May 18, 2015 - per Antenna Power Constraint and Perfect CSI. C. Manikandan1 ..... Springer Science+Business Media New York, 75(2), pp. 1251-1263, 2013.
International Journal of Mathematical Analysis Vol. 9, 2015, no. 31, 1519 - 1528 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.53103
Transceiver Design for SU-MIMO System with Improper Modulations Using per Antenna Power Constraint and Perfect CSI C. Manikandan1 , P. Neelamegam1 , A. Srivishnu1 and B. Sabari Ganesh2 1
School of EEE, SASTRA University, Thanjavur, Tamil Nadu, India 2
Tata Consultancy Services, Chennai, Tamil Nadu, India
c 2015 C. Manikandan et al. This article is distributed under the Creative Copyright Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract This paper addresses the problem of joint transceiver designs for single user-Multiple-input and Multiple-output system(SU-MIMO) employing one dimensional improper modulations such as binary phase shift-keying and M-ary amplitude shift-keying(M-ASK) using per antenna power constraint. Conventionally, most of the transceiver designs are based on the sum power constraint (SPC).However, in practical SPC is not realistic as it does not take into account of power constraint of an individual power amplifier. To solve this problem, under perfect CSI, per antenna power constraint is formulated into an optimization problem giving a novel joint linear transceivers that minimizes the mean square error at the output of the decoder. The simulation results show that the proposed scheme has a near-optimum performance considering practical constraints.
Keywords: MIMO ,PAPC, CSI, TMSE, SPC, BER.
1
Introduction
Multiple-Input Multiple-Output (MIMO) architecture is a multiple-antenna technology for wireless communication systems. By using multiple anten-
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nas the concept of spatial multiplexing can be exploited which is, simultaneously transmitting independent information sequences over multiple antennas, thereby increasing the data rate by the factor of number of antennas used [1]. With MIMO, we can employ spatial diversity which in fact, reduces the bit error rates i.e. by transmitting the same information sequences over multiple antennas. In addition to higher data rates and low bit error rates MIMO technique can be adapted to improve SNR at the receiver. This is achieved by means of adaptive antenna arrays. Using beam-forming techniques, beams can be steered in desired directions nullifying the interference. Some of These techniques do not require the Channel state information at the transmitter(CSIT). But, with the presence of knowledge of CSIT additional gain performance can be achieved [2] and [3]. In practical, to design a precoder or joint transceiver, it requires both channel state information at the transmitter and channel state information at the receiver (CSIR), i.e., at both ends of the MIMO systems. Various performance measurements have been considered to obtain a best design strategy. Among them TMSE condition provided better Bit Error Rate performance [4] and [5].Optimal power allocation is an important challenge posed by industries when transmitting over multiple antennas in a multipleinput multiple-output (MIMO) system. In general, the power is allocated based on the sum power constraint (SPC), at the transmitter.SPC does not take into account the power constraint of individual power amplifier(PA) at each transmit antenna. It is recommended to take the PA into account when designing the transmitters at base stations. The importance of carefully taking the hardware into consideration when designing communication systems has recently been put forward by Industries. The newly proposed p norm constraint jointly meets both the per-antenna power constraint (PAPC) and the Sum Ppower Constraint(SPC) to bound the dynamic range of the power amplifier at each transmit antenna [6]. The improved minimum TMSE design for improper signal constellations was recently proposed in [7] and shown to give superior BER performance than the conventional design in [8]. A Transceiver is designed with Improper modulations using SPC in [9] and Joint optimum precoder and decoder is designed using per antenna constraint in [10].However, to the best of our knowledge, no attention has been paid to the optimum joint linear transceiver design for the SU-MIMO systems which employ one dimensional improper modulation techniques that minimize the sum of symbol estimation errors subject to both sum and individual power constraint, with the perfect CSI at both the transmitter and the receiver.
