- Research
- Open Access

# A novel method for sparse channel estimation using super-resolution dictionary

- Fei Zhou
^{1}Email author, - Jing Tan
^{1}, - Xinyue Fan
^{1}and - Liang Zhang
^{1}

**2014**:29

https://doi.org/10.1186/1687-6180-2014-29

© Zhou et al.; licensee Springer. 2014

**Received:**31 October 2013**Accepted:**12 February 2014**Published:**6 March 2014

## Abstract

Due to the sparse distribution of reflectors in space, wireless channels are commonly sparse. Thus, utilizing the sparsity of channels in the delay-Doppler domain, a channel estimation method based on compressed sensing (CS) theory can reduce the number of pilots. However, because of discrete truncation in the time domain and limited bandwidth, the time delay and frequency shift of non-integer multiple samples can cause energy leakage in the delay and Doppler domain, which seriously reduce the delay-Doppler sparsity of the equivalent channel, thus affecting the accuracy of channel estimation. In this paper, we use an over-complete dictionary based on super-resolution to enhance the sparsity of the equivalent channel and reconstruct a doubly selective channel with greater accuracy. Simulation results demonstrate that the equivalent channel frequency response in the dictionary is sparser than that in the delay-Doppler domain. Compared with the traditional algorithm, the method proposed in this paper can effectively improve the performance of channel estimation.

## Keywords

- Sparse channel estimation
- Doubly selective channel
- Energy leakage
- Super-resolution dictionary

## 1 Introduction

Traditional channel estimation methods on orthogonal frequency division multiplexing (OFDM) systems commonly assume that the channel has rich multipath and require a large number of pilots to obtain more accurate state information of the channel, which seriously reduce the utilization efficiency of the channel. Meanwhile, traditional methods of linear channel estimation already attained optimal estimation performance such as utilization efficiency, so it is difficult to improve them further. To overcome the bottleneck, we need to explore the own characteristic of the channel. More and more experimental evidences show that the sparse distribution of reflectors in space makes the transmission channel sparse.

The research on sparse wireless channel has already begun since the 1990s. Cotter and Rao utilize matching pursuit (MP) algorithm to estimate a small amount of non-zero channel taps in a single-carrier selective channel [1]. MP algorithm on decision feedback equalizer can effectively improve the performance of channel estimation. Compared with MP algorithm, the method proposed by Raghavendra and Giridhar estimates the tap position in a frequency-selective channel based on generalized Akaike information criterion and least squares (LS) algorithm, which greatly reduces the calculation burden of LS algorithm [2]. However, the above results were basically obtained by simulation and lacked relating theory analysis. In recent years, Donoho and Candes et al. propose a novel theory, i.e., compressed sensing, on the basis of functional analysis and approximation theory. The theory suggests that if the signal is sparse in a certain domain, it can be accurately reconstructed by a small amount sampling signal with high probability [3, 4]. Bajwa et al. firstly applied compressed sensing (CS) theory for channel estimation and proposed the concept of compressive channel estimation (CCE) [5]. In the literature [5, 6], Bajwa made a feasibility analysis of CCE and extended it to a doubly selective channel. The literature [7] gives a virtual channel model by Nyquist sampling for physical transmission environment in the angle-delay-Doppler domain and makes a comparison between CCE and LS. Meanwhile, aiming at acoustic OFDM systems, Berger et al. proved that CS channel estimation is superior over the traditional linear channel estimation method, which is reflected in the experimental data, like root-MUSIC and ESPRIT algorithms [8, 9]. Taubock and Hlawatsch transform a doubly selective channel model to a solvable basis pursuit inequality constraint model, utilize the pilot signal as the key measurement that CS reconstruction requires, and analyze the sparsity of the channel parameter in the delay-Doppler domain [10]. However, it was concluded from the analysis that the energy leakage problem caused by discrete truncation of time domain and limited bandwidth obviously deteriorates the channel’s sparsity which limits the performance improvement of CS-based channel estimation methods. To solve the problem, Taubock et al. propose an iterative basis optimization procedure that aims to maximize sparsity [11, 12]. Although the method proposed by Taubock achieves significant performance gains, its basis optimization, which adds additional complexity, has to be performed before data transmission. Aimed at common problems caused by time delay and Dopler frequency shift of non-integer multiple samples, we use a super-resolution over-complete dictionary to improve the performance of channel estimation. The dictionary only increases the run time of the sparse reconstruction procedure with the increase of basis. We find that the over-complete dictionary representation of the channel is much sparser than the classical delay-Doppler representation in most cases, and it can effectively reduce the usage of pilots and improve the estimation performance.

