Sphere Constrained Block DFE with Per-Survivor Intra-Block Processing for CCK Transmission Over ISI Channels

Transcription

1 Sphere Constrained Block DFE with Per-Survivor Intra-Block Processing for CCK Transmission Over ISI Channels Christof Jonietz and Wolfgang H. Gerstacker Institute for Mobile Communications University of Erlangen-Nürnberg D Erlangen, Germany {jonietz, Robert Schober Department of Electrical and Computer Engineering University of British Columbia Vancouver, Canada Abstract In the wireless local area network (WLAN) standard IEEE b, complementary code keying (CCK) modulation has been adopted for the high data rate transmission mode. In this paper, complexity reduction for block decisionfeedback equalization (bdfe), tailored for CCK transmission over frequency-selective channels, is considered. Since the CCK signal may be viewed as a linear block code with respect to the chip phases of the codeword, a trellis diagram with a minimum number of states can be designed that represents the properties of the CCK code set. The Viterbi algorithm (VA) with persurvivor processing is applied to the CCK trellis for decoding and accounting for the inter-chip interference, while inter-codeword interference is canceled by decision feedback. The resulting scheme is denoted as bdfe-ps and has a significantly lower complexity than bdfe with brute-force search. By introducing a sphere constraint on the CCK trellis (SC-bDFE-pS), the complexity of bdfe-ps can be further reduced. Omitting trellis states that violate the sphere constraint, edges that emanate from such states can be pruned, and the average number of metric calculations per CCK trellis segment can be reduced. Simulation results show that the performance of bdfe-ps and SC-bDFEpS, respectively, is essentially equivalent to that of bdfe with brute-force search. I. INTRODUCTION Maximum-likelihood (ML) detection in an integer lattice is a common problem in wireless communications, and is also known as nearest neighbor search or closest point problem for a given point in the lattice [1]. The sphere decoding algorithm solves the ML detection problem by considering only lattice points that reside inside a sphere, resulting in a lower complexity compared to a brute-force search in the whole lattice, e.g. [2] [4]. It has been shown in [2] that turbo coding and iterative processing at the receiver combined with a sphere decoder (SD) can nearly achieve the capacity of a multiple-input multiple-output (MIMO) channel. In [3], joint ML detection and decoding of linear block codes over Gaussian vector channels is studied and a sphere constrained search is performed over lattice points that are valid codewords. A SD for frequency-selective channels is introduced This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under Grant No for uncoded transmission in [4], which can be implemented either with a depth-first or a breadth-first search strategy. The depth-first SD prunes the nodes of a search tree that lie outside the sphere. The ML path is obtained by successively extending those branches which lie inside the sphere. The breadth-first SD in [4] is based on a trellis description of frequency-selective channels. A sphere constraint is imposed to the Viterbi algorithm (VA), which omits states that violate that constraint. In both cases, computational complexity can be reduced by avoiding superfluous metric calculations. In this paper, sphere decoding assisted receivers for complementary code keying (CCK) transmission over frequencyselective fading channels are proposed. CCK is a multidimensional complex-valued modulation scheme, where each codeword consists of 8 quaternary phase-shift keying (QPSK) chips, and has been adopted for the high data rate transmission mode of the wireless local area network (WLAN) standard IEEE b [5]. In rich scattering environments, such as indoor office