used to produce a time-discrete conditional probability density 3442 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 2007 Info. bits X E⊕ B E z j dv,1 . . π . E⊕ B E j dv,k LDGM encoder E dc,1 . . . w U CPM E j B dc,n B ij 'q E % E ' B A I⊕ % . . j c c E⊗E⊕ E BCJR © . . π for CPM s . . B ij ' % E 'q E channel I A B⊕ % j A(t) n(t) Sum-product Decoder Output E E Fig. 1. Schematic of the LDGM system for CPM. A(t) = 1 for AWGN channels. w1 c w2 c w3 c w4 c E VN E π1 E CN E π2 E CPM E channel LDGM codes Fig. 2. An equivalent model of the system. c c c c c c c degree-3 code degree-2 code c c c Fig. 3. An example for calculating the IOWE of repetition codes with different degrees. A. Union Bound From Fig. 1, we can see that the system performance will not change if we assume an interleaver (π2 ) between the LDGM code and CPM due to the random positions of the CNs. Thus, an equivalent structure is shown in Fig. 2. If we assume N CNs, the interleaver π2 has size N . This interleaver is actually not used. We introduce it here only to facilitate the analysis. From Fig. 2, we can see that the system is identical to SCCPM with an LDGM code as the outer code. Hence, the union bound analysis approach for SCCPM can be directly used here if we know the input-output weight enumerator (IOWE) of the LDGM code [5], [14]. From its deﬁnition, we know that the LDGM code is a serial concatenation of repetition codes and SPC codes (only parity bit output). Thus, we can evaluate its IOWE following the SCC [3]. Clearly, when the repetition code is regular, i.e., repeats only m times, its IOWE (denoted as function ARm (·): Rm stands for repeat m times.) is k ratio of the ith code. Thus, assuming that the input weight of the ith component code is wi , the total input weight will be w = i wi . The same holds for its output weight: h = i hi . For every component code, (5) can be used to evaluate the IOWE due to its regularity. Since the input and output of each component code do not affect each other, the overall IOWE (function AR (W, H)) is the product of all component code IOWEs, i.e., AR (W, H) = i ARi (W, H) (6) where ARi (W, H) = ARm (W, H) = w=0 Aw,h W w H h (5) where the capital letters W and H are the dummy variables for the input and output weight of the repetition code, respectively. k is the information bit length (the length of the information bits in one codeword). w is the input weight (the error bits of a k codeword), h = mw is the output weight, and Aw,h = w denotes the number of codewords with input/output weight w/h. A repetition code with different degrees is just a parallel combination of several regular repetition codes (component codes) as is shown in Fig. 3. We again use k to denote the number of information bits. The number of information bits for the ith component code is ki = k · Ri , where Ri is the kRi W wi H hi is the ith wi component codeword IOWE with the input/output weight wi /hi . From Fig. 3, as an example, k2 = 2 for the degree2 repetition codes. From (5), the IOWE is A2 (W, H) = 2 2 2 W 2 H 4 = 1 + 2W H 2 + + W H2 + 2 0 1 W 2 H 4 = (1 + W H 2 )2 . For the degree-3 code, k3 = 2 (2 nodes). In the same way as the degree-2 code, the IOWE is A3 (W, H) = 1 + 2W H 3 + W 2 H 6 = (1 + W H 3 )2 . From (6), the overall IOWE in the ﬁgure is AR (W, H) = A2 (W, H)A3 (W, H) = (1 + W H 3 )2 (1 + W H 2 )2 . Its coefﬁcient for W 2 H 5 is 4, i.e., AR (2, 5) = 4. The 4 possible input ways (weight 2) to produce the output (weight 5) are ({w1 w2 w3 w4 }) {1001}, {1010}, {0101} and {0110}. For the CNs, the input and output of each node (code) also do not affect each other. The overall IOWE is the product of the individual CNs. If we denote the overall IOWE of the CNs as the function AC (H, L), it can be evaluated as: wi N AC (H, L) = i=1 ACi (H, L). (7) XIAO AND AULIN: ON ANALYSIS AND DESIGN OF LOW DENSITY GENERATOR MATRIX CODES FOR CONTINUOUS PHASE MODULATION 3443 where N is the number of check nodes and ACi (H, L) is the IOWE of the individual CN that can be evaluated as [20]: di ACi (H, L) = hi =0 di hi · H hi · L 1+(−1)1+hi 2 . (8) Now hi , li and di are input weight, output weight and degree of the ith CN, respectively. L is the dummy variable for the output weight of the CN. The exponent of L is 0 or 1, depending on whether the input weight hi is odd or even. For example, when the CN degree is 3, ACi (H, L) = 1 + 3HL + 3H 2 + H 3 L. Since the interleaver between the VNs and CNs is random, the concept of a uniform interleaver can be used [3]. Before proceeding, we need to consider the effect of doping. From the system description, we know that each doped CN connects one VN with one edge. We call these edges the doped edges and these VNs the doped VNs. Since each doped VN only connects to one doped CN, the doped edges cannot be considered as purely random as the other edges. Except