On Power Allocation Strategies for Maximum Signal to ... - IEEE Xplore

808

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004

On Power Allocation Strategies for Maximum Signal to Noise and Interference Ratio in an OFDM-MIMO System Antonio Pascual-Iserte, Student Member, IEEE, Ana I. Pérez-Neira, Senior Member, IEEE, and Miguel Angel Lagunas, Fellow, IEEE

Abstract—Orthogonal frequency division multiplexing (OFDM) has been recently established for several systems such as HiperLAN/2 and Digital video/audio broadcasting, due the easy implementation of the modulator/demodulator and the equalizer. Moreover, also increasing interest is currently being put on multiple-input multiple-output (MIMO) channels, based on the use of antenna arrays at both the transmitter and the receiver. Here, we propose two joint beamforming strategies of low computational load for systems combining OFDM and MIMO. The ultimate objective is the maximization of the signal-to-noise and interference ratio (SNIR) over the carriers subject to a total transmit power constraint. Specifically, the maximization of the harmonic SNIR mean and the minimum SNIR over the subcarriers are proposed. The asymptotic behavior of the proposed methods is analyzed to provide a complete comparative and general view of the most relevant and already known transmit power allocation strategies. Finally, a theoretical analysis of the performance degradation of these techniques is carried out for the case in which the channel state information (CSI) is not perfect. Monte Carlo simulation results for the system bit-error rate and performance degradation with imperfect CSI are provided. Index Terms—Antenna arrays, array signal processing, channel estimation, cochannel interference, least mean square methods, minimum bit-error rate (BER), multipath channels, multiple-input multiple-output (MIMO) systems, orthogonal frequency division multiplexing (OFDM), space-time filtering, wireless local area networks (WLANs).

I. INTRODUCTION

R

ECENTLY, an increasing interest has been given to space-time smart antennas to improve the performance of the communication link and the capacity of the scatterer wireless channel, while maintaining the required transmitted power and bandwidth. The simultaneous use of a transmit-receive array of antennas represents a multiple-input multiple-output (MIMO) channel, which can be exploited in several ways attending to different design criteria. For a fixed throughput Manuscript received October 27, 2001; revised April 10, 2003; accepted February 24, 2004. The editor coordinating the review of this paper and approving it for publication is A. F. Molisch. This work was supported in part by the Spanish Government under Project TIC2002-04594-c02-01 (GIRAFA, jointly financed by FEDER) and Project FIT-070000-2003-257 (MEDEA+ A111 MARQUIS). This work was presented in part at the International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 2002. Speech, and signal Processing (ICASSP) 2002. The authors are with the Array and Multichannel Processing Group, Department of Signal Theory and Communications, Universitat Politecnica de Catalunya (UPC), Barcelona 08034, Spain, and also with the Telecommunications Technological Center of Catalonia (CTTC), Barcelona 08034, Spain (e-mail: [email protected], [email protected], [email protected]). Digital Object Identifier 10.1109/TWC.2004.827730

design and when channel state information (CSI) is available at both the transmitter and the receiver, joint beamforming strategies can be applied to maximize the signal-to-noise and interference ratio (SNIR) by choosing the best spatial subchannel or eigenmode for transmission. In [1] this problem is considered, although no co-channel interference (CCI) is assumed. Besides the MIMO channel studies, also the orthogonal frequency division multiplexing (OFDM) [2] modulation has been successfully proposed for many communication and broadcasting systems such as the wireless local area network (WLAN) HiperLAN/2 [3] and the digital video and audio broadcasting (DVB/DAB) systems, due to the easy implementation of the modulator/demodulator and the equalizer. Both MIMO channels and OFDM are suitable for the new user requirements and multimedia applications, which require higher bit rates, capacity and Quality of Service (QoS). In this paper, we analyze the combination of MIMO with OFDM in a fixed throughput design approach. We apply a joint beamforming structure in frequency selective channels and taking into account the contribution of CCI at the receiver side. In [4], this problem is addressed and a technique for designing the beamformers is deduced, so that the SNIR per carrier is maximized, although no power allocation is considered. The approach taken in this paper is a factored design, which means that first the spatial dimension is processed using the best spatial mode, and afterwards the frequency dimension is exploited by means of designing an appropriate power allocation policy. Here, we focus on the power allocation problem and propose new and low complexity strategies. In some previous power allocation techniques, some of the subcarriers may be nulled (see [2] and [5] for examples of OFDM systems considering this fact, although other references will be provided in the following sections of this paper). In that case, the optimum strategy should decide to transmit the information symbols through the remaining active carriers if the throughput is to be maintained. The main problem is that the transmitter must then increase the signalling messages to the receiver to inform about the new bit loading situation. In this paper, we assume that the throughput and the constellation at each frequency are fixed, and so, if a carrier is canceled, then the information allocated to that frequency will be lost. Although the new proposed techniques are ad hoc, they are compared with other already known strategies, showing the improvement in terms of computational load and performance, providing a complete comparative and general view of the state of the art.

