QoS Guaranteed Joint Precoder Design For MIMO Relay Network

QoS Guaranteed Joint Precoder Design For MIMO Relay Network

QoS Guaranteed Joint Precoder Design For MIMO Relay Network

Bhagya Shri Gupta, Khushboo Singh, and Radhika Gour

Department of Information Communication and Technology

ABV-Indian Institute of Information Technology and Management Gwalior, India-474015

AbstractPrecoding designs are used for transmitting the in relay networks where multiple antennas are used at the transmitter and receiver. Applying a minimum-mean-square- error (MMSE) strategy, the objective is to design quality-of- service (QoS) aware joint precoder at source and relay to fulfill minimum signal-to-noise ratio (SNR) requirement at users in amplify-and-forward MIMO relay network. In this approach non-convex optimization problem is solved by dividing into two problems: primary and secondary using primal decomposition which are convex in nature and solvable. Numerical result shows that joint source/relay precoding scheme has better bit error rate (BER) performance as compared with relay precoding scheme while considering minimum SNR constraint and BER performance of both scheme improves with increase in signal-to- noise ratio (SNR).

1. INTRODUCTION

   Relay based technology has attention due to its significant benefits to enhance throughput, increase network coverage and reduce transmission power in cellular network [1]. Moreover, use of multiple antennas in wireless communication has signif- icant advantages to increase the system throughput. Implemen- tation of multiple-input multiple-output (MIMO) technique in recent wireless communication standard such as long- term evolution (LTE) and IEEE 802.16m improves spectral efficiency and transmission reliability [2]. To increase the overall performance single antenna in wireless communication system has been replaced by multiple antennas.

   Furthermore, authors in [3] compared amplify-and-forward (AF) relaying technique with decode-and-forward (DF). Amplify-and-forward (AF) relays are more practical compared to decode-and-forward (DF) since they are low complex and transparent to source and destination [3], [4]. Precoding is mainly used to mitigate the effect of multiuser interference in MIMO system. Extensive work on AF MIMO precoding mainly pays attention on maximizing capacity [5],[6]. How- ever, these papers considered precoding at relay nodes only. Recently, authors have made efforts to propose joint source and relay precoding. Work in [7], proposed joint relay precoder and decoder for cooperative network where single antenna is used at source, relay and destination while [8] employed multiple antennas at both source and relay. Moreover, authors in this work used dirty paper coding to achieve sum rate capacity within transmit power constraints at source and relay and capacity is maximized without any provision of QoS. In [9], authors satisfied minimum signal-to-interference-noise

   ratio (SINR) at users but did not consider any power budget constraints at source and relay. Later, in [10], idea was proposed to design joint precoder at relay and destination to minimize MSE in MIMO relay system having source, relay and destination. In addition to more, Joint relay and destination optimization is done assuming perfect CSI and used Weiner filter in the destination and relay precoder is designed with in transmission power constraints . However, author did not consider any constraint to fulfil minimum SNR.

   Authors in [8]-[10] proposed precoding techniques which mainly focused to minimize MSE within minimum transmis- sion power constraints but did not guarantee any QoS in term of predefind SNR. In the proposed work, however, joint pre- coder design at source and relay is adopted to minimize MSE while satisfying the minimum predefined SNR requirement at users. MMSE is non-convex and complicated function of precoding matrices used at source and relay. Under perfect channel state information (CSI) the problem of a non-convex optimization is solved by deviding into two problems: primary and secondary which are convex and can be solved using standard convex optimization technique.

   Remaining paper is organized as follows: System model and problem formulation is presented in Section II. Section III introduces Proposed joint precoder design. Section IV describes the simulation results of proposed scheme under the SNR constraints. Finally conclusion of paper is given in section V.

2. SYSTEM MODEL AND PROBLEM FORMULATION

   1. System Model

      ×

      ×

      ×

      ×

      This section describes the downlink AF MIMO model consisting of the source S, relay R, and destination D having number of antennas Ms, Mr, ans Md, respectively. Let Ws be the precoding matrix at source S of dimension Ms Ms. Transmitted signal vector at source S is given by x = Wss where s is a Ms 1 vector where each entry represent

      || ||

      || ||

      | |

      | |

      the modulated symbol with unit power, e.g., sk 2 = 1. Transmitted power at source is given as pt = E[ x 2] = tr(WsWsH ).

      The signal received at relay R is given as

      ×

      ×

      where H1 is Ms Mr MIMO Rayleigh channel between source S and relay R and nr is the AWGN vector of dimension

      where Ck is interference noise covariance matrix can be written as

      s

      s

      H

      H

      H

      H

   2. Problem Formulation

   (9)

   Let be the SNR matrix. The objective is to minimize MSE while guaranteeing minimum SNR for users at destination within average transmission power budget, therefore, problem can be formulated as

   min

   }

   }

   {Ws,Wr

   tr{M} (10)

   Fig. 1. System model for MIMO network consisting of Source, Relay and Destination.

   ×

   ×

   Ms 1 with variance E[nrnrH ] = n2I.

