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Improved least mean square algorithm with application to adaptive sparse channel estimation
EURASIP Journal on Wireless Communications and Networking volume 2013, Article number: 204 (2013)
Abstract
Least mean square (LMS)based adaptive algorithms have attracted much attention due to their low computational complexity and reliable recovery capability. To exploit the channel sparsity, LMSbased adaptive sparse channel estimation methods have been proposed based on different sparse penalties, such as ℓ_{1}norm LMS or zeroattracting LMS (ZALMS), reweighted ZALMS, and ℓ_{p}norm LMS. However, the aforementioned methods cannot fully exploit channel sparse structure information. To fully take advantage of channel sparsity, in this paper, an improved sparse channel estimation method using ℓ_{0}norm LMS algorithm is proposed. The LMStype sparse channel estimation methods have a common drawback of sensitivity to the scaling of random training signal. Thus, it is very hard to choose a proper learning rate to achieve a robust estimation performance. To solve this problem, we propose several improved adaptive sparse channel estimation methods using normalized LMS algorithm with different sparse penalties, which normalizes the power of input signal. Furthermore, CramerRao lower bound of the proposed adaptive sparse channel estimator is derived based on prior information of channel taps' positions. Computer simulation results demonstrate the advantage of the proposed channel estimation methods in mean square error performance.
1 Introduction
The demand for highspeed data services has been increasing as emerging wireless devices are widely spreading. Various portable wireless devices, e.g., smartphones and laptops, have generated rapidly, giving rise to massive data traffic [1]. Broadband transmission is an indispensable technique in the nextgeneration wireless communication systems [2, 3]. The broadband channel is described by a sparse channel model in which multipath taps are widely separated in time, thereby creating a large delay spread [4–8]. In other words, most of channel coefficients are zero or close to zero, while only a few channel coefficients are dominant (large value). A typical example of sparse multipath channel is shown in Figure 1, where the number of dominant taps is four while the length is 16.
Traditional least mean square (LMS) is one of the most popular algorithms for adaptive system identification [9], e.g., channel estimation. LMSbased adaptive channel estimation can be easily implemented due to its low computational complexity. In current broadband wireless communication systems, channel impulse response in time domain is often described by a sparse channel model, supported by a few large coefficients. The LMSbased adaptive channel estimation method never takes advantage of the channel sparse structure although its mean square error (MSE) lower bound has a direct relationship with finite impulse response (FIR) channel length. To improve the estimation performance, recently, many algorithms have been proposed to take advantage of the sparse nature of the channel. For example, based on the recent theory of compressive sensing (CS) [10, 11], various sparse channel estimation methods have been proposed in [12–14]. Some of these methods are known to achieve robust estimation, e.g., sparse channel estimation using leastabsolute shrinkage and selection operator [15]. However, these kinds of sparse methods have two potential disadvantages: one is that computational complexity may be very high, especially for tracking fast timevariant channels; the other is that training signal matrices for these CSbased sparse channel estimation methods are required to satisfy the restricted isometry property [16]. It was well known that designing these training matrices is a nondeterministic polynomialtime (NP)hard problem [17].
In order to avoid the two detrimental problems of sparse channel estimation, a variation of the LMS algorithm with ℓ_{1}norm sparse constraint has been proposed in [18]. The ℓ_{1}norm sparse penalty is incorporated into the cost function of the conventional LMS algorithm, which results in the LMS update with a zero attractor, namely zeroattracting LMS (ZALMS) and reweighted ZALMS (RZALMS) [18] which is motivated by the reweighted ℓ_{1}norm minimization recovery algorithm [19]. To further improve the estimation performance, an adaptive sparse channel estimation method using ℓ_{p}norm LMS (LPLMS) algorithm has been proposed [20]. However, there still exists a performance gap between LPLMS and optimal sparse channel estimation. It is worth mentioning that optimal channel estimation is often denoted by the sparse lower bound which is derived in Theorem 4 in Section 3.
Due to the fact that solving the LPLMS algorithm is a nonconvex problem, the algorithm cannot obtain an optimal adaptive sparse solution [10, 11]. Hence, a computationally efficient algorithm is required to obtain a more accurate adaptive sparse channel estimation.
According to the CS theory [10, 11], solving the ℓ_{0}norm sparse penalty problem can obtain an optimal sparse solution. In other words, ℓ_{0}norm sparse penalty on LMS algorithm is a good candidate to achieve more accurate channel estimations. These backgrounds motivate us to use the optimal ℓ_{0}norm sparse penalty on LMS in order to improve the estimation performance.
In addition, since sparse LMSbased channel estimation methods have a common drawback of sensitivity to scaling of random training signal, it is very hard to choose a proper learning rate to achieve a robust estimation performance [21]. To solve this problem, we propose several improved adaptive sparse channel estimation methods using normalized LMS (NLMS) algorithm, which normalizes the power of input signal, with different sparse penalties, i.e., ℓ_{p}norm (0 ≤ p ≤ 1).
