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Channel Estimator and Aliasing Canceller for Equalizing and Decoding Non-Cyclic Prefixed Single-Carrier Block
Transmission via MIMO-OFDM Modem Shau-Yu Cheng, Chueh-An Tsai, and Terng-Yin Hsu
Abstract—Without a cyclic prefix (CP), most single-carrier (SC) trans-missions can not adopt frequency-domain equalizer (FDE) directly. This work utilizes frequency-domain channel estimator (FD-CE) and deci-sion-feedback aliasing canceller (DF-AC) to produce single-FFT SC-FDE. In this way, non-CP single-carrier block transmission (SCBT) can be decoded using sphere decoder of MIMO-OFDM modems to support multimode and backward compatibility under an acceptable complexity in IEEE 802.11 very high throughput (VHT). An -point FFT is sufficient to measure channel frequency responses (CFR) from -sample preambles ( 2). And then, -bit block codes( ) are decodable over frequency domains with DF-AC’s help. Simulations and measurements imply that this work can ensure adequate performance, even if there is no CP existed against the distortions of multipath propagation.
Index Terms—Aliasing canceller, channel estimator, compatibility, multi-mode, single-carrier block transmission (SCBT), single-carrier frequency-domain equalization (SC-FDE).
I. INTRODUCTION
In most of the wireless broadband applications like WiMax, Wifi, UWB and WRAN, compensation for multipath fading is highly pointed in order to make systems work properly. With the help of a cyclic prefix (CP), single-input single-output orthogonal frequency-di-vision multiplexing (SISO-OFDM) and multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) systems can be easily integrated. In addition, this can help estimate channels and equalize packets over frequency domains directly. Although single-carrier block transmission (SCBT) without any CP gains throughput, it causes fast Fourier transformation (FFT) aliasing in frequency-domain equalizer (FDE)—FDE can not assure sufficient performance directly. Thus, one of the major challenge for multi-mode integrations is to make equalizers as compact as possible, i.e.,
Manuscript received July 23, 2008; revised December 29, 2008. First pub-lished October 06, 2009; current version pubpub-lished December 27, 2010.This work was supported in part by the National Science Council of Taiwan, under Grant NSC 98-2220-E-009-004. This work was conducted under “A plan to actively participate in international standard organizations for wireless commu-nications” of the Institute for Information Industry, MOEA.
The authors are with the Department of Computer Science, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: tyhsu@cs.nctu.edu.tw; shauyu. cheng@msa.hinet.net; catsai@csie.nctu.edu.tw).
Digital Object Identifier 10.1109/TVLSI.2009.2031137
Fig. 1. Block diagram of the proposed single-FFT SC-FDE for sharing with an MIMO-OFDM modem.
consolidation of non-CP SCBT, SISO OFDM and MIMO OFDM. Reconfigurable and scalable architectures [1], [2] with heterogeneous units are good solutions to support such operations. However, multiple equalizers are built in designs. A single-carrier frequency-domain equalizer (SC-FDE), being an attractive solution, is developed to eliminate FFT aliasing without a circular property in some approaches [3], [4], e.g., overlap-and-save and overlap-and-add methods. Yet, additional DFT units were included. The user defined formats [5]–[7], inserting CP into single-carrier datum against the multipath propa-gation, were created to improve performance. SC-FDEs with a pair of FFT and IFFT for IEEE 802.16 [8] and IEEE 802.15.3a with an impulse option [9] were developed to demodulate SCBT with CP over time domains. In the case of single-carrier transmissions with pseudo noise (PN) spreading, a block-based SC-FDE with both DFT and IDFT for HIPERLAN-2 [10] was utilized to yield an additional 3-dB gain, and it was also demodulated in the time domain.
The objective of this study is to derive single-FFT processes for sup-porting multimode and backward compatibility under an acceptable complexity in MIMO-OFDM modems, such as IEEE 802.11 very high throughput (VHT), with a non-CP SCBT. Fig. 1 displays the block di-agram of this work, where the sphere decoder (SD) [11]–[13] is widely adopted in MIMO-OFDM modems. All equalizations and decoding are performed over the frequency domain. AnN-point FFT is sufficient to processL-sample preambles and M-bit block codes (M L N=2).
The remainder of this paper is organized as follows. The system as-sumptions with problem statements are addressed in Section II. The proposed single-FFT processes are described in Section III. Perfor-mance evaluations are presented in Section IV. Implementations and complexity are discussed in Section V. Finally, Section VI presents our conclusions.
