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# Sparse channel estimation of MIMO-OFDM systems with unconstrained smoothed *l*_{0}-norm-regularized least squares compressed sensing

*EURASIP Journal on Wireless Communications and Networking*
**volume 2013**, Article number: 282 (2013)

## Abstract

This paper investigates the sparse channel estimation issue of multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. Beginning with the formulation of least squares (LS) solution to sparse MIMO-OFDM channel estimation, a compressed channel sensing (CCS) framework based on the new smoothed *l*_{0}-norm-regularized least squares (*l*_{2}-*Sl*_{0}) algorithm is proposed. Three methods, namely quasi-Newton, conjugate gradient, and optimization in the null and complement spaces of the measurement matrix, are then proposed to solve the *l*_{2}-*Sl*_{0} unconstrained optimization problem. Moreover, the two former are also applied to solve the *l*_{2}-*Sl*_{0} channel estimation. A number of computer simulation-based experiments are conducted showing a better reconstruction accuracy of the *l*_{2}-*Sl*_{0} algorithm as compared with the smoothed *l*_{0}-norm (*Sl*_{0}) algorithm in the presence of noise. The proposed CCS approach can save nearly 25% pilot signals to maintain the same mean square error (MSE) and bit error rate (BER) performances as given by the conventional LS method.

## 1. Introduction

Coherent detection and equalization in multiple input multiple output orthogonal frequency division multiplexing (MIMO-OFDM [1]) systems require channel state information (CSI) at the receiver. In real wireless environments, however, the CSI is not known. Therefore, channel estimation is of crucial importance to MIMO-OFDM systems. In various wireless propagation environments, the channel may consist of only a few dominant propagation (non-zero) paths, even though it has a large propagation delay. Thus, the channel impulse response has a sparse nature [2–4]. However, conventional methods, such as least squares (LS), ignore this prior information about the unknown channel leading to lower spectral efficiency. Recently, sparse channel estimation with an objective of decreasing the training sequence to improve spectral efficiency is becoming a hot research topic.

Previously reported approaches for sparse channel estimation can broadly be categorized into two types, namely the most significant tap (MST) detection and compressed channel sensing (CCS). The MST detection methods [4–6] used a measure to determine if a channel tap was non-zero (‘active’). The disadvantage of this type of methods is that a large number of pilots are needed to render an accurate MST detection and effective channel estimation. The CCS methods are based on the compressed sensing (CS [7]) technology. In [8], the authors formalized the notion of multipath sparsity and proposed the CCS approach. In [9], the orthogonal matching pursuit (OMP) and basis pursuit (BP) algorithms were applied to estimate underwater acoustic channels with large Doppler spread. In [10], the authors proposed an overcomplete basis for doubly selective channels and a metric called localized coherence for selecting training signals to ensure good estimation performance. In [11], a CCS approach for doubly selective channels and a sparsity-enhancing basis expansion with a method for optimizing it were proposed. In [12], two criteria as guiding principles to optimize the pilot pattern for CCS in OFDM systems were proposed. Methods of this type utilize the prior sparse information of the unknown channel and the advantage of CS and thus can improve the spectral efficiency by reducing the number of pilot symbols to be transmitted.

Different from literatures [9–12] that used the existing sparse reconstruction algorithms for CCS in OFDM or single carrier systems, we aim to exploit a novel reconstruction algorithm for CCS in MIMO-OFDM systems. The proposed smoothed *l*_{0}-norm-regularized least squares reconstruction algorithm is named *l*_{2}-*Sl*_{0} in this paper, which differs from the smoothed *l*_{0}-norm reconstruction algorithm (*Sl*_{0}[13]) in two aspects. First, *Sl*_{0} is a constrained optimization problem which is solved in [13] using the steepest descent approach. However, *l*_{2}-*Sl*_{0} is an unconstrained minimization problem which is to be solved in this paper by using three methods, namely quasi-Newton approach, conjugate gradient approach, and optimization in the null and complement spaces of the measurement matrix. Second, unlike the *Sl*_{0} using a fixed step size to control the decrease of the parameter *σ*, which determines the degree of smoothness and the approximation accuracy of *l*_{0}-norm, *l*_{2}-*Sl*_{0} uses a variable one. Simulation results show that the proposed *l*_{2}-*Sl*_{0} reconstruction approach outperforms the *Sl*_{0} approach in the presence of noise, and at the cost of slightly more computational time, the CCS approach using *l*_{2}-*Sl*_{0} in conjunction with conjugate gradient yields a performance slightly better than that of the CCS method using fast iterative shrinkage-thresholding algorithm (FISTA [14]) or orthogonal matching pursuit (OMP [15]) algorithm.