The rest of this paper is organized as follows. Section 1 introduces CS theory and Section 2 introduces the OFDM system model. Section 3 analyzes the energy leakage of the channel in the delay-Doppler domain firstly, then uses the super-resolution dictionary instead of Fourier basis to enhance the sparsity of the channel, and next presents the CS-based channel estimation method. In Section 4, we present numerical results. Finally, Section 5 concludes the paper.

## 2 Compressed sensing theory

*K*-sparsity, or approximate

*K*-sparsity, it can be represented by using

*K*(≤

*N*) non-zero coefficients [3]. Then, a linear measurement value about

**x**can be obtained by selecting appropriate measurement matrix $\Phi \in {\u2102}^{M\times N}(M<N)$, as shown in (1). Only

*M*measurements within

**y**can be utilized to reconstruct the original signal with very high probability.

**y**is far less than that of

**x**, so equation array (1) is underdetermined. However, in view of

**x**which is

*K*-sparse, it is only required to obtain

*K*non-zero coefficients and their position. Candes et al. have proved that if the number of measurement

*M*=

*O*(

*K*log(

*N*)), and the measurement matrix satisfies the constraint of restricted isometry property (RIP), signal

**x**can be reconstructed by solving the

*l*

_{0}-norm minimization in (2) [3].

*N*dimension

*K*-sparsity vectors

**a**all satisfy the following rule:

where *δ*_{
k
}∈(0,1) is a constant.

*l*

_{0}-norm is not convex. Actually, this problem is NP hard and therefore cannot be solved in a reasonable amount of time. By now, there are many different algorithms to solve it. Orthogonal matching pursuit (OMP) becomes a popular way in CS theory because it is simple for computation and easy for implementation [13]. OMP algorithm transforms the problem,

*l*

_{0}-norm minimization, to a relative simple problem shown in (4):

where *ε* is the upper bound of the noise level. The basic idea of OMP algorithm is how to select the column vector of measurement matrix **Φ** by utilizing the greedy iteration way and reconstruct the signal by computing the support vector set on iterative algorithm of parameter **x**[13].

## 3 OFDM system model

### 3.1 System model

where *K* is the number of subcarriers, *L* is the number of transmitted symbol periods, and *N* denotes the symbol duration. *N*_{CP}=*N*−*K* is the guard interval for the CP which is used to avoid the intersymbol interference (ISI). *x*_{l,k} denotes the *l* th symbol transmitted at subcarrier *k*, and discrete transmit pulse *g*[*n*] is 1 on [0,*N*] and 0 otherwise.

*h*(

*t*,

*τ*) includes physical channel

*h*

_{ch}(

*t*,

*τ*), transmitter filter

*f*

_{tr}(

*t*), and received filter

*f*

_{rec}(

*t*), so we have

*T*

_{ s }is given by

where *h*[*n*,*θ*]=*h*(*n* *T*_{
s
},*θ* *T*_{
s
}) and *z*[*n*]=*z*(*n* *T*_{
s
}) is discrete-time noise.

*r*[

*n*] is demodulated, we can obtain

*l*=0,1,…,

*L*−1,

*k*=0,1,…,

*K*−1, and

*γ*[

*n*] is only 1 in [

*N*−

*K*,

*N*−1] and 0 otherwise. Combining (5), (7), and (8), we have

where ${z}_{l,k}=\frac{1}{\sqrt{K}}\sum _{n=-\infty}^{\infty}z\left[n\right]{e}^{-j2\mathrm{\pi k}\left(n-\mathit{\text{lN}}\right)/K}\gamma \left[n-\mathit{\text{lN}}\right]$ denotes the noise or the interference terms. *H*_{l,k} is the system channel coefficients which will be analyzed in the following sections.

### 3.2 Doubly selective fading channel

*P*is the number of multipath components and

*η*

_{ q },

*τ*

_{ q }, and

*v*

_{ q }are the attenuation coefficient, the delay, and the Doppler shift of path

*q*th, respectively.

