TRANSACTIONS OF TIANJIN UNIVERSITY

2016, 22: 352-357 DOI 10.1007/s12209-016-2746-3

Compressed Sensing-Based Multiuser Cooperative Networks* Fu Xiaomei(付晓梅)，Cui Yangran(崔阳然) (School of Electronic Information Engineering, Tianjin University, Tianjin 300072, China) © Tianjin University and Springer-Verlag Berlin Heidelberg 2016

Abstract：To avoid interference, compressed sensing is introduced into multiuser cooperative network. A cooperative compressed sensing and amplify-and-forward(CCS-AF)scheme is proposed, and it is proved that the channel capacity increases compared with the traditional cooperative scheme by considering the CCS-AF transmission matrix as the measurement matrix. Moreover, a new power allocation algorithm among the relays is proposed to improve the channel capacity. Numerical results validate the effectiveness of the proposed scheme. Keywords：compressed sensing; cooperative networks; power allocation; channel capacity

Cooperative communication takes the advantage of wireless transmission to enhance the system channel capacity [1,2]. However, it induces a large amount of interference when the signals are transmitted simultaneously without being allocated orthogonal channels in multiuser cooperative network, which makes it impossible for the sources to transmit at the same time[3,4]. The main idea of compressed sensing(CS)[5,6] is that the signal of interest can be reconstructed by acquiring a relatively small number of samples in the “sparse” domain through an optimization process[7]. With the ability of sampling and compressing simultaneously, CS has been utilized in data compression[8], data acquisition[9] and channel coding[10]. In this paper, CS theory is applied to the traditional multiuser cooperative networks, and a cooperative compressed sensing and amplify-and-forward (CCS-AF) scheme is proposed. In contrast to the traditional cooperative networks(e.g., FDMA), the proposed compressed cooperation scheme can transmit signals simultaneously without allocating orthogonal channels. Moreover, the power allocation of cooperative relays plays an important role in improving the transmission performance [11-13]. An optimal power allocation(OPA)scheme is proposed to improve the channel capacity of our new multiuser cooperative networks.

The rest of this paper is organized as follows. In Section 1, a multiuser cooperative network model based on CS is proposed. In Section 2, the channel capacity is derived, and the optimal power allocation scheme is formulated in Section 3. Numerical results are provided in Section 4. Finally, conclusions are drawn in Section 5.

1

System model

As shown in Fig. 1, the multiuser cooperative network model based on CS consists of N source nodes(Ss), M amplify-and-forward (AF) relay nodes (Rs) and one destination node(D). The transmission consists of two time slots.

Fig. 1

Accepted date: 2015-09-25. *Supported by National Natural Science Foundation of China(No.61571323). Fu Xiaomei , born in 1968, female, Dr, associate Prof. Correspondence to Fu Xiaomei, E-mail:[email protected].

Multiuser cooperative system model based on CS

Trans. Tianjin Univ.

Fu Xiaomei et al: Compressed Sensing-Based Multiuser Cooperative Networks

1.1

Sparse representation of signal Considering a discrete-time signal X with finitelength N, i.e., X  [ x1 , x2 ,, xN ] is the original transmitted data vector, whose degree of sparsity is K, K＜M＜N. This means that the N×N sparse transform basis  is equal to identity matrix. According to the CS theory, the product of measurement matrix and the sparse transform basis must obey the rule of restricted isometry property(RIP)[14]. 1.2 Measurement matrix In the first time slot, the source nodes send their data to the relay nodes simultaneously. The M×N matrix HSR denotes the fast fading of the first time slot transmission channel. The entries of HSR obey Gaussian distribution, i.e., [ H SR ]i , j ~  (0, M -1 ) , which is incoherent with the identity matrix and satisfies the RIP with a high probability as long as M ≥ cK log 2 ( N/K ) . Here c is a positive constant[7]. It is assumed that the perfect channel state information is known at all nodes. Without loss of generality, path loss is considered. Suppose that the distance between source node i( 1 ≤ i ≤ N )and user j( 1 ≤ j ≤ N )is close enough, compared with that between user i and relay m( 1 ≤ m ≤ M ). Hence, the path loss between user i and relay m could be treated as equivalent. The channel matrix A =  H SR , where  =diag{1 , 2 , ,  M } is the path loss from the Ss to the ith relay in the first time slot. Moreover, A still satisfies RIP[15]. Then, the received signal at the relays is

