Power Efficient Pilot Symbol Power Allocation under Time-variant Channels

Transcription

1 Power Efficient Pilot Symbol Power Allocation uner Time-variant Channels Michal Šimko, Paulo S. R. Diniz,QiWang an Markus Rupp Institute of Telecommunications, Vienna University of Technology, Vienna, Austria Department of Electronics - School of Engineering, Universiae Feeral o Rio e Janeiro, Rio e Janeiro, Brazil msimko@nt.tuwien.ac.at Web: Abstract Current Multiple Input Multiple Output (MIMO) Orthogonal Frequency Division Multiplexing (OFDM) base systems for wireless communications enable to ajust power raiate at the pilot symbols. Uner time-invariant channels, a power increase at the pilot symbols results in improve quality of the channel estimate. However, uner time-variant channels, this is not necessarily the case. Due to the ecrease temporal channel correlation, the channel estimation performance becomes saturate. If uner time-variant channels more power is assigne to the ata symbols, such approach increases inter-layer interference ue to the imperfect channel knowlege at the receiver. In this paper, we show how to istribute power among ata an pilot symbols in a power efficient way. Using our propose metho, up to 5% of the transmit power can be save in a 4 4 Long Term Evolution (LTE) ownlink transmission at a Signal to Noise Ratio (SNR) of B. Inex Terms LTE, Power istribution, OFDM, MIMO. I. INTRODUCTION Current systems for cellular wireless communication are esigne for coherent etection. Therefore, channel estimation is a crucial part of a receiver. UMTS Long Term Evolution (LTE) provies the possibility to change the power raiate at the pilot subcarriers relative to that of the ata subcarriers. Clearly, this aitional egree of freeom in the system esign provies potential for optimization. A. Relate Work Many researchers have realize, that the power assigne to the pilot symbols has significant influence on the system s performance [, ]. There has been many attempts to answer the arising question of how to istribute the available power between ata an pilot symbols. However, many of these attempts are limite only to a certain moulation alphabet [], or specific channel estimators [3]. The authors of [4] erive the optimal power istribution between pilot an ata symbols for time-invariant channels uner imperfect channel knowlege. The optimal istribution of power turne out to be inepenent of the Signal to Noise Ratio (SNR) an channel realizations. In [5], the work of [4] was extene to multi enoeb scenarios where the interference from neighboring enoebs was inclue. Due to the LTE pilot symbol esign, the pilot symbols from neighboring enoebs are overlapping with the ata symbols in the enoeb of interest, which complicates the optimization problem. In [6], this work has been extene to answer the question of how to optimally istribute power between pilot an ata symbols uner time-variant channels. The authors showe that performance of Least Squares (LS) an Linear Minimum Mean Square Error (LMMSE) channel estimators saturates with increasing Doppler sprea. Therefore, it might seem that with increasing Doppler sprea, more power shoul be assigne to the ata symbols. This is not true, because this woul increase the inter-layer interference of the Multiple Input Multiple Output (MIMO) system cause by the imperfect channel knowlege. B. Contribution In this paper, we relax our previous constraint of a fixe transmit power [4 6] an introuce only a higher boun for the maximum available transmit power. By oing so, the optimization problem becomes more complicate, but allows to ecrease the total transmit power while keeping the system s performance almost unchange. A novel formulation of the optimization problem elivers a power efficient solution for the power istribution among pilot an ata symbols uner time-variant channels. As with our previous work, all ata, tools, as well implementations neee to reprouce the results of this paper can be ownloae from our homepage [7]. In this paper, we analytically solve the optimization problem of power efficient optimal power allocation for a MIMO system with a Zero Forcing (ZF) equalizer uner imperfect channel state information. In contrast to [4 6], we utilize the actual transmission power as the cost function instea of the post-equalization Signal to Interference an Noise Ratio (SINR), that is use as a boun for efining a solution of interest. The remainer of the paper is organize as follows. In Section II, we escribe the mathematical system moel for transmitting pilots an ata over a MIMO channel. In Section III, we briefly escribe the post-equalization SINR expression for ZF equalizers with imperfect channel knowlege. We formulate the optimization problem for optimal pilot symbol