*δ*denotes the Dirac delta function. We obtain the delay-Doppler spreading function

*S*(

*v*,

*τ*) via Fourier transform.

*h*(

*t*,

*τ*) does not vary in the area of received filter

*f*

_{rec}(

*t*), Equation 12 can be derived from Equation 6.

*ψ*(

*τ*−

*θ*)=

*f*

_{tr}(

*t*)⊗

*f*

_{rec}(

*t*). After discretizing Equation 12, we have

Due to the non-linear relation between *h*[*n*,*θ*] and channel parameters [*η*_{
q
},*v*_{
q
},*τ*_{
q
}], it is difficult to analyze the channel by utilizing Equation 13. It caused us to search for a new model with less parameters.

## 4 Sparse channel estimation using a dictionary

### 4.1 The effect of sparsity caused by energy leakage

*r*

_{l,k}in (8) for all

*l*=0,1,…

*L*−1, the discrete-time received signal

*r*[

*n*] has to be known for

*n*=0,…,

*N*

_{0}−1, where

*N*

_{0}=

*L*

*N*. The discrete time channel impulse response

*h*[

*n*,

*θ*] in Equation 13 can be represented by BEM with a period of

*N*

_{0}[16], so we have

*J*satisfies

*J*/(

*N*

_{0}

*T*

_{ s })≥

*v*

_{max}/2 and

*v*

_{max}/2 denotes the single-sided maximum Doppler shift.

*S*

_{ h }[

*i*,

*θ*] is the discrete delay-Doppler spread function:

_{ N }(

*x*)= sin (

*π*

*x*)/(

*N*sin(

*π*

*x*/

*N*)), ${\eta}_{q}^{\prime}={\eta}_{q}{e}^{\mathrm{j\pi}\left({v}_{q}{T}_{s}-i/{N}_{0}\right)\left({N}_{0}-1\right)}$, and

*θ*=0,…,

*D*−1.

*D*≥

*τ*

_{max}/

*T*

_{ s }denotes the number of discrete time-delay sampling points. It is obvious that if the maximum time-delay satisfies

*τ*

_{max}≤

*N*

_{CP}, the intersymbol interference can be eliminated. And if the ideal filter ${f}_{\text{tr}}\left(t\right)={f}_{\text{rec}}\left(t\right)=\sqrt{1/{T}_{s}}sinc\left(t/{T}_{s}\right)$ is applied, where sin

*c*(

*x*)=sin(

*π*

*x*)/(

*π*

*x*),

*ψ*(

*θ*

*T*

_{ s }−

*τ*

_{ q })≈ sin

*c*(

*θ*−

*τ*

_{ q }/

*T*

_{ s }) can be obtained and applied in Equation 15. So we have

*H*

_{l,k}

*Λ*

_{ q }[

*i*,

*θ*] instead of

*S*

_{ h }[

*i*,

*θ*]. From Equation 11, the delay-Doppler function

*S*(

*v*,

*τ*) consisted of the Dirac function in delay-Doppler point (

*τ*

_{ q },

*v*

_{ q }) which corresponds to reflecting path

*q*and is supposed to be sparse. However, the Dirac function is replaced by sin

*c*(

*x*) and dir

_{ N }(

*x*) in

*S*

_{ h }[

*i*,

*θ*]. Only when

*τ*

_{ q }/

*T*

_{ s }and

*v*

_{ q }

*T*

_{ s }

*N*

_{0}are all integers,

*Λ*

_{ q }[

*i*,

*θ*] can be simplified to the Dirac function. The reason is that sin

*c*(

*x*) and dir

_{ N }(

*x*) are equal to 1 on

*x*=0 and 0 otherwise, as shown in Figure 2. Under any other conditions,

*Λ*

_{ q }[

*i*,

*θ*] is not equal to 0 for any

*i*and

*θ*. In other words, the peak energy of the discrete delay-Doppler function may have leakage to the near delay-Doppler area. Figure 3 demonstrates the leakage effect of

*S*

_{ h }[

*i*,

*θ*] where

*P*=10,

*N*

_{CP}=16,

*K*=25, and

*L*=16.

