Channel estimation with ISFLA based pilot pattern optimization for MIMO OFDM system

Accepted Manuscript Regular paper Channel Estimation with ISFLA based Pilot pattern Optimization for MIMO OFDM System Harjeet Singh, Savina Bansal PII: DOI: Reference:

S1434-8411(17)30713-6 http://dx.doi.org/10.1016/j.aeue.2017.07.024 AEUE 51981

To appear in:

International Journal of Electronics and Communications

Received Date: Revised Date: Accepted Date:

27 March 2017 21 June 2017 18 July 2017

Please cite this article as: H. Singh, S. Bansal, Channel Estimation with ISFLA based Pilot pattern Optimization for MIMO OFDM System, International Journal of Electronics and Communications (2017), doi: http://dx.doi.org/ 10.1016/j.aeue.2017.07.024

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Channel Estimation with ISFLA based Pilot pattern Optimization for MIMO OFDM System 1*

2

Harjeet Singh and Savina Bansal 1*

Ph.D. Research Scholar, Department of Electronics Engineering, I.K G Punjab Technical University, Jalandhar, India

2

Professor, Department of Electronics Engineering, Giani Zail Singh MRSPTU Campus Bathinda, India. [email protected], [email protected]* Abstract In multiple input multiple output orthogonal frequency division multiplexing (MIMO-OFDM) systems, the channel state information should be known by the receiver for obtaining transmitted data. Channel estimation algorithms are used to examine the multipath effects of frequency selective Rayleigh fading channels. In this paper, Compressed Sensing (CS) based channel estimation technique is considered for reconstructing the signal with improved spectral efficiency. It requires transmitting the known pilot data to the receiver for estimating channel information. The optimum pilot patterns are selected through reducing the mutual coherence of measurement matrix. In order to maximize the accuracy of sparse channel estimation and to reduce the computational complexity, an optimization algorithm Improved Shuffled Frog Leaping (ISFL) is proposed. When compared with the traditional estimation methods like least squares (LS), and minimal mean square error (MMSE), 4.7 % of spectral efficiency is increased with ISFLA based channel estimation. Implementation results show that, by using the proposed algorithm, the bit error rate (BER) and Mean Square Error (MER) performance of the system is increased with 1.5 dB and 2 dB respectively. 1

Keywords- OFDM, pilot allocation, Channel estimation, mutual coherence, compressed sensing. 1. Introduction In recent years, compressed sensing has emerged a great role of reconstructing sparse signal from a set of samples in wireless sensor networks. This is based on the concept that, the sparsity of a signal can be destructed to reconstruct it from fewer samples required by the Shannon Nyquist sampling theorem through optimization [1]. Sensing algorithms are implemented to satisfy the metric’s criteria like reliability, complexity and loss in system throughput [2, 3]. Compressed sensing theory reduces the amount of pilot symbols with large number of transmitters in MIMO system [4]. Sparse channel estimation is efficient than conventional algorithms due to the sparse nature of multipath wireless channels. Compressed sampling matching pursuit, orthogonal matching pursuit, and basis pursuit are various sparse recovery algorithms for pilot assisted channel estimation [5, 6]. Comb type and block type are the basic pilot patterns which can be used in channel estimation based on LS, MMSE and Linear MMSE techniques [7-9]. Optimization problem is formulated to minimize a mean square error of upper bound in pilot design [10, 11]. It uses pilot symbol design algorithm and joint pilot placement to increase the accuracy in channel estimation [12-15]. In minimum error rate and maximum sum rate method, different pilot value has been assigned for each transmitter [16-19]. The performances of this estimation, recovery, and data detection and receiver blocks of OFDM relay networks were improved at the presence of CFO and noise [20, 21]. BER of CS based channel estimation is lower in frequency selective fading channel [22, 23]. In traditional channel estimation methods, equidistant pilot placement is commonly used as optimal. It uses random pilot patterns which degrade the performance of channel estimation. In

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case of CS based channel estimation, the pilot placement is based on reducing the mutual coherence of measured matrix. The mutual coherence value obtained for the traditional pilot patterns are high and it increases the bit error rate of the received signal. In this paper, optimized pilot pattern is selected with ISFLA. It improves the channel estimation performance of CS theory by reducing the value of mutual coherence. In this paper a novel channel estimation scheme is proposed to overcome Pilot overhead, and Complexity problems associated with compressed sensing of OFDM system. 2. System model The MIMO-OFDM system is responsible for dividing radio channel into sub channels to enable high speed, reliable broadcasting. It enhances the coverage and capacity against multi path fading. It is commonly used in high speed wireless communication because of its unique properties. High data rate can be achieved through channel state information. Channel estimation techniques are used to obtain this information. The notations used in this paper are described as follows.