for these, all other edges are randomly permuted. We need to separate the undoped edges (connected to the CNs with a degree larger than one) and doped edges while using the uniform interleaver [3] to evaluate the IOWE of the LDGM codes. For convenience, we call the CNs and VNs connected by the doped edges as the doped part of the LDGM codes. The CNs and VNs connected by the undoped edges are called as the undoped part. The doped part connects to the undoped part through the common VNs. When we enumerate the number of cases for the undoped part (from (6)), the results include that of the doped part. The output weight of LDGM codes (l) is the weight from doped CNs (ld ) plus the weight from undoped CNs (lu ), i.e., l = ld + lu . (9) and the same to AC (h, L). After being permuted by π2 , the l error bits produce nin error events in CPM. The probability of N the permutation is 1/ by the uniform interleaver π2 . l We assume that the normalized squared Euclidean distance (NSED) of the error CPM signals from the correct signals o to be d2 . If the outer code rate is Rc , the information bit in o in length is k = N · Rc . We use Al,d2 ,nin to denote the number in of CPM error events with input weight l, NSED d2 and in being concatenated 1 by nin error events. The number of R error information bits is w = i wi . Then, A (w, h) = o (N · Rc )Ri . Thus, the BER of the speciﬁc error i wi event e with parameters w, h, l, nin and din is [3], [5] Pb (e) = M · Q where w M= o N · Rc N nin N l i o (N · Rc )Ri wi T h o d2 Rc SNR in (12) · AC (h, lu ) × Ain 2 l,d in in ,n . (13) Here the Q(.)-function is the complementary Gaussian cumulative distribution function [18]. SNR = Eb /N0 is the signalto-noise ratio (Eb is the transmitted energy of one information bit). By enumerating all possible error events with all possible values of w (and all possible w1 , w2 , · · · with w = i wi ), h, l, nin and din , the BER bound is evaluated as: Pb ≤ w l nin d2 in h For the doped part, it is trivial to show that the input weight equals to the output weight (since there is no code). Then, ld can be evaluated by counting the error weight of the doped VNs. lu is enumerated by the exponent of L of the undoped part by (7). For the undoped part, the size of the uniform interleaver between the VN and CN is T =J −D (10) M ·Q o d2 Rc SNR in (14) where J denotes the edge interleaver size (the number of the edges of VNs), and D is the number of doped CNs. Clearly, the degrees of the doped VNs are reduced by one when removing their doped edges. Then, we can use (6) to calculate the IOWE of VNs. For the undoped CNs, we can form the IOWE of the LDGM codes from the IOWE of the VNs and CNs. The IOWE of the undoped part of LDGM codes can be evaluated as: A(H, L) = h AR (W, h) · AC (h, L) T h (11) where h equals to the output weight of the repetition code minus ld . The exponent of L is lu . Note that with similar notation to [3], AR (W, h) denotes the IOWE of the repetition code with the exponent of H speciﬁc to h (a ﬁxed value), In general, this inequality is very complex to evaluate. Especially, a closed form expression for the CN IOWE is very difﬁcult to ﬁnd. Thus, we turn to the truncated one, i.e., we only consider dominating terms. The validity of the truncated bounds has been widely veriﬁed [3], [5]. Two truncation principles are used for our systems: 1. For a serially concatenated system, the terms with minimal interleaver gain (IG) dominate the union bound [3], [5]. The IG is deﬁned as the exponent by which the BER in the error ﬂoor region (the medium to high SNR region) is decreased with the interleaver length. We only keep the terms with the minimal IG since our system is serially concatenated. These terms can be easily found by checking the exponent of the interleaver size (N and T ). In the following, we use αmax to denote the IG. 2. We only consider the terms with relatively small distance for CPM, i.e. terms with small d2 . in Now we use an example for illustration. A system (called System I for convenience) is deﬁned as follows. The CPM type is MSK. The LDGM code has rate 0.25. Thus, the overall code rate is 0.25 information bits per symbol. The 1 Here concatenation means one error event followed by another immediately. 