*η*

_{ q },

*τ*

_{ q }/

*T*

_{ s }, and

*v*

_{ q }

*T*

_{ s }

*N*

_{0}are random variables.

From Figure 3, the peak energy of *S*_{
h
}[*i*,*θ*] leaks to the near delay-Doppler area, and then its value would fade and approximate to 0 gradually. Therefore, *S*_{
h
}[*i*,*θ*] is approximate sparse, i.e., the channel coefficient *H*_{l,k} is approximate sparse in the delay-Doppler domain. However, the fading of sin*c*(*x*) and dir_{
N
}(*x*) is slow, and a lot of values in *S*_{
h
}[*i*,*θ*] cannot be neglected. It seriously influences the sparsity of the channel and the estimation performance of the channel.

### 4.2 Sparsity enhancement using the super-resolution dictionary

In real wireless channels, the time delay and Doppler frequency shift of non-integer times sampling points exist generally. They seriously influence the sparsity of channel coefficient *H*_{l,k} in the time delay and Doppler domain and do not satisfy the prerequisite condition. Hence, how to avoid the energy leakage is a very important issue in channel estimation. Essentially, the channel energy leakage is actually introduced by channel discrete characterization, and the sparsity of the channel itself does not disappear. Therefore, if we can improve the accuracy of channel discrete characterization, it would greatly reduce the energy leakage. Therefore, improving the discrete accuracy of the channel impulse response reduces the energy leakage [17].

*h*[

*n*,

*θ*] can be represented by BEM with a period of

*λ*

*N*

_{0}, we have

where *J*/(*λ* *N*_{0}*T*_{
s
})≥*v*_{max}/2.

where ${\mathbf{\text{F}}}^{\left(\lambda \right)}=\left[\begin{array}{ccc}1& \cdots & 1\\ {e}^{-\frac{j2\mathrm{\pi J}}{\lambda {N}_{0}}}& \cdots & {e}^{\frac{j2\pi \left(J-1\right)}{\lambda {N}_{0}}}\\ \vdots & \cdots & \vdots \\ {e}^{-\frac{j2\mathrm{\pi J}\left(\lambda {N}_{0}-1\right)}{\lambda {N}_{0}}}& \cdots & {e}^{\frac{j2\pi \left(J-1\right)\left(\lambda {N}_{0}-1\right)}{\lambda {N}_{0}}}\end{array}\right]$.

where ${\mathbf{\text{S}}}_{\theta}^{\left(\lambda \right)}$ corresponds to 2*J*+1 Doppler sample point under *λ* times over-sampling.

**h**

_{ θ }=[

*h*[0,

*θ*],…,

*h*[

*N*

_{0}−1,

*θ*]]

^{ T }, where

*a*

_{ m }=(

*J*+

*m*) mod

*λ*and

*b*

_{ m }=(

*J*−

*m*) mod

*λ*. It is easy to prove Equation 22:

where ${\mathbf{\text{D}}}_{m}^{\left(\lambda \right)}=\text{diag}\left\{{\left[1,{e}^{-\frac{j2\mathrm{\pi m}}{\lambda {N}_{0}}},\dots ,{e}^{-\frac{j2\mathrm{\pi m}\left({N}_{0}-1\right)}{\lambda {N}_{0}}}\right]}^{T}\right\}$ and

*a*

_{ m }+

*b*

_{ m }+1) samples around zero from the critically sampled Doppler spectrum of the

*m*/(

*λ*

*N*) frequency-shifted version of

**h**

_{ θ }. So all samples (

*m*=0,1,…,

*λ*−1) can deduce 2

*J*+1 Doppler frequency shift point. By replacing ${\mathbf{\text{D}}}_{m}^{\left(\lambda \right)}$, ${\mathbf{\text{F}}}_{m}^{\left(\lambda \right)}$, and

**h**

_{ θ }into Equation 23, we have

*v*

_{ q }

*T*

_{ s }

*N*

_{0}=

*n*/

*λ*, ${\text{dir}}_{{N}_{0}}\left(\pi \left(i-\lambda {v}_{q}{T}_{s}{N}_{0}\right)\right)$ can be transformed into a Dirac function. The higher the parameter

