- Email: [email protected]
Performance analysis of zigzag-coded modulation scheme for WiMAX systems
Available online at www.sciencedirect.com
Journal of the Franklin Institute 349 (2012) 2717–2734 www.elsevier.com/locate/jfranklin
Performance analysis of zigzag-coded modulation scheme for WiMAX systems Salim Kahveci Department of Electrical–Electronics Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey Received 5 June 2012; received in revised form 16 August 2012; accepted 9 September 2012 Available online 18 September 2012
Abstract WiMAX is an OFDM-based technology that supports point-to-multi-point broadband wireless access for next-generation radio access such as 4G. In this paper, we evaluate bit-error-rate performance of fixed WiMAX (IEEE 802.16d) and mobile WiMAX (IEEE 802.16e) transmission systems with various modulations and channel coding techniques. In the digital communication phenomenon, coding schemes are broadly used as a term mostly referring to the forward error correction code. The advantage of the forward error correction code is that retransmission of data can often be avoided. The simulation results include performance analysis based on bit-error-rate versus signal-to-noise-ratio plots and spectral efficiency of different modulation and coding schemes according to the IEEE 802.16 general standard. The results show that the proposed zigzag-coded modulation possesses a stronger error correcting capability according to the Reed Solomon with Convolutional code for both fixed WiMAX and mobile WiMAX standards. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The IEEE 802.16 family of standards known as Worldwide Interoperability for Microwave Access (WiMAX) has been designed to facilitate high data rate communication in metropolitan area wireless networks [1,2]. It implements both packet-oriented data transmission and standard mobile telephony, and provides better performance than traditional wireless communication standards, especially for applications requiring high and stable throughput. The first version of the IEEE 802.16 standard operates in the 10–66 GHz frequency band and requires line of sight towers. Later the standard extended its operation E-mail address: [email protected] 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.09.003
2718
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
through different physical layer specification to 2–11 GHz frequency band enabling non-line of sight connections, which require techniques that efficiently mitigate the impairment of adding and multi-path. Convolutional turbo code (CTC) is one of the communication channel encoding schemes under the WiMAX standard, with a variety of optional code rates and flexible block length. CTC is a kind of duo-binary turbo code, possessing many advantages [3,4] compared with classical turbo code: the minimum freedom distance changing and the transmission spectrum efficiency decreases; the choice of code rate becomes more flexible; its interleave depth is half that of classical turbo code, with the shortened decoding delay. Due to the fact that most practical communication systems need processing with higher bandwidth efficiency, bandwidth efficiency can be increased from two aspects: code rate and modulation type. Compared with classical turbo code and CTC, zigzag code has a big advantage, in that it can choose a higher code rate. Modulation is of the M-QAM modulation type, and possesses such advantages as higher bandwidth efficiency and small radiation outside the band. It is robustly applicable in practical communication systems. In this study, we evaluate the bit-error-rate (BER) performance of fixed WiMAX and mobile WiMAX systems that use the various coding and modulation schemes. The results show that proposed zigzag-coded modulation possesses a stronger error-correcting capability according to the conventional structure for the WiMAX standard. In Section 2, we will give a brief overview of the basic Fixed WiMAX, Mobile WiMAX physical layers, and general WiMAX transmission structure. A detailed description of the zigzag-coded modulation is presented in Section 3. In Section 4, we provide simulation results for the different coding and modulation schemes. Finally, we conclude the paper with Section 5. 2. Fixed WiMAX and mobile WiMAX 2.1. Fixed WiMAX (IEEE 802.16d) standard WiMAX uses radio microwave technology to provide wireless internet service to computers and other devices that are equipped with WiMAX compatible chips, for example personal digital assistants (PDAs), cell phones, etc. It works more or less like cellular network technology, because WiMAX technology also involves the use of a basestation to establish a wireless data communication link just as it is required in cellular networks like global system for mobile communication (GSM) and universal mobile telecommunications system (UMTS). The theoretical range of WiMAX is up to 30 mil and achieves data rates up to 75 Mbps [5]. WiMAX operates in a similar manner as wireless fingering (Wi-Fi) but with two very convincing differences compared to Wi-Fi, data rate and data range. An earlier version known as IEEE 802.16a that was updated to IEEE 802.16-2004 (also known as IEEE 802.16d) is a WiMAX standard that supports fixed NLOS. Wireless internet services thus form a point-to-multi-point deployment scenario. The basic goal of IEEE 802.16-2004 standard was to provide a stationary wireless transmission with data rates higher than those provided by digital subscriber line (DSL). This feature makes fixed WiMAX an alternative for cable, DSL and T1. Fixed WiMAX uses Orthogonal Frequency Division Multiplexing (OFDM) for the transmission of data, thus serving a large number
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2719
of users in the division manner in round robin fashion [6]. Some of the silent features of IEEE 802.16d standards are
Design to provide fixed NLOS broad band services to fixed, nomadic and portable users. 256 OFDM PHY with 64-QAM, 16-QAM, QPSK, and BPSK modulation techniques. Support for advanced antenna and adaptive modulation, and coding techniques. Facilitates the use of point-to-point multi-point mesh topology. Low latency for delay-sensitive services, thus improving the quality of service (QoS) parameters. Support for both time-division duplex (TDD) and frequency-division duplex (FDD).
2.2. Mobile WiMAX (IEEE 802.16e) standard IEEE 802.16-2005, formally known as IEEE 802.16e or Mobile WiMAX, is basically an improvement of the IEEE 802.16-2004 standard. It is a bit more complex technology as compared to its predecessor IEEE 802.16-2004 standard. The mobile WiMAX allows the convergence of mobile and fixed broadband networks through a common wide-area broadband radio access technology and flexible network architecture. Some silent features of IEEE 802.16e standards are [7] as follows:
IEEE 802.16-2005 standard offers support both fixed and mobile access over the same infrastructure. Provides an improved coverage range with the use of an adaptive antenna system. It uses scalable OFDM for transmission to carry data-supporting channel bandwidths between 1.25 MHz and 20 MHz with up to 2048 sub-carriers. Provides resistance to multi-path interference by developing fast Fourier transform (FFT) algorithms.
In the case of fixed WiMAX, the number of sub-carriers is fixed to 256. On the other hand, in case of mobile WiMAX, the FFT size is scalable from 128 to 2048 points, thus with the increase in the available bandwidth, the FFT size also increases. 2.3. WiMAX transmission system We consider uplink transmission using the wireless OFDMA physical layer specified in the IEEE 802.16 standard. The assumed OFDM parameters are listed in Table 1. The nominal bandwidth (BW) is assumed to be 20 MHz. Applying a sampling factor of n ¼ 8/7, yields a sampling frequency Fs ¼ 8000 bn BW =8000c ¼ 22:856 MHz. Fs is normalized to become a multiple of 8 KHz. Denoting the useful symbol time by Tb and the length of the cyclic prefix (CP) by Tg, the fraction of G ¼ Tg =Tb was assumed to be G ¼ 1/8, where Tb ¼ 1=Df , Df ¼ Fs =NFFT and Tg ¼ G Tb . A block diagram of the physical layer is depicted in Fig. 1. The binary data after randomization is fed into the forward error correction (FEC) encoder. We consider convolutional codes (CC),
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2720
Table 1 System parameters. BW NFFT Nused n G
20 MHz 2048 1681 8/7 1/8
Fs ¼ 22:856 MHz Df ¼ 11:16 kHz Tb ¼ 89:6 ms Tg ¼ 11:2 ms Ts ¼ Tb þ Tg ¼ 100:8 ms
Fig. 1. IEEE 802.16 uplink structure.
convolutional turbo codes (CTCs), low-density parity-check (LDPC) and especially concatenated zigzag codes. After bitwise interleaving the bits, they are fed into the modulator. Mapping the bits to either QPSK, 16-QAM, 64-QAM or 8-QAM and ring 16-QAM symbols. A subsequent frame formatter is responsible for the assignment of the modulated symbols to sub-carriers. According to the IEEE 802.16 standard, a special sub-carrier allocation pattern is used to account for the specialties dealing with an OFDMA uplink. Subsequently, the OFDM signal in the time domain is computed via the inverse fast Fourier transform (IFFT). Finally, the CP is added. The channel is assumed to be a time-variant multi-path channel, modeling mobile users in an NLOS scenario. The receiver noise is modeled by an additive white Gaussian noise (AWGN) process added to the received signal. Assuming perfect synchronization, the receiver extracts the useful symbol time and therefore removes the CP. After the computation of the frequency domain signal via the FFT, the receiver extracts the user-specific information. With the assumption that the delay spread of the channel is smaller than the CP, and the time variance of the channel during one OFDM symbol is negligible, the received symbols in frequency domain Y ½i are given by Y ½i ¼ H½i X ½i þ N½i,
i ¼ 0, . . . ,Nused 1:
ð1Þ
Here, X ½i is the transmitted symbol, H½i is the complex valued sample of the channel transfer function and N½i is the complex valued noise sample in sub-carrier i. In the following, we assume perfect knowledge of the channel transfer function and therefore also perfect channel estimation [2]. The equalized symbols are fed into the soft output demodulator computing log-likelihood ratios (LLRs) for bits. After deinterleaving the LLRs, they are fed into the FEC decoders using this soft input for decoding.
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2721
3. Concatenated zigzag-coded modulation 3.1. Encoding of zigzag codes 3.1.1. (I,J)-zigzag codes The structure of zigzag code is shown in Fig. 2(a), where node denotes the modulo-2 summation. An (I,J)-zigzag code is a systematic code with I J information bits in the form of an I J matrix D, and I parity bits in the form of a column vector p, where D and
Fig. 2. The structures of zigzag and concatenated zigzag codes.