environments or densely built outdoor areas, multipath propagation causes intersymbol interference (ISI), and equalization is mandatory for high performance. Previously proposed equalization schemes for IEEE b can be found e.g. in [6] [9]. The scheme in [6] consists of an enhanced Rake receiver with a decision-feedback structure embedded in the channel matched filter and a codeword correlator. However, Rake receivers suffer in principle from an error floor for CCK transmission due to the non-orthogonality of the CCK codewords. Hence, for high performance the Rake principle should be avoided for CCK signals. In [7], decision-feedback equalization (DFE) for joint equalization and CCK decoding in a tree diagram for channels with large delay spreads is studied. In [8], combined equalization and decoding of CCK signals is performed on a compound supercode of the CCK code and the channel induced multipath interference using the Fano sequential decoding algorithm. In [9], various equalizers tailored for CCK signals have been presented and it has been shown that a block DFE (bdfe) performs well. This paper is concerned with complexity reduction of bdfe for CCK signals. Since CCK can be interpreted as a linear

2 block encoding scheme with respect to the chip phases, a trellis diagram can be derived from its generator matrix. Optimum decoding of trellis-encoded signals over frequencyselective channels is based on a single finite-state machine, which combines the code and ISI states [10]. However, good performance at lower complexity can be often also achieved with a reduced-state algorithm taking the code states fully into account but treating ISI in a per-survivor fashion in metric calculations [10]. Here, bdfe with full-state CCK decoding and per-survivor processing in order to account for the interchip interference in the current codeword is introduced (bdfeps). Inter-codeword interference is compensated through decision feedback. A sphere constrained version of bdfe-ps (SC-bDFE-pS) is considered to reduce the complexity further. The sphere constraint is applied to the CCK decoder trellis diagram of bdfe-ps. States that violate the sphere constraint are omitted and edges that emanate from such states are pruned, similarly to the breadth-first SD of [4]. Thus, certain unnecessary metric calculations can be avoided. It turns out that computational complexity can be reduced significantly in this way. However, it cannot be guaranteed in general that the proposed bdfe-ps and SC-bDFE-pS, respectively, exactly solve the bdfe detection problem in [9]. Nevertheless, numerical results show that the performance of the novel bdfe-ps and SC-bDFE-pS is essentially equivalent to that of bdfe in [9]. The paper is organized as follows. In Section II, CCK modulation is briefly introduced and the transmission model is established. In Section III, we recapitulate the bdfe detection problem for CCK signals and based on a trellis representation of the CCK code set the novel bdfe-ps and SC-bDFE-pS are introduced. Section IV provides numerical results for the proposed algorithms and conclusions are drawn in Section V. Notation: Bold lower case letters and bold upper case letters denote column vectors and matrices, respectively; ( ) T and ( ) H stand for transpose and Hermitian transpose, respectively; 0 K L is an all-zero matrix of size K L; [x] i is the ith element of vector x; x is the smallest integer x. II. SYSTEM MODEL A. Complementary Code Keying In WLAN IEEE b, CCK has been adopted for the transmission modes with data rates of 5.5 Mbit/s and 11 Mbit/s, respectively. In this paper, only the 11 Mbit/s mode is considered, since it is most relevant for high-rate applications. In this case, consecutive binary input data bits are partitioned into vectors of length N c =8, d =[d 1...d Nc ] T {0, 1} Nc. Based on d, a complex-valued CCK codeword c =[c 1...c Nc ] T is chosen from the CCK code set C with cardinality C =2 8 = 256, where c i =exp(jθ i ), θ i {0,π/2,π,3π/2}, i {1,...,N c }. Hence, each codeword c is composed of N c QPSK chips, where c 1 is transmitted first in time. The chip phase vector θ =[θ 1...