 . T , . ,  . i, j , C mn

,

 . H ,

,

. 1 , . 2  and diag (x) denote the matrix transpose,

Complex modules the (i, j ) th element of the matrix, the set of all complex valued m n matrices, Hermitian transpose, l1 norm, l 2 norm, the null set and a diagonal matrix with x on its main diagonal respectively.

3

Input Stream

Pilot Allocation ISFLA

16 QAM Modulation

Add Cyclic prefix

IFFT

Pilot Insertion

Channel

Channel Estimation

Demodulation Output Stream

Remove Cyclic Prefix

FFT

Fig. 1. Proposed System model The proposed MIMO system shown in Fig. 1 consists of N TR transmit antennas; N RE receive antennas, N number of subcarriers and M number of selected subcarriers for pilot allocation. Initially the input data stream is modulated with 16 QAM modulations and pilot pattern is allocated with ISFLA. After applying Inverse Fast Fourier Transform (IFFT), cyclic prefix is added. The multi path Rayleigh fading channel is used in which Additive White Gaussian Noise (AWGN) is added with the signal. In the receiver, cyclic prefix is removed and Fast Fourier Transform (FFT) is applied. Then the channel is estimated with CS based estimation and demodulated. 2.1. FIR Filter for Sparse channel th

Let us consider a multipath frequency selective fading channel is there between iTR transmitter th

and i RE receiver. With finite length L and impulse response I, FIR (Finite Impulse Response) were designed. The K-sparse channel which has K non-zero elements and K<
TR , i RE

[I i

TR , i RE

(0), I i

TR , i RE

4

(1), ..., I i

TR , i RE

( L  1)]

(1)

Where, I iTR ,iRE (m) 

L 1  q i ,i TR RE l 0

(l )  (m  l ) and qiTR ,iRE (l ) is the complex gain of the

l th tap. The

Number of a subcarrier denoted as S {1, 2, ... , N} . For iTR th transmitter, the parallel stream of data contains modulated data and indicated the Pilot symbol as {xiTR (m)}, m  S .

i {ki TR

TR ,1

, k iTR , 2 , ..., k iTR ,M }  S is the pilot location index, and the group of this index M is

mentioned by m (k iTR ,1 , k iTR , 2 , ..., kiTR ,M ) . Then the pilot symbol is meant as {xiTR (m)}, m  iTR and modulated with IFFT (Inverse Fast Fourier Transform). The received signal of one OFDM symbol in i RE th the receiver is RiRE  C N . After demodulating with FFT (Fast Fourier Transform), OFDM symbol is, RiRE 

NTR  [diag ( xi TR iTR 1

(1), xiTR (2), ... , xiTR ( N ))] V I iTR ,iRE  wiRE

(2)

Where diag ( xiTR (1), xitTR (2), ... , xitTR ( N ) are the diagonal elements and represented as X iTR V is a partial N×N FFT matrix with L columns, I iTR ,i RE is a K-sparse channel impulse vector and wiRE  C N is a noise vector. The noise vector constructed from the element of the additive white

Gaussian variable with variance  2 . AWGN 

Variance of the modulated signal linear SNR

(3)

The variance for the modulated signal {xiTR (m)}, m  iTR is denoted as, Where, s denotes the number of samples, μ denotes the mean signal to noise ratio. 2.2. Channel estimation theory for finding Impulse vector

5

1 S

s  ( x iTR a 0

(m)

1 s 1  ( x iTR (m)   ) . S  1 a 0

and SNR denotes the

MIMO system consists of multiple transmit and multiple receive antenna. For simplifying the process, as a simultaneous evaluation of SISO system are considered. Each transmitter is to be assigned with frequency orthogonal pilot placement that is i   j   (1  i, j  N TR , and i  j) . When i th antenna is transmitting pilot, all the other antennas remain silent to obtain better channel estimation performance. Let x j (m) be the pilot transmitted by the other antenna, then x j (m)  0 (m  i ,1  i, j  NT , and i  j ) . By applying the above assumptions, the pilot symbol

received from iTR th transmitter has derived. ~ ~ ~ ~ RiTR ,iRE  EiTR X iTR V I iTR ,iRE  EiRE wiRE  X iTR ViTR I iTR ,iRE  w iTR ,iRE

