- Email: [email protected]
Compressed sensing based channel estimation used in non-sample-spaced multipath channels of OFDM system
The Journal of China Universities of Posts and Telecommunications April 2015, 22(2): 31–37 www.sciencedirect.com/science/journal/10058885
http://jcupt.xsw.bupt.cn
Compressed sensing based channel estimation used in non-sample-spaced multipath channels of OFDM system Chen Baohao (
), Cui Qimei, Yang Fan
National Engineering Laboratory for Mobile Network Security, Beijing University of Posts and Telecommunications, Beijing 100876, China
Abstract By virtue of an increase in spectral efficiency by reducing the transmitted pilot tones, the compressed sensing (CS) has been widely applied to pilot-aided sparse channel estimation in orthogonal frequency division multiplexing (OFDM) systems. The researches usually assume that the channel is strictly sparse and formulate the channel estimation as a standard compressed sensing problem. However, such strictly sparse assumption does not hold true in non-sample-spaced multiple channels. The authors in this article proposed a new method of compressed sensing based channel estimation in which an over-complete dictionary with a finer delay grid is applied to construct a sparse representation of the non-sample-spaced multipath channels. With the proposed, the channel estimation was formulated as the model-based CS problem and a modified model-based compressed sampling matching pursuit (CoSaMP) algorithm was applied to reconstruct the discrete-time channel impulse response (CIR). Simulation indicates that the new method proposed here outperforms the traditional standard CS-based methods in terms of mean square error (MSE) and bit error rate (BER). Keywords channel estimation, model-based compressed sensing, non-sample-spaced multipath channels
1 Introduction OFDM has been widely employed in current and future wireless communication systems. Typically, the OFDM system employs coherent detection for which channel state information (CSI) is crucial. In OFDM system, the CSI is usually obtained when using pilot tones and many pilot-aided channel estimation methods have been studied. Generally, the least squares (LS) estimation is the simplest channel estimation based on parallel Gaussian channel model in Ref. [1]. The linear minimum mean-squared error (LMMSE) method is the optimum solution for interpolation [2]. However, The practice that the wireless channels are rich multipath and use a large number of pilot tones to obtain accurate CSI, will possibly lead to low spectrum efficiency. The compressed sensing is an emerging sampling theory proposed by Donoho et al. in 2006 [3–4]. Accordingly, if a signal has a spare representation in a certain space, the Received date: 04-05-2014 Corresponding author: Chen Baohao, E-mail: [email protected] DOI: 10.1016/S1005-8885(15)60636-7
signal can be sampled at a rate significantly lower than Nyquist rate, it can be reconstructed with high probability by optimization techniques. Study indicates that the wireless channels tend to exhibit a sparse multipath structure [5] that means that the delay spread of the channel could be very large while the number of significant propagation paths is normally very small. Based on the sparse structure, researchers introduced compressed sensing into channel estimation and obtained extensive achievements [6–8]. The sparse signals efficiently is reconstructed by CS from a small number of measurements to reduce the number of pilot tones required to meet the desired system performance requirement. Usually, the radio channel in a wireless communication system is characterized by multitudinous propagation paths, each is parameterized by different delays and complex amplitudes, resulting in a sparse multipath channel model [9]. Most CS-based channel estimators assume that the channel is strictly sparse in the equivalent discrete-time baseband representation, i.e., the tapped-delay line model with the delay grid spaced at a baseband sampling rate, and
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formulate the channel estimation problem as a standard compressed sensing problem [7, 10]. In practice, however, such strictly sparse assumption on the CIR does not hold true in non-sample-spaced multiple channels. With non-sample-spaced time delay, the path energy will leak to all the other taps and the discrete-time CIR becomes non-strictly sparse in the classical sense [11]. Therefore, the conventional CS-based channel estimators may perform poorly or even completely fail this kind of channels. The article considers the non-strictly sparse discrete-time CIR resulted from the multipath energy leakage in non-sample-spaced multipath channels. In Ref. [12], Berger et al. proved that the baseband channel is compressible and sparsely represented in some dimension. The key ingredient builds a more realistic sparse model that goes beyond the simple sparse nature of physical channel. To approach the channel estimation problem, we propose a new method in following: 1) We apply an over-complete dictionary with a finer delay grid to construct a sparse representation of the non-sample-spaced multipath channels, so the sparse nature of the multipath channel could be revealed. 2) In order to alleviate the high complexity caused by the over-complete dictionary, we formulate the channel estimation as the model-based CS problem [13]. 3) According to the model-based CS theory, we apply a modified model-based CoSaMP algorithm to reconstruct the discrete-time CIR. 4) We perform simulation to investigate the performance of the proposed channel estimation method. The simulation demonstrates the effectiveness of the proposed method in non-sample-spaced multipath channels. Compared with standard CS-based channel estimation technique, such as orthogonal matching pursuit (OMP) based channel estimation [10], the proposed here is of good performance in terms of BER and MSE. The rest is organized as follows. Sect. 2 shows the channel model and OFDM system model. Sect. 3 presents the proposed CS-based method for non-sample-spaced multipath channels estimation. In Sect. 4, performance of the proposed method is shown. Finally, the conclusions are drawn in Sect. 5.