*λ*is, the more the sample of the Doppler spectrum is. And when the position of the sample is closer to the real position, the problem of energy leakage would greatly be reduced. Admittedly, a part of the Doppler spectrum still results in a certain energy leakage; however, its value is very small. Letting $\mathcal{I}$ denote all integer set which satisfied |

*i*−

*λ*

*v*

_{ q }

*T*

_{ s }

*N*

_{0}|≤

*Δ*

*i*in the area of

*i*∈{−

*J*,…,

*J*}, the energy sum of samples in which the distance

*v*

_{ q }

*T*

_{ s }

*N*

_{0}in ${\text{dir}}_{{N}_{0}}\left(\pi \left(i-\lambda {v}_{q}{T}_{s}{N}_{0}\right)\right)$ is larger than

*Δ*

*i*∈{2,3,…} can be given.

*v*

_{ q }

*T*

_{ s }

*N*

_{0}=0.4 and

*v*

_{ q }

*T*

_{ s }

*N*

_{0}=0.5.

**,**[19]. Lastly, equivalent discrete-time baseband channel frequency response is given by

where *D* *T*_{
s
}/*λ*_{delay}≥*τ*_{max}, *λ*_{delay}and *λ*_{Doppler}are the over multiple number of delay and Doppler, respectively, and *A*_{γ,g}(*θ*,*i*/*N*_{0}) is the same as that in Equation 17.

**U**is defined by${\left[\mathbf{\text{U}}\right]}_{\mathit{\text{kL}}+l,\left(i+J\right)D+\theta}={e}^{-j2\pi \left(\mathrm{k\theta}/\mathrm{\lambda K}-\mathit{\text{ni}}/\lambda {N}_{0}\right)}$,${\left[\mathbf{\text{h}}\right]}_{\mathit{\text{kL}}+l}={H}_{l,k}$, and${\left[\mathbf{\text{g}}\right]}_{\left(i+J\right)D+\theta}={S}_{D}\left[i,\theta \right]{A}_{\gamma ,g}^{\prime}\left(\theta ,i/{N}_{0}\right)$. Obviously,

**g**is sparse according to the above analysis. So we can obtain the vector form of Equation 26:

*λ*=1,

**U**is the 2-D Fourier basis as used in Equation 18. Under the same conditions as those in Figure 3, Figure 5 shows the distribution of coefficient

**g**in the dictionary domain.

**g**in Figure 5 is less than that in Figure 3. So the channel coefficient corresponding to

*λ*=2 has a higher sparsity. To analyze the sparsity in the dictionary domain better, we utilize OMP algorithm to solve

*S*-sparse approximation and obtain the most

*S*maximum value in

**g**. From Figure 6, with the increase of

*S*, the mean square error$E\left[\right|\mathbf{\text{h}}-{\widehat{h}}_{S}{|}^{2}]$would decrease gradually and would be close to 10

^{ −1 }, i.e., we can obtain 90% channel energy based on

*S*strongest atoms. If

*λ*is higher, the atoms that satisfy the required mean error are fewer. So we can conclude that if

*λ*is higher, the sparsity of frequency response

**h**is higher in the over-complete dictionary.

### 4.3 The estimation of the sparse channel based on the dictionary domain

Assuming that$\left(l,k\right)\in \mathcal{P}$, where$\mathcal{P}$is the pilot set, the total number of pilot point is$Q=\left|\mathcal{P}\right|$. The number of pilots must satisfy the lowest demand of compressive sensing theory to represent the measurement signal without distortion. The literature [14] gives the limitation of the number of measurement samples required by OMP. *N*_{
m
}≥*K* *S* ln(*N*_{
r
}/*δ*), where *S* denotes sparsity. Commonly, *K*≤20 is reasonable. If *S* is too large, we may also set *K*≈4 and *δ*∈(0,0.36). OMP may reconstruct a signal with 1−2*δ* probability. So we can select the suitable number of pilots, *Q*≥*N*_{
m
}.

**z**

_{ Δ }be the set of${\stackrel{~}{z}}_{l,k}$in$\left(l,k\right)\in \mathcal{P}$. We have

**g**is sparse. So Equation 27 is a standard equation of CS. Measurement matrix

**U**

_{ Δ }is a structured random matrix which is formed by selecting a row vector of unitary matrix corresponding to the pilot point. If we select the position of the pilot uniformly and randomly and

*Q*is large enough, the normalized matrix$\sqrt{1/Q}{\mathbf{\text{U}}}_{\Delta}$has a small constraint isometric constant with very high probability. In other words, we can obtain very good reconstruction performance. So we can obtain Equation 30 from Equation 29.