3444 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 2007 degree proﬁle (node degrees and their ratios [13], [15]) is: the VN {20(20%), 19(5%), 13(28%), 3(47%)} and the CN {1(25%), 3(75%)}. Since the modulation index of MSK is 1/2, only an even input error weight (l) can form error events [1], [5]. The NSEDs for MSK with input weight l = 2k (k is a positive integer) are din = 2(k + j), where j is the number of zeros within single error events. It is easy to verify that J = 2.5N , D = 0.25N and T = 2.25N . Since all VNs are doped, this is a systematic LDGM code in this sense. Now we calculate the bound terms with w = 1, 2, · · · . The SPC codes will not change the odd or even property of the input weight. Only even w can produce error events in MSK since only an odd w will produce an odd l. If w = 2, the minimal value of h = 4 and ld = 2. This happens when both error bits are in the degree-3 VNs. Using (7) and 0.75N 0.75N (3L)4 + (3)2 + (8), AC (4, L) = 4 2 2 0.75N 0.75N 0.75N 3L2 . For 27L2 + 1 2 1 MSK, nin = l/2. Using (13), we can easily calculate the exponent of N . The smallest exponent (in absolute value) is −2. We use αmax to denote this smallest value. It determines the IG of the system. If h > 4, i.e., at least one error bit is not in the degree-3 VN. For example, one bit is in the degree-3 VN, and one is in the degree-13 VN. In the same way, we ﬁnd αmax = −7. Thus, we can ignore the terms with h > 4 for w = 2. Similarly, we can ignore all other terms (with w = 4, 6, · · · ) since they all have much larger IGs in absolute values (much smaller α). For each kept l, multiple error events in CPM can occur with different d2 . in For example, if l = 2, an error event can occur in MSK with d2 = 2, 4, 6, · · · , depending on how many 0s are in inserted between these two error bits. d2 = 2 is the minimal in distance of the System I. We only keep d2 ≤ 100. Finally, in the bounds are shown in Fig. 4 for k = 300 and k = 600. The simulations are also shown for reference. We can see that the bounds are tight in the error ﬂoor region. Another system (System II) is also analyzed. The degree proﬁle is the VN {15(18.33%), 4(2.4%), 3(76.9%), 2(2.37%)} and the CN {3(80%), 1(20%)}. The overall rate is 0.5 bits per symbol. A similar analysis process can be used to calculate the truncated union bound. The simulations and truncated bounds are shown in Fig. 5. B. Design Considerations From the Union Bound Based on the bound analysis, we suggest the following design approaches to achieve a low error ﬂoor for the AWGN channels. 1) From the bound analysis, the error events from the lowest degree VNs will dominate the union bound since they normally have small IGs and small NSEDs. Thus, the ratio of the lowest degree VNs should not be large to have a low error ﬂoor. 2) The LDGM codes with the fully doped VN, or the systematic codes, have l ≥ ld = w (though it might have a higher convergence threshold). Thus, it is preferable in the ﬁnite length system. 3) We can jointly design the degree proﬁle and the CPM 10 −2 10 −3 bound for info. bit length 600 bound for info. bit length 300 simulation for info. bit length 600 simulation for info. bit length 300 BER 10 −4 10 −5 10 −6 1 2 3 4 E /N (dB) b 0 5 6 7 Fig. 4. Simulation results and truncated union bounds for System I over the AWGN channel. 10 0 10 −1 10 −2 simulations for information bit 2000 simulations for information bit 5000 simulations for information bit 10000 bound for information bit 10000 bound for information bit 5000 bound for information bit 2000 10 BER −3 10 −4 10 −5 10 −6 10 −7 0 0.5 1 1.5 Eb/N0(dB) 2 2.5 3 Fig. 5. Simulation results and truncated union bounds for System II over the AWGN channel. scheme. One approach is to design the minimal degree of the VN and its ratio by considering the modulation index for CPM. For example, for MSK (or any CPM system with a modulation index 1/2), the odd input weight (odd input 1s) is not an effective error event [5]. Thus, we shall try more odd degree variable nodes. Especially, we should try to use the degree proﬁle, which has minimum odd degree VN. For example, the system in Fig. 4 has odd minimum degree, and the system in Fig. 5 has even minimum degree. Thus, the BER curves in Fig. 4 decrease faster. Another example is for System III deﬁned by slightly modifying the degree proﬁle for System II. By removing the degree-2 VN, the VN degree now is {15(18.3%), 4(0.4%), 3(81.3%)} (CN degrees are unchanged). The minimal VN degree is 3. The interleaver gain is αmax = −2. Yet for the System II, αmax = −1. Meanwhile, the minimal NSED for System III (d2 = 6) is larger than that for System min II (d2 = 2). Thus, system III has better performance min in the error ﬂoor region. Yet System III has a slightly later convergence by removing the degree-2 VNs (The degree of System II is the optimized result from [14]). The simulation results are shown in Fig. 6. XIAO AND AULIN: ON ANALYSIS AND DESIGN OF LOW DENSITY GENERATOR MATRIX CODES FOR CONTINUOUS PHASE MODULATION A2 E 3445 10 0 10 −1 System II System III 10 −2 10 BER −3 10 −4 10 −5 10 −6 10 −7 0.5 1 1.5 E /N (dB) b 0 2 2.5 3 Fig. 6. Simulation results for System II and System III in AWGN channels. 