2 Channel and system model 2.1
Channel model
Most of pilot-aided channel estimations come out of the
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assumption that the wireless channels are rich multipath. In contrast, a physical channel often exhibits sparseness in CSI. That is, the number of significant propagation paths is very small comparing to the length of the channel. This phenomenon is more prevalent in communication systems operating on wideband and ultra-wideband modes. In this section, we use the popular tapped-delay line model to describe the wireless channel. The CIR in time domain can be given by [14] L −1
h(t ,τ ) = ∑ α l (t )δ(τ − τ l Ts )
(1)
l =0
where α l is the complex gain of the lth path and τ l is the corresponding time delay normalized to the sampling period Ts , L is the total number of the propagation paths. Usually, α l is modeled as complex Gaussian processes with Jake’s power spectrum and all the delay paths are uncorrelated to each other. According to Ref. [11], the continuous time-domain channel can be expressed as the following discrete form after sampling. 1 L −1 − j π ( n + ( N −1)τ l ) sin(πτ l ) h( n) = ∑ α l e N (2) π(τ l − n) N l =0 sin N In Eq. (2), we see that each δ function in the original impulse response will be convoluted with sinc(i) function. For sample-spaced channels, τ l is an integer, and all the energy from the path α l can be mapped to the taps of h(n) . Eq. (2) is rewritten as L −1
h(n) = ∑ α l δ(n − τ l )
(3)
l =0
The sparse structure of wireless channel reflects that the discrete-time CIR h = [h0 , h1 ,..., hLmax −1 ]T contains a small amount of nonzero elements, where Lmax is the length of the channel ( Lmax = Tmax Ts , Tmax is the maximum delay spread). Fig. 1 is the sample-spaced multipath channel model. It shows that the channel power is mainly concentrated on a few significant taps, while the other taps whose energy levels are very weak or approximate zero, can be neglected. On the other hand, however, if the time delays between different paths are not sample period spaced, that means that τ l is not an integer, and the paths energy will possibly leak to all the other taps. Fig. 2 is the non-sample-spaced multipath channel model. In this case, the number of nonzero equivalent baseband taps is much larger than the actual propagation paths. The discrete-time
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Chen Baohao, et al. / Compressed sensing based channel estimation used in…
CIR h is no longer strictly sparse in the classical sense.
FN00 FN0( Lmax −1) ⋯ 1 F= ⋮ ⋮ N ( N −1)0 ( N −1)( Lmax −1) F ⋯ F N N where FNnl = e
−j
2π nl N
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( 5)
.
At the receiver, the received signals and the pilot tones are the given information, proper algorithms can be selected to obtain the estimation of CSI. Under CS framework, the OMP [15] and the algorithm of basis pursuit (BP) [16] is widely used in channel estimation.
3 Channel estimation Fig. 1
Sample-spaced multipath channel
A new CS-based method was proposed for non-sample-spaced multipath channel estimations. Firstly, a representation was adopted to describe the sparse nature of the multipath channel in Sect. 3.1. Then, why and how to formulate the channel estimation as the model-based CS problem was explained in Sect. 3.2. Finally, a modified model-based CoSaMP algorithm was applied for CIR reconstruction in Sect. 3.3. 3.1 Fig. 2
2.2
Non-sample-spaced multipath channel
System model
We consider an OFDM system that consists of N subcarriers, in which, N p subcarriers are reserved as pilot tones with positions represented by k1 , k2 ,..., k Np .
Sparse channel model
In non-sample-spaced multipath channels, the discrete-time CIR is no longer strictly sparse in the classical sense. To formulate the channel estimation as a compressed sensing problem, we need to construct a more realistic sparse model that goes beyond the simple sparse nature of physical channel. We use over-complete dictionary method with a finer delay grid to construct a sparse channel representation. Suppose the delay grid τ i
The duration of an OFDM symbol including cyclic-prefix (CP) is ( N + N cp ) . The CP length Lcp is assumed to be
takes the value in a large equi-spaced grid spans [0, Tmax ]
larger than the maximum delay spread Lmax to avoid the
as,
inter
symbol
interference
X n (1),..., X n ( N − 1)]
T
be
(ISI). the
Let nth
X n = [ X n (0), modulated
complex-valued OFDM symbol for transmission. After passing through the wireless channel depicted in Eq. (1), removing CP and taking fast Fourier transform (FFT), the received signal can be expressed as (4) Yn = X n H + Z n = X n Fh + Z n In Eq. (4), Yn = [Yn (0), Yn (1),..., Yn ( N − 1)]T . Z n denotes zero-mean complex additive white Gaussian noise with variance σ z2 , F is a partial FFT matrix which only includes the first Lmax columns of the standard N × N FFT matrix.