1536-1276/04$20.00 © 2004 IEEE

PASCUAL-ISERTE et al.: ON POWER ALLOCATION STRATEGIES FOR MAXIMUM SNIR IN AN OFDM-MIMO SYSTEM

Fig. 1.

809

General system structure and configuration for a MIMO channel and OFDM modulation.

Fig. 2. Pre-beamforming scheme for the r th transmit antenna including the weight operation, the unitary IFFT operation and the cyclic prefix inclusion. Receiver structure based on beamforming for the k th subcarrier.

In a real system, the CSI is known not to be perfect, i.e., the estimates of the channel and the statistics of the interferences have some error. The sensitivity of the techniques to these errors should be taken into account when applying them in a real communication system. This is the reason why, in this paper, we also carry out a theoretical study of the performance degradation for these algorithms in the presence of imperfect CSI. The paper is organized as follows. In Section II, the system and signal models are presented and the power allocation problem is introduced. Section III treats the power allocation design problem proposing two novel strategies (maximization of the harmonic SNIR mean and maximization of the minimum SNIR) and comparing them with other existing techniques. A theoretical analysis of the expected SNIR degradation when the CSI is imperfect is presented in Section IV. Finally, some simulations results and conclusions are obtained in Section V. II. PROBLEM STATEMENT We consider the use of multiple antennas at both sides of a communication system, configuring a MIMO channel. The front-end design, as shown in Fig. 1, is based on beamforming at

both the transmit antennas array (pre-beamforming) and the receive antennas array (post-beamforming). It is assumed a -carriers OFDM modulation [2], [6], where , represents the information symbol transmitted at the th carrier during the th time-block signal or OFDM symbol. We model , , as zero-mean i.i.d. random variables belonging to a fixed finite alphabet (FA) or signal constellation , where stands with a normalized energy: for the mathematical expectation operator. At the transmitter, the modulation consists of a -points unitary inverse fast Fourier transform (IFFT) after the application of the pre-beamforming represented by the vector , where the superscript stands for transpose. The transmit beamvector represents the beamforming applied to the symbols transmitted through the th carrier. This operation is clearly shown in Fig. 2 for the th transmit antenna. After the IFFT, a cyclic prefix (CP) prefix of samples is applied at the beginning of each OFDM symbol; thus the total number of samples in each OFDM . symbol is The channel is assumed to be time invariant, where represents the taps long impulse response between the th

810


transmit and the th receive antenna, with associated frequency , response at the th carrier, and where we have assumed that , i.e., the channel order is lower than or equal to the length of the CP. as the Under the specified assumptions, let us define received sample at the th carrier and the th receive antenna after the CP removal and the -points unitary fast Fourier transform (FFT), which constitutes the demodulation. All the samples at all the antennas can be collected in a single snapshot . vector The estimate of the information symbol at the th carrier is based on a decision over the soft-estimate , which is the output of the receive beamformer represented by applied to the th carrier. This operation is shown in Fig. 2 and is summarized in the following signal model [2], [4]:

.. .

..

.

.. .