2000 information bits are used. IV. A DAPTIVE R ATE C ODES FOR S LOWLY FADING C HANNELS In practical wireless communications, the transmitted signals are often subject to the distortion from multipath fading. Thus, A(t) in (1) is not a constant any more. Using channel state information (CSI, to be explained below) at both the transmitter and receiver, adaptive modulation [21]-[24] etc., is a powerful technique to combat fading. Perhaps because of the CPM format, such as continuous-phase and constant-envelope, it is difﬁcult to adapt the parameters of CPM according to the channel variations. Hence, adaptive CPM is a relatively unexplored topic. We will investigate rate-adaptive CPM with LDGM codes. The information rate of CPM signals is adapted according to the channel conditions. A. Description of the Adaptive System The schematic of rate-adaptive CPM with LDGM codes is shown in Fig. 7. Rate-adaptive LDGM codes are used as outer codes for CPM. Since LDGM codes are block-oriented, the proposed scheme is also block-oriented, and each block has a transmitted rate. Because of the constant-envelope property of CPM, all signals use the same transmitted symbol energy Es . We assume that the fading is slow enough so that A(t) is approximately constant within each symbol interval (denoted as Ts ). Hence, in the z-th symbol interval, A(t) = A(zTs ) = Az = azi + jazq . Here azi and azq are zero-mean Gaussian distributed, and |Az | = a2 + a2 is Rayleigh distributed zq zi [27]. The autocorrelation and cross-correlation functions of azi and azq are Razi (τ ) = Razq (τ ) = 1 J0 (2πτ fD Ts ) and 2 Razi ,azq (τ ) = 0, respectively. Here fD is the maximum Doppler shift and J0 (·) is the zero-order Bessel function of the ﬁrst kind [27]. Note that the complex channel gains are normalized to have unit energy, and the average received ¯ symbol energy (Es ) equals the transmitted symbol energy, i.e., ¯ Es = Es [27]. Similar to [22], [24], the rate of each block is determined by the instantaneous CSI of the ﬁrst transmitted symbol of the block. Here CSI means the channel gain Az or the z instantaneous received SNR Rz = N0 s . We use a low-rate LDGM code as the mother code [10]. A higher rate code is achieved by puncturing the output of the CNs of the LDGM code. The punctured bits are randomly chosen. Except for doped nodes, any CN can be punctured. A CPM receiver for the fading channels is used to produce the time-discrete CPDF of the CPM signal. A channel estimator calculates the CSI for each symbol. This CSI is sent to the CPM signal receiver and the LDGM decoder. We assume that the CSI estimation is perfect. Thus, the CPDF can be calculated in a similar way for AWGN channels with a small modiﬁcation. We again use the PCA receiver proposed in [11]. A detailed description of the PCA receiver for the slowly fading channels can be found in [19]. Then, the CPDF is sent to the iterative decoders. In the punctured CNs, the input probability values are always set to 0.5 for 0 and 1 (0 if logic-likelihood ratios are used) [10]. Except for this, the punctured code uses the same decoding as the mother code. At the beginning of each transmitted block, the CSI is feedback to the transmitter. The transmitter punctures the output of the LDGM codes according to the CSI. The fading process is slow enough, and the received signal energy of the same block will not change drastically in most cases. As in [21]-[24], we assume that the transmitter and receiver have correct CSI instantaneously. This assumption is ideal and difﬁcult to have in practice. Yet it is a good approximation for slowly fading channels, and denotes the performance upper limit [22]-[24]. The errors in the CSI estimation or feedback can only lead to performance degradation. In the proposed system, many parameters can be modiﬁed to optimize the performance. The most important are the LDGM code degree proﬁle and the adaptation policy. A good adaptive scheme should have a high average throughput (¯) for a given BER and transmitted energy, or equivalently η low transmitted energy for a given BER and η . Due to the ¯ complexity limitation, we use only one mother code. Hence, we use the degree proﬁle being optimal for the mother code. The degree proﬁle being optimal for the mother code may not be optimal for the punctured code. Thus, the approach is suboptimal. Yet it is extremely difﬁcult to ﬁnd the degree proﬁle optimal for all rates (if it exists). The optimization for the mother code is easy to implement, and numerical results show that the approach works well. Given the degree proﬁle, we need to ﬁnd the adaptation policy. This denotes how the transmitter/receiver changes the code rate with the CSI. First, we show how to measure the system performance. To facilitate this, we average the throughput η within a symbol interval. ¯ Hence, η is in information bits per symbol. If the adaptation ¯ policy is ﬁxed, the rate of each block is determined by Rz of the ﬁrst symbol of each transmitted block. Since |Az | is A2 E z Rayleigh distributed, Rz = N0 s is chi-square-distributed with two