θ Nc ] T is determined according to the linear block encoding rule [5] θ = Gϕ + θ, (1) where the generator matrix G is defined as G T = (2) The dibit phases in ϕ =[ϕ 1...ϕ 4 ] T are determined by the dibits {d 1,d 2 }, {d 3,d 4 }, {d 5,d 6 }, and {d 7,d 8 }, respectively, according to the mapping rule {0, 0} 0, {0, 1} π/2, {1, 0} π, and {1, 1} 3π/2. The cover code phase vector θ =[000π 00π 0] T rotates the fourth and seventh chips by π, respectively, in order to optimize the sequence correlation properties and to minimize the DC offset in the codewords [5]. It should be noted that in WLAN IEEE b forward error correction is not employed explicitly. However, CCK has a small coding gain due to the 1/2-rate block encoding of 4 dibit phases into 8 chip phases. B. Transmission Model The CCK encoder maps the binary input symbol sequence d[k] to a codeword sequence c[k] (k: discrete codeword time). The discrete-time received chips are given by r[µ]= q h l=0 h[l] c F [µ l]+n[µ] (3) (µ: discrete chip time), where h[l], l {0, 1,..., q h }, is the discrete-time overall channel impulse response (CIR) of order q h, and n[µ] is a discrete-time white Gaussian noise (WGN) process with variance σ 2 n. {c F[µ]} = {c 1 [1],c 2 [1],c 3 [1],...,c 6 [K],c 7 [K],c 8 [K]} denotes the transmit chip sequence of an entire packet (frame) consisting of K successive codewords c[k] with chips c i [k], i {1,...,N c }. Partitioning the received chip stream into vectors of length N c, we consider received vectors r[k] corresponding to transmitted codewords c[k] instead of single chips r[µ], cf. Fig. 1. Here, Q h r[k]= H[κ] c[k κ]+n[k], (4) κ=0 where H[κ] are finite impulse response (FIR) N c N c matrix filter taps specified below, Q h = q h /N c denotes the order of the matrix filter, and n[k] is a Gaussian vector noise process. The matrices H[κ] are implicitly given by the N c (Q h +1) N c column circulant matrix H, [ H T [0],H T [1],...,H T [Q h ] ] T = H, (5) where the ith column vector h i, i {1,...,N c}, ofh is defined as [ T, h i = 0 1 (i 1), h T, 0 1 (Nc (Q h +1) (q h +1) (i 1))] (6) and h =[h[0]...h[q h ]] T, cf. also [11].

3 n[k] d[k] CCK encoder c[k] H[κ] r[k] y[k] ĉ[k] CCK ˆd[k] F codeword f [κ] decision decoder a fb [k] F b [κ] Fig. 1. Block diagram of discrete-time equivalent baseband transmission model for a block DFE. III. SPHERE CONSTRAINED BLOCK DFE WITH PER-SURVIVOR INTRA-BLOCK PROCESSING In this section, we shortly recapitulate bdfe detection for CCK signals. We then show how the CCK code set can be described by a trellis diagram and introduce a trellis-based bdfe with per-survivor intra-block processing (bdfe-ps) in order to reduce the computational complexity of bdfe. The complexity of bdfe-ps can be further reduced by a sphere constraint on the trellis (SC-bDFE-pS). Finally, the complexity in terms of complex multiplications is discussed for bdfe, bdfe-ps, and SC-bDFE-pS. A. Block DFE (bdfe) Detection of CCK Signals Because block-coded signals are transmitted, DFE has to perform feedback of decisions at codeword level. In [9], bdfe for CCK transmission has been introduced. The feedforward (FF) filter F f [κ] transforms the end-to-end impulse response H[κ] into an overall filter impulse response H ov [κ], which has a higher energy concentration in the front part than H[κ], cf. Fig. 1. In the feedback (FB) section, the postcursor intercodeword interference corresponding to the overall forward impulse response is removed using a FB filter F b [κ], where F b [κ] =H ov [κ] for κ {1,...,Q h } and F b [κ] =0 Nc N c else. F f [κ] and F b [κ] are composed of scalar filters taps f f [l] and f b [l] in a