YR = AX + N 0

(1)

where N 0 is an M × 1 additive white Gaussian noise (AWGN)vector at Rs, and its entries n0i (1 ≤ i ≤ M ) are independent zero-mean Gaussian variables. In the second time slot, the relay nodes amplify the received signal and forward it to the destination node with AF protocol using orthogonal transmissions. H characterizes the channel matrix from Rs to D. Then, the signal at the destination node is YD =  HYR + W0 

 H ( AX + N 0 ) + W0 Equivalently,  y1   y2     yM

  11    0        0

0

 22  0

 

0   0         MM 

(2)

 H11   0     0    A11    A21     A   M1

0 H 22

 0

A12 A22



  

AM 2 

 

        H MM  0

0

A1N   x1   n01     A2 N   x2   n02              AMN   xN   n0 M

   w01     +  w02        w    0M

      

(3) where YD is the signal matrix received at D, and

yi (1≤ i ≤ M ) are the received signals that are transmitted from the ith relay;  is the amplification coefficient matrix, its entries are  ii 

ai PR 2 PS N   Aij + n20 N j 1

(1 ≤ i ≤ M )

where ai is the power allocation coefficient and M

 ai  1,0 ≤ ai ≤1 ; PS is the overall power of the Ss, and i 1

it is allocated among the Ss uniformly; PR is the overall power of the Rs; H is a diagonal matrix and its entries H ii (1 ≤ i ≤ M ) represent the transmission attenuation in the second time slot from the ith relay to D; Aij (1 ≤ i ≤ M ,1 ≤ j ≤ N ) is the path loss and fast fading of the first time slot from the jth source node to the ith relay node; W0 is an M×1 AWGN vector at the D, and the entries w0i (1 ≤ i ≤ M ) are independent zero-mean Gaussian variables. It is known that A =  H SR is incoherent with the sparse transform basis  , i.e., the row vectors of A cannot be sparsely represented by the column vectors of  , and vice versa[16]. Let  =  H  A (4) Then, Eq.(2)can be expressed as YD   X + Z 0

(5)

where Z 0   HN 0  W0 . Since  H is a diagonal matrix , apparently, the row vectors of  cannot be sparsely represented by the column vectors of  . So the measurement matrix is  = HA , which obeys the rule of RIP. 1.3 Reconstruction stage For a sparse vector X and a well-constructed measurement matrix, many reconstruction algorithms have been proposed, such as basis pursuit(BP), orthogonal matching pursuit(OMP).

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The principle of BP is to find a representation of the Assuming  n2   n2 =  = n2 = n2 and  w2   w2 = signal, whose coefficients have minimal l1 norm. As BP  = w2 = w2 . Here  n2 and  w2 represent the channel can stably suppress noise[17], BP is used in this paper, and noise powers of the two time slots, respectively. Because X can be reconstructed from YD by solving the convex the channels of the system are regarded as M parallel optimization problem, i.e., Gaussian channels at the destination node via the SVD theorem, the signal power Pi and the noise power  i2 of the min X  l ith parallel Gaussian channel can be written as (6) 2 s.t. YD -  X  l ≤  H ii PR ai P  i2 Pi  S  2 (13) PS N N 2 where  is an upper bound on the magnitude of the noise ; Aij   n  N j 1 and X  is the reconstructed vector at D. 2 In addition, the relative reconstruction error is used H ii PR ai 2 2 2 i  w n to evaluate the reconstruction performance, i.e., 2 (14) PS N Aij   n2  2 2 N j 1 e  E[|| X - X  || / || X || ] (7) 01

0M

0

02

0M

0

0

0

01

02

1

2

0

0

0

0

2

where E is the expectation operation.

2

Let  i 

Channel capacity

H ii PR

, so the channel capacity is given

2 PS N Aij   n20  N j 1

by M P 1 C   log 2 (1  i2 )  i i 1 2

In the proposed networks, the M×N channel matrix A can be transformed to an M×1 parallel channel vector by using singular value decomposition(SVD). Then Eq.(2)can be rewritten as

M

1

 2 log i 1

2

(1 

(15)

 2 a PS )  2 i i 2i N  w0   n0 i ai

It is known that if f is concave on an interval I and x1, x2,…, xn are in I, then where U and V are unitary matrixes;  is a diagonal max  x    xn f( 1 2 )≥ trix of the singular values of A. Suppose that the transn mitted data vector X  = VX , then f ( x1 )  f ( x2 )    f ( xn ) (16) YD   H (U  V H X   N 0 )  W0  n  H (U  X  N 0 )  W0 (9) which is called Jensen’s inequality. Since f ( x)  log 2 x is YD   H (U V H X  N 0 )  W0