The remainder of the paper is organized as follows: Section 2 formulates the sparse channel estimation problem of MIMO-OFDM systems based on LS, ideal-LS, and compressed sensing. Section 3 presents three sparse reconstruction algorithms using the proposed *l*_{2}-*Sl*_{0} objective function, based on which a new CS-based sparse channel estimation approach is developed. Section 4 comprises a number of experiments showing a better reconstruction accuracy of the *l*_{2}-*Sl*_{0}-based method as compared with the *Sl*_{0} algorithm, and a higher spectrum efficiency of the sparse channel estimation employing *l*_{2}-*Sl*_{0} than that using the LS method. Section 5 concludes this paper by highlighting some of the contributions presented.

## 2. The sparse channel estimation problem of MIMO-OFDM systems

Consider a similar MIMO-OFDM system as described in [16] with *N*_{
T
} transmit and *N*_{
R
} receive antennas. The MIMO channel can be characterized by an array of *L*-tap finite impulse response (FIR) filters given by a number of *N*_{
R
} × *N*_{
T
} matrices **H**(*n*), (*n* = 0,1,…,*L* − 1), whose (*i*_{
R
},*i*_{
T
})-th element ${h}_{{i}_{R},{i}_{T}}\left(l\right),\left(0\le l\le L-1\right)$ represents the *l-* th tap of the channel response between the *i*_{
R
}-th receive antenna and the *i*_{
T
}-th transmit antenna. In the case of uniform sampling, a wireless channel can often be modeled as a sparse channel [17–19], i.e., only a few elements are nonzero in $\left[{h}_{{i}_{R},{i}_{T}}\left(0\right),{h}_{{i}_{R},{i}_{T}}\left(1\right),\dots ,{h}_{{i}_{R},{i}_{T}}\left(L-1\right)\right]$. If the length of the cyclic prefix (CP) is not less than the channel length *L*, the received pilot signal in *i*_{
R
}-th receiver antenna can be written as

where ${\mathbf{X}}_{{i}_{T},\mathit{pilot}}={\left[{X}_{{i}_{T}}\left({k}_{1}\right),\dots ,{X}_{{i}_{T}}\left({k}_{p}\right)\right]}^{T}$ and ${\mathbf{Y}}_{{i}_{R},\mathit{pilot}}={\left[{Y}_{{i}_{R}}\left({k}_{1}\right),\dots ,{Y}_{{i}_{R}}\left({k}_{p}\right)\right]}^{T}$ are the pilot signals in the *i*_{
T
}-th transmit antenna and *i*_{
R
}-th receive antenna, diag(**X**_{1,pilot}) is a diagonal matrix with **X**_{l,pilot} as the main diagonal elements, ${\mathbf{h}}_{{i}_{R}}={\left[{\mathbf{h}}_{{i}_{R},1}^{T},\dots ,{\mathbf{h}}_{{i}_{R},{i}_{T}}^{T},\dots ,{\mathbf{h}}_{{i}_{R},{N}_{T}}^{T}\right]}^{T}$ with ${\mathbf{h}}_{{i}_{R},{i}_{T}}={\left[{h}_{{i}_{R},{i}_{T}}\left(0\right),\dots ,{h}_{{i}_{R},{i}_{T}}\left(L-1\right)\right]}^{T}$, and **n**_{
iR,
pilot
} represents the frequency domain noise. Let **F**_{
L
} be a *K* × *L* matrix formed by the first *L* columns of a *K* × *K* DFT matrix **F**, then **F**_{pilot} can be formed by taking only the rows of **F**_{
L
} associated with the *K*_{
P
} pilot sub-carriers.

By letting $\mathbf{A}={\mathbf{I}}_{{N}_{R}}\otimes \left[\mathit{diag}\left({\mathbf{X}}_{1,\mathit{pilot}}\right){\mathbf{F}}_{\mathit{pilot}},\dots ,\mathit{diag}\left({\mathbf{X}}_{{N}_{T},\mathit{pilot}}\right){\mathbf{F}}_{\mathit{pilot}}\right]$, ${\mathbf{Y}}_{\mathit{pilot}}={\left[{\mathbf{Y}}_{1,\mathit{pilot}}^{T},\dots ,{\mathbf{Y}}_{{N}_{R},\mathit{pilot}}^{T}\right]}^{T}$, $\mathbf{h}={\left[{\mathbf{h}}_{1}^{T},\dots ,{\mathbf{h}}_{{N}_{R}}^{T}\right]}^{T}$, and ${\mathbf{n}}_{\mathit{pilot}}={\left[{\mathbf{n}}_{1,\mathit{pilot}}^{T},\dots ,{\mathbf{n}}_{{N}_{R},\mathit{pilot}}^{T}\right]}^{T}$, where ⊗ represents Kronecker product, we can get

which can be solved by the conventional LS method, giving $\widehat{\mathbf{h}}={\mathbf{A}}^{\u2020}{\mathbf{Y}}_{\mathit{pilot}}$, where † represents the pseudoinverse.