The contributions of this paper are described below. Firstly, we propose an improved adaptive sparse channel estimation method using ℓ_{0}norm least square error algorithm, termed as L0LMS [22]. Secondly, based on algorithms in [18, 20], we propose four kinds of improved adaptive sparse channel estimation methods using sparse NLMS algorithms. Thirdly, CramerRao lower bound (CRLB) of the proposed adaptive sparse channel estimator is derived based on prior information of known positions of nonzero taps. Lastly, various simulation results are given to confirm the effectiveness of our proposed methods.
Section 2 introduces the system model and problem formulation. Section 3 discusses various adaptive sparse channel estimation methods using different LMSbased algorithms. In Section 4, computer simulation results for the MSE performance are presented to confirm the effectiveness of sparsityaware modifications of LMS. Concluding remarks are presented in Section 5.
2 System model and problem formulation
Consider a sparse multipath communication system, as shown in Figure 2.
The input signal x(t) and ideal output signal y(t) are related by
where h = [h_{0}, h_{1}, …, h_{N − 1}]^{T} is Nlength sparse channel vector which is supported by K (K ≪ N) dominant taps, x(t) = [x(t), x(t − 1), …, x(t − N + 1)]^{T} is an Nlength input signal vector of x(t) and z(t) is an additive noise at time t. The objective of the LMStype adaptive filter is to estimate the unknown sparse channel h using x(t) and y(t). According to Equation 1, the n th channel estimation error e(n) is easily written as
at time t, where \tilde{\mathbf{h}}\left(n\right) is the LMS adaptive channel estimator. It was worth noting that x(t) and y(t) are invariant as iterative times. Based on Equation 2, standard LMS cost function can be written as
Hence, the updated equation of LMS adaptive channel estimation is derived as
where μ is the step size of gradient descend. For later use, we define a parameter γ_{max} which is the maximum eigenvalue of the covariance matrix R = E{x(t)x^{T}(t)} of input signal vector x(t). Here, we introduce Theorem 1 in relation to the step size μ. The detailed derivation can also be found in [21].
Theorem 1 The necessary condition of reliable LMS adaptive channel estimation is 0 < μ < 2/γ_{ max }.
Proof At first, the (n + 1)th update coefficient vector \tilde{\mathbf{h}}\left(n+1\right) is written as
Assume that h is a real FIR channel vector; subtracting h from both sides in Equation 5, we can rewrite it as
By defining \mathbf{v}\left(n\right)=\tilde{\mathbf{h}}\left(n\right)\mathbf{h} and according to Equation 1, Equation 6 can be rewritten as
Assume that input signal vector x(t) and estimated FIR channel vector \tilde{\mathbf{h}}\left(n\right) are statistically independent from the other, then x(t) and z(t) are also independent, i.e., E{x(t)z(t)} = 0. Taking the expectation for both sides in Equation 7, we can obtain
The necessary condition of the reliable LMS adaptive channel estimation is given by
where λ_{max} denotes the maximum eigenvalue of R. Hence, to guarantee fast convergence speed and reliable updating of LMS in Equation 4, step size μ should be satisfied:
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3 (N)LMSbased adaptive sparse channel estimation
From Equation 4, we can find that the LMSbased channel estimation method never takes advantage of the sparse structure in channel vector h. To get a better understanding, the LMSbased channel estimation methods can be expressed as
Unlike the conventional LMS method in Equation 11, sparse LMS algorithms exploit channel sparsity by introducing several ℓ_{p}norm penalties to their cost functions with 0 ≤ p ≤ 1. The LMSbased adaptive sparse channel estimation methods can be written as
Equation 12 motivated us to introduce different sparse penalties in order to take advantage of the sparse structure as prior information. Here, if we analogize the updated equation in Equation 12 to the CSbased sparse channel estimation [10, 11], one can find that more accurate sparse channel approximation is adopted, and then better estimation accuracy could be obtained and vice versa. The conventional sparse penalties are ℓ_{p}norm (0 < p ≤ 1) and ℓ_{0}norm, respectively. Since ℓ_{0}norm penalty on sparse signal recovery is a wellknown NPhard problem, only the ℓ_{p}norm (0 < p ≤ 1)based sparse LMS approaches have been proposed for adaptive sparse channel estimation. Compared to conventional two sparse LMS algorithms (ZALMS and RZALMS), LPLMS can achieve a better estimation performance. However, there still exists a performance gap between the LPLMSbased channel estimator and the optimal one. As the development of mathematics continues, a more accurate sparse approximate algorithm to ℓ_{0}norm LMS (L0LMS) is proposed in [22] and analyzed in [23]. However, they never considered any application on sparse channel estimation. In this paper, the L0LMS algorithm is applied in adaptive sparse channel estimation to improve the estimation performance.
