Table of Contents
ISRN Signal Processing
Volume 2011, Article ID 254597, 8 pages
Research Article

Iterative Channel Estimation for Nonbinary LDPC-Coded OFDM Signals

Wireless Systems Engineering and Research (WISER) S.r.l., Via Fiume, 23, 57123 Livorno, Italy

Received 20 December 2010; Accepted 8 March 2011

Academic Editor: C. S. Lin

Copyright © 2011 Giacomo Bacci et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This work deals with the issue of channel estimation in the context of non-binary LDPC-coded OFDM systems over doubly selective multipath channels. In particular, we show how to derive an iterative Wiener-filter-based estimation method using both time and frequency channel correlation and considering the particular characteristics of the channel code. The proposed algorithm can use either soft information or hard decisions fed back by the decoder to refine the channel estimation, so as to improve the system performance at the expense of an increased receiver complexity. Simulation results under typical working conditions are presented to compare the performance of the proposed method with respect to classical techniques.

1. Introduction

The increasing demand for high-speed wireless communications calls for efficient technologies in terms of energy expenditure and bandwidth occupation. In the area of forward error correction (FEC) coding, LDPC codes over high-order finite Galois fields GF(𝑞), termed nonbinary LDPC (NB-LDPC) codes, were shown to bear a potential compared to other techniques [1]. To mention a few, NB-LDPC codes show a lower error floor with respect to their binary counterpart (LDPC codes), while providing a steep waterfall region in terms of word error rate (WER) compared to convolutional turbo codes [2]. Although this feature comes at the expense of an increased complexity at the receiver, NB-LDPC coding can be considered as a viable technology for beyond-4G communication systems [2].

Wideband signals for high-speed digital communications over a wireless link suffer from distortions caused by multipath propagation. The potentiality on NB-LDPC can be fully exploited if we can adopt an iterative channel estimation technique based on the channel decoder information to mitigate the effects of channel selectivity. Recently, there has been a flurry of research in this field using the “turbo” principle [3, 4], that can be applied to ancillary signal detection functions whenever the channel decoder is iterative and/or soft-output. This approach, analogous to what turbo codes do in the field of data detection, is referred to as “turbo” equalization. In [5], an iterative channel estimation is applied to a turbo-coded pilot-aided BPSK system over flat-fading channels. This solution provides a significant improvement in the performance by iteratively estimating the channel, and using the estimate to decode the turbo code. One drawback of the proposed technique is the bit error rate (BER) floor level, which is about an order of magnitude higher than that achieved by a turbo-coded system with ideal BPSK detection. However, the authors suggest the use of the channel estimates from previously decoded frames to significantly reduce this floor. This approach is generalized in [6] to the case of double selectivity, using joint equalization and decoding. The channel estimator proposed in this work exhibits a reduced complexity and it is decoupled from the equalizer module. The proposed method is suitable for high-order modulations, channels with arbitrary power profile, and LDPC. Simulation results show a significant performance improvement in the case of iterative estimation as opposed to non iterative channel estimation.

Similar approaches are considered in [7, 8] assuming the channel fading rate to be high. The work in [7] considers systems where the channel is unknown and time varying, with a fading rate so high that tracking of the channel is required between the training sequences. The proposed algorithm adopts a separate channel estimator that runs in parallel and aids the equalizer to save complexity and to be suitable for a wide range of different equalization techniques. In [8], iterative channel estimation and equalization is derived for frequency-selective Rayleigh fading channels. The authors propose a soft input Kalman channel estimator, derived by restructuring the channel estimation problem as one of Kalman state estimation reflecting the soft information from the decoding process into the statistical description of the channel.