Assuming the positions *l*_{
d
} (d = 0,1,…,D-1, and *l*_{0} < *l*_{1} < … < *l*_{D-1}) of the MST are correctly estimated, Equation 1 can be rewritten as

where ${\mathbf{z}}_{{i}_{R}}={\left[{\mathbf{z}}_{{i}_{R},1}^{T},\dots ,{\mathbf{z}}_{{i}_{R},{i}_{T}}^{T},\dots ,{\mathbf{z}}_{{i}_{R},{N}_{T}}^{T}\right]}^{T}$ with ${\mathbf{z}}_{{i}_{R},{i}_{T}}={\left[{\mathbf{z}}_{{i}_{R},{i}_{T}}\left(0\right),\dots ,{\mathbf{z}}_{{i}_{R},{i}_{T}}\left(D-1\right)\right]}^{T}$, *D* is the number of nonzero taps, and **W**_{
pilot
} can be formed by taking only the *D* columns of **F**_{
pilot
} associated with the nonzero tap positions *l*_{
d
}. Let $\tilde{\mathbf{A}}={\mathbf{I}}_{{N}_{R}}\otimes \left[\mathit{diag}\left({\mathbf{X}}_{1,\mathit{pilot}}\right){\mathbf{W}}_{\mathit{pilot}},\dots ,\mathit{diag}\left({\mathbf{X}}_{{N}_{T},\mathit{pilot}}\right){\mathbf{W}}_{\mathit{pilot}}\right]$ and $\mathbf{z}={\left[{\mathbf{z}}_{1}^{T},\dots ,{\mathbf{z}}_{{N}_{R}}^{T}\right]}^{T}$. We can obtain

When **n**_{
pilot
} is white noise and the positions *l*_{
d
} of MST are correctly estimated, we can obtain the estimate of the MST as $\widehat{\mathbf{z}}={\tilde{\mathbf{A}}}^{\u2020}{\mathbf{Y}}_{\mathit{pilot}}$. We can also obtain the Cramer-Rao bound of the sparse channel estimate $\widehat{\mathbf{h}}$ through setting the elements of the positions *l*_{
d
} equal to $\widehat{\mathbf{z}}$ and other elements equal to zero [4]. The above method to obtain the Cramer-Rao bound of $\widehat{\mathbf{h}}$ is named as ideal-LS for comparison in this paper.

Note that the dimension of **Y**_{
pilot
} is proportional to the number of pilot subcarriers, and Equation 2 is an underdetermined problem when the dimension of **Y**_{
pilot
} is smaller than that of **h**. Therefore, the sparse channel estimation in MIMO-OFDM systems can be viewed as solving an underdetermined linear inverse problem with sparsity constraint, i.e.,

where || · ||_{0} represents the number of nonzero components named as *l*_{0}-norm.

## 3. Sparse channel estimation using *l*_{2}-*Sl*_{0}reconstruction algorithm

The sparse signal reconstruction problem in CS is to estimate a sparse vector **x** ∈ ℂ^{N} from an observed vector **y** ∈ ℂ^{M} based on the linear model

where **w** ∈ ℂ^{M} is unknown noise and **Φ** ∈ *ℂ*^{M × N} is a known measurement matrix, typically with *M* ≪ *N*. This means that the signal **x** is ‘sensed’ by a reduced or ‘compressed’ number of measurements. Therefore, the signal reconstruction problem can be described as the following constrained minimization problem,

where the bound *ϵ* ≥ 0 is used to allow certain error tolerance. In general, *ϵ* is related to the variance of noise **w**. Unfortunately, the problem in Equation 7 is a NP-hard combinatorial problem, whose computational complexity grows exponentially with the increase of the signal size and becomes prohibitive even for signals of moderate sizes. Consequently, several techniques have been proposed to tackle this difficult problem. One of the approaches is the convex relaxation, such as BP [20], which replaces ||**x**||_{0} with ||**x**||_{1} to make the problem easier to solve. Another approach, such as matching pursuit (MP [21]) or OMP, is much faster than BP but is a greedy algorithm and does not have provable reconstruction quality at the level of BP method [22]. Different from the above techniques, the smoothed *l*_{0}-norm approach [13] is to approximate the discontinuous *l*_{0}-norm by a suitable continuous one and then minimize it by an optimization algorithm dedicated to continuous functions. For example, the following continuous function