II. SYSTEMASSUMPTIONS A. System Descriptions
Indoor frequency-selective fading, e.g., IEEE [14] and JTC [15], is assumed to roughly span two symbols ofL-sample preambles and M-bit block codes. The proposed packet format of non-CP SCBT is that datum without FEC are encoded by block code (K code sets con-tainM bits), modulated by QPSK and synchronized via L-sample preambles(L M). The jthreceived signal can be expressed as
rj= HLsj+ HUsj01+ n0 (1)
whererj is thejth 2L 2 1 received vector, n0 is2L 2 1 complex additive white Gaussian noise (AWGN) vector with variance2n, and Sjis thejthL 2 1 transmitted symbol, e.g., preambles, single-carrier (SC) block codes, or multicarrier datum withN 0 L zeros. H2L22L is the circulant Toeplitz matrix with the first column beingh and h = [h0; h1; . . . ; h2L01] denotes a channel impulse response (CIR). Both
HLandHUare the lower triangle (including diagonal entries) matrix
and upper triangle matrix ofH2L22L, respectively
H2L22L= h0 h2L01 1 1 1 1 1 1 h1 h1 h0 . .. . .. ... .. . h1 . .. h2L01 h2L02 .. . ... . .. h0 h2L01 h2L01 h2L02 1 1 1 h1 h0 : (2) B. Problem Statements
Linear convolution of two finite sequences is commonly conducted by multiplying the fast Fourier transforms of these two zero-padding input sequences. In SISO-OFDM and MIMO-OFDM systems, the channel frequency response (CFR) can be obtained from the FFT of the received symbol with CP divided by the FFT of the transmitted sequences [16]
Rk= Hk1 Ck+ nk) ^Hk
RCk
k; k 2 f0; 1; . . . ; N 0 1g (3)
whereN is the FFT size, Ckis thekthelement of the frequency re-sponse of transmitted sequence,Hkis thekthelement of real CFR, and ^Hkis thekth element of estimated CFR. In frequency-selective fading, the CFR is not an identity matrix. One problem associated with single-FFT processes for non-CP SCBT is to extend the FFT size in order to be enough to cover the entire packet. For a finite FFT size, it is impossible to gain a linear convolution of non-CP symbols and channel directly. Another consideration is that most zero-forcing equalizers are adversely affected by noise enhancement due to CFR with zeros or very small values (deep fading) [13], [15].
III. SINGLE-FFT PROCESSES A. Frequency-Domain Channel Estimator
Based on the received vector in (1), the linear convolution of the jthsymbol and the CIR can be deconstructed into two parts: one is
body, denoted byHLsj, and the other isHUsj, caused by interblock interference (IBI). In receivers, the body part of the current symbol convolved with the CIR is mixed with the IBI term of previous symbols in each FFT window. These two components can be separated for linear convolution over the time domain, namely preamble reconstruction. Taking all possibilities of QPSK modulations, the preambles (training symbols) are multiplied by complex coefficient. If “1” is the basis, the other three cases,i, 01 and 0i can be written as c21 1, c31 1, and
c41 1, where c2isi, c3is01 and c4is0i. Two useful combinations
of convolved vectors are y0= c
11 HLb + c21 HUb
y00= c
31 HLb + c41 HUb; c16= c3or c26= c4 (4)
whereb is the L 2 1 base vector of modulated preambles. Via (1), the received signals can be rewritten as
r0= y0+ n0= c
11 HLb + c21 HUb + n0
r00= y00+ n00= c
31 HLb + c41 HUb + n00 (5)
and the body part and the IBI term can be found by HLb = c 1r 0c 1rc c 0c c +0c 1n +c 1nc c 0c c
HUb = c 1r 0c 1rc c 0c c +0c 1n +c 1nc c 0c c :
(6)
In (6), each second term is also AWGN. To reduce AWGN effects, HLb and HUb can be obtained as
HLb = c 1Ec c 0c c[r]0c 1E[r ]=c 1y 0c 1yc c 0c c
HUb = c 1Ec c 0c c[r ]0c 1E[r]=c 1y 0c 1yc c 0c c
(7)
whereE [r0] and E [r00] can be individually calculated through aver-aging the two sets of the received preambles that collect two specific combinations of current and previous symbols defined in (5). After av-eragingr0andr00 and solvingHLb and HUb, the CFR can be
mea-sured via one-tap division. The frequency response is ^
HN[k] = YBN[k]
N[k] (8)
whereYN = FN([ (HLb)T (HUb)T]T)ze,BN = FN(b)ze,FN is anN2N Fast Fourier matrix with entries given by Fp;q= e2pqi=N, ()zerepresents zero padding for the length ofN and N 2L. Fur-thermore,c41 y00 c21 y00 andc11 y000 c31 y0are the functions of preamble reconstruction.
