From Equation 4, we can find that the LMS-based channel estimation method never takes advantage of the sparse structure in channel vector h. To get a better understanding, the LMS-based channel estimation methods can be expressed as
(11)
Unlike the conventional LMS method in Equation 11, sparse LMS algorithms exploit channel sparsity by introducing several ℓp-norm penalties to their cost functions with 0 ≤ p ≤ 1. The LMS-based adaptive sparse channel estimation methods can be written as
(12)
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 CS-based 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 ℓp-norm (0 < p ≤ 1) and ℓ0-norm, respectively. Since ℓ0-norm penalty on sparse signal recovery is a well-known NP-hard problem, only the ℓp-norm (0 < p ≤ 1)-based sparse LMS approaches have been proposed for adaptive sparse channel estimation. Compared to conventional two sparse LMS algorithms (ZA-LMS and RZA-LMS), LP-LMS can achieve a better estimation performance. However, there still exists a performance gap between the LP-LMS-based channel estimator and the optimal one. As the development of mathematics continues, a more accurate sparse approximate algorithm to ℓ0-norm LMS (L0-LMS) is proposed in [22] and analyzed in [23]. However, they never considered any application on sparse channel estimation. In this paper, the L0-LMS 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 LMS-based adaptive sparse channel estimation methods using different sparse penalties.
3.1 LMS-based adaptive sparse channel estimation
The following are the LMS-based adaptive sparse channel estimation methods:
-
ZA-LMS. To exploit the channel sparsity in time domain, the cost function of ZA-LMS [18] is given by
(13)
where ?ZA is a regularization parameter to balance the estimation error and sparse penalty of . The corresponding updated equation of ZA-LMS is
(14)
where ?ZA?=?µ?ZA and sgn(·) is a component-wise function which is defined as
? 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.
-
RZA-LMS. ZA-LMS cannot distinguish between zero taps and non-zero 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 l1-norm minimization recovery algorithm [19], Chen et al. have proposed a heuristic approach to reinforce the zero attractor which was termed as the RZA-LMS [18]. The cost function of RZA-LMS is given by
(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 is then updated as
(16)
where ?RZA?=?µ?RZA?RZA. Equation 16 can be expressed in the vector form as
? Please note that the second term in Equation 16 attracts the channel coefficients whose magnitudes are comparable to 1/?RZA to zeros.
-
LP-LMS. Following the above ideas in [18], LP-LMS-based adaptive sparse channel estimation method has been proposed in [20]. The cost function of LP-LMS is given by
(17)
where ?LP?>?0 is a regularization parameter. The corresponding updated equation of LP-LMS is given as
(18)
where ?LP?=?µ?LP and ?LP?>?0 is a small positive parameter.
? L0-LMS (proposed). Consider l0-norm penalty on LMS cost function to produce a sparse channel estimator as this penalty term forces the channel tap values of to approach zero. The cost function of L0-LMS is given by
(19)
where ?L 0?>?0 is a regularization parameter and denotes l0-norm sparse penalty function which counts the number of non-zero channel taps of . Since solving the l0-norm minimization is an NP-hard problem [17], to reduce computational complexity, we replace it with an approximate continuous function:
(20)
? The cost function in Equation 19 can then be rewritten as
(21)
? The first-order Taylor series expansion of exponential function is given as
(22)
? The updated equation of L0-LMS-based adaptive sparse channel estimation can then be derived as
(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 is also proposed to the updated Equation 23. Finally, the updated equation of L0-LMS-based adaptive sparse channel estimation can be derived as
(24)
where ?
L0
?=?µ?
L0
and is defined as
(25)
for all i ? {1, 2,…, N}.