similar way as H[κ] contains taps h[l], cf. Eqs. (5) and (6). The FF filter f f [l] and the FB filter f b [l] are jointly optimized according to a minimum mean-squared error (MMSE) criterion, cf. [9] and [12] for details. The decision rule of a bdfe is given by [9] ĉ[k]=argmin č[k] C { y [k] H ov [0]č[k] 2}, (7) where y [k] =y[k] a fb [k]. a fb [k] is the feedback vector, which cancels the inter-codeword interference in order to obtain the current codeword decision, and is calculated based on previously decided codewords as Q h a fb [k]= F b [κ]ĉ[k κ]. (8) κ=1 In order to solve the bdfe detection problem in (7) by a brute-force search as was proposed in [9] and [11], C = 256 Euclidean distances have to be calculated, which may be too complex for a practical implementation. In the following, we show how the computational complexity of bdfe can be reduced by utilizing the redundancy in the CCK code set together with the concepts of per-survivor processing and sphere decoding. B. Optimum Decoding of CCK Codewords using a Trellis The decoding of arbitrary linear block codes over the AWGN channel can be performed in a trellis diagram, similarly to convolutional codes [13]. For soft or hard decision decoding, respectively, the BCJR algorithm [14] or the Viterbi algorithm (VA) [15] can be used. In the following, a trellis representation of the CCK code set C is given, which is the basis for the the algorithms presented in the next sections. First, the generator matrix has to be transformed into a trellis-oriented [13] generator matrix, which is an equivalent generator matrix that generates the same codewords. By subtracting and interchanging lower rows and upper rows of the generator matrix G T in the finite field GF(4) [13], a trellis-oriented generator matrix can be obtained as G T m = (9) The construction of a trellis from a trellis-oriented generator matrix is well known from the literature, and we refer to [13] and [16]. In Fig. 2, the resulting trellis diagram for the CCK code set C obtained from (9) is shown. We note that a similar trellis diagram is also given in [8]. The trellis starts in a single state Z C 0, expands into 4, 16, and 64 states, contracts to 16 states, expands again to 64 states, and finally contracts to 16, 4, and 1 states Z C 8. The edges in Fig. 2 represent chip phases θ i {0,π/2,π,3π/2}, such that a CCK codeword corresponds to a unique path in the trellis diagram. Thus, C = 256 different paths result. C. bdfe with Per-Survivor Processing (bdfe-ps) In the following, a bdfe-ps receiver for decoding of CCK signals over frequency-selective channels is considered which has a significantly lower complexity than bdfe. For bdfe-ps, a VA with per-survivor processing is performed on the CCK trellis diagram in order to decode the current codeword and to take into account the inter-chip interference, which is caused by H ov [0]. Since H ov [0] is a

4 Z C 0 Z C 2 Z C 1 θ 1 θ 2 Z C 3 Z C 5 Z C 4 Z C 6 Z C 7 θ 7 θ 8 Z C 8 of the survivor path. However, the VA with per-survivor intrablock processing does not necessarily obtain the ML path in the CCK trellis diagram. In the 4th, 6th, and 7th CCK trellis step, cf. Fig. 2, selection of paths that lead into the same state is performed. However, each CCK chip influences also all future time instants in the codeword due to inter-chip interference. Thus, the ML path may be pruned before the final survivor path is obtained. Consequently, bdfe-ps in principle delivers a suboptimum solution to problem (7). D. Sphere Constrained bdfe with Per-Survivor Processing (SC-bDFE-pS) Fig. 2. θ 3 θ 4 θ 5 θ 6 Trellis diagram of the CCK code set. In the following, bdfe-ps is augmented by a sphere constraint leading to a significantly lower computational complexity compared to bdfe-sp and bdfe with brute-force search. Similarly to the SD for frequency-selective channels with breadth-first search strategy in [4], we apply a sphere constraint to the CCK trellis diagram of bdfe-ps, lower triangular matrix [9], [11], h ov [0] h ov [1] h ov [0]... 