(1) where the superscript stands for complex conjugate transpose. is the MIMO channel frequency response matrix at is the noise plus interferences contribution the th carrier, is its at the receiver and associated covariance matrix, that may depend on the frequency. For binary phase-shift keying (BPSK) modulated carriers, the . As it is shown, the decision operator is frequency selective MIMO channel can be modeled as a collection of parallel flat fading MIMO channels. In the following, and without loss of generality, no dependence on the temporal index is used so as to facilitate the notation. and are available at both the If the estimates of can be detransmitter and the receiver, the beamvectors , signed jointly to maximize the SNIR at the decision stage for the th carrier, whose expression is as follows:

. Our objective is to design the proportional to maximizing the SNIR subject to a carrier power beamvector constraint. The solution of this problem has been deduced in [4] and is based on a maximum eigenvector problem. Here, we summarize the result assuming that the transmit power at the th carrier is equal to

(3) is the maximum eigenvalue, is the normalwhere is a scaled version of the normalized ized eigenvector, and . The SNIR at eigenvector to satisfy the constraint the th carrier taking into account this design can be expressed , which means that the final SNIR deas and on the power pends on the channel through the gain . Therefore, as it can be seen, the only role of MIMO is to . To implement the optimal provide diversity through , the power at each carrier has to be known. However, in a real system the transmit power constraint refers to all the available power for transmitting and not to the power for each carrier. In the following section, we treat this design problem assuming different average cost functions. III. POWER ALLOCATION STRATEGIES Now we consider the power allocation problem correat sponding to the distribution of all the available power the transmit side among the carriers according to the CSI, i.e., and . This global power constraint, the matrices formulated in the frequency domain, is expressed as (4) The power distribution among subcarriers depends on the adopted optimization criterion. This section studies two new criteria to deduce an adequate power allocation policy. These alternatives have a low complexity and are asymptotically related to other classical power allocation techniques. Before presenting these two novel strategies, it is interesting to focus the attention on the simplest technique, which is the uniform power allocation, as shown in the following expression:

(2)

(5)

Note that the CSI could be sent to the transmitter using a digital feedback channel, whose design is, however, outside the scope of this paper. The optimum receive beamvector maximizing the SNIR is the whitened matched filter [4], [6], and where whose expression is does not affect the value of the SNIR and can be chosen, for example, to obtain a normalized amplitude of the equivalent at the detector. When considering channel the design of the transmit weight vector, the power restrictions at the transmitter side should be taken into account. The transmitted power through all the antennas for the th carrier is

It can be verified that this strategy maximizes , i.e., the geometric mean of the SNIR, subject to the global power constraint (4); so, we name this technique as GEOM. As it is widely known, if the transmit power tends to infinity, the uniform power allocation tends to the water-filling solution maximizing the capacity [7]: , where is a constant calculated to satisfy the power constraint (4). The main problem of the uniform power allocation is that it does not cope well with frequency selective channels, or in scenarios in which the noise and interferences are not spectrally white. In the following subsections, we present two novel strategies to distribute the power


Fig. 3. Cumulative density functions of the maximum eigenvalues for a MIMO channel with an angular spread of 30 at the transmitter and 15 at the receiver, and a power delay profile corresponding to model A.

811

means of a simulation consisting in the calculation of the maximum eigenvalues of several realizations of a MIMO random channel taking an angular spread of 30 at the transmitter and 15 at the receiver, and the power delay profile of the model A as described in [9]. As it is shown in [10], this criterion is equivalent to the zero forcing (ZF) solution, in which the mean square error (MSE) averaged over all the carriers is minimized subject to constraints related to the equivalent channel gains, which must be equal to . 1 for all the carriers 1) Asymptotic Behavior of the HARM Technique: In this subsection, we show that the HARM technique is asymptotically related to the power allocation resulting from the minimum MSE (MMSE) criterion with no constraint over the equivalent channel gains. The MMSE is a classical design criterion [6] and it can be extended to the case of a MIMO channel as presented in [11]. Here, we apply it to the case of a multicarrier modulation. The optimization must be carried out by finding the optimum beamvectors at both sides of the communication system. The MMSE design solution is presented as follows:

in a more efficient way and with a low computational load. The techniques are deduced by defining adequate cost functions and constraints. When necessary, and if the Karush–Kuhn–Tucker (KKT) conditions are held, the solution is found using the Lagrange multipliers technique [8], although we do not show explicitly how to apply this method for simplicity reasons. A. Maximization of the Harmonic Mean SNIR (HARM) A possible alternative different from the previous one consists in distributing the total transmit power among the carriers so is maximized. that the harmonic mean of the SNIR (6) The maximization of subject to the global transmit power constraint (4) leads to the following power allocation strategy:

(7)

(8) where is a constant calculated to satisfy the power constraint (4). In (8), it is shown that the power allocated to the carriers do not scale linearly with . Besides, it is possible that, under certain transmit power conditions, the most degraded carriers are . One disadvantage of this strategy canceled is that the parameter must be calculated by means of iterative mechanisms, which increases the computational load when compared to the GEOM or HARM techniques. is high enough and is different from 0 When for all , all carriers are suitable for transmission and the following asymptotic result relating the HARM and MMSE technique holds: (9)

From (7) it is deduced that more power is injected in those carriers where the channel gain presents a fading and/or the level of the noise plus interferences is high. The power allocated to each carrier scales linearly with . It is possible that for a concrete channel realization a deep fading occurs at some subcarrier. In that case, most of the power would be wasted in that frequency. However, this seldom happens in a MIMO channel with a minimum angular and delay spread, since as the number of antennas increases, the maximum eigenvalues also increase. In Fig. 3, the cumulative density functions (CDF) of the maximum eigenvalues are estimated for different antennas configurations, showing that the probability of deep fades is quite low as the number of antennas is increased. This figure is obtained by

The philosophy of the HARM and the MMSE strategies in case of high transmitted power is opposed to the water-filling (maximum capacity), as the first ones inject more power in the most degraded carriers, while the water-filling solution transmits more power in the best frequencies. In an OFDM system, the worst subcarrier is the one that makes the error probability of error rise, and so, it could be useful to force the minimum SNIR over the carriers to be maximum. This is the subject of the next subsection. The resulting power allocation technique is asymptotically optimal in a biterror rate (BER) sense and has the lowest complexity among the presented strategies, except GEOM.

812


B. Maximization of the Minimum SNIR (MAXMIN) The transmit strategy that is proposed hereafter is the selection of the power distribution that presents the best worst case in terms of carrier SNIR, obtaining then a robust maximin solution. Let us assume that an arbitrary power allocation is applied . We suppose, with no loss of generality, that the carriers are numbered so as to have the SNIR ordered in ascending way, as shown in (10):

of the problem can be found by applying the Chernoff upper bound of the probability of error for each subcarrier. The and mentioned Chernoff bound is can be applied as shown in the following expression and deduced in [6] and [12]: e,eff . By making use of this bound, the following power allocation is derived, which we call the minimum effective probability of error (MEPE):

(10)

(13)

Note that this is not the optimal distribution as we can deand increase crease the highest SNIR by lowering and the lowest SNIR, while maintaining the same transmitted power . This process can be applied iteratively until all the subcarriers have the same SNIR and, therefore, this is the solution for which the minimum SNIR over the subcarriers is maximized, as expressed in (11):

(11) From (11), it is deduced that the powers allocated to the subcarriers are proportional to the transmitted power . As in the case of the HARM technique, for the MAXMIN solution the transmitter injects more power in those subcarriers in which the MIMO channel global frequency response presents a fading and/or the level of noise and/or interferences is high. The computational cost associated to this power allocation strategy is the lowest compared to all the other strategies, except GEOM, i.e., it has the lowest complexity among the techniques suitable for frequency selective channels. Additionally, in the case of a scenario in which the noise and interference are Gaussian distributed, this solution is asymptotically the one that minimizes the Chernoff upper bound of the mean probability of error, as it is deduced in the following subsection. 1) Asymptotic Behavior of the MAXMIN Technique: Let us define the effective probability of error as the mean error probability averaged over all the subcarriers, as it is shown in the following expression: (12) is the probability of error associated to the th subwhere carrier. In the case of a scenario where the interferences and the noise are Gaussian distributed, the probability of error corre, sponding to the th subcarrier is: , where depends on the signal constellation applied to the carriers. In the case of BPSK, . The solution of the minimization of the effective BER (12) under the global power constraint (4) is difficult and cannot be found directly, so numerical methods must be utilized in order to apply this design criterion. An approximated solution