B. Decision-Feedback Aliasing Canceller
Based on decision feedback, thejthreceived datum (in frequency domains) can be rewritten as
R0j = Rj0 FN(HU^sj01)ze
= FN(HLsj)ze+ FN(n0)ze (9) whereRj FN(rj)zeand^sj01is the previous decoded symbol. Sim-ilar to MIMO detections, if a maximum-likelihood (ML) decision [17] is applied to decode datum over the frequency domain, the maximum a priori (MAP) equals
^sj = arg min d 2 R
0
j0 FN(HLdj)ze2 (10) where indicates the code set anddjis anM-bit block code. Reducing the complexity of implementations and sharing with a MIMO-OFDM modem, an ML decision is replaced by SD. Although CFR ^HNis es-timated using linear convolution with a preamble reconstruction, the body part and the aliasing term of each ideal code set must be separated for decoding references. Instead of re-transforming to time domains for HLdj andHU^sj01, a separation operator ~G is constructed to extract FN(HLdj)zevia the linear convolution ofdjandh
FN(HLdj)ze= ~GFN(h dj)ze
= ~GDHFN(dj)ze (11) FN(HU^sj)ze= DHFN(^sj)ze0 FN(HL^sj)ze (12) where is the linear convolution and DH = diag ^HN[0] ; ^HN[1] ; . . . ; ^HN[N 0 1] is a diagonal
matrix containing the CFR. After the channel estimation, multiply FN(dj)zebyDH, and pass through the separation operator ~G—A new coefficientFN(HLdj)ze is produced for sphere decoding. In the time domain, the body part is easily derived by taking the first L of linear convolution
HLdj = G 1 (h dj)
G = 0 IL2L 0L2(N0L)
(N0L)2L 0(N0L)2(N0L) : (13)
Then separation operator ~G can be obtained by (11) and (13) ~
Fig. 2. Power delay profiles for JTC and IEEE channel models. (a) and (b): Two random cases of IEEE model (RMS delay spread is 100 ns). (c): JTC indoor office B model. (d): JTC indoor residential C model.
Fig. 3. Simulation of the linear (8, 4) code in IEEE and JTC fading.
Fig. 4. Simulation of CCK in JTC fading—office A, residential C and office B with residual CFOs.
Each entry value of ~G is constant because the matrices of FN,G and F01
N are fixed. After cancellation of the aliasing termFN(HU^sj01)se
of the previous decoded symbol^sj01,sj is decodable over frequency
domains. In the case of decoding the1stdata symbol,FN(HU^sj01)se is acquired prior from the packet header.
IV. PERFORMANCEEVALUATIONS
The WiFi systems and a linear block codes are used to evaluate the proposed SC-FDE (without FEC), where the FFT size is 64 and the number of preambles is 56. Each preamble, modulated by BPSK, is spreading with 11-chip Barker code(L = 11) [14]—two useful com-binations of preambles arefc1 = 1; c2 = 1g and fc3 = 1; c4 =
01g. Only one antenna is utilized to receive the non-CP SCBT in the MIMO-OFDM modem. The basis of our performance is the packet error rate (PER) that is required to be 8% in frequency-selective fading. Fig. 2 displays the power delay profiles of JTC [15] and IEEE 802.11 [14] (Naftali model: Rayleigh fading with phase distributed uniformly) channel models, which are used to measure the system performance in multipath environment. The packet length is 1024 bytes, encoded by linear (8, 4) code [18] with BPSK modulation and complementary code keying (CCK) [14] with QPSK modulation, respectively. The generator matrix of a linear (8, 4) code is given by
H(8;4)= 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 1 1 1 0 0 0 0 : (15)
The minimum hamming distance is 4. In WiFi systems, each CCK codeword (8-bit orthogonal block code) is composed of four phases '1,'2,'3and'4off0; =2; ; 3=2g (M = 8).