Transceiver design for SU-MIMO system
2
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Problem formulation
The conventional SU-MIMO Transceiver designs are derived by minimizing the following TMSE e = E[kˆ s − sk2 ] = E[k(GHF s + Gn) − sk2 ]
(1)
This design criteria is optimum in the systems with proper modulations, such as M-QAM and M-PSK for which E[ssT ] = 0.However, in improper modulation schemes like BPSK and M-ASK, the conventional design gives suboptimum outputs because in improper constellations E[ssT ] 6= 0 .In Improper modulation schemes, sˆ = R(GHF s + Gn)
(2)
With one dimensional improper modulations,The symbol estimation error is defined as follows, e = sˆ − s (3) where sˆ = GHF s + Gn
(4)
With the newly defined error vector,the TMSE can be calculated as follows E[kek2 ] = E[kˆ s − sk2 ]
(5)
= E[kR(GHF s + Gn) − sk2 ]
(6)
= E[k(GHF s + G∗ H ∗ F ∗ s∗ )/2 + (Gn + G∗ n∗ )/2 − sk2 ]
(7)
The TMSE can be calculated as follows, T r E [0.5(GHF s + G∗ H ∗ F ∗ s∗ ) + 0.5(Gn + G∗ n∗ ) − s] H H H H T T T T H H T T H [0.5(S F H G + S F H G ) + (n G + n G ) − s ]
(8)
Taking statistics of the channel,noise and data into consideration, we have E[ssH ] = E[ssT ] = IB , E[nnH ] = σn2 INT andE[n] = E[nnT ] = E[n∗ nH ] = 0 Applying these facts and after some manipulations, we get = T r 0.25(GHF F H H H GH + GHF F T H T GT + G∗ H ∗ F ∗ F H H H GH + G∗ H ∗ F ∗ F T H T GT ) ∗ ∗ ∗ H H H T T T 2 H ∗ T −0.5(GHF + G H F + F H G + F H G ) + IB + 0.25σn (GG + G G ) (9)
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The design goal is to find optimum F and G which minimize the mean square error subject to SPC and per antenna power constraint. Mathematically it can be defined as min E[||e||2 ] subject to [T r(F F H )p ]1/p ≤ α F,G
(10)
Here in the above equationα is a constant and p is also a constant.The value of p is calculated based on the SPC and PAPC.Mathematically p is defined as follows, αB =β β 1/p
αB αB = β ; B 1/p = 1/p B β αB ln β ln B 1/p = ln ; p = αB β ln β ;
(11)
If p=1 then the constant α will be equal to the SPC, β.when p = ∞ then it corresponds to an EPA scheme with the per antenna constraint α = Bβ , where B is the number of bit streams.For p in the interval 1 < p < ∞ , p-norm constraint sufficiently meets both the SPC and PAPC. The formulation in the equation can be referred to as improved minimum TMSE design for SU-MIMO systems employing one dimensional improper modulations with per antenna power constraint. To obtain the solution of the above problem, form the Lagrangian. Lagrangian form, 2 H p 1/p η = E kek + µ [T r(F F ) ] − α (12) µ is the Lagrange multiplier. Taking the derivatives of η with respect to F and G, the associated KarushKuhn-Tucker(KKT) conditions can be derived by using the cyclic property of the trace function. ∂η ∂G
=0
0.25[G∗ (HF F H H H )T + GHF F T H T + GHF F T H T + G∗ H ∗ F ∗ F T H T ] −0.5[F T H T + F T H T ] + 0.25σn2 (G∗ + G∗ ) = 0
(13)
0.25[G∗ H ∗ F ∗ F T H T + 2GHF F T H T + G∗ H ∗ F ∗ F T H T ] −0.5[2F T H T ] + 0.5[σn2 G∗ ] = 0
(14)
Transceiver design for SU-MIMO system