In the field of multicarrier systems, relevant results can be found in [911], which propose joint iterative data detection and channel estimation schemes for orthogonal frequency division multiplexing (OFDM) under double channel selectivity. In [9], pilot symbols are exploited at the first estimation stage. Then, the receiver performance is improved by properly incorporating the soft-information fed back by the decoder into the minimum mean square error (MMSE) channel estimator once it is available after the first iteration. To reduce the computational power, the authors propose an indirect MMSE iterative channel estimation and decoding (ICED) method based on the number of channel taps in the time domain. In [10], the authors adopt a Wiener filtering approach to produce the optimum estimate of channel response in the sense of MMSE. To improve the accuracy of channel estimation, soft information exchange between Wiener-filter-based channel estimator and error-correction decoder is employed. However, according to the authors’ experiments, the iterative information exchange between Wiener-filter-based channel estimator and error-correction decoder does not always improve the receiver performance. In [11], the authors derive an iterative algorithm for joint data-detection and channel-estimation for OFDM systems, which includes iterative decoding at the receiver. This scheme considers a maximum a posteriori-based decoder that works in conjunction with an iterative channel estimator to provide more reliable information on the coded symbols.

The present paper elaborates on the turbo approach to derive an iterative channel estimation algorithm based on the soft output of the NB-LDPC decoder. Wiener filtering in the time and frequency domain is selected due to its simple implementation while producing the optimum estimate of the channel response in the sense of MMSE. To the best of the authors’ knowledge, the impact of iterative channel estimation in the case of an NB-LDPC-coded system has not been investigated in the literature. The main scientific contribution of this document is the rigorous derivation of the proper way to include soft information in the channel estimation procedure. Instead of using ad hoc formulae, we show how the a posteriori average of the received channel symbol is exactly the piece of soft information that is required by the Wiener smoother for channel coefficient estimation. We consider an OFDM system operating over a time-varying frequency selective scenario, as described in Section 2, emphasizing the details that can be exploited to perform the channel estimation task. The proposed iterative estimation technique is derived in Section 3. Section 4 contains some simulation results, whereas some conclusions are drawn in Section 5.

2. System Description

This section describes the system model considered throughout this work, which is sketched in Figure 1. At the transmitter side, the binary source is segmented into words of log2𝑞 bits each, and each word is considered as the natural binary representation of a GF(𝑞) symbol. A sequence 𝑢, =0,,𝐾1, of 𝐾 such symbols represents our source message vector 𝐮 and is encoded into a vector 𝐜, containing 𝑁 GF(𝑞) symbols 𝑐𝑚, 𝑚=0,,𝑁1, by a NB-LDPC encoder with a rate 𝐾/𝑁 [1]. An NB-LDPC code is defined in terms of a very sparse pseudorandom parity check matrix, whose elements belong to a finite Galois field GF(𝑞). The way the encoding process acts is very similar to that employed by binary LDPC codes. The fundamental difference is that all operations are to be intended in the GF(𝑞) domain [2].

Figure 1: Block diagram of the system model.

The transmission takes place over a multicarrier OFDM signal using 𝑁𝑠 subcarriers. We consider an OFDM system, due to its robustness against fast frequency-selective fading, as experienced in advanced systems such as the IEEE 802.16 [12] and LTE [13] standards. To increase the frequency diversity of the signal, our set of encoded vectors c is mapped into quadrature amplitude modulation (QAM) symbols, and interleaved on a carrier basis. The coded symbols 𝑐𝑚, assuming one of the 𝑞 possible values 𝑏𝑔, 𝑔=0,,𝑞1, are modulated onto the OFDM carriers and sent to the receiver. More in detail, the mapping function 𝜇() gathers a number 𝑁𝐺 of coded GF symbols, with 𝑁𝐺=lcm(log2𝑀,log2𝑞)/log2𝑞, and maps them onto a number 𝑁𝑄 of consecutive channel QAM symbols 𝑑𝑘, with 𝑁𝑄=lcm(log2𝑀,log2𝑞)/log2𝑀, where 𝑀 is the QAM constellation order, and lcm() stands for the least common multiple. In practice, this can be performed by expanding the coded symbols 𝑐𝑚 in their binary images (each symbol providing log2𝑞 bits) and grouping them on a QAM symbol basis (i.e., with blocks of log2𝑀 bits). With this approach, the actual mapping that is used to associate QAM symbols to GF(𝑞) symbols has practically no relevance on the coded bit performance—any random mapping gives the same result.