It is easily found that exploitation of more accurate sparse structure information can obtain a better estimation performance. In the following, we investigate sparse LMSbased adaptive sparse channel estimation methods using different sparse penalties.
3.1 LMSbased adaptive sparse channel estimation
The following are the LMSbased adaptive sparse channel estimation methods:

ZALMS. To exploit the channel sparsity in time domain, the cost function of ZALMS [18] is given by
{L}_{\mathrm{ZA}}\left(n\right)=\frac{1}{2}{e}^{2}\left(n\right)+{?}_{\mathrm{ZA}}{\tilde{\mathbf{h}}\left(n\right)}_{1},(13)where ?_{ZA} is a regularization parameter to balance the estimation error and sparse penalty of \tilde{\mathbf{h}}\left(n\right). The corresponding updated equation of ZALMS is
\begin{array}{l}\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)\mu \frac{?{L}_{\mathrm{ZA}}\left(n\right)}{?\tilde{\mathbf{h}}\left(n\right)}\\ \phantom{\rule{3.6em}{0ex}}=\tilde{\mathbf{h}}\left(n\right)+\mathit{\mu e}\left(n\right)\mathbf{x}\left(t\right){?}_{\mathrm{ZA}}\mathrm{sgn}\left(\tilde{\mathbf{h}}\left(n\right)\right),\end{array}(14)where ?_{ZA}?=?µ?_{ZA} and sgn(·) is a componentwise function which is defined as
\mathrm{sgn}\left(\mathbf{h}\right)=\left\{\begin{array}{c}\hfill \begin{array}{c}\hfill 1,h>0\hfill \\ \hfill 0,h=0\hfill \end{array}\hfill \\ \hfill 1,h<0\hfill \end{array}\right..? From the updated equation in Equation 14, the function of its second term is compressing small channel coefficients as zero in high probability. That is to say, most of the small channel coefficients can be simply replaced by zeros, which speeds up the convergence of this algorithm.

RZALMS. ZALMS cannot distinguish between zero taps and nonzero taps as it gives the same penalty to all the taps which are often forced to be zero with the same probability; therefore, its performance will degrade in less sparse systems. Motivated by the reweighted l_{1}norm minimization recovery algorithm [19], Chen et al. have proposed a heuristic approach to reinforce the zero attractor which was termed as the RZALMS [18]. The cost function of RZALMS is given by
{L}_{\mathrm{RZA}}\left(n\right)=\frac{1}{2}{e}^{2}\left(n\right)+{?}_{\mathrm{RZA}}\underset{i=1}{\overset{N}{{\displaystyle ?}}}\phantom{\rule{0.25em}{0ex}}\mathrm{log}\left(1+{?}_{\mathrm{RZA}}\left{h}_{i}\left(n\right)\right\right),(15)where ?_{RZA}?>?0 is the regularization parameter and ?_{RZA}?>?0 is the positive threshold. In computer simulation, the threshold is set as ?_{RZA}?=?20 which is also suggested in [18]. The i th channel coefficient {\tilde{h}}_{i}\left(n\right) is then updated as
\begin{array}{l}{\tilde{h}}_{i}\left(n+1\right)={\tilde{h}}_{i}\left(n\right)\mu \frac{?{L}_{\mathrm{RZA}}\left(n\right)}{?{\stackrel{~}{h}}_{i}\left(n\right)}\\ \phantom{\rule{4em}{0ex}}={\tilde{h}}_{i}\left(n\right)+\mathit{\mu e}\left(n\right)x\left(ti\right){?}_{\mathrm{RZA}}\frac{\mathrm{sgn}\left({\tilde{h}}_{i}\left(n\right)\right)}{1+{?}_{\mathrm{RZA}}\left{\tilde{h}}_{i}\left(n\right)\right},\end{array}(16)where ?_{RZA}?=?µ?_{RZA}?_{RZA}. Equation 16 can be expressed in the vector form as
\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)+\mathit{\mu e}\left(n\right)\mathbf{x}\left(t\right){?}_{\mathrm{RZA}}\frac{\mathrm{sgn}\left(\tilde{\mathbf{h}}\left(n\right)\right)}{1+{?}_{\mathrm{RZA}}\left\tilde{\mathbf{h}}\left(n\right)\right}.? Please note that the second term in Equation 16 attracts the channel coefficients {\tilde{h}}_{i}\left(n\right),i=1,2,\dots ,N whose magnitudes are comparable to 1/?_{RZA} to zeros.