2 power allocation in Section IV. Finally, we present LTE simulation results in Section V an conclue our paper in Section VI. II. SYSTEM MODEL In this section, we briefly point out the key aspects in the LTE stanar that are relevant to this paper an introuce a transmission moel suitable for our further erivation. A receive Orthogonal Frequency Division Multiplexing (OFDM) symbol in the frequency omain at the n r -th receive antenna can be written as ỹ nr = N t n t= H nt,n r x nt + ñ nr, () where H nt,n r C N sub N sub represents the channel matrix in the frequency omain between the n t -th transmit an n r - th receive antennas. The transmitte signal vector is referre to as x nt, the receive signal vector as ỹ nr. The vector ñ nr C Nsub enotes aitive white zero mean Gaussian noise with variance σn on antenna n r. In case of a timeinvariant channel, the channel matrix H nt,n r appears as a iagonal matrix, whereas a time-variant channel forces the channel matrix H nt,n r to become non-iagonal. These noniagonal elements inicate that the subcarriers are not orthogonal anymore, leaing to the so-calle Inter Carrier Interference (ICI). The vector x nt has N sub entries, corresponing to the number of non-zero subcarriers. Let us enote the number of pilot symbols an the number of precoe ata symbols by N p an N, respectively. Specifically, the vector x nt C Nsub in Equation () comprises the precoe ata symbols x,nt C N an the pilot symbols x p,nt P Np from the set of all possible pilot symbols P efine in LTE, at the n t -th transmit antenna place by a suitable permutation matrix P x nt = P [ x T p,n t x T ] T,n t. () On subcarrier k of the ata symbol vector x,nt, the precoing process can be escribe as [x,,k x,nt,k] T = W k [s,k s,k s Nl,k] T, (3) where x,nt,k is a precoe ata symbol at the n t -th transmit antenna port an the k-th subcarrier, W k C Nt N l is a unitary precoing matrix at the k-th subcarrier an s nl,k D is the ata symbol of the n l -th layer at the k-th subcarrier. Here, D is the set of available moulation alphabets. In LTE, three ifferent sets can be use, namely 4 Quarature Amplitue Moulation (QAM), 6 QAM an 64 QAM. In orer to obtain ata symbol vectors x,nt, one has to stack ata symbols x,nt,k obtaine via Equation (3) at a specific antenna n t into a vector. For the erivation of the post-equalization SINR, we will use a MIMO input-output relation at the subcarrier level, given as: y k = H k,k W k s k + n k + H k,l W l s l. (4) } {{ } ICI Matrix H k,l C Nr Nt enotes the MIMO channel matrix between the k-th an l-th subcarrier. This matrix is obtaine by means of the Fourier transformation of the physical channel in the time omain. In this work, we assume Jake s spectrum moel. In LTE, the precoing matrix can be chosen from a finite set of precoing matrices [8]. The vector s k consists of the ata symbols of all layers at the k-th subcarrier. Vector n k represents aitive white zero mean Gaussian noise with variance σn at subcarrier k. We enote the effective channel matrix by G k,k = H k,k W k. (5) Furthermore, the average power transmitte on each of the N l layers is enote by σs. The total power transmitte on one ata position is σ, while that on one pilot position is σ p. When the power is evenly istribute between the ata an pilot symbols, we have: σ s = E { s l,k σ = N t N n t= σp = N t N p n t= } = N l, (6) E { x,nt } =, (7) E { x p,nt } =, (8) where N enotes the number of ata symbols, N p the number of pilot symbols an N l the number of layers. III. POST-EQUALIZATION SINR In this section, we consier a time-variant scenario an briefly escribe an analytical expression for the postequalization SINR of a MIMO system using a ZF equalizer base on imperfect channel knowlege. More etails can be foun in our previous work [6]. If perfect channel knowlege is available at the equalizer, the ZF estimate of the ata symbol s k is given as ŝ k = ( G H ) k,kg k,k G H k,k y k. (9) The ata estimate ŝ k efine in Equation (9) results in a postequalization SINR of the m-th layer given as [9, ] (σn + σici ) eh m σ s ( G H k,k G k,k ) em, () where the vector e m is an N l zero vector with a one on the m-th element. This vector extracts the signal on the corresponing layer m after the equalizer. The variable σici represents the ICI power, that is σici = E H k,l W l s l. () Let us procee to the case of imperfect channel knowlege. We efine the perfect channel as the channel estimate plus the error matrix ue to the imperfect channel estimation H k,k = Ĥk,k + E k,k, ()

3 where the elements of the matrix E k,k are ranom variables, statistically inepenent of each other each with variance σe. Inserting Equation () in Equation (4), the input-output relation changes to ) y k = (Ĥk,k + E k,k W k s k + n k + H k,l W l s l. (3) Since the channel estimation error matrix E k,k is unknown at the receiver, the ZF solution is given again by Equation (9) with channel matrix H k,k replace by its estimate Ĥk,k, that is known at the receiver [] ŝ k = (ĜH k,k Ĝ k,k ) ĜH k,k y k, (4) with matrix Ĝk,k being equal to Ĥk,kW k. Applying a ZF equalizer, Equation (4) leas to the SINR on the m-th layer [6] σ s ). (5) (σn + σici + σ eσ ) eh m (ĜH k,kĝk,k em Note, that in practice variables σ s,σ, σ ICI, an σ n nee to be replace by their estimates. IV. POWER ALLOCATION In this section, we show how to istribute available power among ata an pilot symbols in power efficient way uner time-variant channels. In contrast to [4 6], in the optimization problem formulation, we relax the constraint of a fixe transmit power an efine only a higher boun for it. This enables to optimize the amount of consume power while keeping the system s performance almost unchange. Although the provie results are shown in the