The OFDM signal undergoes a tapped-delay-line fading channel with additive white Gaussian noise (AWGN) and impulse response(𝑡)=𝑁𝑝1𝑖=0𝜌𝑖(𝑡)𝛿𝑡𝜏𝑖,(1) where 𝛿() is the Dirac's delta function, 𝑁𝑝 is the number of multiple paths, 𝜏𝑖 is the delay of the 𝑖th path, and 𝜌𝑖(𝑡) is the realization of the 𝑖th path, with independent Gaussian-distributed real and imaginary parts with zero mean and variance 𝜎2𝑖/2, with 𝜎2𝑖 determined by the channel power delay profile (PDP). Both components of 𝜌𝑖(𝑡) have autocorrelation function 𝑅𝜌𝑖(𝜏)=𝜎2𝑖/2𝑅𝜌(𝜏), which depends on the maximum Doppler frequency 𝑓𝐷. Note that the powers {𝜎2𝑖} are normalized so as to fulfill 𝑁𝑝1𝑖=0𝜎2𝑖=1. The intercarrier interference (ICI) due to the time variation of the channel coefficients within one OFDM symbol block is not considered in this work. This simplifying assumption is consistent with the set of Doppler frequencies that is used in our numerical results (Section 4). The method described below can include the effects of ICI resorting to a Taylor expansion approach [14].

The fast Fourier transform (FFT) operation gives the vector 𝐲[𝑛]=[𝑦1[𝑛],,𝑦𝑁𝑠[𝑛]], where 𝑦𝑘[𝑛]=𝑑𝑘[𝑛]𝑘[𝑛]+𝑤𝑘[𝑛] is the 𝑘th received subcarrier of the 𝑛th OFDM symbol in the frequency domain, with 𝑑𝑘[𝑛] being the QAM symbol mapped on the 𝑘th subcarrier, 𝑤𝑘[𝑛] being the complex AWGN sample in the frequency domain with power 𝜎2𝑤/2 over each component, and 𝑘[𝑛](𝑘Δ𝑓;𝑛𝑇𝑠) being the channel response over the 𝑘th subcarrier of the 𝑛th OFDM symbol, where (𝑓;𝑡)={(𝑡)} is the frequency response of the channel, Δ𝑓 is the subcarrier spacing, and 𝑇𝑠=1/Δ𝑓 is the OFDM symbol duration.

Figure 2 depicts the general overall architecture of the decoder/demodulator considered throughout the paper, expanding the bottom branch of the block diagram of Figure 1. Channel equalization is mandatory to mitigate the effect of channel selectivity. By extracting the pilot carriers embedded in the OFDM format from the vector 𝐲[𝑛], we can obtain a rough estimate of the channel ̂𝐡[𝑛]={𝑘[𝑛]}𝑁𝑠𝑘=1 in conjunction with an estimate of the noise power 𝜎2𝑤 (the latter is used by the soft demapper and possibly by the channel equalizer according to the equalization strategy). The equalizer processes 𝐲[𝑛], and its output is sent to a deinterleaver after removing virtual and pilot carriers. The stream is then subdivided in chunks of 𝑁 samples, corresponding to one codeword, and each chunk is sent to a soft demapper and finally to an NB-LDPC decoder. Decoding techniques for NB-LDPC codes can be borrowed by their binary counterparts by extending all operations to the field GF(𝑞). The considered system adopts a simplified version [15] of the extended min-sum (EMS) algorithm [16] that produces a matrix Λ of a posteriori probabilities (APPs) for all coded GF(𝑞) symbols. Using Λ, we can obtain an estimate ̂𝐮 of the transmitted symbols and the soft/hard information to be fed back to the channel estimator, as described in the next section.

Figure 2: Structure of the NB-LDPC decoding stage using iterative channel estimation.

3. Iterative Channel Estimation

To improve the system performance, we can exploit the information from the NB-LDPC decoder to refine channel estimation. We can use either the soft information Λ or the hard decisions ̂𝐮 (in addition to pilots) to produce a further estimate ̂𝐡[𝑛]. In a recursive fashion, this new estimate is reused by the decoder, and a new decision on the transmitted symbols is taken. This work considers a Wiener-filter-based channel estimator, due to its simplicity of implementation while ensuring an optimum MMSE solution.