where is calculated to satisfy the global power constraint (4). As for the MMSE criterion, in the case of the minimization of the upper bound, the powers allocated to the carriers do not scale linearly with the transmit power . Besides, it is possible that under certain transmit power conditions some carriers are can. As in the case of the MMSE and maxceled imum capacity approaches, the parameter must be calculated by means of iterative techniques, increasing the computational complexity and making the HARM and MAXMIN algorithms more attractive in this sense. is high enough and is different from 0 When for all , all the carriers are suitable for transmission and the following asymptotic result relating the MAXMIN and MEPE technique holds: (14) In case the interferences present in the scenario are not Gaussian, this power allocation does not minimize the upper bound of the probability of error. The optimum technique should then take into account the interferences distribution, although this is difficult in a real and practical system. In Table I, all the power allocation strategies are summarized, including their asymptotic relationships with other designs and techniques. IV. DEGRADATION OF THE SNIR WITH IMPERFECT CSI The previous power allocation techniques were deduced under the assumption that the CSI was perfect, i.e., the estimates of the channel and the covariance matrices of the noise plus interferences had no error. In a real system, this assumption is no longer true, and therefore, a degradation of the performance is expected. In this section we study theoretically which is the expected degradation and give a closed expression of the upper bound of the worst relative SNIR reduction. In the simulations section, the tightness of this upper bound will be shown and evaluated and additional results corresponding to the degradation of the raw BER will be presented. In this section, we focus first on the MAXMIN case, extending then the results to the HARM power allocation technique. Let us introduce as a first step the error model in the estimates of the channel and covariance matrices. We assume that these estimates for each frequency are as follows:

(15)


813

TABLE I SUMMARY OF THE DIFFERENT POWER ALLOCATION TECHNIQUES AND ASYMPTOTIC ANALYSIS

where the matrices and represent the error in the CSI. As a result of differential matrix calculus theory [13], [14], it can be verified that the error in the estimate of the inverse , is apof the covariance matrix, which is represented by proximated as follows (it corresponds to the first order Taylor ): expansion of

the receiver uses the following whitened matched-filter beam. Taking this into account, the former: new SNIR at the th carrier is as follows (expression (2) has been used):

(18) (16) We assume that is hermitian and that , introducing the it can be expressed as: new matrix . The components of the matrices and are modeled as i.i.d. complex and circularly symmetric Gaussian random variables , with variances proportional to and , respectively. The assumed model for is quite known and corresponds to a Maximum Likelihood estimate of the channel in which orthogonal training sequences have been used at the transmitter. In [15], these orthogonal training signals are defined as cyclicly delayed sequences. The Gaussian is based on the central limit theorem, model assumed for as explained in [16], for example. Besides, and for simplicity are indereasons, here we assume that the components of pendent, although the results could be extended to the correlated case. According to the MAXMIN criterion, the power allocated to the th carrier and transmit beamvectors are calculated as follows (we use the superscript to indicate that the design is based on the estimates of the channel and covariance matrices instead of the exact matrices):

The assumption of a receiver with perfect knowledge of the channel is based on the idea that in a real system and a practical implementation, it is known that the estimates of the channel at the transmit side usually have a lower quality than at the receiver. In fact, many other works make use of this assumption, such as [18]. Anyway, the analysis and results that we provide in this paper can be used as a performance bound for a system in which the CSI at the receiver is also noisy. Based on (18), a lower bound of the new SNIR can be obtained as follows (see Appendix I):

(19) where . Based on the Bauer-Fike Theorem about eigenvalue sensitivity to matrix perturbations [17] and taking into account (26) in the Appendix, it is possible to obtain an expression of the inof is terval in which the eigenvalue located as a function of the maximum eigenvalue with no error and the norm-2 of the matrix in the CSI

(17) (20) is the norm-2 of a matrix defined as: and represents the maximum eigenvalue of a diagonalizable positive definite matrix [17]. For siminstead plicity in the notation, in the following we use to denote the maximum eigenvalue. Our goal is to of evaluate which is the new SNIR when using the design of the transmitter according to the available CSI instead of the perfect CSI. When doing this, we assume that the receiver knows perfectly which is the channel and covariance matrices, in addition to the designed transmit beamvectors , which means that where

The transmitter allocates the power according to , so that the carriers with a lower gain are given more power. To obtain the maximum degradation of the SNIR for the th carrier, we assume the following worst-case limit: (21) Taking into account that the SNIR for an ideal MAXMIN system with no error in the CSI is