degrees of freedom [27]. The probability density function (PDF) of Rz is 1 γ fRz (γ) = ¯ exp(− ¯ ), γ ≥ 0. Es /N0 Es /N0 (15) We denote the number of information bits for each symbol as η = f1 (Rz ). The function f1 (·) is determined by the 3446 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 2007 channel Info. bits X E⊕ B E z j dv,1 . . π . E⊕ B E j dv,k LDGM encoder E dc,1 . . . w U ' CPM E j B dc,n i j Output 'q E E B A I⊕ % . . CPDF c c E⊗ E⊕ E CPM E BCJR © . . π for receiver . . CPM s B ij Tz 'q E E A % E ' I A B⊕ E % j Channel E A(t) n(t) ' E estimator Error-Free Feedback channel Sum-product Decoder Fig. 7. Schematic of rate-adaptive CPM with the LDGM system. A(t) is the channel gain due to fading, and n(t) is additive noise. 10 −1 10 −2 10 −3 10 −4 10 −5 −5 0 E /N (dB) s 0 5 10 Fig. 8. The error roof for MSK with punctured LDGM. The simulations are for AWGN channels. adaptation policy. Then, η can be evaluated as ¯ ∞ Then, the SNR region determination can be formulated as an optimization problem, which can be solved by a water-ﬁlling approach. However, since we use a ﬁxed transmitted energy, the received SNR is varying because of fading. The BER is not ﬁxed, and the channel inversion approach in [24] cannot be used. Instead, we use the error roof approach in [22] to determine the Rz regions. Since the number of applicable rates is limited, the approach is a discrete rate adaptation. We simulate the BER-SNR performance curves for the AWGN channels. Without loss of generality, we use System I as the mother code for illustration. Thus, the mother code has a rate 1/4, and the CPM scheme is MSK [1]. We assume that each block has 600 information bits. Since the lowest rate is 1/4, the longest codeword has 2400 symbols. We use 7 SNR regions with rates 1/4, 1/3, 1/2, 7/12, 2/3, 3/4 and 5/6, respectively. The simulation results are shown in Fig. 8. If we set the target BER to 10−3 , then SNRs of simulations (except for the mother code) with the 10−3 BER are the boundaries as in Fig. 8. Thus, the BER is kept below the target BER. In the example, the boundaries (thresholds) of 7 SNR regions are Es = {−2.6, −1.2, 0.7, 3.2, 5.1, 6.5} + C0 (dB) N0 (17) BER η= ¯ 0 f1 (γ)fRz (γ)dγ. (16) With the measurement, we can check the performance of the adaptation policy. In the following, we shall discuss two rate adaptation policies: the discrete rate (DR) adaptation and the quasi-continuous rate (QCR) adaptation. B. Discrete Rate Adaptation In [26], it is shown that Rayleigh fading channels can be modelled as a ﬁnite-state Markov process by partitioning the range of Rz into a ﬁnite number of intervals. A similar approach is used in [22], [24] to perform block-wise adaptive modulation. An adaptation policy needs to determine the Rz regions (number and boundaries), and the rate of each region. In [22], all schemes with different candidate rates are simulated for AWGN channels. The SNR regions are determined by ﬁnding the SNR values that make the systems have the target BER (or lower). The approach is called the error roof [22]. In [24], the transmitted energy is adapted according to CSI by multiplying the transmitted symbol energy Es with 1/Az . Thus, the channel gain Az due to fading is fully compensated. In this way, the received SNR is constant, and the BER values are ﬁxed. This is called channel inversion [23], [24]. where C0 is a positive design constant. Note that we use the transmitted symbol energy (Es ) instead of the transmitted bit energy (Eb ) since the transmitted energy efﬁciency is concerned. Transmitter uses CSI of the ﬁrst symbol to determine the rate of the whole block. Yet each block usually has more than hundreds of symbols. Within each block, the Az s usually are not constant. The received SNRs might be much lower than the boundary SNRs of their region due to deep fading. Though the chance is quite small for the slowly fading assumption, the effect on the BER is signiﬁcant. Then, a higher transmitted SNR is necessary to compensate for the possible deep fading. C0 is not used in [26] since the block is much shorter (only a few symbols). To optimize system performance, different Co s for different regions may produce a better result than using the same Co for all regions. It is hard to ﬁnd them. Thus, we use the same C0 for all regions. The use of C0 will decrease η by ¯ (16) for a given Es . This means a penalty in energy efﬁciency. This penalty seems inevitable due to the system limitation: On the transmitter side, we cannot exploit the channel CSI of every symbol within a block. Yet the gain of the adaptive coding modulation is from the knowledge of the CSI at the transmitter (or called side information in some papers). The XIAO AND AULIN: ON ANALYSIS AND DESIGN OF LOW DENSITY GENERATOR MATRIX CODES FOR CONTINUOUS PHASE MODULATION 3447 20 18 16 14 E /N : dB 12 rate 