LPLMS. Following the above ideas in [18], LPLMSbased adaptive sparse channel estimation method has been proposed in [20]. The cost function of LPLMS is given by
{L}_{\mathrm{LP}}\left(n\right)=\frac{1}{2}{e}^{2}\left(n\right)+{?}_{\mathrm{LP}}{\tilde{\mathbf{h}}\left(n\right)}_{p},(17)where ?_{LP}?>?0 is a regularization parameter. The corresponding updated equation of LPLMS is given as
\begin{array}{l}\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)\mu \frac{?{L}_{\mathrm{LP}}\left(n\right)}{?\tilde{\mathbf{h}}\left(n\right)}\\ \phantom{\rule{4em}{0ex}}=\tilde{\mathbf{h}}\left(n\right)+\mathit{\mu e}\left(n\right)\mathbf{x}\left(t\right){?}_{\mathrm{LP}}\frac{{\tilde{\mathbf{h}}\left(n\right)}_{p}^{1p}\mathrm{sgn}\left\{\tilde{\mathbf{h}}\left(n\right)\right\}}{{?}_{\mathrm{LP}}+{\left\tilde{\mathbf{h}}\left(n\right)\right}^{\left(1p\right)}},\end{array}(18)where ?_{LP}?=?µ?_{LP} and ?_{LP}?>?0 is a small positive parameter.
? L0LMS (proposed). Consider l_{0}norm penalty on LMS cost function to produce a sparse channel estimator as this penalty term forces the channel tap values of \tilde{\mathbf{h}}\left(n\right) to approach zero. The cost function of L0LMS is given by
{L}_{L0}\left(n\right)=\frac{1}{2}{e}^{2}\left(n\right)+{?}_{L0}{\tilde{\mathbf{h}}\left(n\right)}_{0},(19)where ?_{L 0}?>?0 is a regularization parameter and {\tilde{\mathbf{h}}\left(n\right)}_{0} denotes l_{0}norm sparse penalty function which counts the number of nonzero channel taps of \tilde{\mathbf{h}}\left(n\right). Since solving the l_{0}norm minimization is an NPhard problem [17], to reduce computational complexity, we replace it with an approximate continuous function:
{\tilde{\mathbf{h}}\left(n\right)}_{0}\u02dc\underset{i=0}{\overset{N1}{{\displaystyle ?}}}\left(1{e}^{\xdf\left{\tilde{h}}_{i}\left(n\right)\right}\right).(20)? The cost function in Equation 19 can then be rewritten as
{L}_{L0}\left(n\right)=\frac{1}{2}{e}^{2}\left(n\right)+{?}_{L0}\underset{i=0}{\overset{N1}{{\displaystyle ?}}}\left(1{e}^{\xdf\left{\tilde{h}}_{i}\left(n\right)\right}\right).(21)? The firstorder Taylor series expansion of exponential function {e}^{\xdf\left{\tilde{h}}_{i}\left(n\right)\right} is given as
{e}^{\xdf\left{\tilde{h}}_{i}{}_{i}\left(n\right)\right}\u02dc\left\{\begin{array}{cc}\hfill 1\xdf\left{\tilde{h}}_{i}\left(n\right)\right,\hfill & \hfill \mathrm{when}\phantom{\rule{0.25em}{0ex}}\left{\tilde{h}}_{i}\left(n\right)\right=\frac{1}{\xdf}\hfill \\ \hfill 0,\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\right..(22)? The updated equation of L0LMSbased adaptive sparse channel estimation can then be derived as
\begin{array}{l}\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)\\ \phantom{\rule{4.2em}{0ex}}+\mathit{\mu e}\left(n\right)\mathbf{x}\left(t\right){?}_{L0}\xdf\mathrm{sgn}\left(\tilde{\mathbf{h}}\left(n\right)\right){e}^{\xdf\tilde{\mathbf{h}}\left(n\right)},\end{array}(23)where ?_{L 0}?=?µ?_{L 0}. Unfortunately, the exponential function in Equation 23 will also cause high computational complexity. To further reduce the complexity, an approximation function J\left(\tilde{\mathbf{h}}\left(n\right)\right) is also proposed to the updated Equation 23. Finally, the updated equation of L0LMSbased adaptive sparse channel estimation can be derived as
\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)+\mathit{\mu e}\left(n\right)\mathbf{x}\left(t\right){?}_{L0}J\left(\tilde{\mathbf{h}}\left(n\right)\right),(24)where ?_{ L0 }?=?µ?_{ L0 } and J\left(\tilde{\mathbf{h}}\left(n\right)\right) is defined as
J\left(\tilde{\mathbf{h}}\left(n\right)\right)=\left\{\begin{array}{cc}\hfill 2{\xdf}^{2}{\tilde{h}}_{i}\left(n\right)2\xdf\xb7\mathrm{sgn}\left({\tilde{\mathit{h}}}_{i}\left(n\right)\right)\hfill & \hfill \mathrm{when}\phantom{\rule{0.25em}{0ex}}{\tilde{h}}_{i}\left(n\right)=\frac{1}{\xdf}\hfill \\ \hfill 0,\hfill & \hfill \mathrm{otherwise},\hfill \end{array}\right.\phantom{\rule{0.25em}{0ex}}(25)for all i ? {1, 2,…, N}.
