Frequency-Domain Group-based Shrinkage Estimators for UWB Systems
In this work, we propose low-complexity adaptive biased estimation algorithms, called group-based shrinkage estimators (GSEs), for parameter estimation and interference suppression scenarios with mechanisms to automatically adjust the shrinkage factors. The proposed estimation algorithms divide the target parameter vector into a number of groups and adaptively calculate one shrinkage factor for each group. GSE schemes improve the performance of the conventional least squares (LS) estimator in terms of the mean-squared error (MSE), while requiring a very modest increase in complexity. An MSE analysis is presented which indicates the lower bounds of the GSE schemes with different group sizes. We prove that our proposed schemes outperform the biased estimation with only one shrinkage factor and the best performance of GSE can be obtained with the maximum number of groups. Then, we consider an application of the proposed algorithms to single-carrier frequency-domain equalization (SC-FDE) of direct-sequence ultra-wideband (DS-UWB) systems, in which the structured channel estimation (SCE) algorithm and the frequency domain receiver employ the GSE. The simulation results show that the proposed algorithms significantly outperform the conventional unbiased estimator in the analyzed scenarios.
Index Terms–DS-UWB systems, parameter estimation, interference suppression, biased estimation, adaptive algorithm.
In this work, biased estimation algorithms are considered in two common deterministic estimation scenarios in communications engineering, which are parameter estimation and interference suppression -. It is known that under the assumption of AWGN, the least-square (LS) algorithm can provide an efficient solution to these estimation problems and will lead to minimum variance unbiased estimators (MVUE). The unbiasness is usually considered as a good property for an estimator because the expected value of unbiased estimators is the true value of the unknown parameter . However, in some scenarios the LS method is not directly related to the mean square error (MSE) associated with the target parameter vector and it has been found that a lower MSE can be achieved by adding an appropriately chosen bias to the conventional LS estimators ,. Note that some reduced-rank techniques also employ a bias to accelerate the convergence speed -.
A class of biased estimator that has been studied in recent years is known as the biased estimators with a shrinkage factor -. These biased estimation algorithms have shown their ability to outperform the existing unbiased estimators especially in low signal-to-noise ratios (SNR) scenarios and/or with short data records . For these biased estimators -, the complexity is much lower than for MMSE algorithms because the additional number of coefficients to be computed is only one. The motivation for the group-based shrinkage estimator (GSE) is to find a generalized estimator with a number of shrinkage factors that can achieve a better performance and complexity tradeoff than the biased estimator with only one shrinkage factor.
In the parameter estimation scenario,some biased estimators have been proposed to achieve a smaller estimation error than the LS solutions by removing the unbiasedness of the conventional estimators with a shrinkage factor in the parameter estimation scenario. The earliest shrinkage estimators that reduce the MSE over MVUE include the well known James-Stein estimator  and the work of Thompson . Some extensions of the James-Stein estimator have been proposed in -. In , blind minimax estimation (BME) techniques have been proposed, in which the biased estimators were developed to minimize the worst case MSE among all possible values of the target parameter vector within a parameter set. If a spherical parameter set is assumed, the shrinkage estimator obtained is named spherical BME (SBME) .
For the interference suppression scenario, the biased estimators can be employed to achieve a lower estimation error between the estimated filter and the optimal linear LS estimator. The major motivation for adopting the biased algorithms here is to accelerate the convergence rate for the adaptive implementations and provide a better performance with short training data support in long filter scenarios .