The linear causal MMSE estimator that provides an estimate of 𝑘[𝑛] in the time-frequency domain is given by𝑘[𝑛]=𝐿𝑆𝑙=𝐿𝑠=0𝑎𝑘,𝑙𝑠𝑦𝑘𝑙[],𝑛𝑠(2) where the number of coefficients (i.e., the order) of the Wiener filter is (2𝐿+1)×(𝑆+1), {𝑎𝑘,𝑙𝑠}𝑠=0,,𝑆𝑙=𝐿,,𝐿 are the complex-valued Wiener filter coefficients, and 𝑦𝑘[𝑛] is the 𝑘th received subcarrier of the 𝑛th OFDM symbol. The filter coefficients can be rearranged in a (2𝐿+1)×(𝑆+1) matrix 𝐀𝑘 and computed following the MMSE criterion:𝐀𝑘=𝑎𝑘,𝑙𝑠𝑠=0,,𝑆𝑙=𝐿,,𝐿𝐀=argmin𝑘(2𝐿+1)×(𝑆+1)𝔼|||𝑘[𝑛]𝑘[𝑛]|||2,(3) where 𝔼{} denotes expectation. For convenience of notation, it is worth restating the filter (2) as𝑘[𝑛]𝐀=tr𝐻𝐤𝐘𝑘𝐀=vec𝑘𝐻𝐘vec𝑘=𝐚𝐻𝑘𝐲𝑘,(4) where ()𝐻 denotes conjugate transposition; tr() is the trace operator, vec(𝐗) denotes the vectorization of the matrix 𝐗 formed by stacking the columns of 𝐗 into a single column vector, 𝐚𝑘vec(𝐀𝑘), 𝐲𝑘vec(𝐘𝑘), and𝐘𝑘𝑦𝑘𝐿[𝑛]𝑦𝑘+𝐿[𝑛]𝑦𝑘𝐿[]𝑛𝑆𝑦𝑘+𝐿[]=𝑦𝑛𝑆𝑘,𝑙𝑠𝑠=0,,𝑆𝑙=𝐿,,𝐿,(5) with𝑦𝑘,𝑙𝑠𝑦𝑘+𝑙[]𝑛𝑠=𝑑𝑘+𝑙[]𝑛𝑠𝑘+𝑙[]𝑛𝑠+𝑤𝑘+𝑙[]𝑛𝑠(6) is a matrix containing the (2𝐿+1)×(𝑆+1) received samples in the frequency domain centered around the 𝑘th subcarrier of the current OFDM symbol and of the 𝑆 past OFDM symbols. Using (6), 𝐲𝑘 can be rewritten in a more convenient form:𝐲𝑘=𝚫𝑘𝐡𝑘+𝐰𝑘,(7) where Δ𝑘diag(𝐝𝑘), with diag() denoting a diagonal matrix with entries () in the main diagonal, and𝐝𝑘𝐃=vec𝑘,𝐃𝑘=𝑑𝑘,𝑙𝑠,𝑑𝑘,𝑙𝑠=𝑑𝑘+𝑙[],𝐡𝑛𝑠𝑘𝐇=vec𝑘,𝐇𝑘=𝑘,𝑙𝑠,𝑘,𝑙𝑠=𝑘+𝑙[],𝐰𝑛𝑠𝑘𝐖=vec𝑘,𝐖𝑘=𝑤𝑘,𝑙𝑠,𝑤𝑘,𝑙𝑠=𝑤𝑘+𝑙[],𝑛𝑠(8) with 𝑙=𝐿,,𝐿, and 𝑠=0,,𝑆.