2722
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
p are given, respectively, by 2 dð1,1Þ dð1,2Þ 6 6 dð2,1Þ dð2,2Þ D¼6 6 ^ ^ 4 dðI,1Þ dðI,2Þ
... ... & ...
dð1,JÞ
3
7 dð2,JÞ 7 7 ^ 7 5 dðI,JÞ
2 and IJ
pð1Þ
3
6 7 6 pð2Þ 7 6 7 p¼6 7 4 ^ 5 pðIÞ
:
ð2Þ
I1
Thus the code rate of an (I,J)-zigzag code is J=ðJ þ 1Þ. The parity check bits are generated according to ! J X pðiÞ ¼ pði1Þ þ dði,jÞ mod 2, 1rirI, ð3Þ j¼1
with the initial value pð0Þ ¼ 0. For 1rirI, we define ½dði,1Þ,dði,2Þ, . . . ,dði,JÞ,pðiÞ as the ith segment of the (I,J)zigzag code. 3.1.2. Concatenated (I,J)-zigzag codes A powerful code can be obtained by concatenating several component zigzag codes [8]. The structure of a K-dimensional concatenated zigzag code is shown in Fig. 2(b), where the first branch outputs, the information bits D, and the following K branches output the parity check bits from different constituent zigzag encoders. The first constituent code encodes the original data P1 ðDÞ ¼ D, and the remaining K1 constituent codes encode K1 different interleaved versions of D using K1 length-(I J) random interleavers Pk ðÞ, 2rkrK. The kth constituent code generates a parity check vector pk ¼ ½pk ð1Þ, . . . , pk ðIÞT ; and we denote the matrix of I K parity check bits as 2 3 p1 ð1Þ p2 ð1Þ . . . pK ð1Þ 6 7 6 p1 ð2Þ p2 ð2Þ . . . pK ð2Þ 7 7 : P¼6 ð4Þ 6 ^ ^ & ^ 7 4 5 p1 ðIÞ p2 ðIÞ . . . pK ðIÞ IK
Hence the code rate of the K-dimensional concatenated (I,J)-zigzag code is J/(JþK). 3.2. Decoding of concatenated (I,J)-zigzag codes Like that of the LDPC codes, the a posteriori probability (APP) decoding of zigzag codes involves a nonlinear operation at the parity check nodes, which is computationally complex and is less attractive from the implementation point of view. We next describe a lowcomplexity iterative Max-Log-MAP (MLM)-based decoding algorithm for concatenated zigzag codes, which is of the same nature as the min-sum decoding of LDPC codes [8]. For each constituent code, based on the parity check relation (3), the algorithm performs forward and backward recursions and updates the log-likelihood ratio (LLR) of the information bits [8,9]. To combine the LLR of the information bits from all constituent codes, the algorithm performs turbo processing, i.e., the decoders of all constituent codes are placed in a loop. Each constituent decoder uses the output LLR of the information bits
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2723
Fig. 3. The iterative decoding structure of the concatenated zigzag codes. (a) The Turbo decoding structure of the concatenated zigzag codes and (b) The LLR exchange and computation of each constituent decoder.
from the previous decoder to perform the forward and backward recursions and update the LLR of the information bits. The structure of the turbo processing is shown in Fig. 3(a). Note that the LLRs of the parity check bits of each constituent code are not updated in the turbo processing, and thus remain unchanged in the decoding process. In the first turbo iteration, the LLRs of the information bits computed based on the channel outputs and their a priori LLRs are input to the loop (to decoder 1). In the following, we elaborate on the decoding procedure for the K-dimensional concatenated (I,J)zigzag codes. Given the channel output y, the LLR of an information bit is given by ‘a ðdði,jÞÞ ¼ log
Pðdði,jÞ ¼ 0jyÞ Pðdði,jÞ ¼ 0Þ þ log , Pðdði,jÞ ¼ 1jyÞ Pðdði,jÞ ¼ 1Þ
1rirI, 1rjrJ,
ð5Þ
where the first term is the posterior LLR given the channel output, and the second term is the a priori LLR of the information bit. Typically the information bits are assumed equiprobable and therefore the second term is zero. The LLR of a parity check bit is given by ‘a ðpk ðiÞÞ ¼ log
Pðpk ðiÞ ¼ 0jyÞ , Pðpk ðiÞ ¼ 1jyÞ
1rirI, 1rkrK:
ð6Þ
3.2.1. Local processing at each decoder Let f‘ka ði,jÞg be the input LLRs of the information bits of decoder k. The forward and backward recursions for the parity check bit pk(i) are given, respectively, by Fk ðiÞ ¼ ‘a ðpk ðiÞÞ þ W ðFk ði1Þ,‘ka ði,1Þ, . . . ,‘ka ði,JÞÞ, i ¼ 1,2, . . . ,I; Bk ði1Þ ¼ ‘a ðpk ði1ÞÞ þ W ð‘ka ði,1Þ,‘ka ði,2Þ, . . . ,‘ka ði,JÞ,Bk ðiÞÞ, i ¼ I,I1, . . . ,2; ð7Þ with the initialization Fk ð0Þ ¼ Bk ðIÞ ¼ 0, where the function W ðÞ is defined as " # n Y W ðx1 ,x2 , . . . ,xn Þ9 signðxj Þ min jxj j: j¼1
1rjrn
ð8Þ
2724
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
After the forward and backward recursions, the output LLR for the information bit dði,jÞ, denoted as ‘ko ði,jÞ for 1rirI and 1rjrJ, is given by ‘ko ði,jÞ ¼ W ðFk ði1Þ,‘ka ði,1Þ,‘ka ði,2Þ, . . . ,‘ka ði,j1Þ,‘ka ði,j þ 1Þ, . . . ,‘ca ði,JÞ,Bk ðiÞÞ:
ð9Þ
The extrinsic LLR of information bit dði,jÞ, denoted as ‘ke ði,jÞ, is then ‘ke ði,jÞ ¼ ‘ko ði,jÞ‘ka ði,jÞ. In the following we use the term local processing to refer to the recursions and the LLR update given by Eqs. (7)–(9). 3.2.2. Turbo processing across all constituent decoders The K-constituent decoders are placed in a loop shown in Fig. 3(a). At the beginning of the turbo processing, the LLRs of the information bits given by Eq. (5) are fed into decoder 1. Then decoder k þ 1 performs the local processing based on the output of the previous decoder k for 1rkrK1, and decoder 1 performs the local processing based on the output of decoder K-. We call one round of the local processing by decoders 1,2,y,K one iteration. Assume that at iteration n, the input, output, and extrinsic LLRs of the information bit dði,jÞ for decoder k are given by ½‘ka ði,jÞn , ½‘ko ði,jÞn , and ½‘ke ði,jÞn , respectively. For decoder 1, in the first iteration the input LLR is given by ½‘1a ði,jÞ1 ¼ ‘a ðdði,jÞÞ,
ð10Þ
where ‘a ðdði,jÞÞ is given by Eq. (5); and in the subsequent iterations it is updated as ½‘1a ði,jÞn ¼ ½‘Ko ði,jÞn1 ½‘1e ði,jÞn1 ,
n ¼ 2,3, . . . :
ð11Þ
For 2rkrK, the input LLR for decoder k is updated as o ½‘ka ði,jÞn ¼ ½‘k1 ði,jÞn ½‘ke ði,jÞn1 ,
n ¼ 1,2, . . . :
ð12Þ
The initial extrinsic LLR is set to be zero, i.e., ½‘ke ði,jÞ0 ¼ 0 for 1rirI and 1rjrJ. The computation of the extrinsic LLR and the input LLR is illustrated in Fig. 3(b). After a certain number of iterations, the decoder makes hard decisions on the information bits. 3.3. Single-level zigzag-coded modulation for gray mapping Let xk be the transmitted coded symbol at time k. We consider an effective additive white Gaussian noise channel, e.g., the output of a linear equalizer in a single-carrier dispersive optical channel, or the output of a single-tap equalizer in an optical OFDM system [10]. The received signal at time k can be written as yk ¼ xk þ Z, where ZCN ð0,s2 Þ is the white complex Gaussian noise sample. 3.3.1. Single-level coded modulation We consider the quadrature amplitude modulations (QAM) shown in Fig. 4(a), i.e., 4-, 16-, and 64-QAMs. We employ Gray mapping for the M2-QAM (M ¼ 2, 4, 8). For the signal point at ð2iM þ 1,2jM þ 1Þ, 0ri,jrM1, the first-half of the mapping bits are the Gray-M mapping for i, and second-half are the Gray-M mapping for j. The Gray-M mappings of the indices in f0,1, . . . ,M1g are shown in Fig. 4(a) along with the M2-QAM constellations. One concatenated zigzag code is employed. The information bits are encoded and then the coded bits are mapped to the QAM symbols according to the mapping rule shown in Fig. 4(a).
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2725
Fig. 4. Mappings for (a) 4-, 16-, 64-QAM and (b) 8-QAM and ring 16-array.