**and**$\mathbf{\text{x}}=\sqrt{Q/\mathit{\text{KL}}}\mathbf{\text{g}}$. Therefore, we may realize channel equalization based on CS theory. The following are the detailed steps:

- 1.
Obtain the estimation${\stackrel{~}{H}}_{l,k}$of the channel coefficient according to Equation 28, and then combine all${\stackrel{~}{H}}_{l,k}$to form${\stackrel{~}{\mathbf{\text{h}}}}_{\Delta}$.

- 2.
Utilize OMP to obtain an estimation$\stackrel{~}{\mathbf{\text{x}}}$of

**x**based on known${\stackrel{~}{\mathbf{\text{h}}}}_{\Delta}$and**Φ**according to Equation 30. The detail of realization refers to Algorithm 1. After rescaling$\stackrel{~}{\mathbf{\text{x}}}$, we can obtain estimation value$\stackrel{~}{\mathbf{\text{g}}}$, as well the spread coefficient*S*_{ D }[*i*,*θ*] corresponding to the dictionary. - 3.
Compute all channel coefficient$\stackrel{~}{\mathbf{\text{h}}}$by utilizing Equation 27 based on known$\stackrel{~}{\mathbf{\text{g}}}$and

**U**.

The run time of OMP mainly depends on the index set *Λ*_{
k
}selected in the iteration process. We need to select the optimal atom O(*D*(2*J*+1)) from *D*×(2*J*+1) atom. So with the increase of *λ*, the number of time-delay sample *D* and the number of Doppler sample *J* would increase exponentially. However, with the increase of *λ*, the channel sparsity would be enhanced and the iteration times required by OMP would reduce. Lastly, it can save a certain run time.

#### Algorithm 1 **Steps of reconstitution**

## 5 Simulation and analysis

*f*

_{ c }=2 GHz, bandwidth

*B*=10.24 MHz, sub-carrier number

*K*=1,024, the length of cyclic prefix

*N*

_{ CP }=128, and sample period

*T*

_{ s }=0.1 ms. We may utilize the channel simulation tool IlmProp [20] based on the geometrical structure of space to simulate a doubly selective fading channel. The simulated frame for the OFDM block has eight symbols, i.e.,

*L*=8. In the simulated environment, the distance between the transmitter and the receiver is 2,000 m, and 10 reflectors, in which 2 reflectors are distributed within 150 m from the transmitter, form 10 multipath clusters, which satisfy the Gaussian distribution. The random speed of each path is less than 100 m/s and the acceleration is less than 20 m/s

^{ 2 }. Assume that the noise

*z*[

*n*] is additive white Gaussian noise (AWGN), in which the mean is 0 and the variance is

*σ*

^{ 2 }. So the signal-noise ratio (SNR) of the symbol block is given by

In addition, simulation tools are MATLAB 2009 and an Intel Core PC with a 2.8-G processor and 1.5-G RAM.

*λ*=1) [10], iterative optimize basis [12], and over-complete dictionary (DIC) (

*λ*

_{delay}=

*λ*

_{Doppler}=2,

*λ*

_{delay}=

*λ*

_{Doppler}=4). Figure 7 gives the MSE comparison of different algorithms under SNR. Figure 8 shows the bit error ratio (BER) comparison of equalization decoding. Both figures suggest that when we only apply 6.25% resource as pilots, the performance of LS estimation is much bad; the reason is that the distribution of the pilot cannot satisfy the Nyquist sampling criterion. However, when OMP algorithm also applies 6.25% resource for the pilot, the performance of the channel estimation proposed in this paper is better than that of LS estimation that applies 12.5% resource for the pilot. So the utilization efficiency of the spectrum is improved obviously. In addition, OMP algorithm on the over-complete dictionary domain has a better performance than the traditional algorithm on the delay-Doppler domain. When

*λ*=2, the performance on the over-complete dictionary domain approaches that on the iterative optimize basis. And when

*λ*=4, the performance on the over-complete dictionary domain is better than that on the iterative optimize basis. With the increase of the

*λ*, the BER performance of OMP algorithm using over-complete dictionary is close to that of ideal channel estimation. So the higher is

*λ*, the better is the reconstruction performance of OMP and the higher is the estimation accuracy.