1 QCR Adaption DR Adaption Non−Adaption 0.9 0.8 points for curve−fitting Other simulation result Curve−fitting result Rate function 0.7 0 10 8 6 4 2 0.6 s 0.5 0.4 0.3 0 0.35 0.4 0.45 0.5 0.55 0.6 Rate: info. bits per transmitted symbol 0.65 0.7 0.2 −4 −2 0 2 4 6 Es/N0 (dB) 8 10 12 Fig. 9. Comparison of transmitted energy for systems without adaptation, DR adaptation and QCR adaptation schemes. All schemes have Pb ≈ 10−3 . fD Ts = 10−4 . Fig. 10. The curve-ﬁtting process to determine the rate function. The ∗ points from the simulations are used to perform ﬁtting. The simulations are for AWGN channels. BER is ﬁxed to 10−3 . value of C0 can be most easily found by simulations for fading channels. In the example system, we found that C0 ≈ 5dB can give BER ≤ 10−3 . Numerical results are shown in Fig. 9. We can see that the DR adaptation approach can increase the transmitted energy efﬁciency substantially relative to the scheme without adaptation. C. Quasi-Continuous Rate Adaptation: Rate Function Above we discussed the DR adaption approach. Though the approach can improve the energy efﬁciency, it does not exploit the rate-ﬂexible property of the punctured LDGM codes. Here the rate-ﬂexible property means that the rate of LDGM codes can be ﬁnely adjusted by controlling the punctuation. In the example system, the rate 1/4 mother code has 600 VNs and 2400 CNs. Except for the doped CNs, we can puncture all the other 1800 CNs. The range of the CN output bit number is from 600 to 2400. Thus, 600 600 600 the possible rates are: { 2400 , 2399 , 2398 , · · · , 600 , 1}. Rate-1 601 means uncoded CPM. The difference is quite small between two possible neighboring rates. More generally, we assume that the VN length is k and the CN length is N (N > k). Then the possible rates are r= k k k , ,··· , , 1. N N −1 k+1 (18) There are N − k different rates. The largest difference between two neighboring rates is Dmax = 1 − k/(k + 1) = 1/(k + 1). When k increases, Dmax goes to zero, and the code rates become closer. Though the range of the rates still is ﬁnite, it can in practice be regarded as a continuously changing rate. Hence, we call it quasi-continuous rate (QCR) adaption. Since there are many possible rates, it is difﬁcult to produce simulations for all of them. Thus, we will not use the error roof approach as for the DR adaption. A function (denoted as f2 (·)) with Rz as the argument and η as the value is convenient to determine the rate. We call it the rate function. A theoretical derivation of the rate function is extremely difﬁcult. Though the performance bound as being discussed earlier can be used to ﬁnd the relation of the BER and SNR for a given rate, the closed-form equations of the SNR and rate with a given BER for the punctured code is still unknown. A practical approach to ﬁnd the rate function is to draw several SNRrate points in one ﬁgure, and use a curve-ﬁtting approach. All these points should have the same BER (10−3 in the example). For the example system, this approach is shown in Fig. 10. We use 6 points (marked ∗ in the ﬁgure) to perform the curve-ﬁtting. The rate function is assumed to be a highorder polynomial. The ﬁtting result is drawn as the dashed line. Simulation results of other rates (to check the ﬁtting accuracy) are marked as the squares. We can see that the ﬁtting curve closely matches the simulation. Then, the rate function is ⎧ ⎪1/4 Rz < −3.8 + C1 , ⎨ f2 (Rz ) = f2 (Rz − C1 ) C1 − 3.8 ≤ Rz ≤ 7.2 + C1 , ⎪ ⎩ 23/24 Rz > 7.2 + C1 (19) 4 3 2 where f2 (Rz ) = 0.0002Rz − 0.0002Rz − 0.0102Rz + 0.0602Rz + 0.5625 comes from curve-ﬁtting, and C1 is a positive design parameter. Here the highest rate is 23/24. By using C1 , we right shift the SNR-rate curve in the plot. The reason to use C1 is the same as C0 of the DR adaption: The SNR in the receiver is not constant within a transmitted block. We only use CSI of the ﬁrst symbol of a block for adaptation since the system is block-oriented. A deep fade causes the BER performance degradation. Thus, a higher transmitted energy is needed to compensate this. Again, a simple way to determine C1 is by simulations. For the example system, we found that C1 = 5dB is enough for BER ≤ 10−3 . This approach is also suboptimal. The optimal C1 is a function of Rz . Yet, it is very complex to ﬁnd this function. In the realization of the QCR adaptation, we do not have to use (19) directly. Instead, an easier way is to simply modify (19) and calculate the number of punctured bits. In the example system, the number of punctured bits can be evaluated as: Cp = N − k/f2 (Rz ) (20) where x is the largest integer smaller than x. Numerical results for the QCR adaptation are shown in Fig. 9. Here, 3448 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 2007 10 −1 Non−Adaption DR Adaption QCR Adaption 10 −2 10 −3 10 −4 10 −5 0 5 10 E /N (dB) s 0 15 20 25 Fig. 11. Simulation results for the systems without adaptation, DR adaptation and QCR adaptation schemes. All schemes have the average rates 0.5 information bits per symbol. fD Ts = 10−4 . 