0 H ov [0] = , (10) h ov [7] h ov [6]... h ov [0] we start from i =1to calculate the sum cost M(ŽC N c ) with the VA. In the ith CCK trellis step and state ŽC i, a recursion step in the VA with per-survivor processing yields 1 { M(ŽC i )= min M(ŽC i 1 )+ {ŽC i 1 } ŽC i m(ˇθ i, ˆθ i 1 (ŽC i 1 ),...,ˆθ 1 (ŽC i 1 )) }, (11) where the minimization is performed over all trellis branches that emanate from states ŽC i 1 and lead to a successor state. m( ) denotes the branch metric given by Ž C i m(ˇθ i, ˆθ i 1 (ŽC i 1),...,ˆθ 1 (ŽC i 1)) = [y] i [a fb] i h ov [0] exp { } jˇθ i i 1 } h ov [κ] exp {jˆθ i κ (ŽC i 1 ) 2 κ=1 (12) (ˆθ i 1 (ŽC i 1 ),...,ˆθ 1 (ŽC i 1 ): path register content of survivor path of state ŽC i 1 ), i.e., per-survivor processing is applied in order to estimate the inter-chip (intra-codeword) interference term, using state-dependent decisions taken from the path history associated with a previous state ŽC i 1. Inter-codeword interference is taken into account by the decision-feedback term [a fb ] i. An estimate for the transmitted codeword is obtained by reading out the chip phase hypotheses ˆθ 1,...,ˆθ Nc 1 For simplicity, the discrete codeword time k has been dropped in (11) and (12), because a new trellis is set up for each received vector. ˇ stands for hypothetical symbols and states in the VA. M(ŽC i ) R2, i {1,...,N c }. (13) The states with sum costs M(ŽC i ) greater than the squared sphere radius are omitted in the algorithm and the edges that emanate from such states are pruned. Thus, the average number of metric calculations is reduced compared to bdfesp. Problem (11) with sphere constraint (13) has either the desired bdfe-ps solution or no solution, depending on the choice of R. If the sphere radius is too small, all states of the CCK trellis may have been pruned before the algorithm has converged. In this case, the decoding fails and the algorithm restarts with a larger sphere radius given by R 2 +ΔR 2. For the Gaussian noise assumption, the initial squared sphere radius is determined as R 2 = νn c σn, 2 where ν is chosen such that at least one survivor path is found within the CCK trellis with a high probability. Assuming ideal feedback a fb [k], ǫ[k]= y [k] H ov [0]c[k] 2 = n[k] 2 is a chi-square random variable and the probability of a decoding success P dec (ν,n c ) is determined by the chi-square distribution [4]. The complexity of SC-bDFE-pS may be further reduced by introducing an increment Δr 2, by which the sphere radius is successively increased in each trellis step, R 2 i = R 2 i 1 +Δr 2, R 2 0 = R 2, (14) and the sphere constraint is applied on the sum cost as M(ŽC i ) R 2 i. (15) Thus, if Δr 2 is chosen appropriately the probability that at least one path is found within the CCK trellis increases, resulting in a decreased probability for a decoding failure 1 P dec (ν,n c ). However, it is not guaranteed that the final survivor path corresponds to the path which is obtained with bdfe-ps, because the latter may be pruned before the algorithm converges. Hence, a further performance loss cannot be excluded in principle.

5 TABLE I JTC OFFICE-C POWER DELAY PROFILE (CHIP TIME DURATION: T c =90.1 NS) Path Excess Delay Rel. Att. Nr. (ns) (db) NSC-bDFE-pS/NbDFE-pS E. Computational Complexity of bdfe, bdfe-ps, and SCbDFE-pS In the following, the fixed complexity of bdfe with a bruteforce search and bdfe-ps is given. The complexity of SCbDFE-pS is a random variable, because the number of omitted states in the CCK trellis depends on random noise samples, cf. (12). However, a lower complexity bound for SC-bDFE-pS can be specified. For a full lower triangular matrix H ov [0], the complexity for a single CCK trellis is provided in terms of complex multiplications. The