where σ is a small value, has been proposed to approximate ||**x**||_{0} in [13]. In other words, the minimum *l*_{0}-norm solution is then found by minimizing *F*_{σ}(**x**) for a very small value of σ. The parameter σ determines how smooth the function *F*_{σ}(**x**) would be and the accuracy of the approximation. Generally speaking, for larger values of σ, *F*_{σ}(**x**) is smoother and contains less local minima, but the approximation to *l*_{0}-norm is worse. On the other hand, for smaller values of σ, a highly nonsmooth *F*_{σ}(**x**) results, which gives a better approximation to *l*_{0}-norm but a difficult minimization problem. Consequently, the *Sl*_{0} approach used a ‘decreasing’ sequence for σ.

The *Sl*_{0} reconstruction algorithm is typically 2 to 3 orders of magnitude faster than BP, while resulting in the same or better accuracy [13]. However, in the presence of noise, the accuracy of *Sl*_{0} algorithm needs to be improved. Therefore, in the next section, we will propose several improved *Sl*_{0} reconstruction algorithms.

### 3.1 The *l*_{2}-*Sl*_{0}-BFGS reconstruction algorithm for channel estimation

Like *l*_{1}-regularized *l*_{2} approach (*l*_{2}-*l*_{1}[14, 23, 24]) and *l*_{
p
}-regularized *l*_{2} algorithm [25], we use a parameter *λ* > 0 to balance the twin objectives of minimizing both error and sparsity, giving the following unconstrained optimization problem:

The objective function in Equation 9 remains differentiable, and its gradient can be obtained as

where **g** = [*g*_{1}, *g*_{2}, …, *g*_{
N
}]^{T} with *g*_{
i
} being given by

For a fixed value of σ, the problem in Equation 9 is now solved using a quasi-Newton algorithm where an approximation of the inverse of the Hessian is obtained using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update formula [26–28]. As such, the algorithm is referred to as the *l*_{2}-*Sl*_{0}-BFGS algorithm.

The quadratic (*l*_{2}-norm) error term $\frac{1}{2}\left|\right|\mathbf{\Phi x}-\mathbf{y}\left|\right|{}_{2}^{2}$ in Equation 9 is a convex function, but the convex region of the approximate *l*_{0}-norm term ${F}_{\sigma}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^{N}[1-exp\left(\frac{-{x}_{i}^{2}}{2{\sigma}^{2}}\right)}]$ depends on parameter is σ. In general, the greater the value of σ, the larger the convex region is. To see this, we compute the gradient of *F*_{σ}(**x**), denoted as ${\mathbf{g}}^{\text{'}}={\left[{g}_{1}^{\text{'}},{g}_{2}^{\text{'}},\dots ,{g}_{N}^{\text{'}}\right]}^{T}$, whose element is given by

Also, the Hessian of *F*_{
σ
}(**x**) is a diagonal matrix as given by

Therefore, *F*_{
σ
}(**x**) is convex if and only if

Since Equation 14 defines an *N*-dimensional hypercube whose volume is (2*σ*)^{N}, the size of the convex region in the *x*-space is proportional to σ. On the other hand, in order to better approximate the *l*_{0}-norm, σ must be sufficiently small. Consequently, to avoid getting trapped into local minima, we gradually decrease the value of σ, as in the *Sl*_{0} approach. More specifically, for minimum *F*(**x**) at σ_{
i
}, the initial point is **x**_{*}(σ_{i-1}) obtained in the previous iteration, which is near the global optimal solution.

Since a broadband wireless channel response **h** usually consists of a few dominant propagation paths and Equation 2 has a similar form as Equation 6, the estimation of **h** can be viewed as a sparse signal reconstruction in compressed sensing. Thus, we refer to this kind of sparse channel estimation method as CCS. Using Equations 2, 6, and 9, we can obtain the objective function of CCS based on the *l*_{2}-*Sl*_{0} reconstruction algorithm,

From the above analysis, the proposed CCS using the *l*_{2}-*Sl*_{0}-BFGS algorithm can be implemented by the pseudo-code in Algorithm 1.

### Algorithm 1 CCS using the *l*_{2}-*Sl*_{0}-BFGS algorithm

Note that in Algorithm 1, the values of *δ*_{
r
} and *r*_{
J
} are chosen such that 0 < *δ*_{
r
} < 0.1 and 0.5 < *r*_{
J
} < 1. The method of *l*_{2}-*Sl*_{0} uses a variable factor *r*_{
i
} = *r*_{i − 1} + *δ*_{
r
} to control the decrease of the parameter σ. Our idea is to use an ‘increasing’ step size corresponding to the decreasing values of σ.