22 20 18 16 E /N (dB) 14 12 10 8 6 4 Non−Adaption QCR Adaption DR Adaption mainly for the following two reasons. (1) The derivation of the union bound in Section III is based on the AWGN assumption. The bound for fading channels is different [25]. (2) More importantly, to design the adaptive modulation scheme, the SNR-rate function for a ﬁxed BER is used. Yet, the bound analysis is for the SNR-BER function for a ﬁxed rate. Especially, puncturing makes the SNR-rate function more difﬁcult by the bound. In this paper, we have discussed analysis and design issues of LDGM codes for CPM. For a ﬁnite block length, we used the bounds to estimate the error ﬂoor for the AWGN channels. Several design approaches were suggested based on the bound analysis. The VNs with minimal degree were shown to have signiﬁcant effect on the system error ﬂoor. A joint design approach of the CPM modulation index and the minimal VN degree was proposed. For slowly fading channels, a rate-adaptive approach was investigated. Compared to the system without adaptation, the proposed system can substantially increase the energy efﬁciency. A rate function was used to exploit the rate-ﬂexible property of LDGM codes. Hence, we can ﬁnely change the rate of the system with a large block size. To reduce complexity, we left many parameters unchanged. Without considering complexity, system performance can be further improved. For example, we can adapt the CPM transmitter (such as h or M ) if we can pay the price of multiple ﬁlter-banks, and we can ﬁnd different degree proﬁles for each rate if multiple mother codes are used. R EFERENCES BER s 0 0.35 0.4 0.45 0.5 Rate 0.55 0.6 0.65 0.7 Fig. 12. Comparison of transmitted energy for systems without adaptation, DR adaptation and QCR adaptation schemes. All schemes have Pb ≈ 10−3 . fD Ts = 4 × 10−4 . we ﬁx the C0 , C1 and set BER ≤ 10−3 , and modify Es to get different rates. About 1 dB gain can result from the DR adaptation relative to the QCR adaptation. A similar gain from the DR adaption to the continuous-rate adaption is reported and analyzed in [23] for the constant power rate-adaptation scheme. Up to 10 dB gain over the non-adaptive scheme is achieved by the DR adaption. In Fig. 11, we ﬁx the rates by modifying both C0 (or C1 ) and Es , and plot the BERSNR simulation results. Numerical results again highlight the improvement of the DR adaptation over no adaptation, and the QCR adaptation over DR adaptation. In Fig. 12, we show results for faster fading (than that of Fig. 9). Compared to the results of Fig. 9, we can see that the gain of the adaption over the non-adaption scheme reduces. The gain of the QCR adaption over the DR adaption also reduces. The results show that fading should be slow enough for the proposed adaptation systems to work well. V. D ISCUSSION AND C ONCLUSION The analysis and design approaches of Section III are for the AWGN channel only. They cannot be used for fading channels [1] T. Aulin, N. Rydbeck, and C. E. Sundberg, “Continuous phase podulation (CPM). Part I and part II,” IEEE Trans. Commun., vol. 29, pp. 196-225, Mar. 1981. [2] B. 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Garcia-Frias and W. Zhong, “Approaching Shannon performance by iterative decoding of linear codes with low-density generator matrix,” IEEE Commun. Lett., vol. 7 no. 6, pp. 266-268, June 2003. [9] T. Oenning and J. Moon, “A low-density generator matrix interpretation of parallel concatenated single bit parity codes,” IEEE Trans. Magn., pp. 737-741, Mar. 2001. [10] H. Lou and J. Garcia-Frias, “Rate-compatible low-density generator matrix codes,” in Proc. CISS’05, Mar. 2005. [11] P. Moqvist and T. Aulin, “Orthogonalization by principal components applied to CPM,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1838-1845, Nov. 2003. [12] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. 20, pp. 284-287, Mar. 1974. [13] S. ten Brink, G. Kramer, and A. Ashikhmin, “Design of low-density parity-check codes for modulation and detection,” IEEE Trans. Commun., pp. 670-678, Apr. 2004. 