814


for all the carriers (as presented in Section III-B), it is possible to deduce an expression of the maximum degradation of the SNIR at the th carrier. Here, we define the relative degradation ratio parameter as the maximum reduction of the SNIR related to the original SNIR with no error in the CSI, whose approximated expression is as follows (see Appendix II):

(22) We are interested in obtaining an upper-bound expression of averaged over the statistics the mean value of the parameter of the error in the CSI. In order to do it, it is useful to obtain an as follows (see Appendix III): upper bound of

(23) where and stands for the trace operator. By making use of this expression, the final value of the upper bound of the maximum relative degradation parameter averaged over the statistics of the errors in the CSI is calculated as shown in (24), see the equation at the bottom of page. The most important conclusion is that the relative degradation of the SNIR depends linearly on the standard deviation of the noise for high estimation signal-to-moise ratio(SNR), i.e., for low noise levels. In this section, we have focused on the MAXMIN case. These results can be directly applied to the HARM case, obtaining the following expression (see Appendix IV):

(25) and, therefore, the same conclusions as in the MAXMIN case are obtained, as also in this case the parameter depends linearly on .

V. SIMULATIONS RESULTS AND CONCLUSIONS In this section, some simulations and results are presented based on the HiperLAN/2 system, an European standard for WLAN [3]. The modulation specified for this system is a 64 carriers OFDM, although only 52 subcarriers are active, and 4 of them are pilot carriers used for synchronization, channel estima(sampling tion, etc. The number of samples in the CP is frequency: 20 MHz). In the simulations, BPSK subcarrier modulation is assumed and no channel coding is used, and so, the results refer to the raw effective BER as defined in (12). We simulate normalized MIMO channels, where the channel impulse responses have a mean energy equal to 1: , . The channel power delay profiles are those standardto indicate ized by the ETSI in [9]. We use the notation that antennas are available at the transmitter and at the receiver. In the first simulations it is supposed that the transmitter and receiver have perfect estimates of the CSI, including the noise plus interferences covariance matrices. Note that in order to estimate a MIMO channel, it is necessary to transmit different preambles or training sequences from different antennas [15]. The current standards, such as HiperLAN/2, only specify one training sequence, and therefore, the definition of multiple preambles should be introduced in these standards so that the capacities of a system with multiple transmit antennas can be exploited. In Fig. 4, it is shown the maximum eigenvalues for a single realization of a 3 4 MIMO channel following the power delay profile specified for the model A in [9]. The assumed angular spread is 30 at the transmitter and 15 at the receiver. The arrays are linear with half wavelength interelement separation . At the receiver the are no interferences, i.e., only white Gaussian noise. In Fig. 4, it is shown that the global MIMO channel transfer function presents two fading bands. Fig. 5 shows the power allocated to the carriers and the final SNR for each frequency in a low transmit W for the GEOM, MMSE, HARM, power situation and MAXMIN techniques. In the case of low-transmit power, it is shown that the MMSE criterion decides not to transmit in those bands in which the channel presents fading, while injects more power in the bands with a higher frequency response. As expected, the GEOM strategy allocates the same power to all the frequencies and, therefore, the final SNR function is proportional to the maximum eigenvalues. Finally, the HARM

(24)


Fig. 4.

815

Maximum eigenvalues vs subcarrier for a 3 + 4 MIMO channel A.

and MAXMIN techniques inject more power in those frequencies in deeper fading. The MAXMIN presents an equalized behavior for the SNR under the criterion of maximizing the minimum SNR over all the subcarriers. Fig. 6 presents the corresponding results for a high transW . The behavior of the GEOM techmitted power nique is the same as in the previous case. The most important difference refers to the behavior of the MMSE algorithm. In this case more power is injected in those frequencies in which the global MIMO channel frequency response has low values, which is the inverse behavior when compared with the case of low transmitted power. It is also shown that in this case the MMSE technique tends to distribute the power in the same way as the HARM algorithm, as it was theoretically obtained in (9). Finally, it is shown that MAXMIN also transmits more power in the fading bands, with the criterion of maximizing the minimum SNR and getting the same quality in all the subcarriers. From a computational cost point of view, it must be emphasized that the GEOM technique is the simplest technique, followed by MAXMIN, HARM, and finally MMSE. In Figs. 7 and 8, a quantitative evaluation is carried out based on the estimation of the effective raw BER as defined in (12). We make use of two parameters in the simulations, which are , , the SNR defined as: per branch-to-branch and the signal-to-interference ratio (SIR) is the mean additive white per branch-to-branch, where Gaussian noise power at all the receive antennas and is the mean interference power also at the receive antennas. Two kind 2 antennas (angular of channels have been simulated: a 2 spread: 30 at the transmitter and 15 at the receiver) channel C (delay spread: 150 ns) and a 3 4 antennas (with the same angular spread) channel A (delay spread: 50 ns) [9]. When interferences are included in the scenario, it is assumed that they are other OFDM signals transmitted from single-antenna terminals without any pre-processing and using the parameters specified in the HiperLAN/2 standard. The angular spread for the interferences is the same as the angular spread at the