complexity of bdfe with brute-force search in [9] corresponds to N bdfe = C (Nc 2 + N c )/2 +N fb complex multiplications, where the complexity for the feedback vector calculation N fb is given by N fb = Q h Nc 2 complex multiplications, cf. (8). For bdfe-ps (full CCK trellis search using the VA with persurvivor processing) the number of required complex multiplications in the CCK trellis can be obtained as N full = 1332 by observing the CCK trellis in Fig. 2 and (12). Thus, the overall complexity of bdfe-ps is determined as N bdfe-ps = N full + N fb complex multiplications. A lower bound for the computational complexity of SC-bDFEpS can be found by assuming optimum pruning of the edges in the CCK trellis. Optimum pruning means that there is only one edge left in each CCK trellis step. Here, the number of complex multiplications in the CCK trellis can be obtained as N opt = 69. Thus, a lower bound for the complexity of SC-bDFE-pS is given as N LB = N opt + N fb complex multiplications. An average complexity N SC-bDFE-pS of SCbDFE-pS can be obtained by simulations, where N SC-bDFE-pS = N pruned + N fb and N pruned denotes the average number of complex multiplications in the pruned CCK trellis. IV. SIMULATION RESULTS In this section, numerical results for bdfe-ps and SCbDFE-pS are presented. We consider CCK transmission over a block Rayleigh fading channel (random impulse response, possibly varying from packet to packet, but being constant within each packet). The channels are generated according to the JTC Office-C power delay profile [17] depicted in Table I, which corresponds to a severely frequency-selective channel. For continuous-time transmit filtering and receiver input filtering, respectively, a square-root raised cosine frequency response log 10 (E b /N 0 ) [db] bdfe-ps SC-bDFE-pS: R 2 =5, ΔR 2 =5, Δr 2 =0 SC-bDFE-pS: R 2 =4, ΔR 2 =5, Δr 2 =0 SC-bDFE-pS: R 2 =3, ΔR 2 =5, Δr 2 =0 SC-bDFE-pS: R 2 =1, ΔR 2 =5, Δr 2 =0 Lower Complexity Bound of SC-bDFE-pS Fig. 3. Normalized complexity versus E b /N 0 for SC-bDFE-pS with different initial sphere radii and no sphere radius increment (Δr 2 =0), bdfe-ps, and a lower complexity bound for SC-bDFE-pS, respectively. with a roll-off factor of 0.3 is assumed. One packet consists of K = 1000 CCK codewords. Perfect channel knowledge is assumed at the receiver. In the following, the complexity of SC-bDFE-pS is analyzed in terms of the average number of complex multiplications required to detect a single CCK codeword. For the JTC Office-C channel profile Q h =5is valid. Thus, for the calculation of the feedback vector, N fb = Q h N 2 c = 320 complex multiplications are required. In this case, the complexity of bdfe with brute-force search in [9] and bdfe-ps corresponds to N bdfe = C (N 2 c + N c )/2+N fb = 9536 and N bdfe-ps = N full + N fb = 1652 complex multiplications, respectively. Thus, bdfe-ps is only approximately 18 % as complex as bdfe with brute-force search. The lower complexity bound for SC-bDFE-pS is given as N LB = N opt + N fb = 389 complex multiplications, which is only 24 % of the complexity of bdfe-ps in this case. In Fig. 3, the average number of complex multiplications of SC-bDFE-pS normalized to the number of complex multiplications of bdfe-ps, N SC-bDFE-pS /N bdfe-ps, is depicted versus E b /N 0 (E b : average received energy per bit, N 0 : singlesided power spectral density of the underlying continuoustime passband noise process). Additionally, the normalized complexity of bdfe-ps and the lower complexity bound for SC-bDFE-pS are shown (horizontal lines). For SC-bDFE-pS, initial sphere radii of R 2 {1, 3, 4, 5} have been chosen, whereas there has been no sphere radius increment, Δr 2 =0. In case of a decoding failure the sphere radius is increased by ΔR 2 =5each time. Obviously, the average complexity

