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Lowcomplexity WLMMSE channel estimator for MBOFDM UWB systems
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 56 (2015)
Abstract
Widely linear (WL) minimum mean square error (MMSE) channel estimation scheme have been proposed for multiband orthogonal frequency division multiplexing ultrawideband (MBOFDM UWB) systems dealing with noncircular signals. This WLMMSE channel estimation scheme provides significant performance gain, but it presents a high computational complexity compared with the linear one. In this paper, we derive an adaptive WLMMSE channel estimation scheme that significantly reduces the computational complexity. The complexity reduction is done in two stages. The first stage consists of a real evaluation of the WLMMSE channel estimator and the second stage follows with a reducedrank filtering based on the singular value decomposition (SVD). Computational complexity evaluation shows that the proposed lowrank realvalued WLMMSE channel estimator has computation cost comparable with the linear MMSE. Additionally, simulations of the bit error rate (BER) performance show comparable performance with the WLMMSE channel estimator especially at high signaltonoise ratio (SNR).
Introduction
Over the past few years, ultrawideband (UWB) has been promised as an efficient technology for future wireless shortrange high data rate communication [1]. It has received great attention in both academia and industry for applications in wireless communications since 2002 when the Federal Communications Commission (FCC) of the USA reserved the frequency band between 3.1 and 10.6 GHz for indoor UWB applications. This decision has led to the introduction of Institute of Electrical and Electronic Engineering (IEEE) 802.15 high rate alternative physical layer (PHY) Task Group 3a for wireless personal area networks (WPAN). And ever since then, various researches on UWB systems have been conducted all over the world for the final highspeed WPAN standard. One of them, the multiband orthogonal frequency division multiplexing (MBOFDM)based UWB system has been proposed, by the Multiband OFDM Alliance (MBOA), for European Computer Manufacturers Association (ECMA)368 standard to provide low power and high data rate transmission [24].
A collection of problems including channel measurements and modeling, channel estimation, and synchronization have been widely studied by researchers [510]. Clearly, all the performance improvement and capacity increase are based on accurate channel state information. Therefore, the most important component of MBOFDM UWB system, represented by the block diagram of Figure 1, is that blue highlighted channel estimation and equalization component.
In the UWB literature, the channel estimation has mostly been studied for its effect on error rate performance and computational complexity [818]. For example, a minimum mean square error (MMSE) estimator has been proposed by Beek et al., and they have shown that it gives a performance gain over least square (LS) estimator [8]. Moreover, Li et al. have proposed a lowcomplexity channel estimator developed by first averaging the overlapadded (OLA) received preamble symbols within the same band and then applying timedomain least squares method followed by the discrete Fourier transform [9]. Also, a lowsamplingrate scheme, based on multiple observations generated by transmitting multiple pulses, for ultrawideband channel estimation has been proposed in [12]. Furthermore, authors in [18] propose preamblebased improved channel estimation in the presence of both multipleaccess interference (MAI) and narrowband interference (NBI). The aforementioned estimators are applied for strictly linear systems.
Outside the UWB literature, there has been growing interest to widely linear (WL) processing which takes into account both the original values of the signal data as well as their conjugates. For example, Chevalier et al. have presented the widely linear mean square estimation with complex data [19,20]. They have shown that widely linear systems outperform significantly the strictly linear one. Also, Sterle has compared linear and widely linear MMSE transceivers for multiple input multiple output (MIMO) channels [21]. The results have shown that the widely linear MMSE transceiver provides considerable mean square error (MSE) and signal error rate (SER) gains over the linear one. Similarly, Cheng and Haardt have investigated