   After AF operation the signal transmitted from relay is given by

   || ||

   || ||

   ×

   ×

   where Wr is the precoding matrix at relay R of dimension Mr Mr. The transmitted power at relay R is given as pr = E[ x0 2] = tr((WrH1Ws)H WrH1Ws + n2WrH Wr). Therefore, the received signal at destination D is given by

   H

   H

   H 2 H

   H 2 H

   tr{WrH1Ws(WrH1Ws) + n WrWr } pr (13) where min is predefined diagonal SNR matrix to guarantee

   QoS for users at destination.

3. PROPOSED JOINT PRECODING DESIGN

   W

   W

   Lets knowledge of channel matrices H1 and H2 is avail- able at source S and relay R so precoding matrices Ws, Wr and decoding matrix W can be designed jointly to minimize trace of MSE M . To find the optimal decoding matrix W which minimize the MSE expression in (5) apply

   d

   d

   2

   2

   r

   r

   1

   1

   s

   s

   2

   2

   r

   r

   r

   r

   d

   d

   (3)

   tr{M (W, F, G)} = 0 and is written as

   ×

   ×

   ×

   ×

   where H2 is Md Mr MIMO Rayleigh channel between relay R and destination D and nd is the AWGN vector of dimension Md 1 with variance E[ndndH ] = n2I. Let W be the decoding matrix at destination D to recover transmitted signal :

   Equation (14) is the equation of Wiener Filter , substitute

   (14) into (5), and apply matrix inversion lemma, the MSE expression can be written as

   At destination apply minimum mean square error (MMSE) criterion for precoding, expression of MSE is given as M =

   MW=Wopt

   H 1 1

   (15)

   E[(s s)(s s)H]. Substituting (3) and (4), MSE expression can be further written as

   From (7 and (15), the MSE expression of kth user at

   destination is given by

   k,s k

   where

   n

   n

   Upper bound of SNR of kth user in (8) can be given by applying Cauchy Schwarz inequality

   H H 1

   H H 1

   k,s

   k,s

   k

   k

   MSE of kth user is given by from (16) and (17) now SNR of kth user at destination is

   kk k

   kk k

   Mkk = wH H22WrrH11Wss(H22WrrH11Wss)H wkk + wH Cnnwk

   written as

   2 H

   2 H

   n k It can be seen from (15) MSE is a complicated function

   where wk and wk,s are the kth column of decoding matrix

   { }

   { }

   W and precoding matrix Ws respectively and . is real part of variable inside the bracket. From (3) and (4), SNR at kth user is defined as

   of precoding matrices Ws and Wr at source S and relay R and trace of MSE expression is non-convex in Ws and Wr. To find optimal precoder Ws and Wr which diag- onalize the MSE expression in (15), apply singular value decomposition for channel matrices at source S and relay R

   k =

   k

   k

   k

   , k = 1, . . . , Ms (8)

   ×

   ×

   ×

   ×

   2i

   2i

   i

   i

   n

   n

   and 2 are non-diagonal matrices of dimensions Mr Ms and Md Mr, respectively. U1, U2, V1, and V2 are unitary matrices calculated from SVD of H1 and H2. To diagonalize the trace of MSE optimal choices for Ws and Wr is given as

   1. Secondary Problem

      As seen in (25) to (28), design parameter s appears only in (27) and (28), so secondary problem can be written as:

      s

      s

      s

      s

      s

      s

      1

      1

      s

      s

      1

      1

      (19)

      min 1[n21i1i 1]1(2 + 1ii) (29)

      i

      i

      1

      1

      (20)

      i pt

      i

      (30)

      Substituting (19) and (20) into (15), trace of MSE is given as

      This is convex optimization problem and whose proof and solution are given below

      1 r r

      (21)

      Solution: Second derivative of (29) is given by

      2

      2

      2 2

      3 2 2 1

      H 2 1 2

      2 = 2(n1ii i)

      n 1i2i i(1 + i) (31)

      Substitute (19) and (20) into (12) and (13), power budget at source S and relay R is given by

      tr{s} pt (23)

      (29) would be convex in i only when (31) is positive and

      (31) would be positive only when n21ii > i. Therefore,

      Another constraint for secondary problem can be defined as

      n

      n

      1i

      1i

      i > 2 1i (32)

      2 2 H

      2 2 H

      From (22) Problem can be formulated as

      From (22) Problem can be formulated as

      The Lagrangian of secondary problem is written as

      2i

      2i

      n

      n

      L(i) = 1[n21ii 1]1i(2 + 1ii)

      i

      i

      i

      i

      min

      {s,}

      tr{(I + )1} (25)

      +µ[ i pt] i(i 2 1i) (33)

      i

      i

      i

      i

      i

      i

      i

      i

      n

      n

      1i

      1i

      tr{s} pt (27)

      After applying KKT condition, solution of problem is given by

      = 2 1[ + 1 (1 + )µ1] (34)

      i

      where

      n 1i i

      1i 2i i i

      1 1

      n

      n

      1i

      1i

      2i

      2i

      n

      n

      1i

      1i

      (28)

      Constraints (25), (26), and (27) can be verified to be convex

      in and s, however (28) is not convex in and s jointly.

      in and s, however (28) is not convex in and s jointly.