3.2 Improved adaptive sparse channel estimation methods
The common drawback of the above sparse LMSbased algorithms is that they are vulnerable to probabilistic scaling of their training signal x(t). In other words, LMSbased algorithms are sensitive to signal scaling [21]. Hence, it is very hard to choose a proper step size μ to guarantee stability of these sparse LMSbased algorithms even if the step size satisfies the necessary condition in Equation 10.
Let us reconsider the updated equation of LMS in Equation 4. Assuming that the n th adaptive channel estimator \tilde{\mathbf{h}}\left(n+1\right) is the optimal solution, the relationship between the (n = 1)th channel estimator \tilde{\mathbf{h}}\left(n+1\right) and input signal x(t) is given as
where y(t) is assumed to be ideal received signal at the receiver. To solve a convex optimization problem in Equation 26, the cost function can be constructed as [21]
where ξ is the unknown realvalue Lagrange multiplier [21]. The optimal channel estimator at the (n + 1)th update can be found by letting the first derivative of
Hence, it can be derived as
The (n + 1)th optimal channel estimator is given from Equation 29 as
By substituting Equation 30 into Equation 26, we obtain
where e\left(n\right)=y\left(t\right){\tilde{\mathbf{h}}}^{T}\left(n\right)\mathbf{x}\left(t\right) (see Equation 2) and the unknown parameter ξ is given by
By substituting it to Equation 30, the updated equation of NLMS is written as
where μ_{1} is the gradient step size which controls the adaptive convergence speed of NLMS algorithm. Based on the updated Equation 33, for better understanding, NLMSbased sparse adaptive updated equation can be generalized as
where normalized adaptive update term is μ_{1}e(n)x(t)/x^{T}(t)x(t) which replaces the adaptive update μe(n)x(t) in Equation 4. The advantage of NLMSbased adaptive sparse channel estimation it that it can mitigate the scaling interference of training signal due to the fact that NLMSbased methods estimate the sparse channel by normalizing with the power of training signal x(t). To ensure the stability of the NLMSbased algorithms, the necessary condition of step size μ_{1} is derived briefly. The detail derivation can also be found in [21].
Theorem 2 The necessary condition of reliable NLMS adaptive channel estimation is
Proof Since the NLMSbased algorithms share the same gradient step size to ensure their stability, for simplicity, studying the NLMS for a general case. The updated equation of NLMS is given by
where μ_{1} denotes the step size of NLMStype algorithms. Denoting the channel estimation error vector as \mathbf{u}\left(n\right)=\mathbf{h}\tilde{\mathbf{h}}\left(n\right), (n + 1)th update error u(n + 1) can be written as
Obviously, the (n + 1)th update MSE E{u^{2}(n + 1)} can also be given by
To ensure the stable updating of the NLMStype algorithms, the necessary condition is satisfying
Hence, the necessary condition of reliable adaptive sparse channel estimation is μ_{1} satisfying the theorem in Equation 35.
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The following are the improved adaptive sparse channel estimation methods:

ZANLMS (proposed). According to the Equation 14, the updated equation of ZANLMS can be written as
\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)+{\mu}_{1}e\left(n\right)\frac{\mathbf{x}\left(t\right)}{{\mathbf{x}}^{T}\left(t\right)\mathbf{x}\left(t\right)}{?}_{\mathrm{ZAN}}\mathrm{sgn}\left(\tilde{\mathbf{h}}\left(n\right)\right).(40)where ?_{ZAN}?=?µ_{1}?_{ZAN} and ?_{ZAN} is a regularization parameter for ZANLMS.

RZANLMS (proposed). According to Equation 16, the updated equation of RZANLMS can be written as
\begin{array}{l}\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)+{\mu}_{1}e\left(n\right)\frac{\mathbf{x}\left(t\right)}{{\mathbf{x}}^{T}\left(t\right)\mathbf{x}\left(t\right)}\\ \phantom{\rule{5.7em}{0ex}}{?}_{\mathrm{RZAN}}\frac{\mathrm{sgn}\left(\tilde{\mathbf{h}}\left(n\right)\right)}{1+{?}_{\mathrm{RZA}}\left\tilde{\mathbf{h}}\left(n\right)\right}.\end{array}(41)where ?_{RZAN}?=?µ_{1}?_{RZA}?_{RZAN} and ?_{RZAN} is a regularization parameter for RZANLMS. The threshold is set as ?_{ RZAN }?=??_{ RZA }?=?20 which is also consistent with our previous research in [24–27].