Ts 2Ts , ,..., Tmax (6) λ λ which will lead to a dictionary of λ (Tmax Ts ) = λ Lmax
τ i ∈ 0,
entries. For general cases, the delay grid will degenerate into the regular baseband-sampling-interval-spaced model with λ = 1 . But the purpose is to increase the delay resolution, so we only consider the situation of λ > 1 . In Sect. 2, we got that the sampled CIR can be written as Eq. (2). We try to rewrite the CIR as L −1
h(n) = ∑ α l w(τ l )
(7)
l =0
− j π N ( n + ( N −1)τ l ) where w(τ l ) = (1 N ) e ( ) {sin(πτ l ) [sin(π(τ l − n) / N )
sin( (
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n) / N ) ]} . Combining Eq. (6) and Eq. (7), the CIR h(n) is transformed into a vector expression as (8) h = Wα where α is a λ Lmax × 1 vector with ith entry α i = α l if
τ i = τ l and α i = 0 otherwise. Hence α by definition is a L-sparse vector. And W is a Lmax × λ Lmax matrix defined by T 2T W = w(0), w s , w s ,..., w(Tmax ) λ λ
(9)
Combining Eq. (4) and Eq. (8), the received signal is expressed as (10) Yn = X n H + Z n = X n FW α + Z n In frequency domain, the pilot symbols are inserted in different subcarriers. a selection matrix S n was defined. The elements on the pilot positions come from an N dimensional vector. Then, the received signal at the pilot positions can be written as ( 11 ) Yn ,p = X n ,p Fn ,pW α + Z n ,p where
Yn ,p = S nYn ,
X n ,p = S n X n Sn T ,
Fn ,p =S n F ,
Z n ,p =S n Z n . It should be pointed out that the signal model described by Eq. (11) is a compressed sensing formulation, where α is the unknown sparse vector and the measurement matrix is given by Φ = X n ,p Fn,pW (12) The sparse channel model has been proposed above. In this channel model, an over-complete dictionary with a finer delay grid is applied to construct a sparse representation of the non-sample-spaced multipath channels, so the sparse nature of the multipath channel could be revealed. 3.2
Model-based CS framework
With the signal model described by Eq. (11), we can apply the compressed sensing reconstruction algorithm such as OMP algorithm to reconstruct the sparse vector α . However, what should not be neglected is that the size of the dictionary grows as the delay resolution increases. Since the computational complexity substantially increases with the dictionary size, directly using the conventional reconstruction algorithm leads to high complexity, which is not suitable for practical application. To alleviate the problem of high complexity, we apply the method of
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model-based CS [13]. The basic idea of the model-based CS is to restrict the vector α to a subspace of all L-sparse signals. To be specific, let α Ω represent the entries of α to the set of index Ω ⊆ {0,1, 2,..., λ Lmax } , and let Ω C denote the complement of the set Ω . The index corresponding to the nonzero elements of the vector α defines a set called support set of α and it is given by (13) Ω = {ϕ1 , ϕ2 ,..., ϕ L } where ϕi ∈ {0,1, 2,..., λ Lmax } . The elements in the support set satisfies ϕi − ϕ j ≥λ ; ∀i, j
(14)
We collect all such support sets into a collection Ω1 , Ω2 ,..., ΩmL
{
}
where the total L mL = λ L max , L indexed by Lmax
(15)
number of allowed support set is Lmax denotes the binomial coefficient L and L. Specially, the valid support set of
α shall be one of elements in Eq. (15). That is, α shall satisfy α Ω ∈ ℝ L and α Ω = 0 with Ω satisfies C
Eq. (13). The sparse channel estimation problem in the model-based CS framework has been described above. In the following, we will analyze the computational complexity and discuss how to determine the parameter λ. To the general knowledge, the L-sparse signal vector α lives in a L dimensional subspace and the set of all λL L-sparse signal is a union of max subspaces of L ℝ λ Lmax . Directly applying the conventional reconstruction algorithm for the L-sparse signal estimation means we need to search an optimal result in a range with the size of λ Lmax . However, in the model-based CS framework, L
L the number of allowed support set is mL = λ L max . It is L L λL easy to prove that λ L max < max when λ > 1 . L L That is to say, the model-based CS method reduces the computational complexity. For the proposed sparse channel model, a finer delay grid is applied by introduced the parameter λ which
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should be given in advance. In CS theory, in order to recover a good estimation of the sparse signal α from the compressive measurement, the measurement matrix Φ should satisfy the restricted isometry property (RIP) in Ref. [17]. About the RIP, a more general description is N p = O ( L lb(λ Lmax / L)) . That means, λ is a function of
N p , L and Lmax . On the other hand, Berger et al. [12] has proved that a larger λ brings better reconstruction performance. However, it will leads to higher complexity in the mean time. So, it is a trade-off between the performance and complexity to determine the parameter λ. 3.3
CIR reconstruction
We adapt the model-based CoSaMP algorithm [13] to reconstruct the CIR. The used approach is to search the best possible estimation of support set in the subspace of ℝ L , then the support set gives a proper estimate of the sparse vector αˆ . In the initialization, the algorithm sets tentative values for αˆ . Then the values of αˆ will be updated by estimating the support set Ω iteratively. The support set estimation is performed by running D ( e ,Φ , L ) , it refers to the successive interference cancellation scheme [18] widely used in code division multiple access (CDMA) system. The signal residual estimate e is obtained by matched filtering each column of the measurement matrix Φ , e = Φ H r . Each peak of e points toward the approximated location of a large component of α . To improve the approximation accuracy, At each time we identify a large component in α , its contribution is subtracted to the signal residual e , then the appeared interference to other component will be mostly canceled. In order to realize the idea, the method F (Ω ) is applied
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λ Lmax } . Step 3
ϕ ← arg max{ei } . {select the peak of e } i∈I
Ω ← Ω ∪ {ϕ} . {merge supports} Step 5 I ← I \ F (Ω ) , F (Ω ) selects the set of support
Step 4
index in I that are conflict with the ones already in Ω . b = (Φ HΦ ) |ϕ . {select atom correspond to Step 6 support index } Step 7
(
e ← e − eTb b
2 2
) b . {update signal residual}
Step 8 If Ω > L , go to Step 9, otherwise go to Step 3. Step 9 Output: the estimate support set Ω . The support set Ω is a collection that highlights the locations of nonzero elements. The columns of Φ that correspond to these locations were used to form the submatrix ΦΩ , its pseudo-inverse is used to obtain an updated estimation of αˆ . Finally, the illustrated steps are iterated until a halting criterion is fulfilled. In the proposed method, the priori information of the propagation paths number is used to control the iteration. Once we get the estimation of αˆ , we may reconstruct the discrete-time CIR using Eq. (8). The proposed model-based CoSaMP algorithm for channel estimation is summarized as follows: Algorithm 2 Channel estimation with model-based CoSaMP algorithm Step 1 Input: the received signal Yn ,p , measurement matrix Φ , multipath number L. Step 2 Initialization: αˆ 0 = 0 , t = 0 , r = Yn ,p . Step 3 Step 4 Step 5
t = t +1 . e ← Φ H r . {form signal residual estimation} Ω ← D ( e,Φ , L ) . {select the support set}
Step 6
T ← Ω ∪ sup(αˆt −1 ) . {merge supports}
Step 7
b |T ← ΦT†Yn ,p , b |T C ← 0 . {form signal estim-
to select those index that are conflict with the index exist in Ω . Then the complement of F (Ω ) is the feasible one
ation} Step 8
supp ← D ( e, I , L ) , I is an identity matrix.
can be added to Ω in the next iteration. Details of the support set estimation algorithm D ( e ,Φ , L ) were given as follows.