3.3.2. Single-stage demodulation and decoding The decoding is done by first computing the LLRs of the coded bits (including the systematic information bits and the parity check bits), and then feeding the LLRs into the zigzag decoder. Since the decoding of zigzag codes has been discussed in the previous subsection, we present the LLR computation of each coded bit. Let bB . . . b2 b1 be the mapping bits corresponding to symbol x where B ¼ log2 M 2 is the number of mapping bits per QAM symbol. Let Ai,b be the subset of signal points for which the mapping bit bi ¼ b for b ¼ 0,1. Given a received signal point y, the LLR for the mapping bit bi, denoted as ‘ðbi Þ, is given by P Pðbi ¼ 0jyÞ x2A PðyjxÞ i ¼ log P i,0 : ð13Þ ‘ðb Þ ¼ log i Pðb ¼ 1jyÞ x2Ai,1 PðyjxÞ 3.4. Multi-level zigzag-coded modulation for non-gray mapping It is shown in [11,12] that, for the modulation schemes that can be mapped using Gray mapping, such as 4-, 16-, and 64-QAMs, single-level coding and single-stage decoding can approach the channel capacity. However, for the modulation schemes that cannot be mapped using Gray mapping, such as 8-QAM and ring 16-array in Fig. 4(b), such single-level coded
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2726
modulation incurs a significant performance gap to the channel capacity. Note that for the ring 16-array, there is one signal point located at the origin; there are 10 signal points on a circle of radius 1.903, i.e., ð1:903 cosðð2k þ 1Þp=10Þ,1:903 sinðð2k þ 1Þp=10ÞÞ for 0rkr9; and there are five signal points on a circle of radius 1.0, i.e., ðcosð2kp=5Þ,sinð2kp=5ÞÞ for 0rkr4. To approach the channel capacity, we propose a zigzag-coded modulation scheme based on multi-level coding and multi-stage decoding. Specifically, we divide the B mapping PL bits into L levels X1,X2,y,XL where Xi ¼ ðbi,1 ,bi,2 , . . . ,bi,ai Þ for 1rirL and i ¼ 1 ai ¼ B. According to the chain rule of mutual information (MI): IðX1 X2 . . . XL ; Y Þ ¼
L X
IðXi ; Y jX1 X2 . . . Xi1 Þ:
ð14Þ
i¼1
The rate corresponding to the left-hand side (LHS) of (14) can be approached by the multistage successive decoding corresponding to the right-hand side (RHS), in which in 1st stage, the bits X1 are decoded by treating all other bits as random bits; and then at the ith stage the bits Xi are decoded using the decoded bits fXj g0rjri1 and by treating the bits fXj giþ1rjrL as random bits. In the following, we propose a zigzag-coded multi-level coding scheme, where one component code is employed for the mapping bits at each level. We also propose a rate allocation scheme for each component code, as well as the method for selecting the system parameters.
3.4.1. Multi-level zigzag-coded modulation Fig. 5(a) shows the multi-level coding structure consisting of L layers, where each layer is codedPby a concatenated zigzag code. An information bit sequence U of length K ¼ Li¼ 1 Ki is first demultiplexed into L subsequences U1 ,U2 , . . . ,UL of lengths K1 ,K2 , . . . ,KL , respectively. The L subsequences are then encoded by the L concatenated zigzag codes E1 ,E2 , . . . ,EL , respectively, to L codewords c1 ,c2 , . . . ,cL of lengths a1 N,a2 N, . . . ,aL N, where N is the number of channel symbols that U is encoded and mapped to. For 1rirL, let ci ¼ ðci,1 ,ci,2 , . . . ,ci,ai N Þ. Mapping rule: Each channel symbol corresponds to B bits, with ai bits from layer i, i¼ 1,y,L. That is, the bit vector corresponding to symbol j, 1rjrN, is given by bj1,1 , . . . ,bj1,a1 ,bj2,1 , . . . ,bj2,a2 , . . . ,bjL,1 , . . . ,bjL,aL ,
ð15Þ
where bji,k ¼ ci,ðj1Þai þk for 1rkrai and 1rirL. Code rates and spectrum rate: The rate of the zigzag code Ei is Ri ¼
Ki , Nai
1rirL:
ð16Þ
The spectrum rate (i.e., the number of bits in each channel symbol) of the multi-level coding scheme is PL L X Ki ¼ RS ¼ i ¼ 1 ai Ri ð17Þ N i¼1
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2727
Fig. 5. Multi-level encoding and multi-stage decoding with component zigzag codes.
and the coding rate is PL
RS
R ¼ PL
i¼1
ai
¼ Pi ¼L 1
ai Ri
i¼1
ai
:
ð18Þ
3.5. Multi-stage decoding Next we describe a multi-stage decoder for the above multi-level coding scheme, and illustrate it by an example of 8-QAM with L ¼ 2 layers. 3.5.1. Multi-stage decoding The multi-stage decoding structure is shown in Fig. 5(b). When decoding the information bits Ui at stage i, we employ the reencoded codewords f^c j g1rjri1 of the
2728
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
information bits fU^ i g1rirj1 decoded in the previous ði1Þ stages to compute the LLR of the coded bits of the zigzag code Ei. For the LLR computation of the coded bits, the main idea is that the reencoded mapping bits obtained in the previous stages constrain the possible constellation points to a subset of the entire constellation. At stage i, given the reencoded codewords f^c j g1rjri1 , the possible constellation points of the channel symbol j, 1rjrN, corresponding to the coded bits fci,ðj1Þai þm g1rmrai are constrained to a subset of the entire constellation signal points, denoted as Aij . For 1rmrai , let Aij,m,0 and Aij,m,1 denote the subset of Aij containing the signal points for which the mapping bit bi,m ¼ 0 and bi,m ¼ 1, respectively. Similar to Eq. (13), the LLR of bit ci,ðj1Þai þm is given by P ^ PðyjxÞ x2Ai : ð19Þ ‘ðci,ðj1Þai þm Þ ¼ log P j,m,0 ^ PðyjxÞ x2Ai j,m,1
The multi-stage decoding procedure is summarized as follows.
For i ¼ 1 to L, decode information bits U^ i of the zigzag code Ei according to the following three steps: J Compute the LLR of the coded bits of code Ei based on the reencoded codewords f^c j g1rjri1 and the channel output y^ according to Eq. (19). ^ i based on the LLR obtained from the first step, using the J Decode information bits U MLM algorithm described in Section 2.2. ^ i using the zigzag code Ei to codeword c^ i , which is then J Reencode information bits U used to decode the information bits fU^ j giþ1rjrL in the subsequent stages. Multiplex and output the decoded information bits fU^ i g1rirL .
We next give an example of the multi-level coding scheme and the associated multi-stage decoding for 8-QAM. 3.5.2. An example of 8-QAM The 8-QAM constellation considered in this paper is shown in Fig. 4(b) [10,13]. We consider a 2-level coded modulation scheme where the mapping bits are divided into two groups X1 ¼ ðb0 Þ and X2 ¼ ðb2 b1 Þ. Two concatenated zigzag codes are employed, with the codeword lengths N and 2N for the two levels X1 and X2, respectively. The information bits of stream U are divided into substreams U1 and U2, which are then encoded by the two concatenated zigzag codes. The coded bits of the two concatenated zigzag codes are then mapped to 8-QAM symbols using the mapping rule. The decoding procedure is elaborated as follows.
In the first stage, since no codeword has been decoded before, the possible set of constellation points is the entire constellation and thus in Eq. (19) we have A1j,k,0 ¼ fð000Þ,ð010Þ,ð100Þ,ð110Þg and A1j,k,1 ¼ fð001Þ,ð011Þ,ð101Þ,ð111Þg. We decode the information bits U^ 1 based on the obtained LLRs and then reencode U^ 1 to the codeword c^ 1 , which is used to decode U^ 2 . In the second stage, given the reencoded codeword c^ 1 in the first stage, for each transmission symbol the set of constrained signal points A2j ¼ fð000Þ,ð010Þ,ð100Þ,ð110Þg
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2729
if in the reencoded codeword c^ 1 the corresponding mapping bit b0 ¼ 0 and A2j ¼ fð001Þ,ð011Þ,ð101Þ,ð111Þg if in the reencoded codeword c^ 1 the corresponding mapping bit b0 ¼ 1. We then decode the information bits U^ 2 based on the LLRs obtained in Eq. (19). Finally, we multiplex U^ 1 and U^ 2 , and output the decoded information bits.
3.6. Rate allocation based on mutual information Next we propose a rate allocation scheme for the component codes in the multi-level coding scheme. Since the multi-stage decoding fails if the decoding of one component code fails, the performance is determined by the worst P component code. Given the average code rate R [cf. (18)], or the spectrum rate RS ¼ Li¼ 1 ai R, we propose a rate allocation scheme which optimizes the performance of the worst component code based on the MI chain rule (14). 3.6.1. Problem formulation Note that in the chain rule (14) each term in the RHS is an increasing function of the channel SNR denoted as g. Define Ci ðgÞ9IðXi ; Y jX1 X2 . . . Xi1 Þ,
1rirL:
ð20Þ
The threshold SNR that ensures successful decoding for Xi, denoted as gi, is given by Ci ðgi Þ ¼ ai Ri . Our objective is to minimize the maximum SNR threshold among all component codes. The rate allocation problem is then formulated as follows: min
fRi ,gi g1rirL
max gi
1rirL
s:t: Ci ðgi Þ ¼ ai Ri , L X ai Ri ¼ RS :
1rirL, ð21Þ
i¼1
An intuition is that the optimal solution to Eq. (21) is achieved when the threshold SNRs for all codes are the same, i.e., for some g we have Ci ðgÞ ¼ ai Ri for 1rirL, so that there is no bottleneck in the multi-stage decoding system caused by the worst component code. In the following we show that this intuition is correct. 3.6.2. Problem analysis P Define CðgÞ9 Li¼ 1 Ci ðgÞ and gm 9max1rirL gi . Since Ci ðÞ is an increasing function, we have that Cðgm Þ ¼
L X
Ci ðgm ÞZ
i¼1
L X
Ci ðgi Þ ¼
i¼1
L X
ai Ri ¼ RS ;
ð22Þ
i¼1
thus we have gm ZC 1 ðRS Þ. Conversely, for gS ¼ C 1 ðRS Þ, it is easily shown that gi ¼ gS ¼ C 1 ðRS Þ
and
Ri ¼
Ci ðgS Þ , ai
1rirL
ð23Þ
is a feasible solution to Eq. (21). From the above analysis, we conclude that Eq. (23) is the optimal solution to the rate allocation problem (21).