G∗ H ∗ F ∗ F T H T + GHF F T H T − 2F T H T + σn2 G∗ = 0
1523 (15)
Taking complex conjugates on both sides, we get GHF F H H H + G∗ H ∗ F ∗ F H H H + σn2 G = 2F H H H
(16)
Now setting, ∂η ∂F
=0
0.25[(H H GH GHF )∗ + 2H T GT GHF + H T GT G∗ H ∗ F ∗ ] − 0.5[H T GT + H T GT ] +µ[[T r(F F H )p ](1/p)−1 ((F F H )T )p−1 F ∗ ] = 0
(17)
where the partial derivative of [T r(F F H )p ]1/p with respect to F is obtained using the Chain rule of Matrix differentiation and the following property. ∂g(U ) ∂F
) T ∂g(U ) = T r[( ∂g(U ) ∂F ] ∂U
) T r( ∂g(U )= ∂F ∂T r(F p ) ∂F
∂T r(g(U )) ∂F
= p(F T )p−1
Again, taking the complex conjugates of both sides, we get H H GH GHF +H H GH G∗ H ∗ F ∗ +2µ[[T r(F F H )p ](1/p)−1 ((F F H )T )p−1 F ∗ ]∗ = 2H H GH (18) H Next, by post multiplying (16) by G GHF F H H H GH + G∗ H ∗ F ∗ F H H H GH + σn2 GGH = 2F H H H GH
(19)
Next, by pre multiplying (18) by F H F H H H GH GHF +F H H H GH G∗ H ∗ F ∗ +2µF H [[T r(F F H )p ](1/p)−1 ((F F H )T )p−1 F ∗ ]∗ = 2F H H H GH Equating the equations (18) and (19), we get σn2 GGH = 2µF H [[T r(F F H )p ](1/p)−1 ((F F H )T )p−1 F ∗ ]∗ ∗ H p 1/p H T p−1 ∗ H [T r(F F ) ] ((F F ) ) F = 2µF T r(F F H )p considering α = [T r(F F H )p ]1/p we get,
(20)
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σn2 GGH
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α = 2µ T r(F F H )p
(F T F ∗ )p T ∗ F F (F T F ∗ )
∗
Applying trace on both sides, we get µ=
σn2 T r(GGH ) 2α
where, α = [T r(F F H )p ]1/p
(21)
An iterative procedure is developed to find the solutions, and G∗ = GRe − jGIm HF F H H H = ARe + jAIm H ∗ F ∗ F H H H = BRe + jBIm 2F H H H = CRe + CIm
G = GRe + jGIm
(22) (23) (24) (25)
A,B,C represents the terms in the equation AIm + BIm ARe + BRe + σn2 IN R (CRe CIm ) = (GRe GIm ) BIm − AIm ARe − BRe + σn2 IN R (26) The above equation can be rewritten as, −1 ARe + BRe + σn2 IN R AIm + BIm (GRe GIm ) = (CRe CIm ) BIm − AIm ARe − BRe + σn2 IN R (27) Similarly, we define and F ∗ = FRe − jFIm H H GH GH = PRe + jPIm H H GH G∗ H ∗ = QRe + jQIm 2H H GH = RRe + RIm
F = FRe + jFIm
(28) (29) (30) (31)
Assuming k= [[T r(F F H )p ](1/p)−1 ((F F H )T )p−1 ]∗ P,Q,R represents the terms in the equation. RRe PRe + QRe + 2µkIN T QIm − PIm FRe = (32) RIm PIm + QIm PRe − QRe + 2µkIN T FIm The above equation can be rewritten as −1 FRe PRe + QRe + 2µkIN T QIm − PIm RRe = FIm PIm + QIm PRe − QRe + 2µkIN T RIm (33) Based on the above expressions, An iterative approach is obtained for the precoder matrix F and decoder matrix G using per antenna power allocation.
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2.1
Iterative Algorithm
Step1:Initialize F = F0 the upper matrix of F0 is choosen to be scaled identity, while the remaining entries of F0 are set to zero. Step2:Update G using (27). Step3:Update µ using (21). Step4:Update F using (33) if [T r(F F H )p ]1/p > α.Scale such that[T r(F F H )p ]1/p = α. Step5:If [T r((Fi − Fi−1 )(Fi − Fi−1 )H )p ]1/p is sufficiently small(less than 10− 4 ),stop.otherwise go back to Step 2. Here Fi , (Fi−1 ) denotes F in the i-th,(i-1)-th iteration.