LPNLMS (proposed). According to the LPLMS in Equation 18, the updated equation of LPNLMS can be written as
\begin{array}{l}\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)+{\mu}_{1}e\left(n\right)\frac{\mathbf{x}\left(t\right)}{{\mathbf{x}}^{\mathit{T}}\left(t\right)\mathbf{x}\left(t\right)}\\ \phantom{\rule{5.7em}{0ex}}{?}_{\mathrm{LPN}}\frac{{\tilde{\mathbf{h}}\left(n\right)}_{p}^{1p}\mathrm{sgn}\left\{\tilde{\mathbf{h}}\left(n\right)\right\}}{{?}_{\mathrm{LPN}}+{\left\tilde{\mathbf{h}}\left(n\right)\right}^{\left(1p\right)}},\end{array}(42)where ?_{LPN}?=?µ_{1}?_{LPN}/10, ?_{L 0N} is a regularization parameter, and ?_{LPN}?>?0 is a threshold parameter.

L0NLMS (proposed). Based on updated the equation of L0LMS algorithm in Equation 24, the updated equation of L0NLMS algorithm can be directly written as
\tilde{\mathbf{h}}\left(n+1\right)=\tilde{\mathbf{h}}\left(n\right)+{\mu}_{1}e\left(n\right)\frac{\mathbf{x}\left(t\right)}{{\mathbf{x}}^{\mathit{T}}\left(t\right)\mathbf{x}\left(t\right)}{?}_{L0N}J\left(\tilde{\mathbf{h}}\left(n\right)\right),(43)where ?_{L 0N}?=?µ_{1}?_{L 0N} and ?_{L 0N} is a regularization parameter. The sparse penalty function J\left(\tilde{\mathbf{h}}\left(n\right)\right) has been defined as in (25).
3.3 CramerRao lower bound
To decide the CRLB of the proposed channel estimator, Theorems 3 and 4 are derived as follows.
Theorem 3 For an Nlength channel vector h, if μ satisfies 0 < μ < 2/λ_{ max }, then MSE lower bound of LMS adaptive channel estimator isB=\mu {P}_{0}N/\left(2\mu {\lambda}_{\mathit{min}}\right)~\mathcal{O}\left(N\right), where P_{ 0 }is a parameter which denotes unit power of gradient noise and λ_{ min }denotes the minimum eigenvalue of R.
Proof Firstly, we define the estimation error at the (n + 1)th iteration v(n + 1) as
where \mathrm{\Gamma}\left(n\right)=\partial L\left(n\right)/\partial \tilde{\mathbf{h}}\left(n\right)=2\mathbf{x}\left(t\right)e\left(n\right) is a joint gradient error function which includes channel estimation error and noise plus interference error. To derive the lower bound of the channel estimator, two gradient errors should be separated. Hence, assuming Γ(n) can be split in two terms: \mathrm{\Gamma}\left(n\right)=\tilde{\Gamma}\left(n\right)+2\mathbf{w}\left(n\right) where \tilde{\Gamma}\left(n\right)=2\left(\mathbf{R}\tilde{\mathbf{h}}\left(n\right)\mathbf{p}\right) denotes the gradient error and w(n) = [w_{0}(n), w_{1}(n), …, w_{N − 1}(n)]^{T} represents the gradient noise vector [21]. Obviously, E{w(n)} = 0 and
where p = Rh. Then, we rewrite v(n + 1) in Equation 44 as
where the covariance matrix can be decomposed as R = QDQ^{H}. Here, Q is an N × N unitary matrix while D = diag{λ_{1}, λ_{2}, …, λ_{ N }} is an N × N eigenvalue diagonal matrix. We denote \tilde{\mathbf{v}}\left(n\right)={\mathbf{Q}}^{H}\mathbf{v}\left(n\right) and \tilde{\mathbf{w}}\left(n\right)={\mathbf{Q}}^{H}\mathbf{w}\left(n\right) as the rotated vectors, and Equation 46 can be rewritten as
According to Equation 47, the MSE lower bound of LMS can be derived as
Since signal and noise are independent, hence, E\left\{{\tilde{\mathbf{w}}}^{T}\left(n\right)\tilde{\mathbf{v}}\left(n\right)\right\}=\dots =E\left\{{\tilde{\mathbf{w}}}^{T}\left(n\right)\tilde{\mathbf{v}}\left(n\right)\right\}=0, and Equation 48 can be simplified as
For a better understanding, the first term a(n) in Equation 49 can be expanded as
According to Equation 50, Equation 49 can be further rewritten as
where the first term {\text{lim}}_{n\to \infty}{\left(\mathbf{I}\mu \mathbf{D}\right)}^{2n}E\left\{{\tilde{\mathbf{v}}\left(n\right)}_{2}^{2}\right\}\to 0 when 1 − μλ_{ i } < 1. Consider the MSE lower bound of the i th channel taps {b_{ i }; i = 0, 1, …, N − 1}. We obtain
where E\left\{{\left{\tilde{w}}_{i}\left(n\right)\right}^{2}\right\}={\lambda}_{i}{P}_{0} and P_{ 0 } denotes the gradient noise power. For any overall channel, since the LMS adaptive channel estimation method does not use the channel sparse structure information, the MSE lower bound should be cumulated from all of the channel taps. Hence, the lower bound B_{LMS} of LMS is given by
where N is the channel length of h, {λ_{ i }; i = 0, 1, …, N − 1} are eigenvalues of the covariance matrix R and λ_{min} is its minimal eigenvalue.