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2730
3.6.3. Solution using code rate chart We define the code rate chart of the component code i with respect to the SNR g as Ri ðgÞ ¼
Ci ðgÞ , ai
1rirL:
Also define the MI chart of the average code rate with respect to the SNR g as PL L X CðgÞ ai i ¼ 1 Ci ðgÞ RðgÞ ¼ PL ¼ P ¼ Ri ðgÞ, PL L i¼1 i ¼ 1 ai i ¼ 1 ai i ¼ 1 ai
ð24Þ
ð25Þ
which is a weighted average of the code rates of each component code according to the number of mapping bits of each layer. Given the average code rate R, the optimal rate allocation for each component code can be obtained by the following two steps.
Find the SNR g such that RðgÞ ¼ R. Allocate the rate Ri ¼ Ri ðgÞ to component code i for 1rirL.
We show the code rate chart of the 8-QAM using the same system setup for multi-level coding, where the average code rate is given by RðgÞ ¼
1 2 R1 ðgÞ þ R2 ðgÞ: 3 3
ð26Þ
The average code rate is obtained against the SNR Es =s2 where s2 is the power of the complex Gaussian noise. For the average code rate R ¼ 0:7, the SNR g ¼ 6 dB, and thus the rates allocated to the two component codes R 1 and R 2 are 0.5 and 0.8, respectively. See [14] for further information about how the system parameters involved can be classified as the modulation parameters and the code parameters. 4. Simulation results The simulation results are represented and discussed in this section. First we will present the structure of the implemented simulator, and then present the simulation results both in terms of validation of implementation and values for various parameters that characterize the performance of the WiMAX physical layer. We have developed the simulator in VCþþ program. Each block of the transmitter, receiver and channel is written in ‘‘.cpp’’ files. The main procedure call each of the blocks works in the manner a communication system works. The main procedure also contains initialization parameters, input data and delivers results. The parameters that can be set at the time of initialization are the number of simulated symbols, CP length, modulation and coding rate, range of SNR values. The input data stream is randomly generated. BER values for different SNR are stored in text files that facilitate drawing plots. We have presented various BER vs. SNR figures for all the mandatory modulation and coding profiles as specified in the IEEE 802.16 standard on the same channel models. The simulations were carried out for code rates, lengths and modulation schemes including zigzag-coded modulation. We first provide some simulations for the different FEC schemes in a noise channel. In Fig. 6, we show the performance of the general IEEE 802.16 standard. Performance of zigzag-coded scheme rate-3/4 for QPSK and 16-QAM modulation types better than
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2731
Fig. 6. BER performance comparison of different rate/modulation for fixed WiMAX.
Fig. 7. BER performance comparison for fixed WiMAX, R ¼ 1/2 and 3/4, N ¼ 256 and 512, respectively, in the case of 8-QAM.
2732
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
rate-1/2. However, its performance reverses for 64-QAM modulation. For the Turbo/ LDPC codes, coding with lower code-rate has generally larger coding gain for very long codelength such as 65 536 of codelength. The coding gain achievable with Turbo/LDPC is a direct function of the block size used for the encoding process. However, codelength is taken very short in order to decrease the computational complexity in our proposed system. Therefore, coding gain of the proposed coding scheme under the lower code-rate for low-level modulation types may be decreased. In Fig. 7, we depicted the BER versus the SNR for IEEE 802.16d Fixed WiMAX system using 8-QAM. The code lengths were N ¼ 256 and 512 bits, the code rates were R ¼ 1/2 and 3/4, respectively, and the modulation schemes were 8-QAM (QPSK as well as 16-QAM and 64-QAM). It can be seen that the RSþCC performs worst. The performance of the SL system and the ML system code for 8-QAM is quite good. Both ML and SL system need a round 1 dB gain between rate-1/2 and rate-3/4. In Fig. 8, we depicted the BER versus the SNR for IEEE 802.16d Fixed WiMAX system using ring 16-QAM. The code lengths were N ¼ 256 and 512 bits, the code rates
Fig. 8. BER performance comparison for fixed WiMAX, R ¼1/2 and 3/4, N ¼ 256 and 512, respectively, in the case of ring 16-QAM.
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2733
were R ¼ 1/2 and 3/4, respectively, and the modulation schemes were ring 16-QAM (QPSK as well as 16-QAM and 64-QAM). It can be seen that the RSþCC performs worst. Performance increases between rate-1/2 and 3/4 using ring 16-QAM. Furthermore, neither the SL nor ML system has an error floor as LDPC has. In this work, we consider regular LDPC codes which an error floor. The simulated LDPC code is encoded by (512, 256) LDPC code. Finally, we present the simulation results for IEEE 802.16e Mobile WiMAX system. It is modeled on a single user allocating the whole bandwidth. The assumed channel was the well-known COST channel, modeling a mobile user in the typical urban environment moving with a relative velocity of 120 km/h to the base station at a carrier frequency of 2.5 GHz. The results for the different coding schemes are shown in Fig. 9. The rates and code lengths are the same as for the fixed WiMAX. For a BER of 105 it is obtained at a SNR of about 31 dB for RSþCC code with R ¼ 1=2 and QPSK modulation. It can be seen that higher modulation types give a result of an error floor, and BER performance of the proposed Zigzag scheme can be comparable with that of the RSþCC coding.
5. Conclusions FEC techniques are essential for reliable communication over a noisy channel. The effect of error occurring during transmission is reduced by adding redundancy to the data
Fig. 9. BER performance comparison of Mobile WiMAX (velocity¼ 120 km/h).
2734
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
prior to transmission. The redundancy is used to enable the decoder in the receiver to detect and correct errors. Concatenated zigzag codes like CC and RS codes are used efficiently for error detection and correction. The advantage of these codes is that they are easy to encode and process a well-defined mathematical structure, which has lead to very efficient decoding scheme for them. Also, the combination of zigzag code and M-QAM modulation owns a stronger error correcting capability. Therefore, the multi-level zigzag coding approach provides flexible transmission rates because it decouples the dimensionality of the signal constellation from the code rate and multi-level coding offers asymptotic coding gains.
References [1] C. Eklund, R.B. Marks, K.L. Stanwood, S. Wang, Ieee standard 802.16: a technical overview of the wirelessman air interface for broadband wireless access, IEEE Communications Magazine 40 (6) (2002) 98–107. [2] IEEE Std. 802.16-2004, Standard for Local and Metropolitan Area Networks. Part 16: Air Interface for Fixed Broadband Wireless Access Systems, October 2004. [3] C. Berrou, C. Doulland, et al., The Advantages of Non-binary Turbo Codes, 2001, pp. 61–63. [4] O.C.M. Youssou, G. Paul, K. Peter, Enhanced max-log-app and enhanced log-app decoding for DVB-RCS, in: Proceedings of the International Symposium on Turbo Codes, September 2003, pp. 259–262. [5] A. Bajwa, J. Awan, Performance Study of IEEE 802.16d (Fixed WiMAX). M.S. Thesis, Blekinge Institute of Technology, Karlskrona, Sweden, 2008. [6] M.S. Saiful, M.T. Alam, WiMAX: An Analysis of the Existing Technology and Compare with the Cellular Networks, M. S. Thesis, Blekinge Institute of Technology, Karlskrona, Sweden, 2009. [7] WiMAX Forum, Mobile WiMAX-Part 1: A Technical Overview and Performance Evaluation, 2006. [8] L. Ping, X.L. Huang, N. Phamdo, Zigzag codes and concatenated zigzag codes, IEEE Transactions on Information Theory 47 (February) (2001) 800–807. [9] L. Ping, S. Chan, K. Yeung, Iterative decoding of multi-dimensional concatenated single parity check codes, in: Proceedings of the IEEE ICC-98, June 1998, pp. 131–135. [10] C. Gong, X. Wang, Multilevel LDPC-coded high-speed optical systems: efficient hard decoding and code optimization, IEEE Journal of Selected Topics in Quantum Electronics 16 (September/October (5)) (2010) 1268–1279. [11] G. Caire, G. Taricco, E. Biglieri, Bit interleaved coded modulation, IEEE Transactions on Information Theory 44 (May) (1998) 927–946. [12] R.H. Morelos-Zaragoza, M.P.C. Fossorier, S. Lin, H. Imai, Multilevel coded modulation for unequal error protection and multistage decoding-part I: symmetric constellations, IEEE Transactions on Communications 48 (February (2)) (2000) 204–213. [13] I.B. Djordjevic, B. Vasic, Multilevel coding in M-ary DPSK/differential QAM high-speed optical transmission with direct detection, Journal of Lightwave Technology 24 (January (1)) (2006) 420–428. [14] S. Kahveci, C. Gong, X. Wang, Zigzag-coded modulation for high-speed fiber optical channels, Journal of Optical Communications and Networking 4 (May (5)) (2012) 382–391.