the use of widely linear processing for MIMO systems employing filter bank multicarrier/offset quadrature amplitude modulation (FBMC/OQAM) [22]. Moreover, the WL processing has been considered to channel estimation firstly in [23] for multicarrier code division multiple access (MCCDMA) systems and it has been shown via simulations that the asymptotic bias and the MSE are significantly reduced when WL processing is employed. Furthermore, the WL processing has been investigated to suppress inter/intrasymbol interference, multiuser interference, and narrowband interference in a high data rate direct sequence ultrawideband (DSUWB) system [24]. The WL processing has shown that it outperforms the linear one. However, it has been shown that the WL processing presented a high computational cost [2426].
Many studies (see, e.g., [24,2738]) have shown that significant complexity reduction can be achieved by rank reduction techniques. Among these techniques, the eigenvalue decomposition (EVD) and its related singular value decomposition (SVD) techniques have had growing interest. For example, a reducedrank maximum likelihood estimation (RRMLE) has been derived in [27] for the ML estimation of the regression matrix in terms of the data covariances and their Eigen elements. Also, Scharf and Van Veen have proposed lowrank detectors for gaussian random vectors, and they have shown that rank reduction can be performed with no loss in performance [28]. Novel reducedrank scheme based on joint iterative optimization of adaptive filters with a lowcomplexity implementation using normalized least mean square (NLMS) algorithms has been proposed in [29]. The simulation results of the proposed scheme for CDMA interference suppression show a performance significantly better than existing schemes and close to the optimal fullrank MMSE. Moreover, Marzook et al. have proposed a novel channel estimation technique based on SVD technique for time division code division multiple access (TDSCDMA) [30]. The results have shown that the proposed technique improves the performance of the TDSCDMA system with lowrank processing and low computation complexity. Furthermore, channel estimation techniques based on SVD for reducedrank MIMO systems have been proposed in [36]. It has been shown that the SVDbased estimation largely improves the system performance. In all these studies, it has been proved that the rank reduction techniques, especially SVD technique, give a best tradeoff between performance and complexity.
The aim of the paper is twofold. Firstly, based on complextoreal transformation, we propose a realvalued WLMMSE channel estimator algorithm. Secondly, and based on SVD technique, we propose to reduce the rank of the obtained realvalued WLMMSE channel estimator.
The rest of the paper is organized as follows. MBOFDM UWB system model describes the system model. The structure of the proposed lowrank realvalued widely linear minimum mean square error (LrRWLMMSE) channel estimator is presented in channel estimation. Computational complexity evaluation is dedicated to the computational complexity evaluation of the conventional and proposed schemes. Performance evaluations provide the performance evaluations. Finally, some concluding comments are given in ‘Conclusions’.
MBOFDM UWB system model
Multiband OFDM (MBOFDM) is the main solution considered for high rate UWB transmission. Firstly, it was proposed by A. Batra et al. from Texas Instruments for the IEEE 802.15.3a task group [3]. Secondly, it was introduced by the MBOA consortium as their global UWB standard [4]. According to this standard, the available spectrum (3.1 − 10.6 GHz) is divided into 14 subbands. Each subband of 528 MHz offers 480 Mbit/s. In order to introduce multiple accessing capabilities and to exploit the inherent frequency diversity, each OFDM symbol obtained from a 128point inverse fast Fourier transform (IFFT), is transmitted on a different subband as dictated by a timefrequency code (TFC), shown in Figure 2, that leads to band hopping [4]. The architectures of the transmitter and the receiver of a MBOFDM UWB system, based on the specifications of ECMA368 standard, are shown in Figure 1. The modified blocks are those highlighted. These blocks are mainly the rhombicdual carrier modulation (DCM) mapping at the transmitter and the rhombicDCM demapping at the receiver proposed by Hajjaj et al. (unpublished work) for MBOFDM UWB systems. In addition, lowrank realvalued WLMMSE channel estimation and equalization is applied. All MBOFDM UWB system parameters are detailed in Table 1.