(8)

strictly concave on (0, ) , to analyze the bound of channel capacity, Jensen’s inequality is applied to Eq.(15), i.e.,

By decoupling Q with U H , YD   HU H (U  X  N 0 )  W0 

 H  X   HU H N 0  W0

(10)

M

M

Psignal  E [(  H  X )  (  H  X ) H ] 

PS  H H H H  H N

where IM is an M-order unit matrix. —354—

Pi

 i2

)≤

2 M log 2 (1  i 1 i ) 2 M

Pi 2 M Cb  log 2 (1  i 1 i ) 2 M



(11)

(  HU H N 0  W0 ) H ]  0

(1 

(17)

M

Pnoise  E[(  HU H N 0  W0 ) 

0

i 1

2

Then, the upper bound of the information capacity Cb is

The noise power matrix Pnoise from Eq.(10)can be expressed as

 w2   n2 I M  HH H  H

1

 2 log

Then the signal power matrix Psignal from Eq.(10)can be expressed as

Pi



3 3.1

(12)

(18)

Relay power allocation scheme

OPA scheme To maximize the channel capacity, the channel capacity is taken as the optimization parameter, then the optimization problem can be expressed as

Fu Xiaomei et al: Compressed Sensing-Based Multiuser Cooperative Networks

Trans. Tianjin Univ.

Fig. 2 shows the average relative reconstruction error versus the number of cooperative relays for different 0 0 (19) values of sparsity K. The relay power is uniformly alloM ai  1, 0 ≤ ai ≤ 1 s.t.  cated. It can be seen that more relay nodes are needed to i 1 achieve the same reconstruction performance for a larger By using the method of Lagrange multiplier, the function sparsity. Moreover, as long as the relay number is large can be derived as follows enough, the degree of sparsity does not have impact on F (a1 , a2 , , aM ,  )  the relative reconstruction error. M PS  i i 2 ai 1 log (1   )   2 N  20   n20 i ai i 1 2 M P   2a 1 max C  max  log 2 (1  S  2 i i 2 i ) N  w   n  i ai i 1 2

 (a1  a2    aM ) (20) By taking its partial derivatives with respect to ai and let

ai 

F  0 ，then ai can be determined by ai

P 4 i ( n20  S i 2 ) n20 PS PS 2 2 N i  w0   w i 2 ln 2 N 0 N P 2[ i ( n20  S i 2 ) n20 ] N [ w20  n20  ( n20 

PS 2 2 i ) w0 ] N

 Fig. 2 Relative reconstruction error versus the number of cooperative relays for different values of sparsity

Fig. 3 shows the channel capacity performance of the proposed CCS-AF transmission scheme as a function of the number of relays for different values of relay M Since  ai  1, 0 ≤ ai ≤ 1 ,  can be obtained by power PR . It can be seen that the channel capacity ini 1 creases more obviously with larger relay power. summing up Eq.(21)over i, and ai can be solved by substituting  into Eq.(21). If ai is greater than one or less than zero, then a smaller number of cooperative relays M1( cK log 2 ( N/K ) ≤ M 1  M ), which maximizes the channel capacity with uniform power allocation. 3.2 Uniform power allocation(UPA)scheme In the UPA scheme, each relay node forwards with the same power. Thus, the power allocation coefficient is ai  1/ M . The channel capacity can be written as P 2[ i (  S i 2 ) n20 ] N

(21)

2 n0

1  i i 2 PS 1 M ) C   log 2 (1   N  2   2 1 i 1 2 w0 n0 i M

Fig. 3 Average channel capacity versus number of cooperative relays for different values of relay power

M

4

(22)

Simulation results

In the simulation model, Ss and D are located at(0,0)and(1,0)respectively, and Rs range from－0.05 to 1.05 in x-axis. Assume that the Rs are located close to one another and the path loss from the Rs to the Ss is approximately equal. The path loss exponent is 4, and dSR is the normalized distance from Ss to D. The values of PS -2 and PR are 10 W and 10 1 W, respectively. Both  n2 and  w2 are 10 4 W. 0

Fig. 4 and Fig. 5 compare the channel capacity performance of OPA and UPA with different values of PR . It can be seen that the OPA scheme outperforms UPA obviously. In addition, because of the inequality property of Eq.(17), there is a gap between the upper bound of UPA and OPA/UPA. As shown in Fig. 4, when PR is large enough, the capacity gap between the UPA-upper bound and OPA/UPA increases the number of cooperative relays increases as well. Conversely, as shown in Fig. 5, when PR is too small, the gap decreases as the number of cooperative relays increases.

0

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