Using the formulation (4), the optimization problem (3) translates into finding the vector 𝐚𝑘 that satisfies𝜕𝜕𝐚𝑘𝔼|||𝑘[𝑛]𝑘[𝑛]|||2𝐲=2𝔼𝑘𝐲𝐻𝑘𝐚𝑘2𝔼𝑘[𝑛]𝐲𝑘=𝟎(2𝐿+1)(𝑆+1),(9) where 𝟎𝑚 denotes the 𝑚×1 all-zero vector. Hence,𝐚𝑘=𝔼𝐲𝑘𝐲𝐻𝑘1𝔼𝑘[𝑛]𝐲𝑘.(10) To implement the channel estimation (4) in an iterative fashion, we should investigate how to relate the soft information coming from the decoder with the filter coefficients 𝐚𝑘 computed as in (10). Applying the definitions (7) and (8), we get𝔼𝐲𝑘𝐲𝐻𝑘=𝚫𝑘𝐑𝐡𝚫𝐻𝑘+𝐑𝐰,(11) where 𝐑𝐰=𝔼{𝐰𝑘𝐰𝐻𝑘}=𝜎2𝑤𝐈 is the correlation matrix of the AWGN noise, with I denoting the identity matrix, 𝐑𝐡=𝔼{𝐡𝑘𝐡𝐻𝑘} is the channel correlation matrix, and the matrix Δ𝑘=𝔼{Δ𝑘}=diag(𝐝𝑘) contains the expected values for the transmitted QAM symbols.

Considering the Fourier transform of the impulse response (𝑡) defined as in (1) and recalling the statistical independence between different paths, 𝐑𝐡 can be expressed as𝐑𝐡=𝐅𝐻𝐑𝝆𝐅,(12) where𝐅=𝐅𝐈𝑆+1(13) with 𝐅[𝑓𝑖,𝑙]𝑖=0,,𝑁𝑝1𝑙=𝐿,,𝐿, with elements 𝑓𝑖,𝑙=𝑒+𝑗2𝜋𝑙Δ𝑓𝜏𝑖, 𝐈𝑆+1 is the (𝑆+1)×(𝑆+1) identity matrix and denotes the Kronecker product, and𝐑𝝆=𝐑𝝆(𝑓)𝐑𝝆(𝑡)(14) is the correlation matrix of the fading channel, where 𝐑𝝆(𝑓)=diag([𝜎20,,𝜎2𝑁𝑝1]), and 𝐑𝝆(𝑡)=[𝑅𝑢,𝑠]𝑢=0,,𝑆𝑠=0,,𝑆, with elements 𝑅𝑢,𝑠=𝑅𝜌((𝑢𝑠)𝑇𝑠).

Now comes the crucial part of our derivation. The expected symbol values Δ𝑘 are interpreted as a posteriori averages and can be computed in real-time using the soft output from the channel decoder. More in detail, 𝑑𝑘𝑙[𝑛𝑠]𝔼{𝑑𝑘𝑙[𝑛𝑠]} can be obtained exploiting the APP matrices computed by the NB-LDPC decoder on the (𝑛𝑠)th OFDM symbol. Recalling that the decoder operates code block by code block, Λ=[𝝀0,,𝝀𝑁1], where 𝝀𝑚, 𝑚=0,,𝑁1, is a vector containing the APP of each coded symbol 𝑐𝑚[𝑛𝑠]𝐜[𝑛𝑠] for any possible symbol 𝑏𝑔,𝑔=0,,𝑞1 in the GF(𝑞): 𝝀𝑚={𝜆𝑚,𝑔}𝑞1𝑔=0, with 𝜆𝑚,𝑔=Pr{𝑐𝑚[𝑛𝑠]=𝑏𝑔𝐲[𝑛𝑠]}. In general, the QAM constellation order 𝑀 is different from the Galois field order 𝑞. For the sake of brevity, let us assume here that 𝑀=𝑞 (our results can be extended to the case 𝑀𝑞 using the methods discussed in [17]). Under this hypothesis, after interleaving and multicarrier modulation, the mapped QAM symbol 𝑑𝑘𝑙[𝑛𝑠] has a one-to-one correspondence with the GF(𝑞) symbol 𝑐𝑚[𝑛𝑠]. Using the output from the decoder, we can obtain a soft-mapped symbol𝑑𝑘𝑙[]=𝑛𝑠𝑞1𝑔=0𝜆𝑚,𝑔𝑏𝜇𝑔,(15) where 𝜇() is the bijective mapping that assigns an 𝑀-QAM symbol to a GF(𝑞) coded symbol. In the case of a hard-decision approach, we can replace (15) with𝑑𝑘𝑙[]𝑛𝑠=𝜇̂𝑢𝑚,(16) where ̂𝑢𝑚=𝑏𝑔(𝑚), with 𝑔(𝑚)=argmax𝑔𝜆𝑚,𝑔.