At the baseband transmitter, the bits from information sources are first scrambled into a pseudo random sequence. The resulting scrambled sequence is then encoded using convolutional encoder of rate R = 1/3 and constraint length K = 7, interleaved via bit interleaver and converted into the l ^{th} OFDM symbol of N _{D} = 100 tones. The N _{D} energycarrying tones are modulated using rhombicDCM. Then, a total of N _{P} = 12 pilot tones, N _{G} = 10 guard tones and 6 null tones are inserted into the OFDM symbol. An IFFT is used to transform the block of N _{IFFT} (N _{IFFT} = N _{D} + N _{P} + N _{G} + 6 nulls) tones into timedomain. The duration for the OFDM symbol is T _{IFFT} = 242.42 ns. Then, a zero padding sequence of length N _{ZP} = 32 and duration T _{ZP} = 60.61 ns is added to the resulting timedomain OFDM symbol, to eliminate the intersymbol interference (ISI) caused by the multipath propagation, and a guard interval of length 5 and duration T _{GI} = 9.47 ns is added to the end of the OFDM symbol, to ensure the frequency hopping. Thus, each transmitted OFDM symbol has a duration T _{S} = T _{IFFT} + T _{ZP} + T _{GI} = 312.5 ns and includes N _{S} = N _{IFFT} + N _{ZP} + 5 = 165 subcarriers.
The useful discretetime OFDM signal model is defined by:
where n = 0,…, N _{IFFT} −1, D, P and G are the k ^{th} data, pilot, and guard subcarriers of the l ^{th} OFDM symbol, respectively, and the functions q _{ D }, q _{ P }, and q _{ G } define a mapping functions for data, pilot, and guard subcarriers, respectively.
The baseband of the receiver, in general, consists of similar blocks of the baseband in the transmitter but in the reverse order [2].
The RFtransmitted signal is obtained by converting the baseband signal into continuoustime waveforms via a digitaltoanalog converter (DAC) and then upconverting it to the appropriate center frequency, as:
where Re(⋅) represents the real part of the signal, Nsym is the number of symbols, f _{ c } (m) is the center frequency for the m ^{th} frequency band, q(l) is a function that maps the l ^{th} symbol to the appropriate frequency band, and x(t) is the complex baseband signal representation for the l ^{th} symbol. x(t) must satisfy the following property: x(t) = 0 for t ∉[0, T_{S}].
The radiated signal S _{RF} (t) is transmitted over the UWB channel proposed for the IEEE 802.15.3a standard [40]. The impulse response of the multipath UWB channel model is described as:
where \( \left\{{\alpha}_{k,l}^i\right\} \) and \( \left\{{\tau}_{k,l}^i\right\} \) are the gain and the delay of the k ^{th} multipath component relative to the l ^{th} cluster arrival time \( \left({T}_l^i\right) \), respectively, {χ _{ i }} is the lognormal shadowing of the i ^{th} channel realization.
The IEEE 802.15.3a task group also defined four different configurations (CM1, CM2, CM3, and CM4) identified by the propagation scenarios line of sight (LOS) and nonline of sight (NLOS), and the distance between the transmitting and receiving antennas. The main characteristics of the UWB channel models are listed in Table 2.
The discretetime UWB channel is modeled as a N _{ h }tap finiteimpulseresponse (FIR) filter whose channel frequency response (CFR) of the l ^{th} OFDM symbol on its corresponding subband is given by:
where (⋅)^{T} denotes the transposition operation.
At the receiver side, the received signal r _{RF} (t), which is the sum of the output of the channel and additive white Gaussian noise (AWGN) w(t), is first filtered via bandpass filter and downconverted to baseband, and then the zero padding is removed using OLA method [41]. Then, the unitary fast Fourier transform (FFT) is applied to transform the discrete combined signal into frequency domain as:
where X(l) = diag{X(l, 0), X(l, 1), …, X(l, N _{IFFT} − 1)} represents the transmitted data, Y(l) = [Y(l, 0), Y(l, 1), …, Y(l, N _{IFFT} − 1)]^{T} denotes the received data, and \( W(l)={\left[\begin{array}{cccc}\hfill W\left(l,0\right),\hfill & \hfill W\left(l,1\right),\hfill & \hfill \dots, \hfill & \hfill W\left(l,{N}_{IFFT}1\right)\hfill \end{array}\right]}^T \) indicates the additive noise component, of the l ^{th} OFDM symbol.
For the sake of simplicity, we refer, in this paper, to Y(l), X(l), H(l), and W(l) as Y, X, H, and W, respectively.