{prune residual estimate} Step 9 αˆ |supp ← b |supp , αˆ |suppC ← 0 . {update signal
Algorithm 1 Support set estimation algorithm Step 1 Input: the measurement matrix Φ , signal residual estimate e , multipath number L, signal dimension λ Lmax . Step 2
Initialization: Ω = ∅ , b = 0 , I = {0,1, 2,...,
estimate} Step 10
r ← Yn ,p − Φαˆ t .{update measurement residual}
Step 11
If Ω > L , go to Step 12, otherwise go to
Step 3. Step 12 Output: the sparse vector αˆ ← αˆ t .
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The Journal of China Universities of Posts and Telecommunications
4 Simulation and analysis The proposed method was performed to estimate the CSI in non-sample-spaced multipath channels. Simulation to investigate the performance of the proposed channel estimation method was carried out. It concentrated on comparison between the proposed method and two other channel estimation methods. They are the discrete Fourier transformation (DFT) method (a representative linear channel estimation method) [11] and the OMP-based method (a conventional standard CS-based channel estimation method) [10]. In order to evaluate the performance, we adopt the MSE and BER to quantize the channel estimation errors. In the simulation, an OFDM system with N = 1 024 subcarriers, N cp = 72 CP length and 16 quadrature
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reduced. Compared with the simple adoption of OMP-based method, the proposed method with a finer delay grid enjoys superior performance in over all SNR ranges. The reason for this gain is because the proposed method captures the larger components of the discrete-time CIR and identifies the smaller components. In addition, the advantage that less pilot tones requirement is reflected in the proposed method. The worthy observation is that, by using only 4.5% pilot tones, the performance of the proposed method achieves is comparable to what the DFT method using 9.8% pilot tones do.
amplitude modulation 16 QAM modulation was employed. The multipath channel has L = 6 underlying physical multipath components. Each excess delay value τ i is uniformly chosen from the set given in Eq. (6) and the path gain α l is a complex circularly symmetric Gaussian random variable. We set N p = 46 pilot tones in each OFDM symbol and their locations in frequency domain are randomly determined. As the channel maximum delay spread is unknown to the estimator, we assumed that the channel length Lmax was same as the CP length. According to the RIP condition from CS theory, the delay grid parameter was chosen as λ = 3 . Moreover, we assumed that the channel state did not change during an OFDM symbol, each of OFDM symbols was generated independently. Detail of the simulation parameters was summarized in Table 1. Table 1
Fig. 3
MSE performangce comparisons
Fig. 4 shows the BER performance of the three kinds of channel estimation methods. The BER performance obtained with the ideal CSI was included in the figure for reference. It can be seen that the BER performance shows a similar trend with the MSE performance. Moreover, the proposed method performs quite closely to the situation with ideal CSI.
System parameters
Parameter Number of subcarriers N Number of pilots N p Pilots position Number of propagation paths L Channel length Lmax
Value 1 024 46 Randomly 6
Modulation
72 16QAM
Channel estimation method SNR/dB
DFT/OMP/Proposed method 0~20
Fig. 3 shows the MSE performance of the DFT method, the OMP-based method and the proposed method with the signal to noise ratio (SNR) ranging from 0 dB to 20 dB. As shown in Fig. 3, with increase in SNR, the MSE of the three kinds of channel estimation methods are all gradually
Fig. 4 BER performance comparisons
5 Conclusions The authors proposed a CS-based channel estimation method for OFDM system in non-sample-spaced multipath
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channels in which an over-complete dictionary with a finer delay grid was applied to construct a sparse representation of the non-sample-spaced multipath channels. With the proposed sparse channel model, the channel estimation was formulated as the model-based CS problem and a modified model-based CoSaMP algorithm was applied to reconstruct the discrete-time CIR. Simulation demonstrates that the proposed outperforms the conventional standard CS-based channel estimation evidently. Meanwhile, the performance of the proposed method is as good as the liner estimation but with less pilot tones. Acknowledgements This work was supported by the National Science and Technology Major Project (2012ZX03001039-002).