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(ICC’05), May 2005, pp. 805-809. [20] A. Abbasfar, D. Divsalar, and K. Yao, “Maximum likelihood decoding analysis of accumulate-repeat-accumulate codes,” in Proc. IEEE Globecom, Nov. 2004, pp. 514-519. [21] B. Vucetic, “An adaptive coding scheme for time-varying channels,” IEEE Trans. Commun., vol. 39, pp. 653-663, May 1991. [22] S. M. Alamouti and S. Kallel, “Adaptive trellis-coded multiple-phaseshift keying for Rayleigh fading channel,” IEEE Trans. Commun., vol. 42, pp. 2305-2314, June 1994. [23] S. T. Chung and A. Goldsmith, “Degrees of freedom in adaptive modulation: A uniﬁed view,” IEEE Trans. Commun., vol. 49, pp. 15611571, Sep. 2001. [24] S. Vishwanath and A. Goldsmith, “Adaptive turbo-coded modulation for ﬂat-fading channels,” IEEE Trans. Commun., vol. 51, pp. 964-972, June 2003. [25] T. Duman and M. Salehi, “The union bound for turbo-coded modulation systems over fading channels,” IEEE Trans. Commun., vol. 47, pp. 14951502, Oct. 1999. [26] H. Wang and N. Moayeri, “Finite-state Markov channel–A useful model for radio communication channels,” IEEE Trans. Veh. Technol., vol. 44, no. 1, pp. 163-171, Feb. 1995. [27] J. Proakis, Digital Communications, 4th ed. New York: McGraw-Hill Press, 2000. ACKNOWLEDGMENT The authors would like to thank the associate editor for kind help, and anonymous reviews for careful reading and valuable comments. Ming Xiao (S’02) was born in the SiChuan Province, P. R. China, on May 22nd, 1975. He received Bachelor and Master degrees in Engineering from the University of Electronic Science and Technology of China, ChengDu in 1997 and 2002, respectively. From 1997 to 1999, he worked as an assistant engineer for ChinaTelecom. From 2000 to 2002, he also held a position in the SiChuan communications administration. From September 2002 to now, he is a Ph.D student in the computer engineering department of Chalmers University of Technology, Sweden. Tor Aulin (S’77-M’80-SM’83-F’99) was born in Malmö, Sweden, on September 12, 1948. He received the M.S. degree in electrical engineering from the University of Lund, Lund, Sweden, in 1974 and the Dr. Techn. (Ph.D.) degree from the Institute of Telecommunication Theory, University of Lund, in November 1979. He became a Docent at the University of Lund in 1981 and worked at this institute as a Postdoctoral Fellow. During this period he was also a Visiting Scientist at the ECSE Department at Rensselaer Polytechnic Institute, Troy, NY. Following this he spent one year at the European Space Agency (ESA), the European Space Research and Technology Centre (ESTEC) in Noordwijk, the Netherlands, as an ESA Research Fellow. In 1983 he became a Research Professor (Docent) in Information Theory at Chalmers University of Technology, Göteborg, Sweden. In 1991 he formed the Telecommunication Theory Group there and also became a Docent in Computer Engineering in 1995. During the fall of 1995 he was a Visiting Fellow at the Telecommunications Engineering Department, Australian National University, Canberra, ACT, Australia. He was a Visiting Professor at City University of Hong Kong in 2004, and in 2005 he was a Research Scholar at the University of Southern California (USC) in Los Angeles, CA, USA. During 2005 he also spent several months working at the Communication Systems Department at Lund University, Lund, Sweden. Some of his research interests are communication theory, combined modulation/coding strategies (such as CPM and TCM), analysis of general sequence detection strategies, digital radio channel characterization, digital satellite communication systems, and information theory. During recent years the potential of these have been considered for iterative decoding in concatenated versions. This is also the case for such schemes integrated into Multiple Access strategies (TCMA, Trellis Code Multiple Access and its CPM counterpart). Joint source/channel coding also falls into this concept. His company, AUCOM, has performed several advanced theoretical studies as a consultant to some of the major international organizations dealing with developing and operating satellite communication systems, e.g., INTELSAT and ESA. He has also performed theoretical study contracts for Saab and Volvo. Nokia has trusted him as an Internal Lecturer and he has performed numerous studies for Ericsson in the area of digital radio transmission, the latter resulting in a patent. He has authored and published some 200 technical papers and has also authored the book Digital Phase Modulation (Plenum, 1986) as a result of his extensive research in this area at that time. He has organized and chaired several sessions at international symposia/conferences organized by, e.g., IEEE, and is an EAMEC representative within the Communications Society of the IEEE. He has been an Editor for the IEEE Transactions on Communications in the area of communication theory and coding for a decade. He is also (since 30 years) on the Communication Theory Committee within IEEE COMSOC. In December 1997 Dr. Aulin was awarded the Senior Individual Grant at a ceremony in Stockholm, Sweden, handed over by the Prime Minister of Sweden. This has thereafter been repeated in 2004. Dr. Aulin has two papers among the best (Best-of-the-Best) published during the ﬁrst 50 years of the IEEE COMSOC, selected in connection with their 50th anniversary in 2002. Dr. Aulin also has an academic degree as a solo cellist.