Wiener filter (WF) based on LMMSE criterion, employing the secondorder statistics of the channel conditions, is considered for channel estimation [8]. However, since the proposed MBOFDM UWB system is dealing with noncircular signals, it requires widely linear processing to take into account all the secondorder statistics of the channel conditions and the received signal. Therefore, the received signal, in the frequency domain, is augmented with its conjugate as:
Channel estimation
The MBOFDM UWB systems employ framebased transmission. As shown in Figure 3, each MBOFDM UWB frame, which is referred to PPDU (PLCP protocol data unit), is composed of three components: the PLCP (Physical Layer Convergence Protocol) preamble, the PLCP header, and the PSDU (PLCP service data unit) [3]. The PLCP preamble consists of three distinct portions: packet synchronization sequence, frame synchronization sequence, and the channel estimation sequence. The latter portion is used to estimate the channel frequency response. The channel estimation sequence is followed by the PLCP Header which contains the data rate, the data length, the transport mode, the preamble type, and the MAC Header. The last component of the PPDU is the PSDU, which contains the actual information data (coming from higher layers).
We focus in this section on channel estimation for MBOFDM UWB systems. There are two main methods: the first is based on the channel estimation sequence (CES) of the PLCP preamble and the second is based on pilot signals insertion into each OFDM symbol. Here, even assuming that the UWB channel is invariant over the transmission period of an OFDM frame, the channel estimation can be performed using the channel training sequence.
In the end of the MBOFDM UWB system model, we have shown the relation between the augmented output \( \underset{\bar{\mkern6mu}}{Y} \) and the augmented transmitted symbol \( \underset{\bar{\mkern6mu}}{X} \) as:
This relation can be equally written for the N (N = 112) CSES data subcarriers of a single training OFDM symbol as:
where \( \overset{\smile }{\underset{\bar{\mkern6mu}}{X}} \) is 2 N × 2 N diagonal matrix with diagonal entries the N nonzero transmitted data augmented with its conjugate, \( \overset{\smile }{\underset{\bar{\mkern6mu}}{Y}},\overset{\smile }{\underset{\bar{\mkern6mu}}{H}} \) and \( \overset{\smile }{\underset{\bar{\mkern6mu}}{W}} \) are 2 N × 1 related subblocks of \( \overset{\smile }{\underset{\bar{\mkern6mu}}{Y}},\overset{\smile }{\underset{\bar{\mkern6mu}}{H}} \) and \( \underset{\bar{\mkern6mu}}{W} \), respectively.
Augmented LS estimator
Let \( \widehat{\underset{\bar{\mkern6mu}}{H}} \) be the estimate of \( \overset{\smile }{\underset{\bar{\mkern6mu}}{H}} \). The expression of augmented LS estimation is:
which minimizes \( \left(\overset{\smile }{\underset{\bar{\mkern6mu}}{Y}}\overset{\smile }{\underset{\bar{\mkern6mu}}{X}}\overset{\smile }{\underset{\bar{\mkern6mu}}{H}}\right)H\left(\overset{\smile }{\underset{\bar{\mkern6mu}}{Y}}\overset{\smile }{\underset{\bar{\mkern6mu}}{X}}\overset{\smile }{\underset{\bar{\mkern6mu}}{H}}\right) \)
WLMMSE estimator
We denote the widely linear estimator of the augmented transfer function of channel \( \overset{\smile }{\underset{\bar{\mkern6mu}}{H}} \) by:
Based on the orthogonality principle for linear estimation \( E\left[\left({\widehat{\underset{\bar{\mkern6mu}}{H}}}_{\mathrm{WLMMSE}}\overset{\smile }{\underset{\bar{\mkern6mu}}{H}}\right){\overset{\smile }{\underset{\bar{\mkern6mu}}{Y}}}^H\right]=0 \) [42], the optimal Ă is given by:
Hence, the WLMMSE estimator of \( \overset{\smile }{\underset{\bar{\mkern6mu}}{H}} \) can be written as:
where the augmented autocovariance matrix of \( \overset{\smile }{\underset{\bar{\mkern6mu}}{Y}} \) is given by:
and the augmented crosscovariance matrix is defined as:
In strictly linear case, the LMMSE estimator is given by:
Lowrank realvalued WLMMSE estimator
We denote by H _{ R }, X _{ R }, W _{ R }, and Y _{ R } the real composite representation of \( \overset{\smile }{H},\overset{\smile }{X},\overset{\smile }{W} \), and \( \overset{\smile }{Y} \), respectively:
where H _{ r } and H _{ i } are the real and imaginary vectors of \( \overset{\smile }{H} \)
where X _{ r } and X _{ i } are the real and imaginary matrices of \( \overset{\smile }{X} \)
where W _{ r } and W _{ i } are the real and imaginary vectors of \( \overset{\smile }{W} \)
where Y _{ r } and Y _{ i } are the real and imaginary vectors of \( \overset{\smile }{Y} \).
Then,
The lineartocomplex transformation gives the following relations