References 1. Edfors O, Sandell M, van de Beek J J, et al. OFDM channel estimation by singular value decomposition. IEEE Transactions on Communications, 1998, 46(7): 931−939 2. Zhu M, Awoseyila A B, Evans B G. Low-complexity time-domain channel estimation for OFDM systems. Electronics Letters, 2011, 47(1): 60−62 3. Donoho D L. Compressed sensing. IEEE Transactions on Information Theory, 2006, 52(4): 1289−1306 4. Candes E J, Tao T. Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 2006, 52(12): 5406−5425 5. Muqaibel A H, Alkhodary M T. Practical application of compressive sensing to ultra-wideband channels. IET Communications, 2012, 6(16): 2534−2542 6. Berger C R, Wang Z H, Huang J Z, et al. Application of compressive sensing to sparse channel estimation. IEEE Communications Magazine,
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2010, 48(11): 164−174 7. Cheng P, Gui L, Rui Y, et al. Compressed sensing based channel estimation for two-way relay networks. IEEE Wireless Communications Letters, 2012, 1(3): 201−204 8. Hu D, Wang X D, He L H. A new sparse channel estimation and tracking method for time-varying OFDM systems. IEEE Transactions on Vehicular Technology, 2013, 62(9): 4648−4653 9. Yang B, Letaief K B, Cheng R S, et al. Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling. IEEE Transactions on Communications, 2001, 49(3): 467−479 10. Xue J, Zeng W J, Cheng E. A fast algorithm for sparse channel estimation via orthogonal matching pursuit. Proceedings of the 73rd Vehicular Technology Conference (VTC-Spring’11), May 15−18, 2011, Budapest, Hungary. Piscataway, NJ, USA: IEEE, 2011: 5p 11. Wang Y, Li L, Zhang P, et al. Channel estimation for OFDM systems in non-sample-spaced multipath channels. Electronics Letters, 2009, 45(1): 66−68 12. Berger C R, Zhou S L, Preisig J C, et al. Sparse channel estimation for multicarrier underwater acoustic communication: from subspace methods to compressed sensing. IEEE Transactions on Signal Processing, 2010, 58(3): 1708−1721 13. Baraniuk R G, Cevher V, Duarte M F, et al. Model-based compressive sensing. IEEE Transactions on Information Theory, 2010, 56(4): 1982−2001 14. Pan C Y, Dai L L. Time domain synchronous OFDM based on compressive sensing: a new perspective. Proceedings of the IEEE Global Communications Conference (GLOBECOM’12), Dec 3−7, 2012, Anaheim, CA, USA. Piscataway, NJ, USA: IEEE, 2012: 4880−4885 15. Tropp J A, Gilbert A C. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on Information Theory, 2007, 53(12): 4655−4666 16. Bruckstein A M, Donoho D L, Elad M. From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Review, 2009, 51(1): 34−81 17. Candes E J, Wakin M B. An introduction to compressive sampling. IEEE Signal Processing Magazine, 2008, 25(2): 21−30 18. Patel P, Holtzman J. Analysis of a simple successive interference cancellation scheme in a DS/CDMA system. IEEE Journal on Selected Areas in Communications, 1994, 12(5): 796−807
(Editor: Wang Xuying)
http://jcupt.xsw.bupt.cn
Compressed sensing based channel estimation used in non-sample-spaced multipath channels of OFDM system Chen Baohao (
), Cui Qimei, Yang Fan
National Engineering Laboratory for Mobile Network Security, Beijing University of Posts and Telecommunications, Beijing 100876, China
Abstract By virtue of an increase in spectral efficiency by reducing the transmitted pilot tones, the compressed sensing (CS) has been widely applied to pilot-aided sparse channel estimation in orthogonal frequency division multiplexing (OFDM) systems. The researches usually assume that the channel is strictly sparse and formulate the channel estimation as a standard compressed sensing problem. However, such strictly sparse assumption does not hold true in non-sample-spaced multiple channels. The authors in this article proposed a new method of compressed sensing based channel estimation in which an over-complete dictionary with a finer delay grid is applied to construct a sparse representation of the non-sample-spaced multipath channels. With the proposed, the channel estimation was formulated as the model-based CS problem and a modified model-based compressed sampling matching pursuit (CoSaMP) algorithm was applied to reconstruct the discrete-time channel impulse response (CIR). Simulation indicates that the new method proposed here outperforms the traditional standard CS-based methods in terms of mean square error (MSE) and bit error rate (BER). Keywords channel estimation, model-based compressed sensing, non-sample-spaced multipath channels
1 Introduction OFDM has been widely employed in current and future wireless communication systems. Typically, the OFDM system employs coherent detection for which channel state information (CSI) is crucial. In OFDM system, the CSI is usually obtained when using pilot tones and many pilot-aided channel estimation methods have been studied. Generally, the least squares (LS) estimation is the simplest channel estimation based on parallel Gaussian channel model in Ref. [1]. The linear minimum mean-squared error (LMMSE) method is the optimum solution for interpolation [2]. However, The practice that the wireless channels are rich multipath and use a large number of pilot tones to obtain accurate CSI, will possibly lead to low spectrum efficiency. The compressed sensing is an emerging sampling theory proposed by Donoho et al. in 2006 [3–4]. Accordingly, if a signal has a spare representation in a certain space, the Received date: 04-05-2014 Corresponding author: Chen Baohao, E-mail: [email protected] DOI: 10.1016/S1005-8885(15)60636-7
signal can be sampled at a rate significantly lower than Nyquist rate, it can be reconstructed with high probability by optimization techniques. Study indicates that the wireless channels tend to exhibit a sparse multipath structure [5] that means that the delay spread of the channel could be very large while the number of significant propagation paths is normally very small. Based on the sparse structure, researchers introduced compressed sensing into channel estimation and obtained extensive achievements [6–8]. The sparse signals efficiently is reconstructed by CS from a small number of measurements to reduce the number of pilot tones required to meet the desired system performance requirement. Usually, the radio channel in a wireless communication system is characterized by multitudinous propagation paths, each is parameterized by different delays and complex amplitudes, resulting in a sparse multipath channel model [9]. Most CS-based channel estimators assume that the channel is strictly sparse in the equivalent discrete-time baseband representation, i.e., the tapped-delay line model with the delay grid spaced at a baseband sampling rate, and
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The Journal of China Universities of Posts and Telecommunications
formulate the channel estimation problem as a standard compressed sensing problem [7, 10]. In practice, however, such strictly sparse assumption on the CIR does not hold true in non-sample-spaced multiple channels. With non-sample-spaced time delay, the path energy will leak to all the other taps and the discrete-time CIR becomes non-strictly sparse in the classical sense [11]. Therefore, the conventional CS-based channel estimators may perform poorly or even completely fail this kind of channels. The article considers the non-strictly sparse discrete-time CIR resulted from the multipath energy leakage in non-sample-spaced multipath channels. In Ref. [12], Berger et al. proved that the baseband channel is compressible and sparsely represented in some dimension. The key ingredient builds a more realistic sparse model that goes beyond the simple sparse nature of physical channel. To approach the channel estimation problem, we propose a new method in following: 1) We apply an over-complete dictionary with a finer delay grid to construct a sparse representation of the non-sample-spaced multipath channels, so the sparse nature of the multipath channel could be revealed. 2) In order to alleviate the high complexity caused by the over-complete dictionary, we formulate the channel estimation as the model-based CS problem [13]. 3) According to the model-based CS theory, we apply a modified model-based CoSaMP algorithm to reconstruct the discrete-time CIR. 4) We perform simulation to investigate the performance of the proposed channel estimation method. The simulation demonstrates the effectiveness of the proposed method in non-sample-spaced multipath channels. Compared with standard CS-based channel estimation technique, such as orthogonal matching pursuit (OMP) based channel estimation [10], the proposed here is of good performance in terms of BER and MSE. The rest is organized as follows. Sect. 2 shows the channel model and OFDM system model. Sect. 3 presents the proposed CS-based method for non-sample-spaced multipath channels estimation. In Sect. 4, performance of the proposed method is shown. Finally, the conclusions are drawn in Sect. 5.