(4)

Where, EiTR  C M N obtained by selecting M rows from N×N identity matrix that is it selects elements

at

the

pilot

locations,

and

indices

belong

to

~

iTR , EiTR X iTR  EiTR X iTR EiTR T EiTR , X iTR  EiTR X iTR EiTR T is a P×P diagonal matrix and P pilot

symbols {xiTR (k iTR ,1 ), xiTR (k iTR , 2 ), ... , xiTR (k iTR ,l )} ~ w iTR ,iRE  EiTR wiRE is a noise vector of

are

elements

of

the

diagonal

matrix,

~ i th transmitters pilot location and ViTR  EiTR V . For

~ (a, b) th element ViTR is  j 2kiTR ,ab / N ~ 1 [ViTR ]a,b  ( )e ; a = 1,2,…, M; b = 1,2,…, L; K iTR ,a  iTR N

(5)

I iTR ,iRE is considered for each receive antenna i RE . After omitting receive antenna index i RE , the

pilot symbol received from iTR th transmitter becomes,

~ ~ ~ ~ ~ RiTR  X iTR ViTR I iTR  w iTR  U iTR I iTR  wiTR

6

(6)

~

~

Where, UiTR  X iTR ViTR . Above equation (4) is considered as a reconstruction problem hence it is needed to calculate the K-sparse impulse vector I iTR . Compressed sensing based channel estimation theory is used to estimate I iTR . The minimization problem can be determined by recovering I iTR  C L from (4). IˆiTR  arg min I iTR IiTR

1

~ s.t RiTR  U iTR I iTR

2



(7)

M L Where  is an error tolerance and measurement matrix U iTR  C are constructed by solving

(5) minimization problem. From the designed for U iTR m, mutual coherence estimated as,  {U iTR } 

 i H j max

i  j , j  L, i  j

i

2

.

(8)

j 2

Where,  j is the i th column vector of the matrix U iTR . From I iTR , the deviation IˆiTR can be obtained by IˆiTR  I iTR

2 2



(   ) 2 1  {U iTR } (4 K  1)

(9)

~ in equation (4) and   B . For a K-sparse signal, the better Where, B is the variance of noise w iTR

~ approximation of I iTR obtained by minimizing {U iTR } . Pilot symbol X TR and their location iTR

can be used to determine the measurement matrix U iTR from (3) and (4). In the MIMO-OFDM system, by minimizing {U iTR } to improve the performance of CS based channel estimation, the pilot pattern can be optimized. 2.3. Reducing mutual coherence for measurement matrix

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By reducing mutual coherence {U iTR } , channel estimation performance can be enhanced. From the updated N×N FFT matrix (5) and the pilot symbol received from iTR th the transmitter (6), P×L matrix U iTR constructed for (a, b) th element.





~ ~ [U iTR ] a,b  X iTR ViTR a,b

1

~

~

By applying X iTR  xiTR (k iTR , a ) and ViTR  [U iTR ]a,b 

1 N

xiTR (k iTR , a )e

N

 j 2k iTR , ab / N

)e

(10)

 j 2k jTR , ab / N

, Eqn. (10) becomes,

; a = 1,2,…, M; k iTR ,a  iTR ; b = 1,2,…, L

(11)

The matrix U iTR has the subset of iTR , iTR  1, 2, ..., N TR and locations xiTR (kiTR , a ), a = 1, 2, …, M . When we assume that all pilot symbols are same, the value of pilot symbol will not affect the mutual coherence of the matrix. Average distribution of pattern U jTR obtained for processing random variables which have described as, [ FiTR ] a, b 

1

e

 j 2k iTR , ab / N

M

; a =1, 2,…, M;kiTR ,aiTR ;b=1, 2, …, L

(12)

From N number of a total subcarrier, M subcarriers utilized for transmitting pilot symbols. When selecting the subcarriers, it is significant to ensure that the matrix U iTR has minimum mutual coherence {U iTR } . Subcarrier Selection is defined as,

 m(iTR ) M 1i, j  L, i  j i TR

min

max

1  j 2 (i  j ) / N e M

(13)

To obtain optimal subset of pilot pattern, random search procedure is proposed. This procedure creates required number of subsets from the pilot indices set S {1, 2, ... , N} . Optimal pattern is selected from the pilot indices which is associated with low mutual coherence. In MIMO system

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it is essential to optimize subsets iTR , iTR  1, 2, ..., N TR with i   j   (1  i, j  N TR , and i  j ) Minimal IˆiTR  I iTR

2

can be obtained after ensuring that the value of mutual coherence is small.