Journal of the Franklin Institute 349 (2012) 2717–2734 www.elsevier.com/locate/jfranklin
Performance analysis of zigzag-coded modulation scheme for WiMAX systems Salim Kahveci Department of Electrical–Electronics Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey Received 5 June 2012; received in revised form 16 August 2012; accepted 9 September 2012 Available online 18 September 2012
Abstract WiMAX is an OFDM-based technology that supports point-to-multi-point broadband wireless access for next-generation radio access such as 4G. In this paper, we evaluate bit-error-rate performance of fixed WiMAX (IEEE 802.16d) and mobile WiMAX (IEEE 802.16e) transmission systems with various modulations and channel coding techniques. In the digital communication phenomenon, coding schemes are broadly used as a term mostly referring to the forward error correction code. The advantage of the forward error correction code is that retransmission of data can often be avoided. The simulation results include performance analysis based on bit-error-rate versus signal-to-noise-ratio plots and spectral efficiency of different modulation and coding schemes according to the IEEE 802.16 general standard. The results show that the proposed zigzag-coded modulation possesses a stronger error correcting capability according to the Reed Solomon with Convolutional code for both fixed WiMAX and mobile WiMAX standards. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The IEEE 802.16 family of standards known as Worldwide Interoperability for Microwave Access (WiMAX) has been designed to facilitate high data rate communication in metropolitan area wireless networks [1,2]. It implements both packet-oriented data transmission and standard mobile telephony, and provides better performance than traditional wireless communication standards, especially for applications requiring high and stable throughput. The first version of the IEEE 802.16 standard operates in the 10–66 GHz frequency band and requires line of sight towers. Later the standard extended its operation E-mail address: [email protected] 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.09.003
2718
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
through different physical layer specification to 2–11 GHz frequency band enabling non-line of sight connections, which require techniques that efficiently mitigate the impairment of adding and multi-path. Convolutional turbo code (CTC) is one of the communication channel encoding schemes under the WiMAX standard, with a variety of optional code rates and flexible block length. CTC is a kind of duo-binary turbo code, possessing many advantages [3,4] compared with classical turbo code: the minimum freedom distance changing and the transmission spectrum efficiency decreases; the choice of code rate becomes more flexible; its interleave depth is half that of classical turbo code, with the shortened decoding delay. Due to the fact that most practical communication systems need processing with higher bandwidth efficiency, bandwidth efficiency can be increased from two aspects: code rate and modulation type. Compared with classical turbo code and CTC, zigzag code has a big advantage, in that it can choose a higher code rate. Modulation is of the M-QAM modulation type, and possesses such advantages as higher bandwidth efficiency and small radiation outside the band. It is robustly applicable in practical communication systems. In this study, we evaluate the bit-error-rate (BER) performance of fixed WiMAX and mobile WiMAX systems that use the various coding and modulation schemes. The results show that proposed zigzag-coded modulation possesses a stronger error-correcting capability according to the conventional structure for the WiMAX standard. In Section 2, we will give a brief overview of the basic Fixed WiMAX, Mobile WiMAX physical layers, and general WiMAX transmission structure. A detailed description of the zigzag-coded modulation is presented in Section 3. In Section 4, we provide simulation results for the different coding and modulation schemes. Finally, we conclude the paper with Section 5. 2. Fixed WiMAX and mobile WiMAX 2.1. Fixed WiMAX (IEEE 802.16d) standard WiMAX uses radio microwave technology to provide wireless internet service to computers and other devices that are equipped with WiMAX compatible chips, for example personal digital assistants (PDAs), cell phones, etc. It works more or less like cellular network technology, because WiMAX technology also involves the use of a basestation to establish a wireless data communication link just as it is required in cellular networks like global system for mobile communication (GSM) and universal mobile telecommunications system (UMTS). The theoretical range of WiMAX is up to 30 mil and achieves data rates up to 75 Mbps [5]. WiMAX operates in a similar manner as wireless fingering (Wi-Fi) but with two very convincing differences compared to Wi-Fi, data rate and data range. An earlier version known as IEEE 802.16a that was updated to IEEE 802.16-2004 (also known as IEEE 802.16d) is a WiMAX standard that supports fixed NLOS. Wireless internet services thus form a point-to-multi-point deployment scenario. The basic goal of IEEE 802.16-2004 standard was to provide a stationary wireless transmission with data rates higher than those provided by digital subscriber line (DSL). This feature makes fixed WiMAX an alternative for cable, DSL and T1. Fixed WiMAX uses Orthogonal Frequency Division Multiplexing (OFDM) for the transmission of data, thus serving a large number
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2719
of users in the division manner in round robin fashion [6]. Some of the silent features of IEEE 802.16d standards are
Design to provide fixed NLOS broad band services to fixed, nomadic and portable users. 256 OFDM PHY with 64-QAM, 16-QAM, QPSK, and BPSK modulation techniques. Support for advanced antenna and adaptive modulation, and coding techniques. Facilitates the use of point-to-point multi-point mesh topology. Low latency for delay-sensitive services, thus improving the quality of service (QoS) parameters. Support for both time-division duplex (TDD) and frequency-division duplex (FDD).
2.2. Mobile WiMAX (IEEE 802.16e) standard IEEE 802.16-2005, formally known as IEEE 802.16e or Mobile WiMAX, is basically an improvement of the IEEE 802.16-2004 standard. It is a bit more complex technology as compared to its predecessor IEEE 802.16-2004 standard. The mobile WiMAX allows the convergence of mobile and fixed broadband networks through a common wide-area broadband radio access technology and flexible network architecture. Some silent features of IEEE 802.16e standards are [7] as follows:
IEEE 802.16-2005 standard offers support both fixed and mobile access over the same infrastructure. Provides an improved coverage range with the use of an adaptive antenna system. It uses scalable OFDM for transmission to carry data-supporting channel bandwidths between 1.25 MHz and 20 MHz with up to 2048 sub-carriers. Provides resistance to multi-path interference by developing fast Fourier transform (FFT) algorithms.
In the case of fixed WiMAX, the number of sub-carriers is fixed to 256. On the other hand, in case of mobile WiMAX, the FFT size is scalable from 128 to 2048 points, thus with the increase in the available bandwidth, the FFT size also increases. 2.3. WiMAX transmission system We consider uplink transmission using the wireless OFDMA physical layer specified in the IEEE 802.16 standard. The assumed OFDM parameters are listed in Table 1. The nominal bandwidth (BW) is assumed to be 20 MHz. Applying a sampling factor of n ¼ 8/7, yields a sampling frequency Fs ¼ 8000 bn BW =8000c ¼ 22:856 MHz. Fs is normalized to become a multiple of 8 KHz. Denoting the useful symbol time by Tb and the length of the cyclic prefix (CP) by Tg, the fraction of G ¼ Tg =Tb was assumed to be G ¼ 1/8, where Tb ¼ 1=Df , Df ¼ Fs =NFFT and Tg ¼ G Tb . A block diagram of the physical layer is depicted in Fig. 1. The binary data after randomization is fed into the forward error correction (FEC) encoder. We consider convolutional codes (CC),
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2720
Table 1 System parameters. BW NFFT Nused n G
20 MHz 2048 1681 8/7 1/8
Fs ¼ 22:856 MHz Df ¼ 11:16 kHz Tb ¼ 89:6 ms Tg ¼ 11:2 ms Ts ¼ Tb þ Tg ¼ 100:8 ms
Fig. 1. IEEE 802.16 uplink structure.
convolutional turbo codes (CTCs), low-density parity-check (LDPC) and especially concatenated zigzag codes. After bitwise interleaving the bits, they are fed into the modulator. Mapping the bits to either QPSK, 16-QAM, 64-QAM or 8-QAM and ring 16-QAM symbols. A subsequent frame formatter is responsible for the assignment of the modulated symbols to sub-carriers. According to the IEEE 802.16 standard, a special sub-carrier allocation pattern is used to account for the specialties dealing with an OFDMA uplink. Subsequently, the OFDM signal in the time domain is computed via the inverse fast Fourier transform (IFFT). Finally, the CP is added. The channel is assumed to be a time-variant multi-path channel, modeling mobile users in an NLOS scenario. The receiver noise is modeled by an additive white Gaussian noise (AWGN) process added to the received signal. Assuming perfect synchronization, the receiver extracts the useful symbol time and therefore removes the CP. After the computation of the frequency domain signal via the FFT, the receiver extracts the user-specific information. With the assumption that the delay spread of the channel is smaller than the CP, and the time variance of the channel during one OFDM symbol is negligible, the received symbols in frequency domain Y ½i are given by Y ½i ¼ H½i X ½i þ N½i,
i ¼ 0, . . . ,Nused 1:
ð1Þ
Here, X ½i is the transmitted symbol, H½i is the complex valued sample of the channel transfer function and N½i is the complex valued noise sample in sub-carrier i. In the following, we assume perfect knowledge of the channel transfer function and therefore also perfect channel estimation [2]. The equalized symbols are fed into the soft output demodulator computing log-likelihood ratios (LLRs) for bits. After deinterleaving the LLRs, they are fed into the FEC decoders using this soft input for decoding.
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2721
3. Concatenated zigzag-coded modulation 3.1. Encoding of zigzag codes 3.1.1. (I,J)-zigzag codes The structure of zigzag code is shown in Fig. 2(a), where node denotes the modulo-2 summation. An (I,J)-zigzag code is a systematic code with I J information bits in the form of an I J matrix D, and I parity bits in the form of a column vector p, where D and
Fig. 2. The structures of zigzag and concatenated zigzag codes.