Fig. 3 plots the PER of the linear (8, 4) code in both JTC and IEEE 802.11 fading channels. Compared with the case of AWGN, our SNR losses are around3:6 dB 5.6 dB, depending on fading environ-ments. In Fig. 4, the proposed scheme has improved performances com-pared with some time-domain extents [19], [20]. The simulation results of channel model of JTC indoor office A indicates that the proposed scheme performs better than previous study [19], because the nonlinear equalization employs the sphere decoding with the search of minimum Euclidean distance over transmitted symbols. In the case of JTC indoor residential B, a CFO of 50 ppm with 1-ppm and 2-ppm residual errors caused by automatic frequency control (AFC) is induced in systems. In contrast with the ICI equalizer [20], this proposal also yields an im-provement of 20-dB, indicating that the impact of residual CFOs does not make significant performance degradation. By transferring RF sig-nals to MATLAB via USB for real-time measurements before VLSI implementations, XilinxDSP Development Kits with on-board 14-bit A/Ds, 14-bit D/As and FPGA (2-million gates and 50 MHz) are con-nected to an in-house 2.4 GHz 22 2 RF module with 20-MHz band-width to transmit and record real wireless packets. In this way, a soft-ware-defined radio (SDR) platform is shown in Fig. 5. The received EVM of QPSK is about021 dB.
V. IMPLEMENTATIONS ANDCOMPLEXITY
Figs. 6 and 7 show the block diagrams of the proposed FD-CE and single-FFT SC-FDE with DF-AC, respectively. Five key modules are derived in the FD-CE: (1) a preamble reconstruction with pattern recog-nition for pre-processing linear convolution before FFT; (2) a look-up table (LUT) for storing the ideal frequency responses of preamble; (3) complex multipliers with conjugate output for calculating the linear convolution; and, (4) two averages for reducing AWGN effects. The single-FFT SC-FDE with DF-AC (Fig. 7) contains five major building blocks: (1) an DF-AC to eliminate the FFT aliasing of non-CP sym-bols; (2) a sphere decoder to decode datum over the frequency domain without noise enhanced; (3) an LUT to save ideal frequency responses
Fig. 5. SDR platform for SCBT measurements.
Fig. 6. Block diagram of the proposed FD-CE.
Fig. 7. Block diagram of the single-FFT SC-FDE with DF-AC.
ofM-bit block codes as decoding references; and, (4) complex multi-pliers (shared with FD-CE) to generate new references for sphere de-coding.
Fig. 8 displays the detail architecture of a 42 4 MIMO-OFDM modem with the proposed SC-FDE supporting two kinds of packet formats in WiFi systems: (1) DSSS-CCK and (2) MIMO-OFDM. The MIMO-OFDM modem was implemented via hardware-description language (HDL) and mapped on FPGA of a XilinxDSP Development Kit. In Fig. 8, one of the four FFTs based on variable-length FFT architecture [21] supports 32 and 64 points, transferring the recon-structed data to the frequency domain. The preamble reconstruction extracts both body and ISI terms of Barker code to reconstruct linear convolution before FFT. By multiplying the frequency re-sponse of reconstructed data with the reciprocal of the frequency response of an ideal preamble (Barker code; pre-stored in ROM1), the channel frequency response (CFR) is extracted. Once the CFR is obtained, the ALU uses the four complex multiplier banks to calculate DHFN(dj)ze andFN(HLdj)ze = ~GDHFN(dj)ze for all dj, wheredj is the frequency response of an ideal CCK code and ~G is the matrix operator (both saved in the ROM2). For all CCK codes,
each output of DHFN(dj)ze and the body (FN(HLdj)ze) and aliasing terms(FN(HUdj)ze) are stored in SRAM1 and SRAM2,
respectively. If the received signals are data symbols, CE flag is set to 0. Then, the received signals subtract from an aliasing term with the feedback code index in SRAM2 to get the body term of the received signals. Finally, the sphere decoder with four SD engines [22] decodes the body term of received signals and feeds back the decision for next decoding.