Fig. 5. GEOM, MMSE, HARM, and MAXMIN techniques. Low transmitted power (P = 1). Power allocation and SNR.

receiver for the desired signal. From the point of view of the BER it is shown that in an acceptable margin, the MAXMIN and HARM techniques have the best performance, and as the transmitted power is higher or the interference levels are lower, MAXMIN tends to perform better than HARM. In case that there is no interference and the noise is Gaussian, this result is justified by the fact that the MAXMIN technique minimizes asymptotically the upper-bound of the BER (14). However, from the simulations it is concluded that even in case that there are non-Gaussian interferences, MAXMIN performs better than HARM for high SNR and SIR. From the computational cost point of view, GEOM is the best one, although its performance is the worst when compared with the other power allocation strategies, due to the fact that no power allocation is carried out. The following simplest algorithm is MAXMIN, concluding that this algorithm presents a good performance—complexity tradeoff.

816


Fig. 7. 2 + 2 antennas. Channel C. Performance evaluation in the presence of one interference (SIR per branch-to-branch = 5 dB) and no interferences. Effective BER vs SNR per branch-to-branch.

0

Fig. 6. GEOM, MMSE, HARM, and MAXMIN techniques. High transmitted power (P = 100). Power allocation and SNR.

In the previous simulations, it was assumed that the CSI was perfect at both the transmitter and the receiver. If there is some noise in the estimation process, a degradation of the techniques is expected. Figs. 9 and 10 analyze this degradation when the noise in the estimate of the channel is white and Gaussian and the estimate of the covariance matrix is perfect, assuming no interference. This approach has been taken because then the results are more clearly presented and the conclusions that are obtained from the simulations do not change even if interferences are considered and errors in the estimate of the covariance matrix are also introduced. In these simulations, we , define the estimation SNR as i.e., measures the relative estimation noise power compared to the channel energy, which is assumed to be 1. In Fig. 9, the worst relative degradation of the SNR averaged over the channel statistics is shown. We compare the degradation

Fig. 8. 3 + 4 antennas. Channel A. Performance evaluation in the presence of two interferences for two different situations of SNR per branch-to-branch: 5 and 10 dB. Effective BER vs SIR per branch-to-branch.

obtained theoretically as explained in Section IV with the one obtained by simulations. As it can be seen, the obtained upper-bound is very tight, specially for high estimation SNR. Besides, the slope of the curves tends to be 1/2 when the estimation SNR increases, as it was expected from the theoretical analysis where it was shown that the performance degradation was proportional to the standard deviation of parameter the estimation noise. 2 In Fig. 10, the degradation of the raw BER in a 2 antennas configuration with no interferences is obtained for different values of estimation SNR. As it can be observed, the degradation is important as the estimation SNR decreases, specially for values lower than 25–20 dB approximately. For these values of estimation SNR, the worst relative degradation is around 0.1, as seen in Fig. 9. Further work is to be done on


817

APPENDIX I DEDUCTION OF THE LOWER BOUND OF THE SNIR IN THE MAXMIN CASE Let us obtain, as a first step, a first-order approximation of as a function of the CSI available at the , , as it is expressed in the following transmitter equation:

(26) Fig. 9. Relative degradation of the SNR for different estimation SNR. Comparisons between the estimated relative degradation and the theoretical upper bound.

is an hermitian matrix. Based on these definitions, where the SNIR with imperfect CSI is reformulated as shown in (27) (expression (2) has been used).