The relation between the augmented vector \( \overset{\smile }{\underset{\bar{\mkern6mu}}{Y}} \) and the real vector Y _{ R } can be described by: \( \overset{\smile }{\underset{\bar{\mkern6mu}}{Y}}=T{Y}_R \) where \( T=\left[\begin{array}{cc}\hfill I\hfill & \hfill jI\hfill \\ {}\hfill I\hfill & \hfill jI\hfill \end{array}\right] \) is unitary up to a factor of 2: TT ^{H} = T ^{H} T = 2I, and I is the identity matrix.

The relation between the augmented vector \( \overset{\smile }{\underset{\bar{\mkern6mu}}{W}} \) and the real vector W _{ R } can be described by \( \overset{\smile }{\underset{\bar{\mkern6mu}}{W}}=T{W}_R \).

The relation between the augmented vector \( \overset{\smile }{\underset{\bar{\mkern6mu}}{H}} \) and the real vector H _{ R } can be described by \( \overset{\smile }{\underset{\bar{\mkern6mu}}{H}}=T{H}_R \).
We assume that \( \overset{\smile }{H} \) and \( \overset{\smile }{W} \) are uncorrelated as in all stochastic filtering applications, i.e., \( \overset{\smile }{H}{\overset{\smile }{W}}^H=\overset{\smile }{W}{\overset{\smile }{H}}^H=0 \).
Then, the augmented autocovariance matrix of \( \overset{\smile }{Y} \) can be expressed as:
where \( {R}_{Y_R{Y}_R} \) is the autocorrelation matrix of Y _{ R }.
Then,
where \( {R}_{H_R{H}_R} \) is the autocorrelation matrix of H _{ R }.
The augmented crosscovariance matrix can be determined as:
where
By inserting the expression of \( {\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{\underset{\bar{\mkern6mu}}{H}}\overset{\smile }{\underset{\bar{\mkern6mu}}{Y}}},{\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{\underset{\bar{\mkern6mu}}{Y}}\overset{\smile }{\underset{\bar{\mkern6mu}}{Y}}} \), and \( \overset{\smile }{\underset{\bar{\mkern6mu}}{Y}} \) into (12), the resulting realvalued WLMMSE (RWLMMSE) estimator of \( \underset{\bar{\mkern6mu}}{H} \) is denoted by:
The LS estimator of \( \overset{\smile }{\underset{\bar{\mkern6mu}}{H}} \) is:
where \( {H}_{L{S}_R}=\left(\begin{array}{c}\hfill {H}_{L{S}_r}\hfill \\ {}\hfill {H}_{L{S}_i}\hfill \end{array}\right) \), and \( {H}_{L{S}_r} \) and \( {H}_{L{S}_i} \) are the real and imaginary vectors of \( {\overset{\smile }{H}}_{LS} \).
Then,
where
To further reduce the computational complexity, we adopt the lowrank approximation technique based on SVD [35]. The SVD of \( {R}_{H_R{H}_R} \) is denoted by \( {R}_{H_R{H}_R}=U\varLambda {U}^H \), where U is a matrix with orthonormal columns u_{0}, u_{1}, . . ., u_{2N1} and Λ is a diagonal matrix, containing the singular values λ _{0} ≥ λ _{1} ≥ … ≥ λ _{2N − 1} ≥ 0 on its diagonal. Since some singular values of the matrix Λ are negligible, by selecting r significant singular values, we can obtain optimal rank r channel estimator.
The proposed rank r channel estimator is given by:
where Δ _{ r } is a diagonal matrix containing the values:
Computational complexity evaluation
In this section, we analyze the performance of our proposed estimator in terms of computational complexity. It is shown that the multiplication of two N × N complex matrixes requires 3 N ^{3} + 2 N ^{2} real additions and 4 N ^{3} real multiplications [43].

