A novel method for sparse channel estimation using super-resolution dictionary
© Zhou et al.; licensee Springer. 2014
Received: 31 October 2013
Accepted: 12 February 2014
Published: 6 March 2014
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.
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 . 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 . 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) . In the literature [5, 6], Bajwa made a feasibility analysis of CCE and extended it to a doubly selective channel. The literature  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 . 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
where δ k ∈(0,1) is a constant.
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.
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. NCP=N−K is the guard interval for the CP which is used to avoid the intersymbol interference (ISI). xl,k denotes the l th symbol transmitted at subcarrier k, and discrete transmit pulse g[n] is 1 on [0,N] and 0 otherwise.
where h[n,θ]=h(n T s ,θ T s ) and z[n]=z(n T s ) is discrete-time noise.
where denotes the noise or the interference terms. Hl,k is the system channel coefficients which will be analyzed in the following sections.
3.2 Doubly selective fading channel
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
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 Hl,k is approximate sparse in the delay-Doppler domain. However, the fading of sinc(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 Hl,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 .
where J/(λ N0T s )≥vmax/2.
where corresponds to 2J+1 Doppler sample point under λ times over-sampling.
where D T s /λdelay≥τmax, λdelayand λDopplerare the over multiple number of delay and Doppler, respectively, and Aγ,g(θ,i/N0) is the same as that in Equation 17.
4.3 The estimation of the sparse channel based on the dictionary domain
Assuming that, whereis the pilot set, the total number of pilot point is. The number of pilots must satisfy the lowest demand of compressive sensing theory to represent the measurement signal without distortion. The literature  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 1−2δ probability. So we can select the suitable number of pilots, Q≥N m .
Obtain the estimationof the channel coefficient according to Equation 28, and then combine allto form.
Utilize OMP to obtain an estimationof x based on knownand Φ according to Equation 30. The detail of realization refers to Algorithm 1. After rescaling, we can obtain estimation value, as well the spread coefficient S D [i,θ] corresponding to the dictionary.
Compute all channel coefficientby utilizing Equation 27 based on knownand 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(2J+1)) from D×(2J+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
In addition, simulation tools are MATLAB 2009 and an Intel Core PC with a 2.8-G processor and 1.5-G RAM.
The run time of different algorithms
Extra time (s)
The required run time about different algorithms
Extra time (s)
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.
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).
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