### 3.2 The *l*_{2}-*Sl*_{0}-CG reconstruction algorithm for channel estimation

The Hessian matrix of the objective function *F*(**x**) in Equation 9 can be computed as

where ∇^{2}*F*_{
σ
}(**x**) is computed using Equation 13. Since the gradient and Hessian matrix of *F*(**x**) can be efficiently evaluated using the closed-form formula in Equations 10 and 16, it is convenient to apply the conjugate gradient method to solve the *l*_{2}-*Sl*_{0} optimization problem. The algorithm is thus referred to as the *l*_{2}-*Sl*_{0}-CG algorithm.

In the *k*-th iteration of the conjugate gradient technique, **x**_{
k
} is updated as

The conjugate direction **d**_{
k
} is computed as

and the *k*-th step size *α*_{
k
} is computed using

Where **g**_{
k
} is the gradient vector computed using Equation 10 and **H**_{
k
} is the Hessian matrix obtained using Equation 16 at **x** = **x**_{
k
}, respectively. The proposed CCS using *l*_{
2
}*-Sl*_{0}*-* CG algorithm can be implemented by the pseudo-code in Algorithm 2.

### Algorithm 2 CCS using *l*_{2}*-Sl*_{0}*-* CG algorithm

### 3.3 Signal reconstruction via optimization in null and complement spaces of Φ

Let **Φ** = **UΣV**^{T} be the singular value decomposition (SVD) of **Φ** where **U**_{M×M} and **V**_{N×N} are unitary matrices, and **Σ** = [**S**, **0**]_{M × N} with **S** = diag(*s*_{1}, …, *s*_{
M
}) being a diagonal matrix composed by the singular values of **Φ**. Let **V**=[**V**_{
r
},**V**_{
n
}], where the columns of **V**_{
n
} span the null space of **Φ** and the columns of **V**_{
r
} span the orthogonal complement of the null space. Using **V**_{
n
} and **V**_{
r
}, a signal **x** of length *N* can be expressed as

Where **α** and **β** are vectors of length *M* and *N* − *M*, respectively. Applying the SVD of **Φ**, the *l*_{2}-norm term in Equation 9 can be simplified as [25]

Where *S*_{
i
} is the *i*-th singular value of **Φ**, α_{
i
} and ${\tilde{y}}_{i}$ are the *i*-th component of **α** and $\tilde{\mathbf{y}}={\mathbf{U}}^{T}\mathbf{y}$, respectively. Using Equations 20 and 21, the optimization problem in Equation 9 can be recast as

Where *V*_{
r,j
} and *V*_{
n,j
} are the *j*-th row of **V**_{
r
} and that of **V**_{
n
}, respectively.

An iterative algorithm to solve the optimization problem in Equation 22 is proposed as follows. In the *k*-th iteration of the optimization process, signal **x**^{(k)} is updated as

where

and the step size *μ*^{(k)} > 0 is determined by the inexact line search method of Roger Fletcher [26]. Assuming that the updating vectors ${\mathbf{d}}_{r}^{\left(k\right)}$ and ${\mathbf{d}}_{n}^{\left(k\right)}$ are written as

which are determined by minimizing *F*(**α,β**) along each of the directions defined by the column vectors of [**V**_{
r
},**V**_{
n
}]. Therefore, ${\mathbf{d}}_{r}^{\left(k\right)}$ and ${\mathbf{d}}_{n}^{\left(k\right)}$ become descent directions of *F*(**α,β**), and in the case of real **Φ** and **x**, ${d}_{r,i}^{\left(k\right)}$ can be calculated via iteration as

Where *x*_{
j
} is the *j*-th component of vector **x**^{(k)}, ${v}_{r}^{\left(j,i\right)}$ is the (*j*,*i*)-th component of matrix **V**_{
r
}, *α*_{
i
} is the *i*-th component of vector **α**^{(k)}, and ${\left({d}_{r,i}^{\left(k\right)}\right)}^{\left(p\right)}$ is the *p*-th iteration value of ${d}_{r,i}^{\left(k\right)}$ with the initialization value ${\left({d}_{r,i}^{\left(k\right)}\right)}^{\left(0\right)}=0$. Similarly, ${d}_{n,i}^{\left(k\right)}$ in Equation 25 is given by

where ${v}_{n}^{\left(j,i\right)}$ is the (*j*,*i*)-th component of matrix **V**_{
n
}, and ${\left({d}_{n,i}^{\left(k\right)}\right)}^{\left(q\right)}$ is the *q*-th iteration value of ${d}_{n,i}^{\left(k\right)}$ with the initialization value ${\left({d}_{n,i}^{\left(k\right)}\right)}^{\left(0\right)}=0$. The derivation of Equations 26 and 27 is given in the Appendix. In addition, the computation of ${d}_{r,i}^{\left(k\right)}$ using Equation 26 requires vector **α**^{(k)} to be computed first as

The reconstruction algorithm via optimization in null and complement spaces of measurement matrix **Φ** is referred to hereafter as the *l*_{
2
}*-Sl*_{0}*-* NC algorithm, which can be implemented by the pseudo-code in Algorithm 3.