Using similar manipulations, we can obtain𝔼𝑘[𝑛]𝐲𝑘=𝚫𝑘𝐅𝐻𝐑𝝆̃𝐟,(17) where ̃𝐟 is the central column of the matrix 𝐅.

We now have all the ingredients to provide an estimate ̂𝐡[𝑛] of the channel coefficients according to (4) and (10). The estimation method starts from an initial estimate of the channel coefficients based on pilot subcarriers only. Using this initial estimate, the error-correction decoder decodes the received signal and produces soft information of the coded GF(𝑞) symbols. The iterative process between the channel estimator and the decoder then proceeds by exchanging soft information using a turbo approach. This process constitutes an outer iteration of the turbo architecture sketched in Figure 2, which aims at minimizing the mean-square error of the channel estimate at the expense of an increased complexity in terms of computational power at the receiver. The number of outer iterations (i.e., the number of iterative channel estimation, as opposed to the inner iterations performed by the EMS algorithm at the channel decoder) must be then optimized, as briefly discussed in Section 4.

To reduce the computational demand, we can exploit the properties of the correlation matrices to avoid the explicit real-time inversion of the matrix 𝔼{𝐲𝑘𝐲𝐻𝑘} as in (10). Since 𝔼{𝐲𝑘𝐲𝐻𝑘} is a block-Toeplitz matrix, we can use the Levinson-Durbin recursion [18] to significantly reduce the number of operations. To further simplify the computation of the Wiener filter coefficients, we can also apply the Woodbury formula [19] to (10) to get, using (11) and (17),𝐚𝑘=𝚫𝐻𝑘1𝐑𝐡+𝜎2𝑤𝚫𝐻𝑘𝚫𝑘11𝚫𝑘1𝚫𝑘𝐅𝐻𝐑𝝆̃𝐟.(18) In the case of phase shift keying (PSK) (i.e., equal-energy constellations), in which Δ𝐻𝑘Δ𝑘=𝐈, (18) can be rewritten as𝐚𝑘=𝚫𝐻𝑘1𝝋(19) with 𝝋[𝐑𝐡+𝐑𝐰]1𝐅𝐻𝐑𝝆̃𝐟. Equation (19) shows two interesting properties: (i) it contains the vector 𝝋, which does not depend on the data, and can thus be computed offline once for ever before filtering starts; and (ii) the data-dependent matrix [Δ𝐻𝑘]1 requires the inversion of a diagonal matrix, thus yielding a negligible impact in terms of computational load. In the case of unequal-energy (e.g., QAM) constellations, the equality in (19) does not hold anymore, and therefore (19) represents a suboptimal solution, whose impact in terms of performance loss with respect to (10) will be evaluated by means of numerical simulations.

As a final remark, it is worth emphasizing that the procedure described above to compute the Wiener filter coefficients (10), using the relations (11) and (17), is valid for any coded OFDM transmission over a doubly selective channel. When using a different FEC technique, the only modification in fact occurs in computing the data-dependent matrix Δ𝑘, whose elements 𝑑𝑘[𝑛] (i.e., the expected values of the transmitted symbols) depend on the output of the channel decoder.