To the best of our knowledge, biased estimators with shrinkage factors are rarely implemented into real-world signal processing and have not been considered in the frequency domain for communication systems. One possible reason is that some assumptions required for the signal model may not be satisfied. For example, in time-hopping UWB (TH-UWB) systems, the multiple access interference (MAI) cannot be accurately approximated by a Gaussian distribution for some values of the the signal-to-interference-plus-noise ratio (SINR) . Another possible reason is that the existing shrinkage-based estimators usually require some statistical information such as the noise variance and the norm of the actual parameter vector. In our previous work  and , adaptive biased estimation algorithms with only one shrinkage factor have been proposed to fulfill the tasks of interference suppression and parameter estimation. In this work, a novel biased estimation technique, named group-based shrinkage estimators (GSE), is proposed. In this algorithm, the target parameter vector is divided into several groups and one shrinkage factor is calculated for each group. Least mean square (LMS)-based adaptive estimation algorithms are then developed to calculate the shrinkage factors. The GSE estimators are able to improve the performance of the recursive least squares (RLS) algorithm that recursively computes the LS estimator. In DS-UWB systems, the estimation tasks are usually very challenging because the environments include dense multipath. In this work, in order to test the proposed algorithms, we consider applications of DS-UWB systems with SC-FDE. Specifically, we concentrate on the channel estimation and interference suppression with the proposed algorithms. The MSE performance of the proposed GSE schemes is then analyzed, a lower bound of the MSE performance is derived and the relationship between the number of groups and the lower bound is set up. Simulations show that with an additional complexity that is only linearly dependent on the size of the parameter vector and the number of groups, the proposed biased GSE algorithms outperform the conventional RLS algorithm in terms of MSE in low SNR scenarios and/or with short data support. It should be noted that the proposed GSE estimator can be employed for applications where a high estimation accuracy is required. These include localization in wireless sensor networks  and in dense cluttered environments with UWB technology . The proposed estimators can also be employed into emergent multicast and broadcast systems , such as the orthogonal frequency-division multiplexing (OFDM) based multi-user multiple-input multiple-output (MIMO) systems as specified in the IEEE 802.11ac standard and the 3GPP long-term-evolution (LTE) systems.
The main contributions of this work are summarized as follows:
Novel GSE schemes are proposed to improve the performance of the frequency domain RLS algorithms in the applications of parameter estimation and interference suppression in DS-UWB systems.
LMS based adaptive algorithms are developed for both scenarios to adjust the shrinkage factors.
The MSE analysis is carried out which indicates a lower bound of the proposed estimator and the relationship between the lower bound and the number of groups.
The performance of the proposed biased estimators is examined in multiuser SC-FDE for DS-UWB systems with the IEEE 802.15.4a channel model, convolutional and low-density parity-check (LDPC) codes.
The rest of this paper is structured as follows. In Section II, we first review the LS solution for the parameter estimation scenario and present the structured channel estimation (SCE) problem in SC-FDE of DS-UWB systems. Then, the signal model of the frequency domain receiver design for DS-UWB systems that represents the interference suppression scenario, is presented. The proposed GSE scheme and its adaptive implementations for the parameter estimation scenario and the interference suppression scenario are developed in Section III and Section IV, respectively. The MSE analysis is shown in Section V. The simulation results are shown in Section VI. Section VII draws the conclusions.
Ii System model
In this section, we introduce the channel estimation and receiver design tasks in the frequency-domain of DS-UWB systems with SC-FDE that represent the parameter estimation scenario and the interference suppression scenario, respectively.
Ii-a Problem statement for the parameter estimation scenario
The linear model for the parameter estimation scenario can be expressed as:
where the training data matrix and the received signal are given, is AWGN with zero mean and variance . In this scenario, the typical target is to estimate the parameter vector that leads to the minimum MSE. The MSE consists of the estimation variance and the squared bias and is given by
where represents expectation of a random variable.
The conventional LS algorithm estimates the parameter by minimizing
Assuming that the matrix is a full rank matrix, the LS solution is given by
Under the assumption of AWGN with zero mean and variance , the LS estimator is a MVUE that leads to a minimum MSE
we define , where represents the trace operator .
The objective of introducing the biased estimator in the parameter estimation scenario is to achieve a lower MSE than the unbiased estimator, which can be expressed as
Although the objective shown here is similar to MMSE algorithms that is to achieve an MSE as small as possible. It should be noted that the biased algorithm developed in this work adopts a different strategy from MMSE algorithms.