context of the current LTE stanar, the presente concept can be applie to any MIMO OFDM base system. Furthermore, we will limit our iscussion only to an LS channel estimator. Note, that base on the results shown in [6], all concepts can be easily applie also to an LMMSE channel estimator. We introuce power ajusting factors c p an c for the pilot an ata symbols, respectively. A variable p off is efine as the power offset between the power of the pilot symbols an the ata symbols, enote by c = p off c p. (6) If we increase the power at the pilot symbol by c p, the noise an ICI epenent parts of the Mean Square Errors (MSEs) of an LS channel estimator ecreases by the same factor c p. Therefore, the new MSE after power ajustment can be state as ( ) σ σ e = c n + σici e c +, (7) p where c e an are real constants that etermine the performance of the channel estimators [6]. Plugging the variables c an c p into Equation (5), we obtain the SINR expression at layer m with ajuste power of the pilot symbols σ sc (σn + σici + σ eσ c ) eh m ( ). (8) G H k,k G k,k em Inserting Equation (7) into Equation (8) an simplifing the expression, we obtain the SINR: N l (σn + σici ) eh m ( G H k,k G k,k ) ( em f ( ), c p,c) + (9) for which the power allocation function f ( c p,c ) is given as f ( c p,c ) = c + c e c. () p The constant is proportional to the channel saturation coefficient an is given as = σn + σici. () Note, that Equation () is inepenent of channel realization, noise variance, ICI power an even user velocity. Let us focus on the term in Equation (9). This term is always positive. It thus becomes obvious that it causes the overall limitation of the post-equalization SINR. Let us consier a situation for the moment when f ( c p,c). In this case, the post-equalization SINR is mainly etermine by the value of, an almost inepenent of the choice of c p an c. Therefore, a formulation of the optimization problem similar to [6] oes not lea to power efficient values of power istribution among pilot an ata symbols: minimize f ( c p,c ) () c p,c subject to c pn p + c N N + N p In this paper, in contrast to the previous approach we rather try to minimize the actual transmit power while keeping the power allocation function small enough, so that it oes not worsen the post-equalization SINR compare to the case when the whole available power is use. The optimization problem can be efine as follows: minimize c N + c pn p (3) c p,c subject to f ( c p,c ) <a + f ( c p, c ) <c p < N + N p N p <c < N + N p N c pn p + c N N + N p The first conition from Equation (3) constrains the power allocation function, so that it oes not become larger than the variable multiplie by a real constant a plus a power

4 allocation function evaluate at c p an c, which are the optimal values erive in [6]. The purpose of the constant a is to ensure that the value of the power allocation function is much smaller than the variable in case of channel estimator saturation f ( c p,c). The secon an thir conitions from Equation (3) warrant, that at least some power is assigne to the pilot an ata symbols, respectively. At the same time, the power assigne to the ata an pilot symbols is not larger than the maximum available power. The last conition upper bouns the sum transmit power of the pilot an ata symbols by the maximum available power. We also assume, that N p +N is constant. This assumption is fulfille in common systems for wireless communications. Due to the simplicity of the cost function, the state problem was solve using a full search metho. Figure isplays an example of cost function [B] c p Fig.. Cost function c N + c pnp for a SISO system applying an LS channel estimator at a user velocity of 5 km/h an SNR = B. a cost function c N + c pn p for SISO transmission scheme utilizing an LS channel estimator at user spee of 5 km/h an SNR = B. In this case, by the solving optimization problem efine in Equation (3), variables c p an c were chosen by algorithm as c p.6 an c.7, which results in p off 5.6 B an actual transmit power saving of aroun %. In this example, we set a = 6, this warranties postequalization SINR loss of maximum.6 B. V. SIMULATION RESULTS In this section, we present simulation results an iscuss the performance of LTE system using ifferent pilot symbol powers. All results are obtaine with the LTE Link Level Simulator version r89 [, 3], which can be ownloae from The Vienna LTE simulator is a Matlab implementation of all physical layer proceure like coing, rate matching [4], synchronization [5, 6], timing estimation [7] etc. All ata, tools an scripts are available online in orer to allow other researchers to reprouce the results shown in the paper [7]. Table I shows the most important simulator settings. Figure shows throughput over user velocity for various number of utilize antennas for user velocity at SNR= B. c TABLE I SIMULATOR SETTINGS FOR FAST FADING SIMULATIONS Parameter Value Banwith.4 MHz Number of transmit antennas,, 4 Number of receive antennas,, 4 Receiver type ZF Transmission moe Open-loop spatial multiplexing Channel type ITU VehA [8] The blue ashe line epicts an LTE system with no power istribution among pilot an ata symbols. The re continuous line represents a system with our novel propose power