2

2.4. Proposed pilot allocation method Set of pilot indices are selected from the number of subcarriers is a significant issue in channel estimation. Here Improved Shuffled Frog Leaping (ISFL) pilot allocation technique is proposed to get pilot patterns iTR , iTR  1, 2, ..., NTR S  {1, 2, ..., N} . At the same time the value of mutual coherence {U iTR }, iTR = 1, 2, …, N TR should be small. The basic concept of ISFL algorithm is to initialize random solutions from the set of solution space. The aim for initializing random solution is to seek for the best value to minimize the mutual coherence. The proposed ISFL is defined initially to get the core pilot pattern 1  {d1 , d 2 , ..., d M } . It is assumed that the other pilot pattern should satisfy iTR  {d1  iTR  1, d 2  iTR  1,..., d M  iTR  1}, iTR  2,..., N TR . It can be described as,  m(1) M 1i , j  M ,i  j da1

min

max

1  j 2da(i  j ) / N e s.t. min (d i 1  d i )  NTR P 1i M 1

(14)

Let 1  {d1 , d 2 , ..., d M } be the set of pilot locations, and the distance between this locations is

Y  {D1 , D2 ,..., DM 1 with Di  d i 1  d i . The pilot distance determines the value of mutual coherence based on 1  {d1 , d 2 , ..., d M } . The individual Y specifies the unique pilot location 1 M 1

by assuming d1  1 and the last location index d M  1   Di . Large pilot distance earns pilot i 1

patterns with tremendous value of mutual coherence. 2.4.1. Improved Shuffled Frog Leaping Algorithm

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The efficiency of channel estimation is based on the type of pilot pattern used. ISFLA is an optimization algorithm for selecting the suitable pilot pattern. The step by step procedure of ISFLA is described as follows. (i) Initialization: The total populations of P frogs initialized, and they selected randomly from the solution space. P assigned to 256. The i th solution for the problem is represented as

S i  (si1 , si 2 ,..., siD ) . (ii) The fitness value for each frog is calculated from the fitness function. The fitness can be defined as [3], 0.6.N  d M and d M  N  N TR  1 is used to get small fitness value.  1 ,   f (Y )     1 ,  1000

if 0.6.N  d M  N  NTR  1

(15)

else

Where, N is the number of subcarriers, d M is the last location index, M is the number of subcarriers allocated to pilot symbol and NTR is the total number of transmitters. (iii) Grouping: The whole population is divided into m groups and each group is created with n frogs (P=m×n). The first frog is assigned to the first group, the second frog is assigned to the second group, the m th frog is assigned to the m th group, the (m  1) th frog is assigned back to the first group and so on. (iv) Local evolution: X b and X w are calculated for each group using fitness of the frog. From the whole population, the group best X g is calculated. The position of X w is updated from the position of best frog in each group. The current position of the X w can be updated by

new X w  current X w  r  ( X b  X w )

10

(16)

Where, current X w is the current position of the worst frog, C denotes the factor X b is the position of best frog, X w is the position of worst frog and r ranges between the values 0 and 1. If it causes improvement in worst frog, then the position of X w is updated and new X w from the group is calculated again. Otherwise X b is replaced by X g and the position of X w is recalculated. If there is no improvement occurs, then the random position is generated for X w . (iv) Global shuffling: All the frogs are shuffled and rearranged in descending order according to the fitness function. Then the frogs are divided into groups and continue with step (iii). (v) Local updating of frog’s position and the shuffling process of whole population is continued until the required convergence criteria have been achieved. The proposed pilot allocation scheme select individuals with best fitness value from the total population. The searching ability of this scheme is improved with minimal mutual coherence. 3. Performance analysis The performance of the proposed pilot allocation technique was evaluated with Monte Carlo simulations. A sparse Rayleigh channel with exponential power delay profile is modeled by a six-nonzero tap MIMO-FIR filter of length L = 50. Gray Mapping is used which minimizes the number of bits that can be demodulated with error. The elements of each tap were 4×4 random vector and they are identically distributed complex Gaussian variables with zero mean. To enable frequency orthogonal pilot placement to each antenna, the 4×4 MIMO OFDM channel estimation is decoupled into 16 SISOs. From set S  1,2,..., N , 24 elements are selected randomly to get a subset of pilot location. For implementing CS based channel estimation, the numbers of transmitting and receiving antennas are assigned to 4 and 24 pilots are allocated for each transmitter.