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Theorem 4 For an Nlength sparse channel vector h which consists of K nonzero taps, if μ satisfies 0 < μ < 2/λ_{ max }, then the MSE lower bound of the sparse LMS adaptive channel estimator is {B}_{S}=\mu {P}_{0}K/\left(2\mu {\lambda}_{\mathrm{min}}\right)~\mathcal{O}\left(K\right).
Proof From Equation 53, we can easily find that the MSE lower bound of the adaptive sparse channel estimator has a direct relationship with the number of nonzero channel coefficients, i.e., K. Let Ω denote the set of nonzero taps' position, that is, h_{ i } ≠ 0, for i ϵ Ω and h_{ i } = 0 for others. We can then obtain the lower bound of the sparse LMS as
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4 Computer simulations
In this section, we will compare the performance of the proposed channel estimators using 1,000 independent Monte Carlo runs for averaging. The length of sparse multipath channel h is set as N = 16, and its number of dominant taps is set as K = 1, 2, 4, and 8. For simplicity, the number of dominant taps is written in the form of a set as K ϵ {1, 2, 4, 8}. The values of the dominant channel taps follow the Gaussian distribution, and the positions of dominant taps are randomly selected within the length of h and is subjected to {\mathbf{h}}_{2}^{2}=1. The received signaltonoise ratio (SNR) is defined as 10\phantom{\rule{0.25em}{0ex}}\mathrm{log}\left({E}_{0}/{\mathrm{\sigma}}_{n}^{2}\right), where E0 = 1 is the received signal power and the noise power is given by {\mathrm{\sigma}}_{n}^{2}={10}^{\mathrm{SNR}/10}. Here, we compare their performance with three SNR regimes: {5, 10, 20 dB}. All of the step sizes and regularization parameters are listed in Table 1. It is worth noting that the (N)LMSbased algorithms can exploit more accurate sparse channel information at higher SNR environment. Hence, all of the parameters are set at a direct ratio in relation with noise power. For example, in the case of SNR = 10 dB, the parameters of LMSbased algorithm are matched with the parameters which are given in [20]. Hence, the propose regulation parameter method can adaptively exploit the channel sparsity under different SNR regimes.
The estimation performance between the actual and estimated channels is evaluated by MSE standard which is defined as
where E{∙} denotes the expectation operator, and h and \tilde{\mathbf{h}}\left(n\right) are the actual channel vector and its estimator, respectively.
In the first experiment, we evaluate the estimation performance of LP(N)LMS as a function of p ϵ {0.3, 0.5, 0.7, 0.9} which are shown in Figures 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14 in three SNR regimes, i.e., {5, 10, 20 dB}. The parameter is set as ϵ_{LP} = ϵ_{LPN} = 0.05 which is suggested in [20]. In the case of low SNR regime, e.g., SNR = 5 dB, MSE performance curves are depicted in Figures 3, 4, 5, and 6, with different K ϵ {1, 2, 4, 8}. One can find that LP(N)LMS algorithm is not stable if we set p = 0.3 because their estimation performances are even worse than the other cases, i.e., p ϵ {0.5, 0.7, 0.9}. In the case of intermediate SNR regime, e.g., SNR = 10 dB, MSE performance curves are depicted in Figures 7, 8, 9, and 10, with different K ϵ {1, 2, 4, 8}. As shown in Figure 7, to estimate a very sparse channel (K = 1), the LP(N)LMS algorithm with p = 0.3 can obtain a better estimation performance than the others, i.e., p ϵ {0.5, 0.7, 0.9}. However, if K > 1, the LP(N)LMS algorithm is no longer stable as shown in Figures 8, 9, and 10, whose estimation performance is even worse than that in (N)LMS. In the case of high SNR regime, e.g., SNR = 20 dB, likewise, the LP(N)LMS algorithm achieves a better estimation performance than the others only when K ≤ 2 as shown in Figures 11 and 12. Hence, one can find that the stability of the LP(N)LMS algorithm with p = 0.3 depends highly on both SNR and K. If the algorithm adopts p ϵ {0.5, 0.7, 0.9} as shown in Figures 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14, fortunately, there is no obvious relationship between stability and either SNR or K. In order to trade off stability and estimation performance of the LP(N)LMS algorithm, it is better to set the value of the sparse penalty as p = 0.5. For one thing, the LP(N)LMS algorithm with p = 0.5 can always achieve a better estimation performance than the cases with p ϵ {0.7, 0.9}. In another case, even though the LP(N)LMS algorithm with p = 0.3 can obtain a better estimation performance in certain circumstances, its stability depends highly on SNR and K. Hence, in the following simulation results, p = 0.5 is considered for LP(N)LMSbased adaptive sparse channel estimation.