2 Channel and system model 2.1
Channel model
Most of pilot-aided channel estimations come out of the
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assumption that the wireless channels are rich multipath. In contrast, a physical channel often exhibits sparseness in CSI. That is, the number of significant propagation paths is very small comparing to the length of the channel. This phenomenon is more prevalent in communication systems operating on wideband and ultra-wideband modes. In this section, we use the popular tapped-delay line model to describe the wireless channel. The CIR in time domain can be given by [14] L −1
h(t ,τ ) = ∑ α l (t )δ(τ − τ l Ts )
(1)
l =0
where α l is the complex gain of the lth path and τ l is the corresponding time delay normalized to the sampling period Ts , L is the total number of the propagation paths. Usually, α l is modeled as complex Gaussian processes with Jake’s power spectrum and all the delay paths are uncorrelated to each other. According to Ref. [11], the continuous time-domain channel can be expressed as the following discrete form after sampling. 1 L −1 − j π ( n + ( N −1)τ l ) sin(πτ l ) h( n) = ∑ α l e N (2) π(τ l − n) N l =0 sin N In Eq. (2), we see that each δ function in the original impulse response will be convoluted with sinc(i) function. For sample-spaced channels, τ l is an integer, and all the energy from the path α l can be mapped to the taps of h(n) . Eq. (2) is rewritten as L −1
h(n) = ∑ α l δ(n − τ l )
(3)
l =0
The sparse structure of wireless channel reflects that the discrete-time CIR h = [h0 , h1 ,..., hLmax −1 ]T contains a small amount of nonzero elements, where Lmax is the length of the channel ( Lmax = Tmax Ts , Tmax is the maximum delay spread). Fig. 1 is the sample-spaced multipath channel model. It shows that the channel power is mainly concentrated on a few significant taps, while the other taps whose energy levels are very weak or approximate zero, can be neglected. On the other hand, however, if the time delays between different paths are not sample period spaced, that means that τ l is not an integer, and the paths energy will possibly leak to all the other taps. Fig. 2 is the non-sample-spaced multipath channel model. In this case, the number of nonzero equivalent baseband taps is much larger than the actual propagation paths. The discrete-time
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CIR h is no longer strictly sparse in the classical sense.
FN00 FN0( Lmax −1) ⋯ 1 F= ⋮ ⋮ N ( N −1)0 ( N −1)( Lmax −1) F ⋯ F N N where FNnl = e
−j
2π nl N
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( 5)
.
At the receiver, the received signals and the pilot tones are the given information, proper algorithms can be selected to obtain the estimation of CSI. Under CS framework, the OMP [15] and the algorithm of basis pursuit (BP) [16] is widely used in channel estimation.
3 Channel estimation Fig. 1
Sample-spaced multipath channel
A new CS-based method was proposed for non-sample-spaced multipath channel estimations. Firstly, a representation was adopted to describe the sparse nature of the multipath channel in Sect. 3.1. Then, why and how to formulate the channel estimation as the model-based CS problem was explained in Sect. 3.2. Finally, a modified model-based CoSaMP algorithm was applied for CIR reconstruction in Sect. 3.3. 3.1 Fig. 2
2.2
Non-sample-spaced multipath channel
System model
We consider an OFDM system that consists of N subcarriers, in which, N p subcarriers are reserved as pilot tones with positions represented by k1 , k2 ,..., k Np .
Sparse channel model
In non-sample-spaced multipath channels, the discrete-time CIR is no longer strictly sparse in the classical sense. To formulate the channel estimation as a compressed sensing problem, we need to construct a more realistic sparse model that goes beyond the simple sparse nature of physical channel. We use over-complete dictionary method with a finer delay grid to construct a sparse channel representation. Suppose the delay grid τ i
The duration of an OFDM symbol including cyclic-prefix (CP) is ( N + N cp ) . The CP length Lcp is assumed to be
takes the value in a large equi-spaced grid spans [0, Tmax ]
larger than the maximum delay spread Lmax to avoid the
as,
inter
symbol
interference
X n (1),..., X n ( N − 1)]
T
be
(ISI). the
Let nth
X n = [ X n (0), modulated
complex-valued OFDM symbol for transmission. After passing through the wireless channel depicted in Eq. (1), removing CP and taking fast Fourier transform (FFT), the received signal can be expressed as (4) Yn = X n H + Z n = X n Fh + Z n In Eq. (4), Yn = [Yn (0), Yn (1),..., Yn ( N − 1)]T . Z n denotes zero-mean complex additive white Gaussian noise with variance σ z2 , F is a partial FFT matrix which only includes the first Lmax columns of the standard N × N FFT matrix.