➢ The calculation of \( {\widehat{\underset{\bar{\mkern6mu}}{H}}}_{\mathrm{WLMMSE}} \) requires:

12 N ^{3} + 8 N ^{2} real additions and 12 N ^{3} real multiplications to calculate \( {\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{H}\overset{\smile }{Y}} \),

24 N ^{3} + 20 N ^{2} real additions and 24 N ^{3} real multiplications to calculate \( {\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{Y}\overset{\smile }{Y}} \),

8 N ^{3} operations to calculate the inverse of \( {\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{Y}\overset{\smile }{Y}} \),

48 N ^{3} + 8 N ^{2} operations to calculate the product of \( {\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{H}\overset{\smile }{Y}} \) by \( {\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{Y}\overset{\smile }{Y}}^{1} \),

and 64 N ^{2} operations to calculate the product of \( {\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{H}\overset{\smile }{Y}}{\underset{\bar{\mkern6mu}}{C}}_{\overset{\smile }{Y}\overset{\smile }{Y}}^{1} \) by \( \overset{\smile }{\underset{\bar{\mkern6mu}}{Y}} \).

Then, the computational complexity of \( {\widehat{\underset{\bar{\mkern6mu}}{H}}}_{\mathrm{WLMMSE}} \) is given by:

➢ The calculation of Ĥ _{LMMSE} requires:

3 N ^{3} + 2 N ^{2} real additions and 3 N ^{3} real multiplications to calculate \( {C}_{\overset{\smile }{H}\overset{\smile }{Y}} \),

6 N ^{3} + 5 N ^{2} real additions and 6 N ^{3} real multiplications to calculate \( {C}_{\overset{\smile }{Y}\overset{\smile }{Y}} \),

N ^{3} operations to calculate the inverse of \( {C}_{\overset{\smile }{Y}\overset{\smile }{Y}} \),

6 N ^{3} + 3 N ^{2} operations to calculate the product of \( {C}_{\overset{\smile }{H}\overset{\smile }{Y}} \) by \( {C}_{\overset{\smile }{Y}\overset{\smile }{Y}}^{1} \),

and 8 N ^{2} operations to calculate the product of \( {C}_{\overset{\smile }{H}\overset{\smile }{Y}}{C}_{\overset{\smile }{Y}\overset{\smile }{Y}}^{1} \) by \( \overset{\smile }{Y} \).

Then, the computational complexity of Ĥ _{LMMSE} is given by:
Therefore, the computational complexity of WLMMSE estimator requires about five times more operations than the LMMSE estimator.

➢ The calculation of \( {\widehat{\underset{\bar{\mkern6mu}}{H}}}_{\mathrm{RWLMMSE}} \) requires:

8 N ^{3} operations to calculate the product of T by \( {R}_{H_R{H}_R} \),

4 N ^{2} operations to calculate the addition of \( {R}_{H_R{H}_R} \) and ξI,

8 N ^{3} operations to calculate the inverse of \( \left({R}_{H_R{H}_R}+\xi I\right) \),

8 N ^{3} operations to calculate the product of \( T{R}_{H_R{H}_R} \) by \( {\left({R}_{H_R{H}_R}+\xi I\right)}^{1} \),

and 4 N ^{2} operations to calculate the product of \( T{R}_{H_R{H}_R}{\left({R}_{H_R{H}_R}+\xi I\right)}^{1} \) by \( {\widehat{H}}_{L{S}_R} \).

Then, the computational complexity of \( {\widehat{\underset{\bar{\mkern6mu}}{H}}}_{\mathrm{RWLMMSE}} \) is given by:

➢ The calculation of \( {\underset{\bar{\mkern6mu}}{\widehat{H}}}_{Lr\mathrm{RWLMMSE}} \) requires:

4N ^{2} r + 2Nr operations to calculate \( U\left(\begin{array}{cc}\hfill {\varDelta}_r\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill \end{array}\right){U}^H \),

8 N ^{3} operations to calculate the product of T by \( U\left(\begin{array}{cc}\hfill {\varDelta}_r\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill \end{array}\right){U}^H \),

and 4 N ^{2} operations to calculate the product of \( TU\left(\begin{array}{cc}\hfill {\varDelta}_r\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill \end{array}\right){U}^H \) by \( {\widehat{H}}_{L{S}_R} \).