In the second experiment, we compare all the (N)LMSbased sparse adaptive channel estimation methods with different SNR regimes: {5, 20 dB} as well as K ϵ {1, 8}, as shown in Figures 15, 16, 17, and 18, respectively. In the case of low SNR regime (e.g., SNR = 5 dB), the MSE curves show that NLMSbased methods achieve a better estimation performance than LMSbased ones as shown in Figure 15. Let us take ZANLMS and ZALMS as examples. As shown in the figure, each performance curve of ZANLMS is much lower than that of ZALMS. That is to say, the proposed ZANLMS achieves a better estimation performance than the traditional ZALMS. Similarly, other proposed NLMStype methods, i.e., RZANLMS and LPNLMS, also achieve better estimation performances than their corresponding sparse LMS types. To further confirm the stability and effectiveness of our proposed methods, sparse (N)LMSbased estimation methods are also evaluated in the case of 20 dB as well as with different K, respectively.
As the SNR increases, the obvious performance advantages of NLMSbased methods are vanishing. Hence, compared with LMSbased methods, we can conclude that NLMSbased methods not only work more reliably for unknown signal scaling, but also work more stably for noise interference, especially in low SNR environment.
In addition, the simulation results also show that (N)LMSbased methods have an inverse relationship with the number of nonzero channel taps. In other words, for a sparse channel, (N)LMSbased methods can achieve a better estimation performance and vice versa. Let us take K = 1 and 8 as examples. In Figures 15 and 16, when the number of nonzero taps is K = 1, performance gaps are bigger than in the case where K = 8, as shown in Figures 17 and 18. Hence, these simulation results also show that the estimation performance of adaptive sparse channel estimation is also affected by the number of nonzero channel taps. When the channel is no longer sparse, the performance of these proposed methods will reduce to the performance of (N)LMSbased methods.
5 Conclusions
In this paper, we have investigated various (N)LMSbased adaptive sparse channel estimation methods by enforcing different sparse penalties, e.g., ℓ_{p}norm and ℓ_{0}norm. The research motivation originated from the fact that LMSbased channel estimation methods are sensitive to the scaling of random training signal and easily causing the estimation performance unstable. Unlike LMSbased methods, the proposed NLMSbased methods have avoided the uncertain signal scaling and normalized the power of input signal with different sparse penalties.
Initially, we proposed an improved adaptive sparse channel estimation method using ℓ_{0}norm sparse constraint LMS algorithm and compared it with ZALMS, RZALMS, and LPLMS. The proposed method was based on the CS background that ℓ_{0}norm sparse penalty can exploit a more accurate channel sparsity.
In addition, to improve the robust performance and increase the convergence speed, we proposed NLMSbased adaptive sparse channel estimation methods using different sparse penalties, i.e., ZANLMS, RZANLMS, LPNLMS, and L0NLMS. For example, ZANLMS can achieve a better estimation than ZALMS. The proposed methods exhibit faster convergence and better performance which are confirmed by computer simulations under various SNR environments.
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Acknowledgements
The authors would like to thank Dr. Koichi Adachi of the Institute for Infocomm Research for his valuable comments and suggestions as well as for the improvement of the English expression of this paper. The authors would like to extend their appreciation to the anonymous reviewers for their constructive comments. This work was supported by a grantinaid for the Japan Society for the Promotion of Science (JSPS) fellows (grant number 24∙02366).
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Gui, G., Adachi, F. Improved least mean square algorithm with application to adaptive sparse channel estimation. J Wireless Com Network 2013, 204 (2013). https://doi.org/10.1186/168714992013204
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DOI: https://doi.org/10.1186/168714992013204
Keywords
 Least mean square
 Normalized LMS
 Adaptive sparse channel estimation
 ℓ_{p}Norm normalized least mean square
 ℓ_{0}Norm normalized least mean square
 Compressive sensing