      Optimal solution of such problem can not be found directly, therefore it would be better to use decomposition methods to solve this problem.

      Decomposition is an approach to solve a problem by break- ing it into smaller problems and solve each problem separately [11]. When there are multiple variables and constraints in any optimization problem, it can be solved by decomposing into two smaller problems: primary and secondary. Primary prob- lem then accords the secondary problem using joint variables. In primal decomposition problem optimization is done over variables, e.g., minu,vf(u, v) = minv minu f(u, v). This

      method decomposes (25) to (28) into primary problem with

      method decomposes (25) to (28) into primary problem with

      µi = [ 2 11i(1 + i)]2[pt 2 1i]2.

      i

      i

      i

      i

   2. Primary Problem

   Optimal solution of i obtained in secondary problem is substituted in constraints (28) and optimization problem then is given as:

   min (1 + i)1 (35)

   i

   i

   i i,min (36)

   n

   n

   1i

   1i

   2i

   2i

   i

   i

   i

   i

   t

   t

   n

   n

   1i

   1i

   i

   i

   [ 2 11 (1 + )]2[p 2 1 ]1

   i

   i

   i

   i

   i

   i

   i

   i

   n

   n

   2i

   2i

   i

   i

   r

   r

   a design parameter. In secondary problem solution of s is

   found as a function of , e.g. s = f (). Now, in primarys problem for a given s obtained from secondary problem,

   found as a function of , e.g. s = f (). Now, in primarys problem for a given s obtained from secondary problem,

   + 2 1 p

   i

   i

   (37)

   i

   i

   optimal opt is calculated.

   Let 1i, 2i, i and i are the diagonal (i, i)th entry of matrices 1, 2, and s respectively.

   similar to secondary problem, it can be easily varied that (35) is also convex in nature [12] and can be solved efficiently using available software packages [13].

4. RESULT AND DISCUSSION

   The performance of joint precoding scheme is analyzed in this section with the help of simulation results.To solve the primary problem which is convex in nature, interior point method is used. Relay and Destination can estimate the channel using pilot signals transmitted by Source and relay and fed it to Source by using feedback channel directly or through relay node.

   1. Simulation Setup

      × ×

      × ×

      The channel matrices H1 and H2 are considered as Rayleigh fading channels, i.e., elements of each channel matrix is independent identically distributed and having zero mean. Transmission power at source and relay is given by pt = Ms (SNR)s and pr = Mr (SNR)r. Noise power at source and relay is taken 1, i.e., unit variance. Average SNR at source and relay are same, i.e., (SNR)s = (SNR)r and Ms = Mr = Md = M . Minimum SNR requirement of 0 dB is assumed at destination for QoS aware transmission.

   2. Simulation Result

   Fig. 2. BER performance of joint precoding scheme.

   Fig. 2 shows the BER performance of proposed joint precoding design. Here results of BER versus average (SNR)s have been plotted for M = 2 and M = 3 with various SNR requirement (in dB) min = 0 and min = 4. Results show that in both case M = 2 and M = 3 for a given min, BER decrease with increase in (SNR)s. Furthermore, for a given (SNR)s, if min increases BER performance also improves but upper bound for minis given by (17). Reason for this is explained as if optimal value of calculated from proposed scheme does not satisfy the minimum SNR requirement min, transmission will be seized and outage may occur. Therefore, higher min means higher or higher SNR

   Fig. 3. BER Comparison of proposed precoding scheme with relay precoding scheme.

   is given to user, therefore, BER performance is also improved.

   M

   M

   Fig. 3 compares the proposed joint precoding design with relay only precoding in which source precoder given in (19) is fixed by s = pt I. Moreover, to understand the behavior of proposed design, it has been simulated for both ZF (Zero Forcing) and MMSE receiver. It is clear from figure proposed precoded design outperforms the relay only prcoding design because the proposed design finds the optimal precoders at both source and relay. However, in relay only precoding scheme, optimal value of precoder is calculated at relay node only and source node precoding is dene by assuming that all antennas are having equal power. It can be seen in Fig. 3, for BER = 103, performance of proposed precoding (MMSE) scheme has SNR gain of 5 dB from relay only precoding scheme.

5. CONCLUSION

This paper presents a QoS guaranteed joint precoding design scheme for MIMO relay network. Proposed precoding method has been designed to minimize the MSE while satisfying the minimum SNR constraint for users at destination. Since MSE is a complicated function of precoding matrices, and non-convex in nature. To solve this non-convex problem, it is divided into primary and secondary problem which are now convex optimization problem and can be solved easily. Comparison is also done for proposed joint precoding method with relay precoding method where all antennas are having equal power at source and it can be concluded that proposed scheme has better BER performance than relay only precoded scheme. In the proposed work optimal precoders are obtained with in transmission power budget while satisfying QoS for each user.

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