Ts 2Ts , ,..., Tmax (6) λ λ which will lead to a dictionary of λ (Tmax Ts ) = λ Lmax
τ i ∈ 0,
entries. For general cases, the delay grid will degenerate into the regular baseband-sampling-interval-spaced model with λ = 1 . But the purpose is to increase the delay resolution, so we only consider the situation of λ > 1 . In Sect. 2, we got that the sampled CIR can be written as Eq. (2). We try to rewrite the CIR as L −1
h(n) = ∑ α l w(τ l )
(7)
l =0
− j π N ( n + ( N −1)τ l ) where w(τ l ) = (1 N ) e ( ) {sin(πτ l ) [sin(π(τ l − n) / N )
sin( (
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The Journal of China Universities of Posts and Telecommunications
n) / N ) ]} . Combining Eq. (6) and Eq. (7), the CIR h(n) is transformed into a vector expression as (8) h = Wα where α is a λ Lmax × 1 vector with ith entry α i = α l if
τ i = τ l and α i = 0 otherwise. Hence α by definition is a L-sparse vector. And W is a Lmax × λ Lmax matrix defined by T 2T W = w(0), w s , w s ,..., w(Tmax ) λ λ
(9)
Combining Eq. (4) and Eq. (8), the received signal is expressed as (10) Yn = X n H + Z n = X n FW α + Z n In frequency domain, the pilot symbols are inserted in different subcarriers. a selection matrix S n was defined. The elements on the pilot positions come from an N dimensional vector. Then, the received signal at the pilot positions can be written as ( 11 ) Yn ,p = X n ,p Fn ,pW α + Z n ,p where
Yn ,p = S nYn ,
X n ,p = S n X n Sn T ,
Fn ,p =S n F ,
Z n ,p =S n Z n . It should be pointed out that the signal model described by Eq. (11) is a compressed sensing formulation, where α is the unknown sparse vector and the measurement matrix is given by Φ = X n ,p Fn,pW (12) The sparse channel model has been proposed above. In this channel model, an over-complete dictionary with a finer delay grid is applied to construct a sparse representation of the non-sample-spaced multipath channels, so the sparse nature of the multipath channel could be revealed. 3.2
Model-based CS framework
With the signal model described by Eq. (11), we can apply the compressed sensing reconstruction algorithm such as OMP algorithm to reconstruct the sparse vector α . However, what should not be neglected is that the size of the dictionary grows as the delay resolution increases. Since the computational complexity substantially increases with the dictionary size, directly using the conventional reconstruction algorithm leads to high complexity, which is not suitable for practical application. To alleviate the problem of high complexity, we apply the method of
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model-based CS [13]. The basic idea of the model-based CS is to restrict the vector α to a subspace of all L-sparse signals. To be specific, let α Ω represent the entries of α to the set of index Ω ⊆ {0,1, 2,..., λ Lmax } , and let Ω C denote the complement of the set Ω . The index corresponding to the nonzero elements of the vector α defines a set called support set of α and it is given by (13) Ω = {ϕ1 , ϕ2 ,..., ϕ L } where ϕi ∈ {0,1, 2,..., λ Lmax } . The elements in the support set satisfies ϕi − ϕ j ≥λ ; ∀i, j
(14)
We collect all such support sets into a collection Ω1 , Ω2 ,..., ΩmL
{
}
where the total L mL = λ L max , L indexed by Lmax
(15)
number of allowed support set is Lmax denotes the binomial coefficient L and L. Specially, the valid support set of
α shall be one of elements in Eq. (15). That is, α shall satisfy α Ω ∈ ℝ L and α Ω = 0 with Ω satisfies C
Eq. (13). The sparse channel estimation problem in the model-based CS framework has been described above. In the following, we will analyze the computational complexity and discuss how to determine the parameter λ. To the general knowledge, the L-sparse signal vector α lives in a L dimensional subspace and the set of all λL L-sparse signal is a union of max subspaces of L ℝ λ Lmax . Directly applying the conventional reconstruction algorithm for the L-sparse signal estimation means we need to search an optimal result in a range with the size of λ Lmax . However, in the model-based CS framework, L
L the number of allowed support set is mL = λ L max . It is L L λL easy to prove that λ L max < max when λ > 1 . L L That is to say, the model-based CS method reduces the computational complexity. For the proposed sparse channel model, a finer delay grid is applied by introduced the parameter λ which
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should be given in advance. In CS theory, in order to recover a good estimation of the sparse signal α from the compressive measurement, the measurement matrix Φ should satisfy the restricted isometry property (RIP) in Ref. [17]. About the RIP, a more general description is N p = O ( L lb(λ Lmax / L)) . That means, λ is a function of
N p , L and Lmax . On the other hand, Berger et al. [12] has proved that a larger λ brings better reconstruction performance. However, it will leads to higher complexity in the mean time. So, it is a trade-off between the performance and complexity to determine the parameter λ. 3.3
CIR reconstruction
We adapt the model-based CoSaMP algorithm [13] to reconstruct the CIR. The used approach is to search the best possible estimation of support set in the subspace of ℝ L , then the support set gives a proper estimate of the sparse vector αˆ . In the initialization, the algorithm sets tentative values for αˆ . Then the values of αˆ will be updated by estimating the support set Ω iteratively. The support set estimation is performed by running D ( e ,Φ , L ) , it refers to the successive interference cancellation scheme [18] widely used in code division multiple access (CDMA) system. The signal residual estimate e is obtained by matched filtering each column of the measurement matrix Φ , e = Φ H r . Each peak of e points toward the approximated location of a large component of α . To improve the approximation accuracy, At each time we identify a large component in α , its contribution is subtracted to the signal residual e , then the appeared interference to other component will be mostly canceled. In order to realize the idea, the method F (Ω ) is applied
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λ Lmax } . Step 3
ϕ ← arg max{ei } . {select the peak of e } i∈I
Ω ← Ω ∪ {ϕ} . {merge supports} Step 5 I ← I \ F (Ω ) , F (Ω ) selects the set of support
Step 4
index in I that are conflict with the ones already in Ω . b = (Φ HΦ ) |ϕ . {select atom correspond to Step 6 support index } Step 7
(
e ← e − eTb b
2 2
) b . {update signal residual}
Step 8 If Ω > L , go to Step 9, otherwise go to Step 3. Step 9 Output: the estimate support set Ω . The support set Ω is a collection that highlights the locations of nonzero elements. The columns of Φ that correspond to these locations were used to form the submatrix ΦΩ , its pseudo-inverse is used to obtain an updated estimation of αˆ . Finally, the illustrated steps are iterated until a halting criterion is fulfilled. In the proposed method, the priori information of the propagation paths number is used to control the iteration. Once we get the estimation of αˆ , we may reconstruct the discrete-time CIR using Eq. (8). The proposed model-based CoSaMP algorithm for channel estimation is summarized as follows: Algorithm 2 Channel estimation with model-based CoSaMP algorithm Step 1 Input: the received signal Yn ,p , measurement matrix Φ , multipath number L. Step 2 Initialization: αˆ 0 = 0 , t = 0 , r = Yn ,p . Step 3 Step 4 Step 5
t = t +1 . e ← Φ H r . {form signal residual estimation} Ω ← D ( e,Φ , L ) . {select the support set}
Step 6
T ← Ω ∪ sup(αˆt −1 ) . {merge supports}
Step 7
b |T ← ΦT†Yn ,p , b |T C ← 0 . {form signal estim-
to select those index that are conflict with the index exist in Ω . Then the complement of F (Ω ) is the feasible one
ation} Step 8
supp ← D ( e, I , L ) , I is an identity matrix.