Then, the computational complexity of \( {\underset{\bar{\mkern6mu}}{\widehat{H}}}_{Lr\mathrm{RWLMMSE}} \) is given by:
Thus, the computational complexity of \( {\widehat{\underset{\bar{\mkern6mu}}{H}}}_{\mathrm{RWLMMSE}} \) requires a total of 24 N ^{3} + 8 N ^{2} operations, which is about 5 times less than that of \( {\widehat{\underset{\bar{\mkern6mu}}{H}}}_{\mathrm{WLMMSE}} \) and slightly less than that of Ĥ _{LMMSE}. Consequently, the realvalued WLMMSE channel estimator reduces the computational complexity by 80% and accomplishes a computational complexity comparable with that of LMMSE channel estimator. Furthermore, by applying the lowrank Wiener filter technique, the complexity of \( {\widehat{\underset{\bar{\mkern6mu}}{H}}}_{\mathrm{Lr}\hbox{} \mathrm{RWLMMSE}} \) is significantly reduced mainly for small values of r.
Performance evaluations
In this section, we compare the bit error rate (BER) of the proposed lowrank RWLMMSE channel estimator with that of WLMMSE channel estimator. The MBOFDM UWB system parameters are summarized in Table 1. In our simulations, we adopt the CM1 and CM2 UWB channel models of the standard IEEE 802.15.3a.
In Figures 4 and 5, the BER performance of the proposed channel estimator for MBOFDM UWB system for various values of rank r are evaluated in terms of signaltonoise ratio (SNR) in CM1 and CM2, respectively. The used modulation is the rhombicDCM with a rhombic deformation coefficient ε = 0.5. In these figures, the legends WLMMSE, LrRWLMMSE (r = 16), LrRWLMMSE (r = 32), and LrRWLMMSE (r = 64) present the estimators based on WLMMSE, LrRWLMMSE with rank r = 16, LrRWLMMSE with rank r = 32, and LrRWLMMSE with rank r = 64, respectively. In both figures, it is observed that the performance of the LrRWLMMSE estimator is comparable with that of WLMMSE at low SNR and slightly under the WLMMSE estimator’s performance at high SNR. For LrRWLMMSE estimator, the performance degrades slightly as the rank r decreases. This is due to the loss of channel information when the rank of the channel correlation matrix is reduced. These results are further proved by the Figures 6 and 7 which give the performance of the WLMMSE and LrRWLMMSE with rank r = 32 for different values of rhombic deformation coefficient.
Conclusions
In this paper, we have presented a lowcomplexity channel estimator for MBOFDM UWB system. We have investigated the WLMMSE channel estimator algorithm, and we have proceeded with realvalued algorithm followed by lowrank approximation of such channel estimator algorithm. The proposed lowrank realvalued WLMMSE channel estimator algorithm has reduced the computational complexity while maintaining the performance improvement of the WLMMSE one. As a consequence, the proposed LrRWLMMSE channel estimator algorithm has provided not only BER performance improvement but also lower computation cost.
Abbreviations
 AWGN:

additive white Gaussian noise
 BER:

bit error rate
 CDMA:

code division multiple access
 CES:

channel estimation sequence
 DAC:

digitaltoanalog converter
 DCM:

dual carrier modulation
 DS:

direct sequence
 DSUWB:

direct sequence ultrawideband
 EVD:

eigenvalue decomposition
 FCC:

Federal Communications Commission
 IFFT:

inverse fast Fourier transform
 LOS:

line of sight
 Lr:

low rank
 LS:

least square
 MAI:

multipleaccess interference
 MB:

multiband
 MBOA:

Multiband OFDM Alliance
 MC:

multicarrier
 MIMO:

multiple input multiple output
 MMSE:

minimum mean square error
 MSE:

mean square error
 NBI:

narrowband interference
 NLMS:

normalized least mean square
 NLOS:

nonline of sight
 OFDM:

orthogonal frequency division multiplexing
 OLA:

overlapadded
 PLCP:

physical layer convergence protocol
 PPDU:

PLCP protocol data unit
 PSDU:

PLCP service data unit
 RR:

reducedrank
 RRMLE:

reducedrank maximum likelihood estimation
 RWLMMSE:

realvalued WLMMSE
 SER:

signal error rate
 SNR:

signaltonoise ratio
 SVD:

singular value decomposition
 UWB:

ultrawideband
 WF:

Wiener filtering
 WL:

widely linear
 WPAN:

wireless personal area networks
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Keywords
 Widely linear estimation
 Computational complexity
 Low rank
 Real valued
 MBOFDM
 UWB