Ii-B System model for the SCE: parameter estimation scenario
Here, we consider the channel estimation problem of a synchronous downlink block-by-block transmission DS-UWB system based on SC-FDE with users. The block diagram of the parameter estimation scenario is shown as branch (a) in Fig. 1. For notational simplicity, we assume that a -by- Walsh spreading code is assigned to the -th user. The spreading gain is , where and denote the symbol duration and chip duration, respectively. At each time instant , a data vector is transmitted by the -th user. We define the signal after spreading as , where the block diagonal matrix () performs the spreading of the data block and its first column is constructed by the spreading code zero-padded to the length of . In order to prevent inter block interference (IBI), a cyclic-prefix (CP) is added and the length of the CP is assumed to be larger than the length of the channel impulse response (CIR). With the insertion of the CP at the transmitter and its removal at the receiver, the equivalent channel is denoted as a circulant Toeplitz matrix , whose first column is composed of a vector zero-padded to length , where is the equivalent CIR. At the receiver, a chip matched-filter (CMF) is applied and the received sequence is then sampled at chip-rate and organized in an -dimensional vector. This signal then goes through the discrete Fourier transform (DFT). The frequency-domain received signal is given by
where represents the AWGN, is a diagonal matrix whose diagonal vector is defined as and its -th entry is given by , where . represents the DFT matrix and its -th entry is where . By defining a matrix that contains the first columns of the DFT matrix , we obtain the following relationship
In unstructured channel estimation (UCE), the vector is directly estimated, while in the structured channel estimation (SCE), the fact that is taken into account and the vector is the parameter vector to be estimated. The concept of SCE was proposed in , where the SCE shows a better performance than the UCE. In , adaptive MMSE detection schemes for SC-FDE in multi-user DS-UWB systems based on SCE are developed, where the estimated is adaptively calculated based on RLS, least-mean squares (LMS) and the conjugate gradient (CG) algorithm for the detection and the RLS version performs the best. The purpose of developing biased estimation in this scenario is to further improve the performance of the RLS algorithm in terms of the MSE.
We consider user 1 as the desired user and omit the subscript of this user for simplicity. Note that the frequency domain received signal can be expressed as
where we define a diagonal matrix , the noise and interference vector consists of the MAI and the noise and is assumed to be AWGN. As shown in (7), the SCE problem is an implementation example of the parameter estimation problem where a given matrix is defined as . The LS solution of is given by
where , and is the forgetting factor. Then the LS solution can be computed recursively by the following RLS algorithm 
where is the -dimensional error vector.
In Section III, a novel biased estimation algorithm called group-based shrinkage estimator (GSE) is incorporated into the unbiased LS estimator that is able to improve the estimation performance in terms of the MSE.
Ii-C System model for the frequency domain receiver design: interference suppression scenario
The block diagram of the interference suppression scenario is shown as branch (b) in Fig. 1. For each time instant , an -dimensional data vector is transmitted by the -th user. After the spreading, the -dimensional transmit signal is given by
where the spreading matrix , , is a circulant Toeplitz matrix and its first column consists of the spreading codes and zero-padding . The equivalent -dimensional expanded data vector is
where is the transpose. Using this signal expression we can obtain a simplified frequency domain receiver design. At the receiver, a CMF is applied and the received sequence is then sampled at chip-rate and organized in an -dimensional vector. After the DFT, the received signal is given by
where is the AWGN and represents the DFT matrix. Since both and are circulant Toeplitz matrices, their product also has the circulant Toeplitz form. This feature makes a diagonal matrix. Hence, we have
We can further expand as 
where denotes the DFT matrix and is structured as
where denotes the -by- identity matrix. Finally, the frequency domain received signal is given by
Note that the expression in (15) is an implementation example of the interference suppression scenario where the unknown matrix for each time instant is given by . To fulfill the interference suppression task, an MMSE filter can be developed via the following cost function:
The MMSE solution is given by 
where the matrix is
where denotes the identity matrix. Note that the matrix consists of times diagonal matrices , where . Hence, we take a closer look at the product of and :
where , , are diagonal matrices. Hence, the product of and can be converted into a product of a diagonal matrix () and , where the diagonal entries of are , , and equal the sum of all entries in the -th row of matrix . Finally, we express the MMSE design as
where is an equivalent filter with taps.