6 NSC-bDFE-pS/NbDFE-pS BER log 10 (E b /N 0 ) [db] bdfe-ps SC-bDFE-pS: R 2 =5, ΔR 2 =1, Δr 2 =0 SC-bDFE-pS: R 2 =5, ΔR 2 =4, Δr 2 =2 SC-bDFE-pS: R 2 =5, ΔR 2 =3, Δr 2 =2 SC-bDFE-pS: R 2 =5, ΔR 2 =2, Δr 2 =2 SC-bDFE-pS: R 2 =5, ΔR 2 =1, Δr 2 =2 SC-bDFE-pS: R 2 =3, ΔR 2 =1, Δr 2 =2 Lower Complexity Bound of SC-bDFE-pS Fig. 4. Normalized complexity versus E b /N 0 for SC-bDFE-pS for different radius increments Δr 2, bdfe-ps, and a lower complexity bound for SCbDFE-pS, respectively. of SC-bDFE-pS is similar to that of bdfe-ps for low E b /N 0 values. This is because the sphere radius has to be selected relatively large due to the high noise variance. The average complexity of SC-bDFE-pS decreases for increasing E b /N 0 values and for high E b /N 0 values it reaches a floor. The average complexity of SC-bDFE-pS becomes smaller when the initial sphere radius is decreased from R 2 =5to R 2 =3. This is related to the fact that the number of pruned edges in the CCK trellis increases for a decreasing sphere radius. For high E b /N 0 values, the average complexity of SC-bDFE-pS corresponds to the lower complexity bound for optimal trellis pruning in most cases. However, for an initial sphere radius of R 2 =1, a higher complexity results compared to bdfe-ps at low E b /N 0 values because the probability for a decoding failure 1 P dec (ν,n c ) increases as the sphere radius decreases. Therefore, in contrast to SC-bDFE-pS with R 2 {3, 4, 5}, SC-bDFE-pS with R 2 =1 also does not reach the lower complexity bound for high E b /N 0. We conclude that the initial sphere radius has to be chosen appropriately for a low complexity over the whole E b /N 0 range. For a relatively small initial sphere radius, the probability for decoding failure is high which results in a high number of restarts of the algorithm. If the initial radius is too large, the number of omitted states in the CCK trellis is small and the complexity reduction is limited. Furthermore, the complexity of SC-bDFE-pS may exceed the complexity of bdfe-ps for low E b /N 0 values, when the initial sphere log 10 (E b /N 0 ) [db] bdfe (brute-force search) [9] bdfe-ps SC-bDFE-pS: R 2 =5, ΔR 2 =1, Δr 2 =2 SC-bDFE-pS: R 2 =3, ΔR 2 =1, Δr 2 =2 PA ML detection Fig. 5. BER versus E b /N 0 for SC-bDFE-pS, bdfe-ps, bdfe, and ML sequence detection, respectively. radius is not chosen appropriately. In Fig. 4, the normalized complexity of SC-bDFE-pS versus E b /N 0 is depicted. The initial sphere radius is R 2 =5and is increased by ΔR 2 {1, 2, 3, 4} after a decoding failure. Additionally, SC-bDFE-pS with R 2 =3, ΔR 2 =1, Δr 2 =0 is shown. It can be observed that the complexity of SC-bDFEpS can be further reduced by introducing a radius increment of Δr 2 =2for each CCK trellis step. For ΔR 2 =1and without sphere radius increment (Δr 2 =0), the complexity of SC-bDFE-pS is even greater than that of bdfe-ps for low E b /N 0 values. Also, the lower bound is not reached for high E b /N 0 values in this case. In contrast, the complexity of SCbDFE-pS with ΔR 2 =1and Δr 2 =2corresponds to 75 % of that of bdfe-ps for E b /N 0 0 db, and for high E b /N 0 values the lower complexity bound is reached. For SC-bDFEpS with R 2 =3, ΔR 2 =1, and Δr 2 =2, the complexity is lower than that for R 2 =5for E b /N 0 < 5.5 db and for E b /N 0 =0dB a complexity of 63 % results compared to that of a bdfe-ps. This is (again) related to the increasing number of pruned edges in the CCK trellis when the initial sphere radius decreases. For high E b /N 0 values, the complexity of SC-bDFE-pS with R 2 =3does not reach the lower bound and is larger than that with R 2 =5, again due to decoding failures. The performance of bdfe in [9], bdfe-ps, and SC-bDFEpS in terms of bit error rate (BER) versus E b /N 0 is depicted in Fig. 5. Based on the pairwise error probability of the CCK codewords, a performance approximation (PA) of the BER for highly complex optimum ML sequence detection [11] is also shown. The performance of SC-bDFE-pS with all parameters considered in this paper (see Figs. 3 and 4) is essentially equal