3
Results and Analysis
The simulation results of the above problem formulation are shown in this section, the numbers of transmit and receive antennas are set to be NT = NR = 4.In all figures, the signal-to-noise ratio is defined as SNR= PσT2 ,the value n of SNR is set to be 26.016 dB.It has to be pointed that number of iterations needed for obtain the precoder and decoder matrices depends on the value of the objective function which reduces at every iteration and the proposed algorithm converging point.It was observed that 4 to 6 iterations were adequate enough in all simulations.
(a)
(b)
Figure 1: Performance comparison of SPC and PAPC (a)when p=4.12 for bit streams,B=4 (b)when p=2.36 for bit streams,B=4 First, figure 1(a)compares the performance of the transceiver design in [9] with that of proposed tranceiver design for BPSK and 4-ASK modulations when perfect CSI is available at both the transmitter and the receiver.It is
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(a)
(b)
Figure 2: Performance comparison of SPC,PAPC and PAPC with different values of B (a)when p=4.12 for bit streams,B=3 (b) when p=4.12 for bit streams B=4 and B=3 clear from the figure that performance of proposed per antenna power constraint(PAPC) is near to the optimum SPC. Figure 1(b) compares the performance similar to figure 1(a) with p=2.36 instead of p=4.12 ,As the value of p moves from p =4.12 to p =2.36,the performance of proposed per antenna power constraint(PAPC) further improved For the figure 1 (a)value of p is calculated as follows by considering B=4,PAPCα = 1.1W ,SPCβ = 3.16W we get p=
ln(3.16) ln( (1.1)(4) ) 3.16
= 4.12.
The PAPC, α can be chosen from the interval [ Bβ , β] For figure 1 (b) value of p is calculated as follows by considering B=4,PAPCα = 2.8W ,SPCβ = 6.31W we get p=
ln(6.31) ln( (2.8)(4) ) 6.31
= 2.36
In figure 2(a) compares the performace of the conventional transceiver design with that of proposed transceiver design for the case of bit streams B=3 instead of B=4.It is observed both realistic PAPC and un-realistic SPC gaining better BER performance than for case B =4 In figure 2(b) compares the performance of the proposed transceiver design with bit streams B=4 and B=3. The purpose of this performance analysis
Transceiver design for SU-MIMO system
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is that, the proposed design enjoys even further gain if the number of data streams are reduced from B=4 to B=3. When B=3 is used, the performance improves because of the Spatial diversity of MIMO systems.
4
Conclusion
In this paper, a joint linear transceiver with one dimensional improper modulations for MIMO channels is obtained using the p-norm constraint for perfect CSIT.The performance of the proposed power allocation scheme using p-norm constraint has a performance closer to the optimal SPC for small values of ’p’.Hence the proposed scheme gives a near optimum solution when considering the dynamic range of the power amplier at each stage.Finally, it is pointed out that the design proposed in this work can be extended to the Multi-User MIMO(MU-MIMO)systems.
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[7] Xiao, P., and Sellathurai, M,”Improved linear transmit processing for single-user and multi-user MIMO communications systems” IEEE Transactions on Signal Processing, 58(3), pp. 1768-1779, 2010. http://dx.doi.org/10.1109/tsp.2009.2037347 [8] Ding, M., and Blostein, S,”MIMO minimum total MSE transceiver design with imperfect CSI at both ends” IEEE Transactions on Signal Processing, 57(3), pp. 1141-1150, 2009. http://dx.doi.org/10.1109/tsp.2008.2008542 [9] M. Raja., P. Muthuchidambaranathan., Ha H. Nguyen,”Transceiver Design for MIMO Systems with Improper Modulations” Wireless Personnal Communications, Springer Science+Business Media, LLC. 68(2), pp. 265-280, 2011. http://dx.doi.org/10.1007/s11277-011-0451-z [10] A. Merline, S. J. Thiruvengadam,”Design of Optimal Linear Precoder and Decoder for MIMO Channels with Per Antenna Power Constraint and Imperfect CSI” Wireless Personnal Communications, Springer Science+Business Media New York, 75(2), pp. 1251-1263, 2013. http://dx.doi.org/10.1007/s11277-013-1421-4 Received: March 31, 2015; Published: May 18, 2015