The expression shown in (19) enables us to design an -dimensional receive filter rather than an -by- matrix form receive filter. The estimated data vector can be expressed as
where and is the weight vector of the adaptive receiver. Since and are fixed, we consider the equivalent -by- received data matrix as and express the estimated data vector as .
Ii-D LS solution and adaptive RLS algorithm for the interference suppression scenario
Here, we detail the LS and RLS designs for the frequency domain multiuser receiver . The cost function for the development of the LS estimation is given by
The LS design of the linear receiver can be expressed as
where the matrix is defined as and represents the vector . Note that, the data vector can be expressed as
where is the measurement error vector and is the optimum tap-weight vector of the receiver (optimum in the MSE sense). Assuming that is white and Gaussian with zero mean and covariance of , then the LS solution in (22) is a MVUE . Now, let us have a look at the following MSE:
Defining , we have 
where is the variance of the measurement error.
In the interference suppression scenario, it is possible to introduce the biased estimation to reduce the MSE between the optimal receive filter and the LS estimator . Note that, for the interference suppression scenario, the typical objective is to minimize the overall performance criterion which is determined as , rather than to minimize . The main motivation to introduce the bias in the interference suppression scenario is to provide an initial improvement for the overall performance when the adaptive filtering techniques are employed and the training data are limited. This can also help with tracking problems and with robustness against interference.
The LS solution of the receiver can be computed recursively by the RLS adaptive algorithm. We employ the RLS update equation that is proposed in 
where and . Note that is an -by- symmetric sparse matrix in which the number of nonzero elements equals . Hence, the complexity of each adaptation by using this algorithm is .
Iii Proposed GSE for parameter estimation scenario
Iii-a Proposed GSE: Optimal Solution
It is known that the biased estimator with a shrinkage factor can be expressed as
where is the LS estimator of the parameter vector and is the biased estimator with a shrinkage factor, is a real-valued variable and is defined as the real-valued shrinkage factor that is larger than 0 but smaller than 1 (i.e., ).
Actually, for the parameter estimation scenario, the MMSE estimators with the following expression can also be considered as a biased estimator,
where and is a full-rank matrix. As in  and , such MMSE channel estimators are developed for MIMO and OFDM systems, respectively. Although these MMSE estimators can achieve a much lower MSE than the LS estimator especially in low SNR regime, they experience much higher complexity than the biased estimator with only one shrinkage factor. In , the proposed scaled LS channel estimator can be considered as a biased estimator with only one shrinkage factor, which outperforms the conventional LS estimator while it requires a much lower complexity than the MMSE estimator. The basic idea of the following proposed group-based shrinkage estimator (GSE) is to find a solution with a better tradeoff between the complexity and the performance than the MMSE estimator and the biased estimator with only one shrinkage factor.
The proposed GSE can be expressed as follows:
where is a block diagonal matrix that is constructed from the elements of as well as zeros, is the number of groups, we define the -dimensional column vectors and . The scalar is a real-valued variable and is defined as the shrinkage factor for the -th group of coefficients that is larger than 0 but smaller than 1, where . Here we propose to use a uniform group size for the -dimensional parameter vector, hence the size of each group is . If the length of the parameter vector divided by the group size is not an integer, we can perform zero-padding in the parameter estimation vector to fulfill this requirement. If any statistical knowledge of the parameter vector is given, the group size could be different from each other. But this approach will introduce a higher complexity because we need to select the size of each group and choose a suitable one. In this work, for notational simplicity, we will focus on the low complexity uniform group size approach.