efficient power istribution among pilot an ata symbols. The amount of use power when utilizing our propose power efficient power istribution is shown in Figure 3. In case when no power istribution is applie, the whole available transmit power is utilize. For example, consiering a 4 4 transmission system at a user spee of 5 km/h, when using our propose power istribution algorithm more than 5% of the total transmit power can be save compare to a system with no power istribution, while achieving the same throughout. For the simulate curves we calculate the 95% confience intervals, which are plotte in gray color. Their size inicates a high quality of the simulation results. At lower user velocities, the system utilizing power istribution outperforms the system without any power istribution. This is consistent with results from [4 6]. At higher user velocities the system with power efficient power istribution experiences small throughput loss, at the same time utilizes less transmit power than the system not utilizing power istribution. Note, that the throughput loss can be further ecrease by changing the variable a in the first conition of Equation (3). In the shown simulation we set a = 6, this warranties post-equalization SINR loss of maximum.6 B. If no SINR loss is esire, the variable a has to be set to zero, in this case, no power savings are achieve. Figure 3 epicts the actual transmit power for various antenna setups over user velocity at SNR = B when using our propose solution. With increasing user velocity also the amount of power use for the transmission ecreases. This behavior can be explaine by the fact, that the performance of the channel estimators with increasing Doppler sprea become saturate ue to the low temporal channel correlation. Therefore, less power is raiate at the pilot symbols an at the same time the power at the ata pilots is limite, since it woul only increase the inter-layer interference cause by the imperfect channel knowlege. In the previous problem formulation [6], the whole available power ha to be use, however by increasing the power raiate at the ata symbols, inter-layer interference woul be increase, therefore the power raiate at the pilot symbols is increase, with no effect on the quality of the channel estimate ue to the saturation effect with increasing user velocity. The saturation effect is more relevant with increasing SNR, therefore the transmit power is lower with increasing SNR.

5 throughput [Mbit/s] x x 4x4 no power istribution power efficient power istribution user spee [km/h] Fig.. Throughput comparison of various LTE systems without any power istribution an with power efficient power istribution at SNR = B over user velocity. power usage [%] x4 x x user spee [km/h] Fig. 3. Percentage of power use for the transmission of LTE system for various numbers of transmit antennas plotte over user velocity utilizing our propose power efficient power istribution at SNR= B. VI. CONCLUSION In this paper, we answere the question of how to istribute available power among pilot an ata symbols uner timevariant channels in an power efficient way. Compare to the previous work, instea of utilizing the post-equalization SINR as the cost function, in this paper, we utilize the total consume power as the cost function an the post-equalization SINR as one of the conitions of the optimization problem. Up to 5% of the raiate power can be save using 4 4 MIMO LTE ownlink transmission system at SNR = B. This power saving can be explaine by the saturation effect of the state-of-the-art channel estimators (e.g., LS, LMMSE) uner time-variant channels. It simply oes not pay off to raiate more power at the pilot symbols at high Doppler sprea since the state-of-the-art channel estimators show saturation effects at high user velocities. At the same time, increase power at ata subcarriers woul lea to increase inter-layer interference ue to the imperfect channel knowlege. ACKNOWLEDGMENTS The authors woul like to thank the LTE research group for continuous support an lively iscussions. This work has been fune by the Christian Doppler Laboratory for Wireless Technologies for Sustainable Mobility, KATHREIN- Werke KG, an A Telekom Austria AG. The financial support by the Feeral Ministry of Economy, Family an Youth an the National Founation for Research, Technology an Development is gratefully acknowlege. REFERENCES [] C. Novak an G. Matz, Low-complexity MIMO-BICM receivers with imperfect channel state information: Capacity-base performance comparison, in Proc. of SPAWC, Marrakech (Morocco), June. [] E. Alsusa, M. W. Baias, an Yeonwoo Lee, On the Impact of Efficient Power Allocation in Pilot Base Channel Estimation Techniques for Multicarrier Systems, in Proc. of IEEE PIMRC 5, Sept. 5, vol., pp [3] J. Wang, O. Yu Wen, H. Chen, an S. Li, Power Allocation between Pilot an Data Symbols for MIMO Systems with MMSE Detection uner MMSE Channel Estimation, EURASIP Journal on Wireless Communications an Networking, Jan.. [4] M. Šimko, S. Penl, S. Schwarz, Q. Wang, J. C. Ikuno, an M. Rupp, Optimal Pilot Symbol Power Allocation in LTE, in Proc. 74th IEEE Vehicular Technology Conference (VTC-Fall), San Francisco, USA, Sept.. [5] M. 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