The bottlenecks of the data path are the computations of FN(HLdj)ze = ~GDHFN(dj)ze and DHFN(dj)ze for all
dj, wheredj is the ideal frequency response of thejth CCK code and ~G is the matrix operator (both saved in the ROM2). Because only the code set with zero phase ones needs to be calculated and ~G is the circular matrix with the first column[~g0; ~gN01; . . . ; ~g2; ~g1], (~g = 0
ifi is even; ~gi = ai+ bii and ~gN0i = ai0 bii for 1 i 15), the computing complexity of DHFN(dj)ze and FN(HLdj)ze
of 256 CCK codes can be reduced to32364 and 9332364 complex multiplications, respectively. To balance with processing time, clock rate and complexity, there are 32 complex multipliers to build 4 multiplier banks, shared with MIMO OFDM and SCBT. At the top right of Fig. 8, it shows the computing order of 32 complex multipliers forFN(HLdj)ze in DSSS-CCK modes.DHFN(dj)ze and each body/aliasing term are then stored in SRAM1 and SRAM2, respec-tively. The processing time of DHFN(dj)ze and FN(HLdj)ze is 12.8 us or 13 Barker codes at 50-MHz clock rate. Due to packet detection and synchronizations, the number of available preambles for preamble reconstruction is about 27.
Furthermore, the classical SD for MIMO detection includes two stages: 1) the preprocessing of QR decomposition and 2) a tree search of depth-first branch and bound (B&B) algorithm. In order to support both MIMO-OFDM and SCBT modes, three parts of SD have been modified, which are the preprocessing, metric computation and tree structure. Instead of the preprocessing of QR decomposi-tion in MIMO-OFDM mode, the preprocessing of SCBT decoding is to calculate the body term (FN(HLdj)ze) and aliasing term (FN(HUdj)ze) by using the shared multiplier banks. The metric
computation of SCBT decoding which is simpler than that of MIMO detection requires subtraction between two complex signals’ vectors. Furthermore, unlike each parent node to extendQ possible child nodes in the tree structure ofQ-QAM MIMO detection, the parent node can only extend one child node except the root node in the tree structure of SCBT decoding. If the partial Euclidean distance of current node is larger than the upper bound, a dead end is declared and the next start node of backward recursion is always one of the child nodes of root. Because the behavior of tree punching also follows the depth-first B&B algorithm in SCBT mode, the SD search engine does not require additional modifications. The main overhead of such SD is the control unit to handle all data paths in VLSI implementations. The clock rate and the number of SD engines are dominated by 64-QAM MIMO detection of a 42 4 MIMO-OFDM modem that four SD engines with 150-MHz clock rate are utilized in this design. The decoding latency of CCK using a 150-MHz SD engine is less than 720 ns, so it met DSSS-CCK decoding requirements (CCK symbol duration is 720 ns). By partitioning CCK into four sub-sets decoded via four parallel SD engines, we can reduce the decoding latency to 40 cycles or we can slow down the clock rate to 100 MHz.
The hardware complexity is summarized in Table I. The 0.3% for combination logics, 33.3% for SRAM and 99% ROM are only used in DSSS-CCK mode, which indicates the additional cost making a 42 4 MIMO-OFDM modem support non-CP SCBT is memory (SRAMs and ROMs). Although the total ROM size for the DSSS-CCK mode is 7760 bytes, the major cost of ROMs is just the address decoder. Most combi-national logics are shared with MIMO-OFDM and DSSS-CCK modes,
Fig. 8. Detail architecture and complexity of a 4 4 MIMO-OFDM modem with the proposed SC-FDE (top right for the rule of signal multiplications of ~G; dark squares for the elements being currently computed, and light squares for the elements after calculated).
TABLE I
VLSI COMPLEXITY OF A4 4 MIMO-OFDM MODEM
e.g., channel estimators, FFTs, complex multipliers and SD engines. Hence, the additional complexity compared with conventional multi-mode systems is acceptable.
VI. CONCLUSION
The contributions of this work are to design and implement the pro-posed FD-CE and DF-AC for providing the equalization and decoding of non-CP SCBT over frequency domains. Only one N-point FFT (M L N=2) is required to make the SC-FDE integrate with an MIMO-OFDM modem efficiently, ensuring IEEE 802.11VHT mul-timode and backward compatibility. Evaluations of a linear (8, 4) code and WiFi in frequency-selective fading indicate that all results satisfy 8% PER to assure adequate performance within acceptable complexity. Moreover, this work is also corresponded to Giga-bit wireless specifi-cation discussed in IEEE 802.15.3c to support OFDM and non-OFDM packet accesses.
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