### Algorithm 3 The *l*_{2}*-Sl*_{0}*-* NC reconstruction algorithm

## 4. Simulation results

In this section, the reconstruction performance of the proposed approach (*l*_{2}*-Sl*_{0}) is evaluated by computer simulations. The spectral efficiency of the CCS using *l*_{2}*-Sl*_{0} algorithm is also discussed. More specifically, the *l*_{2}*-Sl*_{0} algorithm includes *l*_{2}*-Sl*_{0}-BFGS, *l*_{2}*-Sl*_{0}*-* CG, and *l*_{2}*-Sl*_{0}*-* NC in the scenario where **y**, **Φ** ,**x** are real-valued. While in complex-valued scenarios, *l*_{2}*-Sl*_{0} only means *l*_{2}*-Sl*_{0}*-* BFGS and *l*_{2}*-Sl*_{0}*-* CG. Note that Equations 26 and 27 are obtained only in the case where **y**, **Φ**, **x** are real-valued. Namely, the *l*_{2}*-Sl*_{0}*-* NC is not suitable to reconstruct complex signals. In all the experiments, the initial value of *r* is set to 0.5 for both *Sl*_{0} and *l*_{2}*-Sl*_{0} algorithms. The values of *δ*_{
r
} and *r*_{
J
} required by the *l*_{2}*-Sl*_{0} algorithm are chosen as 0.05 and 0.7, respectively.

In experiments 1 and 2, the signal length and the number of measurements are set to *N* = 1,000 and *M* = 400, respectively. A *K*-sparse source **x** was artificially created as follows: (1) set **x** to a zero vector of length *N*, (2) generate a vector **z** of length *K* assuming that each element *z*_{
i
} is a random value drawn from the normal distribution *N*(0,1) in the real-valued scenario or from *N*(0,1/2)+*jN*(0,1/2) in the complex-valued scenario, and (3) randomly select *K* components of **x** and set them to **z**. Each element of the measurement matrix **Φ** is randomly generated using the normal distribution *N*(0,1) or *N*(0,1)+*jN*(0,1), and each row is normalized to unity. Then, the mixtures are generated using the noisy model **y**=**Φx**+**w**, where **w** is an additive white Gaussian noise with covariance matrix *σ*_{
w
}**Ι**_{
M
} (**I**_{
M
} stands for the M × M identity matrix). To evaluate the estimation accuracy, the signal-to-noise ratio (SNR) defined as $20log\left(\left|\right|\mathbf{x}\left|\right|{}_{2}/\left|\right|\mathbf{x}-\widehat{\mathbf{x}}\left|\right|{}_{2}\right)$ is used, where **x** and $\widehat{\mathbf{x}}$ denote the true value and its estimate, respectively.

In experiment 1, we compare the reconstruction performance of *l*_{2}*-Sl*_{0} with that of *Sl*_{0}. Figures 1 and 2 show the reconstruction SNR at different powers of noise σ_{w} in real and complex signal scenarios, respectively. For each value of σ_{
w
}, the reconstruction SNR is averaged over 100 runs. It is seen that *l*_{2}*-Sl*_{0} produces a better SNR than *Sl*_{0}, which shows the robustness of *l*_{2}*-Sl*_{0} against noise. The objective function of *l*_{2}*-Sl*_{0} algorithm in Equation 15 comprises the quadratic error term $\frac{1}{2}\left|\right|\mathbf{Ah}-{\mathbf{Y}}_{\mathrm{pilot}}\left|\right|{}_{2}^{2}$ which permits a small perturbation. Therefore, the *l*_{2}*-Sl*_{0} algorithm has a larger capability to reconstruct sparse signal in the presence of noise than *Sl*_{0}. For smaller values of σ, *F*_{σ}(**x**) contains more local minima. Therefore, the decrease of *σ* should not be too quick in the *Sl*_{0} and *l*_{2}*-Sl*_{0} algorithms. Moreover, unlike *Sl*_{0} using a fixed step size to control the decrease of the parameter σ, *l*_{2}*-Sl*_{0} uses a variable one, and the step size *δ*_{
r
} slightly increasing with the reduction of σ may also help the *l*_{2}*-Sl*_{0} to improve its estimation accuracy.