4. Simulation Results

The signal format used to evaluate the link-level performance of the iterative estimation is based on the IEEE 802.16e time division duplex (TDD) downlink frame [20]. The relevant system parameters are number of subcarriers 𝑁𝑠=1024, with 720 subcarriers allocated to information symbols and 120 used as pilots, sampling frequency 𝑓𝑠=11.2 MHz, subcarrier spacing Δ𝑓=𝑓𝑠/𝑁𝑠10 kHz, OFDM symbol duration 𝑇𝑠=1/Δ𝑓100μs, and carrier frequency 𝑓0=3.5 GHz. The Galois field order is 𝑞=64, and the used parity check matrices are those derived in the FP7-DAVINCI project “Design and Versatile Implementation of Nonbinary Wireless Communications Based on Innovative LDPC Codes” [21]. The encoding scheme adopts a codeword length 𝑁=360 and a coding rate 1/2. The I/Q modulation considers an 𝑀=64QAM constellation using standard Gray mapping. The 24-tap ITU modified veh.-A channel profile [22] is considered to model the frequency selectivity, and the Clarke model 𝑅𝜌𝑖(𝜏)=𝜎2𝑖𝐽0(2𝜋𝑓𝐷𝜏) is used for the time selectivity, where 𝐽0() is the zero-order Bessel function of the first kind. A zero-forcing (ZF) strategy is adopted at the equalizer.

Figure 3 shows the experimental WER as a function of the signal-to-noise ratio (SNR) in terms of energy per symbol 𝐸𝑆 and noise power spectral density (PSD) 𝑁0. The relative speed between the transmitter and the receiver is assumed to be 𝑣=120 km/h, which yields 𝑓𝐷388 Hz. The solid line reports the performance of perfect channel state information (CSI), whereas the dashed line represents a fictitious situation in which all the transmitted symbols are known to the receiver (all-pilot case) and are used to perform channel estimation. Wiener filtering in the time domain considering three subsequent OFDM symbols is used to smooth the effects of the noise. Since the only errors affecting the channel estimate are due to the AWGN, this represents some sort of experimental lower bound to the WER performance. The dash-dotted line is obtained using a least-squares pilot-aided method, to serve as an experimental upper bound for the WER performance. Dark and light lines report the results of the proposed solution when soft information and hard decisions are used, respectively. A Wiener filter with 𝐿=2 and 𝑆=2 is employed. Four different configurations are reported: 3 × 10 (upper triangles), 3 × 30 (lower triangles), 6 × 30 (circles), and 10 × 30 (square markers), where the first and the second parameter are the number of outer and inner iterations, respectively. As expected, increasing the number of outer and/or inner iterations yields better performance at the expense of an increased computational complexity. Note that in all configurations, the proposed algorithm allows us to reduce the gap with the all-pilot case with respect to the pilot-aided case: in the soft-based scenario, the gap reduces from 1.2 dB to 0.4 dB on average, whereas it becomes about 0.5 dB in the hard-based case. Interestingly, the performance of hard-directed estimation is quite similar to the soft-based one. We can thus adopt the hard-based approach to reduce the complexity at the expense of an acceptable performance degradation. Note also that a number >6 of outer iterations does not produce substantial WER improvement.

Figure 3: WER performance as a function of 𝐸𝑆/𝑁0 (𝑀=𝑞=64, 𝑣=120km/h, 𝐿=2, and 𝑆=2).

Figure 4 reports the WER performance of the channel estimator when no history is considered (𝑆=0) and when three OFDM symbols are considered to improve the estimation accuracy (𝑆=2). As can be easily argued, the impact of the filter memory S heavily depends on the coherence time of the considered channel. To this aim, we consider a lower relative speed 𝑣=10 km/h, which implies a higher coherence time (𝑓𝐷32.4 Hz). Dark and light lines depict the cases 𝑆=2 and 𝑆=0, respectively, when soft information is used. Averaging over three (𝑆=2) channel realizations allows us to reduce the effects of AWGN (the gain is around 0.4 dB). The impact of the channel coherence time is also confirmed by better performance with respect to the 𝑣=120 km/h case: when 6 outer iterations are employed, the difference between all-pilot and turbo-aided configurations reduces from 0.4 to 0.1 dB when 𝑆=2, whereas it is substantially unchanged when 𝑆=0. Note also that when 𝑣=120 km/h, using 𝑆=0 and 𝑆=2 yields almost the same performance due to shorter channel coherence time (the case 𝑆=0 is not reported in Figure 3 for the sake of presentation).