7 and also equal to that of bdfe and bdfe-ps. Therefore, only two curves for SC-bDFE-pS with R 2 =5, ΔR 2 =1, Δr 2 =2 and R 2 =3, ΔR 2 =1, Δr 2 =2 are shown. It can be concluded that with bdfe-ps and SC-bDFE-pS a significant complexity reduction can be achieved compared to bdfe with brute-force search without sacrificing performance if the parameters of SC-bDFE-pS are chosen appropriately. V. CONCLUSION Complexity reduction for block decision-feedback equalization (bdfe) tailored for complementary code keying (CCK) transmission over frequency-selective channels has been considered. Based on a CCK trellis diagram the Viterbi algorithm with per-survivor intra-block processing (bdfe-ps) has been introduced, which has only 18 % of the complexity of bdfe with brute-force search for a typical WLAN scenario. A significant further complexity reduction can be achieved by introducing a sphere constraint to the CCK trellis (SCbDFE-pS), resulting in a complexity of 63 % and 24 % of that of a bdfe-ps for low and high signal-to-noise ratios, respectively. bdfe-ps and SC-bDFE-pS yield essentially the same performance as bdfe with brute-force search. REFERENCES [1] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, Closest Point in Lattices, IEEE Transactions on Information Theory, vol. 48, no. 48, pp , August [2] B. Hochwald and S. t. Brink, Achieving Near-Capacity on a Multiple- Antenna Channel, IEEE Transactions on Communications, vol. 51, no. 3, pp , March [3] H. Vikalo and B. Hassibi, On Joint Detection and Decoding of Linear Block Codes on Gaussian Vector Channels, IEEE Transactions on Signal Processing, vol. 54, no. 9, pp , September [4] H. Vikalo, B. Hassibi, and U. Mitra, Sphere Constrained ML Detection for Frequency Selective Channels, IEEE Transactions on Communications, vol. 54, no. 7, pp , July [5] IEEE Standard b, Part 11: Wireless LAN Medium Access Control and Physical Layer Specifications Higher Speed Physical Layer Extension in the 2.4 GHz Band, [6] M. Webster, R. Nelson, K. Halford, and C. Andren, Rake Receiver with Embedded Decision Feedback Equalizer, May 2001, United States Patent, No.: US 6,233,273 B1. [7] M. Ghosh, Joint Equalization And Decoding for Complementary Code Keying (CCK) Modulation, in Proceedings of IEEE International Conference on Communications, Paris, June [8] C. Heegard, S. Coffey, S. Gummadi, E. Rossin, M. Shoemake, and M. Wilhoyte, Combined Equalization and Decoding for IEEE b Devices, IEEE Journal on Selected Areas in Communications, vol. 21, no. 2, pp , Feb [9] C. Jonietz, W. Gerstacker, and R. Schober, Transmission and Reception Concepts for WLAN IEEE b, IEEE Transactions on Wireless Communications, vol. 5, no. 12, pp , Dec [10] P. Chevillat and E. Eleftheriou, Decoding of Trellis-Encoded Signals in the Presence of Intersymbol Interference and Noise, IEEE Transactions on Communications, vol. 37, no. 7, pp , July [11] C. Jonietz, W. Gerstacker, and R. Schober, Receiver Concepts for WLAN IEEE b, in Proceedings of the 5th Int. ITG Conference on Source and Channel Coding (SCC), Erlangen, Jan. 2004, pp [12] J. Cioffi, G. Dudevoir, M. Eyuboglu, and G. Forney, Jr., MMSE Decision-Feedback Equalizers and Coding Parts I and II, IEEE Transactions on Communications, vol. 43, no. 10, pp , Oct [13] R. McEliece, On the BCJR Trellis for Linear Block Codes, IEEE Transactions on Information Theory, vol. 42, no. 4, pp , July [14] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate, IEEE Transactions on Information Theory, vol. 20, no. 3, pp , March [15] J. Proakis, Digital Communications. Boston, USA: McGraw Hill, [16] C. Schlegel and L. Perez, Trellis and Turbo Coding. New Jersey, USA: Wiley-IEEE Press, [17] Joint Technical Committee on Wireless Access, Final Report on RF Channel Characterization, Sept