In experiment 2, the *l*_{2}*-Sl*_{0} algorithms are tested using *N* = 1,000, *M* = 400, and various *K* sparse signals with *σ*_{
ω
} = 0.01, to examine the algorithms' performance for signals of different sparsity levels. The results obtained are plotted in Figures 3 and 4 with **y**, **Φ**, **x** being real and complex values, respectively. It is observed that the performance of the *l*_{2}*-Sl*_{0} algorithm is better than the *Sl*_{0} algorithm in most cases. In real-valued scenario, the *l*_{2}*-Sl*_{0}*-* BFGS, *l*_{2}*-Sl*_{0}*-* CG, and *l*_{2}*-Sl*_{0}*-* NC are comparable for *K* smaller than 130, but *l*_{2}*-Sl*_{0}*-* BFGS performs better for *K* between 130 and 210. In addition, when *K* is smaller than 90, the final SNR of the *Sl*_{0} algorithm increases with the rise of sparsity *K*. This is because the initial estimate ${\widehat{\mathbf{x}}}_{0}$ is set to the minimum *l*_{2}-norm solution of **y** = **Φx** + **w**, which has few zero elements and is far away from the actual signal with many zero elements, and ${\widehat{\mathbf{x}}}_{0}$ is gradually close to the actual signal with the rise of sparsity *K*. However, this phenomena is not obvious in *l*_{2}-*Sl*_{0} algorithm, since the initial estimate ${\widehat{\mathbf{x}}}_{0}$ is set to zeros in *l*_{2}-*Sl*_{0}-CG and *l*_{2}-*Sl*_{0}-NC, which is near the actual solution for a small value of *K*. Because of ${\widehat{\mathbf{x}}}_{0}$ being set to zeros and the thresholds of *δ*_{1} and *δ*_{2} being not small enough for the value sparsity *K* above 230, the *l*_{2}-*Sl*_{0}-NC performs the worst among the algorithms tested.

Next, we investigate the accuracy and the spectral efficiency of CCS method using *l*_{2}-*Sl*_{0}. We consider a MIMO-OFDM system with two transmit and two receive antennas (*N*_{
T
} = *N*_{
R
} = 2). The number of subcarriers is 512, and the QPSK modulation is used. The length of cyclic prefix is 20, which equals the length of wireless channel impulse response. In experiment 3, a Rayleigh channel modeled by a 4-tap MIMO-FIR filter is assumed, in which each tap corresponds to a 2 × 2 random matrix whose elements are i.i.d. complex Gaussian variables with zero mean and unit variance, and the position *l*_{
d
} of MSTs is {2, 6, 13, 19}. The estimation performance is evaluated in terms of the bit error rate (BER) and mean square error (MSE) given by *MSE(Δh)* =$\frac{{\displaystyle {\sum}_{i=1}^{M}\left|\right|{\widehat{\mathbf{h}}}_{i}-{\mathbf{h}}_{i}\left|\right|{}_{2}^{2}}}{{\displaystyle {\sum}_{i=1}^{\mathit{M}}\left|\right|{\mathbf{h}}_{i}\left|\right|{}_{2}^{2}}}$, where *M* represents the number of simulations and **h**_{
i
} and ${\widehat{\mathbf{h}}}_{i}$ represent the actual and the estimated channels from the *i*-th simulation, respectively.

In experiment 3, we investigate the performance and required computational time of the CCS using *l*_{2}-*Sl*_{0}-BFGS, *l*_{2}-*Sl*_{0}-CG, and *Sl*_{0} reconstruction algorithms with 30 pilot signals in each transmit antenna. The simulation consists of 2,000 Monte Carlo runs. Moreover, their performance is compared with those of the CCS using OMP and FISTA. OMP is the most popular one in the type of greedy reconstruction algorithm, and FISTA is the most fast one in the type of *l*_{2}-*l*_{1} reconstruction algorithm. Figures 5 and 6 show the MSE and BER plots resulting from the above five CCS methods and the conventional LS method, respectively. As can be seen, the CCS method using *l*_{2}-*Sl*_{0}-CG only needs 30 pilot signals to obtain the approximate performance of the LS method using 40 pilot signals which implies that the CCS using *l*_{2}-*Sl*_{0}-CG can save nearly 25% pilot signals. This merit of CCS is due to the prior sparse information of the wireless channel utilized and the efficient reconstruction of sparse signals from a very limited number of measurements allowed by CS. In addition, the CCS applying *l*_{2}-*Sl*_{0}-CG or *l*_{2}-*Sl*_{0}-BFGS outperforms the CCS using *Sl*_{0} more obviously than that in experiment 1, which shows that the *l*_{2}-*Sl*_{0} has a larger capability to reconstruct sparse signal than *Sl*_{0} in the case when each row of measurement matrix is not normalized to unity. Since the *l*_{2}-*Sl*_{0} algorithm is halted after a fixed number of iterations, furthermore, the fixed number does not depend on the sparsity of the signal directly; it is convenient to set the number in practical applications.