Figure 4: WER performance as a function of 𝐸𝑆/𝑁0 (𝑀=𝑞=64, 𝑣=10 km/h, 𝐿=2, variable 𝑆, soft information).

In Figure 5, we evaluate the impact of a priori knowledge of the channel statistics on the system performance. As emerged from Section 3, (12) requires prerequisite information, such as correlation in time and frequency domain. To this aim, we simulate a scenario in which the channel taps are computed using the 24-tap ITU veh.-A channel model, whereas the 6-tap ITU veh.-A channel model [22] is assumed at the receiver. In this case, the mismatch is mild, since the coherence time is assumed to be known based on the correct Doppler shift, and the coherence bandwidth is similar. However, the average PDP and the number of paths are modified. Figure 5 considers 𝑣=10 km/h and a soft-based Wiener filter with 𝐿=2 and 𝑆=1. In this case, we notice a performance loss of about 0.15 dB and 0.3 dB in the cases 6 × 30 and 3 × 30, respectively.

Figure 5: WER performance in the case of imperfect knowledge of a priori channel statistics (𝑣=10 km/h, 𝐿=2, and 𝑆=1).

Finally, to measure the performance loss of the suboptimal solution (19), Figure 6 compares the WER performance as a function of the SNR. The solution (10), exploiting the joint time-frequency correlation, is shown with circular markers, whereas the suboptimal method (19) is depicted with triangular markers. Both soft-based (dark lines) and hard-directed (light lines) approaches are reported, adopting a Wiener filter with parameters 𝐿=2 and 𝑆=1, and assuming 𝑣=10 km/h. For the sake of presentation, we report here only the case with 6 × 30 iterations. As can be seen, the loss of the proposed suboptimal solution with respect to the optimal derivation is roughly 0.5 dB. However, it is worth noting that even in the case of the hard-decision approach, the suboptimal iterative solution performs better than the pilot-aided solution, at the expense of a mild increase in the receiver complexity.

Figure 6: WER performance in the case of suboptimal estimation (𝑀=𝑞=64, 𝑣=10 km/h, 𝐿=2, and 𝑆=1).

5. Conclusion and Perspectives

This paper derived an iterative channel estimation method suitable for OFDM systems with nonbinary LDPC encoding. The proposed approach exploits the channel decoder information, at both soft and hard level to derive a joint time-frequency Wiener filter that minimizes the mean square error of the channel estimate. The novelty of this approach lies in (i) showing how to include the a posteriori soft information of any FEC channel decoder in the estimator without resorting to some heuristic formulation and (ii) pairing the turbo estimator with nonbinary LDPC codes, exploiting the properties of 𝑞-ary Galois fields and 𝑀-QAM constellations.

The benefits of this method have been evaluated using numerical simulations, in which the word error rate (WER) performance as a function of the signal-to-noise ratio (SNR) has been compared to the case of perfect channel knowledge and pilot-aided techniques. Simulation results have shown a significant improvement with respect to the pilot-aided scenario, at the expense of an increased complexity at the receiver. The improvement of turbo techniques versus pilot-aided estimator can be quantified to be about 0.5÷1 dB in terms of SNR. This is especially true for large constellations and with low mobility. In addition, we also noticed that the performance degradation due to channel mismatch and/or hard-detected information and/or suboptimal filtering are per se almost negligible when considered individually. The cumulative effect of the three, on the contrary, may degrade the overall performance by a nonnegligible amount.

As a final remark, we observe that the techniques investigated in this paper, evaluated for the downlink of an IEEE 802.16e link, are applicable to the uplink as well. The issue of the additional complexity of turbo techniques is not so stringent for the uplink (since the algorithm would be implemented in the base station) even with today's technology, whilst at the moment the implementation impact for a mobile station must be evaluated versus the potential benefit.


This work is supported by INFSCO-ICT-216203 DAVINCI “Design And Versatile Implementation of Nonbinary wireless Communications based on Innovative LDPC codes” ( funded by the European Commission under the Seventh Framework Programme (FP7).


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