2722
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
p are given, respectively, by 2 dð1,1Þ dð1,2Þ 6 6 dð2,1Þ dð2,2Þ D¼6 6 ^ ^ 4 dðI,1Þ dðI,2Þ
... ... & ...
dð1,JÞ
3
7 dð2,JÞ 7 7 ^ 7 5 dðI,JÞ
2 and IJ
pð1Þ
3
6 7 6 pð2Þ 7 6 7 p¼6 7 4 ^ 5 pðIÞ
:
ð2Þ
I1
Thus the code rate of an (I,J)-zigzag code is J=ðJ þ 1Þ. The parity check bits are generated according to ! J X pðiÞ ¼ pði1Þ þ dði,jÞ mod 2, 1rirI, ð3Þ j¼1
with the initial value pð0Þ ¼ 0. For 1rirI, we define ½dði,1Þ,dði,2Þ, . . . ,dði,JÞ,pðiÞ as the ith segment of the (I,J)zigzag code. 3.1.2. Concatenated (I,J)-zigzag codes A powerful code can be obtained by concatenating several component zigzag codes [8]. The structure of a K-dimensional concatenated zigzag code is shown in Fig. 2(b), where the first branch outputs, the information bits D, and the following K branches output the parity check bits from different constituent zigzag encoders. The first constituent code encodes the original data P1 ðDÞ ¼ D, and the remaining K1 constituent codes encode K1 different interleaved versions of D using K1 length-(I J) random interleavers Pk ðÞ, 2rkrK. The kth constituent code generates a parity check vector pk ¼ ½pk ð1Þ, . . . , pk ðIÞT ; and we denote the matrix of I K parity check bits as 2 3 p1 ð1Þ p2 ð1Þ . . . pK ð1Þ 6 7 6 p1 ð2Þ p2 ð2Þ . . . pK ð2Þ 7 7 : P¼6 ð4Þ 6 ^ ^ & ^ 7 4 5 p1 ðIÞ p2 ðIÞ . . . pK ðIÞ IK
Hence the code rate of the K-dimensional concatenated (I,J)-zigzag code is J/(JþK). 3.2. Decoding of concatenated (I,J)-zigzag codes Like that of the LDPC codes, the a posteriori probability (APP) decoding of zigzag codes involves a nonlinear operation at the parity check nodes, which is computationally complex and is less attractive from the implementation point of view. We next describe a lowcomplexity iterative Max-Log-MAP (MLM)-based decoding algorithm for concatenated zigzag codes, which is of the same nature as the min-sum decoding of LDPC codes [8]. For each constituent code, based on the parity check relation (3), the algorithm performs forward and backward recursions and updates the log-likelihood ratio (LLR) of the information bits [8,9]. To combine the LLR of the information bits from all constituent codes, the algorithm performs turbo processing, i.e., the decoders of all constituent codes are placed in a loop. Each constituent decoder uses the output LLR of the information bits
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2723
Fig. 3. The iterative decoding structure of the concatenated zigzag codes. (a) The Turbo decoding structure of the concatenated zigzag codes and (b) The LLR exchange and computation of each constituent decoder.
from the previous decoder to perform the forward and backward recursions and update the LLR of the information bits. The structure of the turbo processing is shown in Fig. 3(a). Note that the LLRs of the parity check bits of each constituent code are not updated in the turbo processing, and thus remain unchanged in the decoding process. In the first turbo iteration, the LLRs of the information bits computed based on the channel outputs and their a priori LLRs are input to the loop (to decoder 1). In the following, we elaborate on the decoding procedure for the K-dimensional concatenated (I,J)zigzag codes. Given the channel output y, the LLR of an information bit is given by ‘a ðdði,jÞÞ ¼ log
Pðdði,jÞ ¼ 0jyÞ Pðdði,jÞ ¼ 0Þ þ log , Pðdði,jÞ ¼ 1jyÞ Pðdði,jÞ ¼ 1Þ
1rirI, 1rjrJ,
ð5Þ
where the first term is the posterior LLR given the channel output, and the second term is the a priori LLR of the information bit. Typically the information bits are assumed equiprobable and therefore the second term is zero. The LLR of a parity check bit is given by ‘a ðpk ðiÞÞ ¼ log
Pðpk ðiÞ ¼ 0jyÞ , Pðpk ðiÞ ¼ 1jyÞ
1rirI, 1rkrK:
ð6Þ
3.2.1. Local processing at each decoder Let f‘ka ði,jÞg be the input LLRs of the information bits of decoder k. The forward and backward recursions for the parity check bit pk(i) are given, respectively, by Fk ðiÞ ¼ ‘a ðpk ðiÞÞ þ W ðFk ði1Þ,‘ka ði,1Þ, . . . ,‘ka ði,JÞÞ, i ¼ 1,2, . . . ,I; Bk ði1Þ ¼ ‘a ðpk ði1ÞÞ þ W ð‘ka ði,1Þ,‘ka ði,2Þ, . . . ,‘ka ði,JÞ,Bk ðiÞÞ, i ¼ I,I1, . . . ,2; ð7Þ with the initialization Fk ð0Þ ¼ Bk ðIÞ ¼ 0, where the function W ðÞ is defined as " # n Y W ðx1 ,x2 , . . . ,xn Þ9 signðxj Þ min jxj j: j¼1
1rjrn
ð8Þ
2724
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
After the forward and backward recursions, the output LLR for the information bit dði,jÞ, denoted as ‘ko ði,jÞ for 1rirI and 1rjrJ, is given by ‘ko ði,jÞ ¼ W ðFk ði1Þ,‘ka ði,1Þ,‘ka ði,2Þ, . . . ,‘ka ði,j1Þ,‘ka ði,j þ 1Þ, . . . ,‘ca ði,JÞ,Bk ðiÞÞ:
ð9Þ
The extrinsic LLR of information bit dði,jÞ, denoted as ‘ke ði,jÞ, is then ‘ke ði,jÞ ¼ ‘ko ði,jÞ‘ka ði,jÞ. In the following we use the term local processing to refer to the recursions and the LLR update given by Eqs. (7)–(9). 3.2.2. Turbo processing across all constituent decoders The K-constituent decoders are placed in a loop shown in Fig. 3(a). At the beginning of the turbo processing, the LLRs of the information bits given by Eq. (5) are fed into decoder 1. Then decoder k þ 1 performs the local processing based on the output of the previous decoder k for 1rkrK1, and decoder 1 performs the local processing based on the output of decoder K-. We call one round of the local processing by decoders 1,2,y,K one iteration. Assume that at iteration n, the input, output, and extrinsic LLRs of the information bit dði,jÞ for decoder k are given by ½‘ka ði,jÞn , ½‘ko ði,jÞn , and ½‘ke ði,jÞn , respectively. For decoder 1, in the first iteration the input LLR is given by ½‘1a ði,jÞ1 ¼ ‘a ðdði,jÞÞ,
ð10Þ
where ‘a ðdði,jÞÞ is given by Eq. (5); and in the subsequent iterations it is updated as ½‘1a ði,jÞn ¼ ½‘Ko ði,jÞn1 ½‘1e ði,jÞn1 ,
n ¼ 2,3, . . . :
ð11Þ
For 2rkrK, the input LLR for decoder k is updated as o ½‘ka ði,jÞn ¼ ½‘k1 ði,jÞn ½‘ke ði,jÞn1 ,
n ¼ 1,2, . . . :
ð12Þ
The initial extrinsic LLR is set to be zero, i.e., ½‘ke ði,jÞ0 ¼ 0 for 1rirI and 1rjrJ. The computation of the extrinsic LLR and the input LLR is illustrated in Fig. 3(b). After a certain number of iterations, the decoder makes hard decisions on the information bits. 3.3. Single-level zigzag-coded modulation for gray mapping Let xk be the transmitted coded symbol at time k. We consider an effective additive white Gaussian noise channel, e.g., the output of a linear equalizer in a single-carrier dispersive optical channel, or the output of a single-tap equalizer in an optical OFDM system [10]. The received signal at time k can be written as yk ¼ xk þ Z, where ZCN ð0,s2 Þ is the white complex Gaussian noise sample. 3.3.1. Single-level coded modulation We consider the quadrature amplitude modulations (QAM) shown in Fig. 4(a), i.e., 4-, 16-, and 64-QAMs. We employ Gray mapping for the M2-QAM (M ¼ 2, 4, 8). For the signal point at ð2iM þ 1,2jM þ 1Þ, 0ri,jrM1, the first-half of the mapping bits are the Gray-M mapping for i, and second-half are the Gray-M mapping for j. The Gray-M mappings of the indices in f0,1, . . . ,M1g are shown in Fig. 4(a) along with the M2-QAM constellations. One concatenated zigzag code is employed. The information bits are encoded and then the coded bits are mapped to the QAM symbols according to the mapping rule shown in Fig. 4(a).
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2725
Fig. 4. Mappings for (a) 4-, 16-, 64-QAM and (b) 8-QAM and ring 16-array.