We use the CPU time as a measure of complexity. The simulations are performed in MATLAB R2009b environment using an Intel Core i3, 2.53-GHz processor with 2 GB of memory, and under Microsoft Windows XP operating system. The results shown in Figure 7 indicate that the CCS using *l*_{2}-*Sl*_{0} requires more computational time than that using other algorithms tested. The *l*_{2}-*Sl*_{0} algorithm needs an iterative process to find the optimal solution at each value of *σ*; therefore, the running time of *l*_{2}-*Sl*_{0} is longer than that of others tested. However, at the cost of slightly more computational time, the CCS using *l*_{2}-*Sl*_{0}-CG yields slightly better performance than the CCS using OMP or FISTA, and the threshold value for termination iteration in the *l*_{2}-*Sl*_{0} algorithm is easier to be set. More specifically, it is shown in Algorithm 2 that *l*_{2}-*Sl*_{0}-CG applies a constant value *L* to stop the iteration, and the constant value is independent of the sparsity of signal and the power of noise. However, the valid threshold values for termination iteration in the OMP and FISTA algorithms always depend on the power of noise or the sparsity of signal, which are both quite difficult to estimate beforehand in practical applications.

In experiment 4, we investigate the BER of the CCS using 30 pilot signals in each transmit antenna under different channel sparsities, namely for different numbers of MSTs. Moreover, the position *l*_{
d
} of MST is randomly selected in each Monte Carlo simulation. Figure 8 shows the BER plots of CCS using *l*_{2}-*Sl*_{0}-CG and *l*_{2}-*Sl*_{0}-BFGS algorithms. The figure shows that a better BER performance can be expected in general for less number of MSTs. In addition, when the length of channel response is 20, the CCS using *l*_{2}-*Sl*_{0}-CG and that using *l*_{2}-*Sl*_{0}-BFGS are found to yield acceptable BERs for up to 8 and 4 MSTs, respectively.

## 5. Conclusion

In this paper, a new approach for sparse channel estimation of MIMO-OFDM systems based on compressed sensing has been presented. The new approach uses a smoothed *l*_{0}-norm-regularized least squares (*l*_{2}-*Sl*_{0}) objective function and solves the optimization problem by three reconstruction algorithms: quasi-Newton, conjugate gradient (CG), and optimization in the null and complement spaces of measurement matrix (ONCS). The better reconstruction accuracy of the *l*_{2}-*Sl*_{0} as compared with the *Sl*_{0} algorithm and the higher spectrum efficiency of the CCS using *l*_{2}-*Sl*_{0}-CG or *l*_{2}-*Sl*_{0}-BFGS as compared with the conventional LS method have been shown by computer simulations.

## Appendix

### Derivation of Equations 26 and 27

Suppose that **e**_{
i
} is the *i*-th column of an *M* × *M* identity matrix, and the vectors **α** and **β** in Equation 22 are fixed. At point **α**, a line search along direction **e**_{
i
} is carried out by solving the one-dimensional optimization problem

Where *x*_{
j
} is the *j*-th component of vector **x**, and ${v}_{r}^{\left(j,i\right)}$ is the (*j*,*i*)-th component of matrix **V**_{
r
}. By equating the derivative ∂*F*(**α** + *d*_{r,i}**e**_{
i
}, **β**)/∂*d*_{r,i} to zero, for real **Φ** and **x**, we can obtain

Note that *d*_{
r,i
} can be solved via iterations with the initial value of *d*_{
r,i
} being set to zero in the right side of (30). In a similar manner, *d*_{
n,i
} can be obtained as

where ${v}_{n}^{\left(j,i\right)}$ is the (*j*,*i*)-th component of matrix **V**_{
n
}.

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## Acknowledgements

We express our thanks to the anonymous reviewers for their valuable comments to improve the quality and the presentation of this paper. This work is supported by the National Natural Science Foundation of China under grant nos. 61372122 and 61302104 and the Basic Research Program of Jiangsu Province under grant no. BK2011756.

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Ye, X., Zhu, W., Zhang, A. *et al.* Sparse channel estimation of MIMO-OFDM systems with unconstrained smoothed *l*_{0}-norm-regularized least squares compressed sensing.
*J Wireless Com Network* **2013, **282 (2013). https://doi.org/10.1186/1687-1499-2013-282

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### Keywords

- Sparse channel estimation
- Smoothed
*l*_{0}-norm *l*_{2}-norm- MIMO-OFDM