The goal is to minimize the MSE defined by
and we have where
and , assuming that all the elements in this equivalent noise vector are independent and identically distributed (i.i.d.) random variables. Hence, we have
Note that equals the variance of the equivalent noise times the length of the group. We also have
Finally, the optimal solution of the vector that minimizes (30) is given by
and we have
Note that this equation is a general expression for different numbers of groups. The complexity of this algorithm is very low because the inverse matrix required to calculate the is a diagonal matrix. Hence, this estimator combined with the conventional RLS algorithm will only introduce an additional complexity that is linear in the length of the parameter vector and the number of groups . If the group size equals , then the GSE converges to the biased estimator with only one shrinkage factor. In the following section, adaptive algorithms will be developed to compute the best GSE with a given group size.
Iii-B Proposed GSE: Adaptive Algorithms
It should be noted that the optimal solution of the biased estimator requires some prior knowledge of the system, which is the matrix and the scalar term for calculation of the vector . In addition, the LS channel estimator is also required. The LS channel estimator can be recursively calculated by the RLS adaptive algorithm that is detailed in Section II-B. In this work, we propose LMS-based adaptive algorithms that enable us to estimate the vector without prior knowledge of the channel and the noise variance. Substituting (34) and (33) into (30) and considering the MSE cost function as a function of , we can obtain a new cost function
The gradient of with respect to is given by
Note that, because is a real-valued vector, there is a factor of 2 for this gradient. In what follows, this factor is absorbed into the step size of gradient-type recursions. Hence, the LMS-based update equation of the vector for the -th time slot can be expressed as
where is the step size of the LMS algorithm and the estimated gradient vector is given by
Here, is the estimated equivalent noise variance and the diagonal matrix is defined as the estimator of the matrix , the main diagonal vector of this matrix is defined as . In this work, we adopt the instantaneous estimator as , where is the RLS channel estimator and represent the time averaged channel estimator. Note that is a diagonal matrix with its -th diagonal element equals .
Hence, the elements in the optimal solution can be expressed as . If we use the matrix to replace the matrix , the estimated optimal solution becomes
Recall that we assume that the shrinkage factors for each group are larger than zero but smaller than one. It can be found that if , this assumption no longer holds. In addition, if , the shrinkage factor converges to zero, and the biased estimator actually converges to the unbiased estimator. So we should constrain the values of into the range .
In order to determine the diagonal matrices for each time instant, two approaches are developed in this work. In the first approach, which is named estimator based (GSE-EB) method, the matrices are replaced by the diagonal matrices , where is the RLS estimator of . Note that, when the number of groups is only one, the GSE-EB method will lead to an optimal shrinkage factor that has the same expression as the SBME that is proposed in . However, the knowledge of the noise variance is not required in our work. In the second approach, which is named automatic tuning (GSE-AT) method, an LMS-based algorithm is proposed to update the diagonal matrices .
For the GSE-EB method, the estimation of is only determined by the RLS estimator. If the effective spreading codes and the channel information is known, the RLS algorithm can be initialized efficiently. Here, we consider a general scenario where all these quantities are unknown and the initialization of the RLS algorithm is an all zero vector, which means the beginning stage of the RLS algorithm is not very accurate. In order to improve the convergence rate of the proposed GSE schemes, we develop the following GSE-AT algorithm. For each time instant, is firstly set to as in the GSE-EB algorithm, then we consider as the variables of the MSE cost function, where are the diagonal elements of the diagonal matrix and . Then we develop an LMS adaptive recursion to further adapt these values and improve the estimation accuracy for each time instant. Let us reexpress the MSE cost function as shown in (30) as follows. Here, we omit the time index for simplicity
where is also a function of . Expanding this cost function, we have
Hence, for each group, the corresponding can be obtained by the following equation
where is the iteration index and is the estimator of the gradient of the function (43) with respect to , which is given by
In order to obtain a low complexity solution, for the GSE-AT algorithm, we set the iteration index to 1, which means for each time instant we only update the values of once. As pointed out previously, the values of