3.3.2. Single-stage demodulation and decoding The decoding is done by first computing the LLRs of the coded bits (including the systematic information bits and the parity check bits), and then feeding the LLRs into the zigzag decoder. Since the decoding of zigzag codes has been discussed in the previous subsection, we present the LLR computation of each coded bit. Let bB . . . b2 b1 be the mapping bits corresponding to symbol x where B ¼ log2 M 2 is the number of mapping bits per QAM symbol. Let Ai,b be the subset of signal points for which the mapping bit bi ¼ b for b ¼ 0,1. Given a received signal point y, the LLR for the mapping bit bi, denoted as ‘ðbi Þ, is given by P Pðbi ¼ 0jyÞ x2A PðyjxÞ i ¼ log P i,0 : ð13Þ ‘ðb Þ ¼ log i Pðb ¼ 1jyÞ x2Ai,1 PðyjxÞ 3.4. Multi-level zigzag-coded modulation for non-gray mapping It is shown in [11,12] that, for the modulation schemes that can be mapped using Gray mapping, such as 4-, 16-, and 64-QAMs, single-level coding and single-stage decoding can approach the channel capacity. However, for the modulation schemes that cannot be mapped using Gray mapping, such as 8-QAM and ring 16-array in Fig. 4(b), such single-level coded
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2726
modulation incurs a significant performance gap to the channel capacity. Note that for the ring 16-array, there is one signal point located at the origin; there are 10 signal points on a circle of radius 1.903, i.e., ð1:903 cosðð2k þ 1Þp=10Þ,1:903 sinðð2k þ 1Þp=10ÞÞ for 0rkr9; and there are five signal points on a circle of radius 1.0, i.e., ðcosð2kp=5Þ,sinð2kp=5ÞÞ for 0rkr4. To approach the channel capacity, we propose a zigzag-coded modulation scheme based on multi-level coding and multi-stage decoding. Specifically, we divide the B mapping PL bits into L levels X1,X2,y,XL where Xi ¼ ðbi,1 ,bi,2 , . . . ,bi,ai Þ for 1rirL and i ¼ 1 ai ¼ B. According to the chain rule of mutual information (MI): IðX1 X2 . . . XL ; Y Þ ¼
L X
IðXi ; Y jX1 X2 . . . Xi1 Þ:
ð14Þ
i¼1
The rate corresponding to the left-hand side (LHS) of (14) can be approached by the multistage successive decoding corresponding to the right-hand side (RHS), in which in 1st stage, the bits X1 are decoded by treating all other bits as random bits; and then at the ith stage the bits Xi are decoded using the decoded bits fXj g0rjri1 and by treating the bits fXj giþ1rjrL as random bits. In the following, we propose a zigzag-coded multi-level coding scheme, where one component code is employed for the mapping bits at each level. We also propose a rate allocation scheme for each component code, as well as the method for selecting the system parameters.
3.4.1. Multi-level zigzag-coded modulation Fig. 5(a) shows the multi-level coding structure consisting of L layers, where each layer is codedPby a concatenated zigzag code. An information bit sequence U of length K ¼ Li¼ 1 Ki is first demultiplexed into L subsequences U1 ,U2 , . . . ,UL of lengths K1 ,K2 , . . . ,KL , respectively. The L subsequences are then encoded by the L concatenated zigzag codes E1 ,E2 , . . . ,EL , respectively, to L codewords c1 ,c2 , . . . ,cL of lengths a1 N,a2 N, . . . ,aL N, where N is the number of channel symbols that U is encoded and mapped to. For 1rirL, let ci ¼ ðci,1 ,ci,2 , . . . ,ci,ai N Þ. Mapping rule: Each channel symbol corresponds to B bits, with ai bits from layer i, i¼ 1,y,L. That is, the bit vector corresponding to symbol j, 1rjrN, is given by bj1,1 , . . . ,bj1,a1 ,bj2,1 , . . . ,bj2,a2 , . . . ,bjL,1 , . . . ,bjL,aL ,
ð15Þ
where bji,k ¼ ci,ðj1Þai þk for 1rkrai and 1rirL. Code rates and spectrum rate: The rate of the zigzag code Ei is Ri ¼
Ki , Nai
1rirL:
ð16Þ
The spectrum rate (i.e., the number of bits in each channel symbol) of the multi-level coding scheme is PL L X Ki ¼ RS ¼ i ¼ 1 ai Ri ð17Þ N i¼1
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2727
Fig. 5. Multi-level encoding and multi-stage decoding with component zigzag codes.
and the coding rate is PL
RS
R ¼ PL
i¼1
ai
¼ Pi ¼L 1
ai Ri
i¼1
ai
:
ð18Þ
3.5. Multi-stage decoding Next we describe a multi-stage decoder for the above multi-level coding scheme, and illustrate it by an example of 8-QAM with L ¼ 2 layers. 3.5.1. Multi-stage decoding The multi-stage decoding structure is shown in Fig. 5(b). When decoding the information bits Ui at stage i, we employ the reencoded codewords f^c j g1rjri1 of the
2728
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
information bits fU^ i g1rirj1 decoded in the previous ði1Þ stages to compute the LLR of the coded bits of the zigzag code Ei. For the LLR computation of the coded bits, the main idea is that the reencoded mapping bits obtained in the previous stages constrain the possible constellation points to a subset of the entire constellation. At stage i, given the reencoded codewords f^c j g1rjri1 , the possible constellation points of the channel symbol j, 1rjrN, corresponding to the coded bits fci,ðj1Þai þm g1rmrai are constrained to a subset of the entire constellation signal points, denoted as Aij . For 1rmrai , let Aij,m,0 and Aij,m,1 denote the subset of Aij containing the signal points for which the mapping bit bi,m ¼ 0 and bi,m ¼ 1, respectively. Similar to Eq. (13), the LLR of bit ci,ðj1Þai þm is given by P ^ PðyjxÞ x2Ai : ð19Þ ‘ðci,ðj1Þai þm Þ ¼ log P j,m,0 ^ PðyjxÞ x2Ai j,m,1
The multi-stage decoding procedure is summarized as follows.
For i ¼ 1 to L, decode information bits U^ i of the zigzag code Ei according to the following three steps: J Compute the LLR of the coded bits of code Ei based on the reencoded codewords f^c j g1rjri1 and the channel output y^ according to Eq. (19). ^ i based on the LLR obtained from the first step, using the J Decode information bits U MLM algorithm described in Section 2.2. ^ i using the zigzag code Ei to codeword c^ i , which is then J Reencode information bits U used to decode the information bits fU^ j giþ1rjrL in the subsequent stages. Multiplex and output the decoded information bits fU^ i g1rirL .
We next give an example of the multi-level coding scheme and the associated multi-stage decoding for 8-QAM. 3.5.2. An example of 8-QAM The 8-QAM constellation considered in this paper is shown in Fig. 4(b) [10,13]. We consider a 2-level coded modulation scheme where the mapping bits are divided into two groups X1 ¼ ðb0 Þ and X2 ¼ ðb2 b1 Þ. Two concatenated zigzag codes are employed, with the codeword lengths N and 2N for the two levels X1 and X2, respectively. The information bits of stream U are divided into substreams U1 and U2, which are then encoded by the two concatenated zigzag codes. The coded bits of the two concatenated zigzag codes are then mapped to 8-QAM symbols using the mapping rule. The decoding procedure is elaborated as follows.
In the first stage, since no codeword has been decoded before, the possible set of constellation points is the entire constellation and thus in Eq. (19) we have A1j,k,0 ¼ fð000Þ,ð010Þ,ð100Þ,ð110Þg and A1j,k,1 ¼ fð001Þ,ð011Þ,ð101Þ,ð111Þg. We decode the information bits U^ 1 based on the obtained LLRs and then reencode U^ 1 to the codeword c^ 1 , which is used to decode U^ 2 . In the second stage, given the reencoded codeword c^ 1 in the first stage, for each transmission symbol the set of constrained signal points A2j ¼ fð000Þ,ð010Þ,ð100Þ,ð110Þg
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2729
if in the reencoded codeword c^ 1 the corresponding mapping bit b0 ¼ 0 and A2j ¼ fð001Þ,ð011Þ,ð101Þ,ð111Þg if in the reencoded codeword c^ 1 the corresponding mapping bit b0 ¼ 1. We then decode the information bits U^ 2 based on the LLRs obtained in Eq. (19). Finally, we multiplex U^ 1 and U^ 2 , and output the decoded information bits.
3.6. Rate allocation based on mutual information Next we propose a rate allocation scheme for the component codes in the multi-level coding scheme. Since the multi-stage decoding fails if the decoding of one component code fails, the performance is determined by the worst P component code. Given the average code rate R [cf. (18)], or the spectrum rate RS ¼ Li¼ 1 ai R, we propose a rate allocation scheme which optimizes the performance of the worst component code based on the MI chain rule (14). 3.6.1. Problem formulation Note that in the chain rule (14) each term in the RHS is an increasing function of the channel SNR denoted as g. Define Ci ðgÞ9IðXi ; Y jX1 X2 . . . Xi1 Þ,
1rirL:
ð20Þ
The threshold SNR that ensures successful decoding for Xi, denoted as gi, is given by Ci ðgi Þ ¼ ai Ri . Our objective is to minimize the maximum SNR threshold among all component codes. The rate allocation problem is then formulated as follows: min
fRi ,gi g1rirL
max gi
1rirL
s:t: Ci ðgi Þ ¼ ai Ri , L X ai Ri ¼ RS :
1rirL, ð21Þ
i¼1
An intuition is that the optimal solution to Eq. (21) is achieved when the threshold SNRs for all codes are the same, i.e., for some g we have Ci ðgÞ ¼ ai Ri for 1rirL, so that there is no bottleneck in the multi-stage decoding system caused by the worst component code. In the following we show that this intuition is correct. 3.6.2. Problem analysis P Define CðgÞ9 Li¼ 1 Ci ðgÞ and gm 9max1rirL gi . Since Ci ðÞ is an increasing function, we have that Cðgm Þ ¼
L X
Ci ðgm ÞZ
i¼1
L X
Ci ðgi Þ ¼
i¼1
L X
ai Ri ¼ RS ;
ð22Þ
i¼1
thus we have gm ZC 1 ðRS Þ. Conversely, for gS ¼ C 1 ðRS Þ, it is easily shown that gi ¼ gS ¼ C 1 ðRS Þ
and
Ri ¼
Ci ðgS Þ , ai
1rirL
ð23Þ
is a feasible solution to Eq. (21). From the above analysis, we conclude that Eq. (23) is the optimal solution to the rate allocation problem (21).