can be added to Ω in the next iteration. Details of the support set estimation algorithm D ( e ,Φ , L ) were given as follows.
{prune residual estimate} Step 9 αˆ |supp ← b |supp , αˆ |suppC ← 0 . {update signal
Algorithm 1 Support set estimation algorithm Step 1 Input: the measurement matrix Φ , signal residual estimate e , multipath number L, signal dimension λ Lmax . Step 2
Initialization: Ω = ∅ , b = 0 , I = {0,1, 2,...,
estimate} Step 10
r ← Yn ,p − Φαˆ t .{update measurement residual}
Step 11
If Ω > L , go to Step 12, otherwise go to
Step 3. Step 12 Output: the sparse vector αˆ ← αˆ t .
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The Journal of China Universities of Posts and Telecommunications
4 Simulation and analysis The proposed method was performed to estimate the CSI in non-sample-spaced multipath channels. Simulation to investigate the performance of the proposed channel estimation method was carried out. It concentrated on comparison between the proposed method and two other channel estimation methods. They are the discrete Fourier transformation (DFT) method (a representative linear channel estimation method) [11] and the OMP-based method (a conventional standard CS-based channel estimation method) [10]. In order to evaluate the performance, we adopt the MSE and BER to quantize the channel estimation errors. In the simulation, an OFDM system with N = 1 024 subcarriers, N cp = 72 CP length and 16 quadrature
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reduced. Compared with the simple adoption of OMP-based method, the proposed method with a finer delay grid enjoys superior performance in over all SNR ranges. The reason for this gain is because the proposed method captures the larger components of the discrete-time CIR and identifies the smaller components. In addition, the advantage that less pilot tones requirement is reflected in the proposed method. The worthy observation is that, by using only 4.5% pilot tones, the performance of the proposed method achieves is comparable to what the DFT method using 9.8% pilot tones do.
amplitude modulation 16 QAM modulation was employed. The multipath channel has L = 6 underlying physical multipath components. Each excess delay value τ i is uniformly chosen from the set given in Eq. (6) and the path gain α l is a complex circularly symmetric Gaussian random variable. We set N p = 46 pilot tones in each OFDM symbol and their locations in frequency domain are randomly determined. As the channel maximum delay spread is unknown to the estimator, we assumed that the channel length Lmax was same as the CP length. According to the RIP condition from CS theory, the delay grid parameter was chosen as λ = 3 . Moreover, we assumed that the channel state did not change during an OFDM symbol, each of OFDM symbols was generated independently. Detail of the simulation parameters was summarized in Table 1. Table 1
Fig. 3
MSE performangce comparisons
Fig. 4 shows the BER performance of the three kinds of channel estimation methods. The BER performance obtained with the ideal CSI was included in the figure for reference. It can be seen that the BER performance shows a similar trend with the MSE performance. Moreover, the proposed method performs quite closely to the situation with ideal CSI.
System parameters
Parameter Number of subcarriers N Number of pilots N p Pilots position Number of propagation paths L Channel length Lmax
Value 1 024 46 Randomly 6
Modulation
72 16QAM
Channel estimation method SNR/dB
DFT/OMP/Proposed method 0~20
Fig. 3 shows the MSE performance of the DFT method, the OMP-based method and the proposed method with the signal to noise ratio (SNR) ranging from 0 dB to 20 dB. As shown in Fig. 3, with increase in SNR, the MSE of the three kinds of channel estimation methods are all gradually
Fig. 4 BER performance comparisons
5 Conclusions The authors proposed a CS-based channel estimation method for OFDM system in non-sample-spaced multipath
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channels in which an over-complete dictionary with a finer delay grid was applied to construct a sparse representation of the non-sample-spaced multipath channels. With the proposed sparse channel model, the channel estimation was formulated as the model-based CS problem and a modified model-based CoSaMP algorithm was applied to reconstruct the discrete-time CIR. Simulation demonstrates that the proposed outperforms the conventional standard CS-based channel estimation evidently. Meanwhile, the performance of the proposed method is as good as the liner estimation but with less pilot tones. Acknowledgements This work was supported by the National Science and Technology Major Project (2012ZX03001039-002).
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(Editor: Wang Xuying)