*λ*=4 is the best, its complexity is also the highest, as shown in Table 1. Considering the reconstruction time only, iterative optimize basis is the optimal except for LS estimation. However, the method based on iterative optimize basis needs extra 163.8541 s to compute the optimal basis. But, in the channel estimation algorithm proposed in this paper, we can produce the dictionary by FFT. The method only enlarges the size of the reconstructed atomic set and does not need extra computation. By comparing the required time between

*λ*=4 and

*λ*=2, it can be concluded that the sparsity of the channel coefficient in the dictionary domain would be sparer with the increase of parameter

*λ*. Meanwhile, the sparsity of the channel coefficient can also be influenced by the physical path and lower than the number of reflectors. Therefore, when

*λ*=2, the channel estimation method based on the super-resolution dictionary domain has a certain advantage on the algorithm performance and computation complexity.

**The run time of different algorithms**

Algorithm | Reconstruction (s) | Extra time (s) |
---|---|---|

LS (6.25%) | 0.0455 | 0 |

LS (12.5%) | 0.0372 | 0 |

OMP_DFT | 0.4282 | 0 |

OMP_OPT | 0.3017 | 163.8541 |

OMP_DIC ( | 0.9661 | 0 |

OMP_DIC ( | 2.9548 | 0 |

*λ*=4, the required resource of pilots is only about 4%. If

*λ*=2, the percentage is about 5%. However, the method based on Fourier basis needs about 7.5% resource. Figure 10 shows the comparison under the different numbers of pilots. With the increase of pilot number, BER would be close to the performance of ideal channel estimation. In other words, under the same performance of MSE or BER, the pilot number required by DIC is less than that by DFT and OPT. Hence, the sparse channel estimation based on the dictionary domain can effectively reduce the number of pilots and improve the spectrum efficiency.

^{ 2 }, so the Doppler influence is not much serious. The other parameters are the same as those in the first simulation. Figures 11 and 12 show that when

*λ*

_{delay}=4,

*λ*

_{Doppler}=1, the performance of channel estimation is better than that of others. The accuracy performance on

*λ*

_{delay}=

*λ*

_{Doppler}=2 is almost the same as that on

*λ*

_{delay}=2,

*λ*

_{Doppler}=1; however, the former run time is about twice the latter, as shown in Table 2. Compared with that of the DFT and OPT, the run time of dictionary basis on

*λ*

_{delay}=2,

*λ*

_{Doppler}=1 is only slightly more than that of Fourier basis; however, the former performance is obviously better than the latter. Considering that the Doppler influence is not much serious, we only over-sample in the time-delay domain, so we can obtain an optimal selection in both computation complexity and performance. Similarly, we can also apply the same method in the Doppler domain.

**The required run time about different algorithms**

Algorithm | Reconstruction (s) | Extra time (s) |
---|---|---|

DFT | 0.4543 | 0 |

OPT | 0.3210 | 164.4662 |

DIC ( | 0.5343 | 0 |

DIC ( | 0.9310 | 0 |

DIC ( | 0.8603 | 0 |

## 6 Conclusions

This paper proposes a novel estimation method of a sparse and doubly selective channel based on CS theory. The method can reduce the problem of energy leakage caused by discrete truncation and the limited bandwidth by over-sampling in the delay-Doppler domain, enhance the sparsity of the equivalent channel in the dictionary domain, and then improve the performance of channel estimation. The results show that although the method based on the over-complete dictionary needs more computation, the estimation accuracy is improved obviously and the pilot resource is reduced very much. Lastly, compared with the increase of spectrum utilization, it is worth for more complexity.

## Declarations

### Acknowledgements

This work was supported by the Fundamental and Frontier Research Project of Chongqing (cstc2013jcyjA40034), National Natural Science Foundation of China (61301126), Program for Changjiang Scholars and Innovative Research Team in University (IRT1299), and special fund of Chongqing key laboratory (CSTC).

## Authors’ Affiliations

## References

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