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Table 1: System Parameters Parameter

Range

Number of transmitters N TR

4

Number of pilots for each transmitter

24

Guard interval

1/4

Number of receivers N RE

4

Number of Subcarriers N

512

Signal Constellation

16 QAM modulation

Data symbol coding

Gray code

Length of FIR filter L

50

Tap

6

For optimal pilot allocation, the value of mutual coherence with random Search procedure [15] is 0.2562 and it uses 5 10 5 iterations. In [24] the pilot patterns were obtained by reducing the largest element in the mutual coherence set and using Genetic algorithm with shift mechanism. By using these two methods the obtained value of mutual coherence is 0.2915, 0.2971, 0.3055, 0.3065 and 0.2187. For reducing the value of mutual coherence existing methods [24] take 50000 and 5000 iterations respectively. By using ISFLA, it has only 500 iterations and the optimal pilot pattern iTR , iTR  1, 2, ..., N TR was selected with the mutual coherence of 0.1872. With reduced number of iterations, the total execution time gets reduced.

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Fig. 2: The value of Mutual coherence with delay spread=30, 50 and 60 Fig. 2 shows the difference in mutual coherence value with varying delay spread. When the delay spread of the channel is 50, the pilot pattern is optimized with the mutual coherence 0.1872. This value can be reduced by setting delay spread 30. The mutual coherence 0.2102 is obtained with the delay spread 60. Different algorithms used for pilot allocation and their mutual coherence value were shown in table 2. Fig. 3 (a) shows the performance of mutual coherence with different algorithms such as Improved Shuffled Frog Leaping Algorithm (ISFLA), Genetic Algorithm (GA), Reducing Largest Element (RLE) and Random Search Procedure (RSP). Table 2: Mutual coherence for different algorithm Algorithm

Mutual Coherence

Reducing the largest element

0.2915, 0.2971, 0.3055, 0.3065

Genetic algorithm with shift mechanism

0.2187

Random Search Procedure

0.2562

Improved Shuffled Frog Leaping Algorithm

0.188

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Fig. 3. Comparison of mutual coherence with (a) Pilot allocation algorithms (b) No of iterations Fig. 3 (b) shows the reduced value of mutual coherence with the number of iterations. Mutual coherence below 0.188 obtained after 400 iterations. To produce a more accurate estimate, the estimator does not account for information, and it makes a difference in estimation. Hence, as the difference between the estimator and the evaluated value were determined and the mean square error calculated as N MC

MSE  (1 / N MC ) 

n 1

N RE NTR  iRE 1 iTR

(1 / N TR N RE ). 

Iˆ (n) iTR ,iRE  I (n) iTR , iRE

2

2

/ I i(n) , i TR RE 2 2

(17)

Where, N MC is the number of repetitions, NTR is the number of transmitters, N RE are the number of receivers, Iˆ( n ) iTR ,iRE are the estimated channel course and I ( n ) iTR , iRE the native medium vector.

Fig. 4. MSE Vs SNR

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Fig. 5. MSE performance with varying number of transmitting and receiving antennas Fig. 4 shows the MSE performance of proposed channel estimation with NTR=4 and NRE=4. In between 10 dB to 15 dB SNR, the MSE of 10 3 obtained for proposed method. But in the case of LS and MMSE channel estimation, the MSE exceeds 10 3 . The rate of MSE decreased with increase in SNR. If the number of transmitting antenna is increased to 6, then the BER performance will be reduced to 10-3.5 for 20 dB SNR value as shown in fig 5. If the receiving antenna is increased to 6, then the BER is increased to 10 -5 for 20 dB SNR value. There is a slight variation in the BER, if both transmitting and receiving antennas are increased to 6. At 10 dB SNR, the pilot overhead ratio of LS and MMSE approaches are 18.60% and 19.76%. But for the proposed approach the pilot overhead is reduced to 17%. At 20 dB SNR, the pilot overhead of proposed approach is 17.5% and for LS and MMSE this values are increased to 18.80% and 20 %. BER calculates the number of altered bits of the data stream over the communication channel. Hence it defined as the number of bit error per unit times and measured as, BER 

Ne Nb

(18)

Where, N e is the total number error and N b is the total number of bits sent. For different pilot patterns, the BER value shown in fig. 6 (a). With 10 dB SNR, the algorithms ISFLA, GA, RSP, and RLE generate the BER values 0.1, 0.3, 0.4 and 0.4 respectively. The BER performance of 15

LS, MMSE, and CS is shown in Fig. 6 (b). For the proposed method, the BER is between 10 4 to

10 3 at SNR gain of 20 dB. When compared with proposed method, LS and MMSE produces BER above 10 3 which shows the efficiency of proposed channel estimation.