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2730
3.6.3. Solution using code rate chart We define the code rate chart of the component code i with respect to the SNR g as Ri ðgÞ ¼
Ci ðgÞ , ai
1rirL:
Also define the MI chart of the average code rate with respect to the SNR g as PL L X CðgÞ ai i ¼ 1 Ci ðgÞ RðgÞ ¼ PL ¼ P ¼ Ri ðgÞ, PL L i¼1 i ¼ 1 ai i ¼ 1 ai i ¼ 1 ai
ð24Þ
ð25Þ
which is a weighted average of the code rates of each component code according to the number of mapping bits of each layer. Given the average code rate R, the optimal rate allocation for each component code can be obtained by the following two steps.
Find the SNR g such that RðgÞ ¼ R. Allocate the rate Ri ¼ Ri ðgÞ to component code i for 1rirL.
We show the code rate chart of the 8-QAM using the same system setup for multi-level coding, where the average code rate is given by RðgÞ ¼
1 2 R1 ðgÞ þ R2 ðgÞ: 3 3
ð26Þ
The average code rate is obtained against the SNR Es =s2 where s2 is the power of the complex Gaussian noise. For the average code rate R ¼ 0:7, the SNR g ¼ 6 dB, and thus the rates allocated to the two component codes R 1 and R 2 are 0.5 and 0.8, respectively. See [14] for further information about how the system parameters involved can be classified as the modulation parameters and the code parameters. 4. Simulation results The simulation results are represented and discussed in this section. First we will present the structure of the implemented simulator, and then present the simulation results both in terms of validation of implementation and values for various parameters that characterize the performance of the WiMAX physical layer. We have developed the simulator in VCþþ program. Each block of the transmitter, receiver and channel is written in ‘‘.cpp’’ files. The main procedure call each of the blocks works in the manner a communication system works. The main procedure also contains initialization parameters, input data and delivers results. The parameters that can be set at the time of initialization are the number of simulated symbols, CP length, modulation and coding rate, range of SNR values. The input data stream is randomly generated. BER values for different SNR are stored in text files that facilitate drawing plots. We have presented various BER vs. SNR figures for all the mandatory modulation and coding profiles as specified in the IEEE 802.16 standard on the same channel models. The simulations were carried out for code rates, lengths and modulation schemes including zigzag-coded modulation. We first provide some simulations for the different FEC schemes in a noise channel. In Fig. 6, we show the performance of the general IEEE 802.16 standard. Performance of zigzag-coded scheme rate-3/4 for QPSK and 16-QAM modulation types better than
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2731
Fig. 6. BER performance comparison of different rate/modulation for fixed WiMAX.
Fig. 7. BER performance comparison for fixed WiMAX, R ¼ 1/2 and 3/4, N ¼ 256 and 512, respectively, in the case of 8-QAM.
2732
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
rate-1/2. However, its performance reverses for 64-QAM modulation. For the Turbo/ LDPC codes, coding with lower code-rate has generally larger coding gain for very long codelength such as 65 536 of codelength. The coding gain achievable with Turbo/LDPC is a direct function of the block size used for the encoding process. However, codelength is taken very short in order to decrease the computational complexity in our proposed system. Therefore, coding gain of the proposed coding scheme under the lower code-rate for low-level modulation types may be decreased. In Fig. 7, we depicted the BER versus the SNR for IEEE 802.16d Fixed WiMAX system using 8-QAM. The code lengths were N ¼ 256 and 512 bits, the code rates were R ¼ 1/2 and 3/4, respectively, and the modulation schemes were 8-QAM (QPSK as well as 16-QAM and 64-QAM). It can be seen that the RSþCC performs worst. The performance of the SL system and the ML system code for 8-QAM is quite good. Both ML and SL system need a round 1 dB gain between rate-1/2 and rate-3/4. In Fig. 8, we depicted the BER versus the SNR for IEEE 802.16d Fixed WiMAX system using ring 16-QAM. The code lengths were N ¼ 256 and 512 bits, the code rates
Fig. 8. BER performance comparison for fixed WiMAX, R ¼1/2 and 3/4, N ¼ 256 and 512, respectively, in the case of ring 16-QAM.
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
2733
were R ¼ 1/2 and 3/4, respectively, and the modulation schemes were ring 16-QAM (QPSK as well as 16-QAM and 64-QAM). It can be seen that the RSþCC performs worst. Performance increases between rate-1/2 and 3/4 using ring 16-QAM. Furthermore, neither the SL nor ML system has an error floor as LDPC has. In this work, we consider regular LDPC codes which an error floor. The simulated LDPC code is encoded by (512, 256) LDPC code. Finally, we present the simulation results for IEEE 802.16e Mobile WiMAX system. It is modeled on a single user allocating the whole bandwidth. The assumed channel was the well-known COST channel, modeling a mobile user in the typical urban environment moving with a relative velocity of 120 km/h to the base station at a carrier frequency of 2.5 GHz. The results for the different coding schemes are shown in Fig. 9. The rates and code lengths are the same as for the fixed WiMAX. For a BER of 105 it is obtained at a SNR of about 31 dB for RSþCC code with R ¼ 1=2 and QPSK modulation. It can be seen that higher modulation types give a result of an error floor, and BER performance of the proposed Zigzag scheme can be comparable with that of the RSþCC coding.
5. Conclusions FEC techniques are essential for reliable communication over a noisy channel. The effect of error occurring during transmission is reduced by adding redundancy to the data
Fig. 9. BER performance comparison of Mobile WiMAX (velocity¼ 120 km/h).
2734
S. Kahveci / Journal of the Franklin Institute 349 (2012) 2717–2734
prior to transmission. The redundancy is used to enable the decoder in the receiver to detect and correct errors. Concatenated zigzag codes like CC and RS codes are used efficiently for error detection and correction. The advantage of these codes is that they are easy to encode and process a well-defined mathematical structure, which has lead to very efficient decoding scheme for them. Also, the combination of zigzag code and M-QAM modulation owns a stronger error correcting capability. Therefore, the multi-level zigzag coding approach provides flexible transmission rates because it decouples the dimensionality of the signal constellation from the code rate and multi-level coding offers asymptotic coding gains.
References [1] C. Eklund, R.B. Marks, K.L. Stanwood, S. Wang, Ieee standard 802.16: a technical overview of the wirelessman air interface for broadband wireless access, IEEE Communications Magazine 40 (6) (2002) 98–107. [2] IEEE Std. 802.16-2004, Standard for Local and Metropolitan Area Networks. Part 16: Air Interface for Fixed Broadband Wireless Access Systems, October 2004. [3] C. Berrou, C. Doulland, et al., The Advantages of Non-binary Turbo Codes, 2001, pp. 61–63. [4] O.C.M. Youssou, G. Paul, K. Peter, Enhanced max-log-app and enhanced log-app decoding for DVB-RCS, in: Proceedings of the International Symposium on Turbo Codes, September 2003, pp. 259–262. [5] A. Bajwa, J. Awan, Performance Study of IEEE 802.16d (Fixed WiMAX). M.S. Thesis, Blekinge Institute of Technology, Karlskrona, Sweden, 2008. [6] M.S. Saiful, M.T. Alam, WiMAX: An Analysis of the Existing Technology and Compare with the Cellular Networks, M. S. Thesis, Blekinge Institute of Technology, Karlskrona, Sweden, 2009. [7] WiMAX Forum, Mobile WiMAX-Part 1: A Technical Overview and Performance Evaluation, 2006. [8] L. Ping, X.L. Huang, N. Phamdo, Zigzag codes and concatenated zigzag codes, IEEE Transactions on Information Theory 47 (February) (2001) 800–807. [9] L. Ping, S. Chan, K. Yeung, Iterative decoding of multi-dimensional concatenated single parity check codes, in: Proceedings of the IEEE ICC-98, June 1998, pp. 131–135. [10] C. Gong, X. Wang, Multilevel LDPC-coded high-speed optical systems: efficient hard decoding and code optimization, IEEE Journal of Selected Topics in Quantum Electronics 16 (September/October (5)) (2010) 1268–1279. [11] G. Caire, G. Taricco, E. Biglieri, Bit interleaved coded modulation, IEEE Transactions on Information Theory 44 (May) (1998) 927–946. [12] R.H. Morelos-Zaragoza, M.P.C. Fossorier, S. Lin, H. Imai, Multilevel coded modulation for unequal error protection and multistage decoding-part I: symmetric constellations, IEEE Transactions on Communications 48 (February (2)) (2000) 204–213. [13] I.B. Djordjevic, B. Vasic, Multilevel coding in M-ary DPSK/differential QAM high-speed optical transmission with direct detection, Journal of Lightwave Technology 24 (January (1)) (2006) 420–428. [14] S. Kahveci, C. Gong, X. Wang, Zigzag-coded modulation for high-speed fiber optical channels, Journal of Optical Communications and Networking 4 (May (5)) (2012) 382–391.