3.2 Improved adaptive sparse channel estimation methods
The common drawback of the above sparse LMS-based algorithms is that they are vulnerable to probabilistic scaling of their training signal x(t). In other words, LMS-based 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 LMS-based 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 is the optimal solution, the relationship between the (n = 1)th channel estimator and input signal x(t) is given as
(26)
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]
(27)
where ξ is the unknown real-value 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
(29)
The (n + 1)th optimal channel estimator is given from Equation 29 as
(30)
By substituting Equation 30 into Equation 26, we obtain
(31)
where (see Equation 2) and the unknown parameter ξ is given by
(32)
By substituting it to Equation 30, the updated equation of NLMS is written as
(33)
where μ1 is the gradient step size which controls the adaptive convergence speed of NLMS algorithm. Based on the updated Equation 33, for better understanding, NLMS-based sparse adaptive updated equation can be generalized as
(34)
where normalized adaptive update term is μ1e(n)x(t)/xT(t)x(t) which replaces the adaptive update μe(n)x(t) in Equation 4. The advantage of NLMS-based adaptive sparse channel estimation it that it can mitigate the scaling interference of training signal due to the fact that NLMS-based methods estimate the sparse channel by normalizing with the power of training signal x(t). To ensure the stability of the NLMS-based algorithms, the necessary condition of step size μ1 is derived briefly. The detail derivation can also be found in [21].
Theorem 2
The necessary condition of reliable NLMS adaptive channel estimation is
(35)
Proof Since the NLMS-based 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
(36)
where μ1 denotes the step size of NLMS-type algorithms. Denoting the channel estimation error vector as (n + 1)th update error u(n + 1) can be written as
(37)
Obviously, the (n + 1)th update MSE E{u2(n + 1)} can also be given by
(38)
To ensure the stable updating of the NLMS-type algorithms, the necessary condition is satisfying
(39)
Hence, the necessary condition of reliable adaptive sparse channel estimation is μ1 satisfying the theorem in Equation 35.
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The following are the improved adaptive sparse channel estimation methods:
-
ZA-NLMS (proposed). According to the Equation 14, the updated equation of ZA-NLMS can be written as
(40)
where ?ZAN?=?µ1?ZAN and ?ZAN is a regularization parameter for ZA-NLMS.
-
RZA-NLMS (proposed). According to Equation 16, the updated equation of RZA-NLMS can be written as
(41)
where ?RZAN?=?µ1?RZA?RZAN and ?RZAN is a regularization parameter for RZA-NLMS. The threshold is set as ?
RZAN
?=??
RZA
?=?20 which is also consistent with our previous research in [24–27].
-
LP-NLMS (proposed). According to the LP-LMS in Equation 18, the updated equation of LP-NLMS can be written as
(42)
where ?LPN?=?µ1?LPN/10, ?L 0N is a regularization parameter, and ?LPN?>?0 is a threshold parameter.
-
L0-NLMS (proposed). Based on updated the equation of L0-LMS algorithm in Equation 24, the updated equation of L0-NLMS algorithm can be directly written as
(43)
where ?L 0N?=?µ1?L 0N and ?L 0N is a regularization parameter. The sparse penalty function has been defined as in (25).
3.3 Cramer-Rao lower bound
To decide the CRLB of the proposed channel estimator, Theorems 3 and 4 are derived as follows.
Theorem 3 For an N-length channel vector h, if μ satisfies 0 < μ < 2/λ
max
, then MSE lower bound of LMS adaptive channel estimator is, 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
(44)
where 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: where denotes the gradient error and w(n) = [w0(n), w1(n), …, wN − 1(n)]T represents the gradient noise vector [21]. Obviously, E{w(n)} = 0 and
(45)
where p = Rh. Then, we rewrite v(n + 1) in Equation 44 as
(46)
where the covariance matrix can be decomposed as R = QDQH. Here, Q is an N × N unitary matrix while D = diag{λ1, λ2, …, λ
N
} is an N × N eigenvalue diagonal matrix. We denote and as the rotated vectors, and Equation 46 can be rewritten as
(47)
According to Equation 47, the MSE lower bound of LMS can be derived as
(48)
Since signal and noise are independent, hence, , and Equation 48 can be simplified as
(49)
For a better understanding, the first term a(n) in Equation 49 can be expanded as
(50)
According to Equation 50, Equation 49 can be further rewritten as
(51)
where the first term 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
(52)
where 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 BLMS of LMS is given by
(53)
where N is the channel length of h, {λ
i
; i = 0, 1, …, N − 1} are eigenvalues of the covariance matrix R and λmin is its minimal eigenvalue.
□
Theorem 4 For an N-length sparse channel vector h which consists of K non-zero taps, if μ satisfies 0 < μ < 2/λ
max
, then the MSE lower bound of the sparse LMS adaptive channel estimator is .
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 non-zero channel coefficients, i.e., K. Let Ω denote the set of non-zero 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
(54)
□