Fig. 6. BER Comparison for (a) different pilot patterns, (b) Varying SNR values Fig. 7 (a) and Fig. 7 (b) shows the reduced in BER and MSE with reduce in mutual coherence. In Fig. 7(a), MSE is slightly increased after 0.196 of mutual consistency and the BER 1.23 at the mutual consistency of 0.192. Least square channel estimation method is implemented in [25]. The number of transmit and receive antenna was fixed at 2. 4-PSK modulations were used with 16 states trellis structure. On the average of SNR  25 dB was estimated within 10-20 iteration. Minimum mean square error estimation was implemented in [26].

Fig. 7. Comparison mutual Coherence with (a) MSE (b) BER 16

The total number of sub carriers selected is 32 with 16 PSK modulations. With PSK it provides SNR above 6dB compared with DPSK. Due to error propagation, the SNR exceeds 20 dB in MMSE. In proposed scheme, 24 pilots assigned to each transmitter with 512 subcarriers. The measure of the periodical prefix is 8 and symbol duration is 1.13 ms with the data rate of 4 bits per symbol. As shown in Fig. 4, Fig. 6, Fig. 7(a), and Fig. 7(b), the BER and MSE of channel estimation of CS based technique has decreased with minimal computational complexity. Because the number of pilots used in CS is smaller than LS and sparseness of the medium is also low. As to the computational complexity, the number of complex multiplications in LSE channel estimation and MMSE channel estimation are o( PLSE LNTR N RE ) and, respectively, where I is the number of iterations and it is approximately equal to the sparseness L of the channel. Since PLSE is much larger than PMMSE and K is very small, the two methods of channel estimation have the same order of computational complexity. The proposed channel estimation schemes require the same number of complex multiplications as the LSE and MMSE scheme does. The computational complexity of the proposed approach is o( IPMMSE LNTR N RE ) which is same as the computational complexity of LSE and MMSE. In a word because of employing the optimal pilot locations, the ISFLA compressed sensing based MIMO channel estimation is superior to the LSE and MMSE schemes in terms of BER, MSE and computational complexity. 4. Conclusion This paper presented compressed sensing based estimation of sparse channel in MIMO OFDM system. Here the input signal was modulated with 16QAM modulation. The optimum pilot pattern was selected with ISFLA by reducing the mutual coherence. AWGN was used for adding noise in multipath Rayleigh fading channel. The channel was estimated with CS based channel

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estimation and demodulated. Compared with the traditional channel estimation techniques like LS, and MMSE, the performance of proposed pilot allocation for CS based channel estimation was higher in terms of spectral efficiency. Implementation results show that the proposed pilot allocation for CS based method provides better performance than traditional channel estimation methods with respect to MSE and BER. The channel estimation performance can be further improved through optimal pilot allocation by increasing the number of iterations in optimization algorithms. References [1] Zhen Gao, Linglong Dai, Wei Dai, Byonghyo Shim, Zhaocheng Wang. Structured compressive sensing-based spatio-temporal joint channel estimation for FDD massive MIMO. IEEE Transactions on Communications 2016; 64(2):601-617. [2] Zhijin Qin, Yue Gao, Clive Parini G. Data-assisted low complexity compressive spectrum sensing on real-time signals under sub-nyquist rate. IEEE Transactions on Wireless Communications 2016; 15(2): 1174-1185. [3] Xueyun He, Rongfang Song, Wei-Ping Zhu. Pilot Allocation for Distributed-CompressedSensing-Based Sparse Channel Estimation in MIMO-OFDM Systems. IEEE Transactions on Vehicular Technology 2016; 65(5): 2990-3004. [4] Kee-Hoon Kim, Hosung Park, Jong-Seon No, Habong Chung, Dong-Joon Shin. Clipping Noise Cancelation for OFDM Systems Using Reliable Observations Based on Compressed Sensing. IEEE Transactions on Broadcasting 2015; 61(1): 111-118. [5] Aamir Ishaque, and Gerd Ascheid. Efficient MAP-based estimation and compensation of phase noise in MIMO-OFDM receivers. AEU-International Journal of Electronics and Communications 2013; 67(12): 1096-1106.

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