(27)

+

Fig. 10. Raw BER degradation in a 2 2 antennas configuration with no interferences and MAXMIN power allocation. BER vs SNR for different estimation SNR.

the design of robust power allocation strategies, i.e., power allocation algorithms not sensitive to errors in the CSI. By means of analysis such as the ones presented in Figs. 9 and 10, important conclusions about the real application of MIMO techniques can be obtained. As an example, using the curves in Fig. 9, a minimum quality threshold in the degradation can be defined, and the corresponding transmit power during the estimation period can be calculated. In case of implementing a digital feedback channel to transmit the CSI from the receiver to the transmitter, the channel estimates must be quantized. Taking into account the results on performance degradation, the minimum number of bits so as to carry out the quantization can be found. As a general conclusion, in this paper several joint beamforming techniques with a low computational complexity have been derived, and the corresponding performance degradation in real systems has been analyzed.

If the error in the CSI is small when compared to the real channel and covariance matrix, it is possible to obtain a worst case bound of the SNIR assuming that the second term in the expression (27) is positive and maximum. The maximum value is known to be [17]: of

(28)

APPENDIX II DEDUCTION OF THE EXPRESSION OF IN THE MAXMIN CASE We assume the worst-case situation at the th carrier as described in (21). Taking into account this consideration and (28), the upper bound for the SNIR degradation is obtained as shown

818


in (29), where the expression of the SNIR for the MAXMIN has been used. case

and the fact that they are zero-mean and independent between , them: , , .

(29) (33) By applying the definition of the parameter tain the result shown in (30).

(22) we obis calculated by evaluating indentations the two elements in (33). The first one is calculated as shown in (34).

(34) can be ex-

The second term panded as shown in expression (35).

(30) By means of first order approximations in the Taylor series , , it is possible to obtain expansion of ). the expression (31) (assuming

(35) At this point, it is useful to use the identity: [14] and the statistical assump, tions: . By making use of this, the expression (36) is obtained, based on the fact that the last two terms in (35) are canceled due to the statistical assumption: .

(31)

APPENDIX III DEDUCTION OF THE UPPER BOUND OF An upper bound of

can be found as follows:

(36) (32) where the last inequality corresponds to the Jensen’s theorem for concave functions [8]. In order to evaluate as shown in (33), we make use of the following relationships taking into account the statistics of the errors in the estimates of the channel and covariance matrices

Note that, for simplicity in the notation, we have not made use of the index in (35) and (36). Collecting these expressions into (32), the following result is obtained:

(37) where

.


APPENDIX IV DEDUCTION OF THE EXPRESSION OF

IN THE

HARM CASE

In the HARM case, the equations and relationships shown in (38) hold (see Section III-A).

(38) Using the first order Taylor approximation , , the following result is obtained (assuming ):

(39)

819

[9] “ETSI EP BRAN 3ERI085B: Channel models for HIPERLAN/2 in different indoor scenarios,” ETSI, Mar. 1998. [10] A. Pascual-Iserte, A. I. Pérez-Neira, and M. A. Lagunas, “Pre- and postbeamforming in MIMO channels applied to HIPERLAN/2 and OFDM,” in Proc. IST Mobile & Wireless Communications Summit, Barcelona, Spain, Sept. 2001, pp. 3–8. [11] J. Yang and S. Roy, “On joint transmitter and receiver optimization for multiple-input-multiple-output (MIMO) transmission systems,” IEEE Trans. Commun., vol. 42, pp. 3221–3231, Dec. 1994. [12] E. N. Onggosanusi, B. D. Van Veen, and A. M. Sayeed, “Efficient signaling schemes for wideband space-time wireless channels using channel state information,” IEEE Trans. Commun., submitted for publication. [13] G. S. Rogers, Matrix Derivatives, Lecture Notes in Statistics. New York: Marcel Dekker, 1980, vol. 2. [14] J. R. Magnus and H. Neudecker, Matrix Differential Calculus With Applications in Statistics and Econometrics. New York: Wiley, 1999. [15] Y. G. Li, “Simplified channel estimation for OFDM systems with multiple transmit antennas,” IEEE Trans. Wireless Commun., vol. 1, pp. 67–75, Jan. 2002. [16] J.-P. Delmas, “Asymptotic normality of sample covariance matrix for mixed spectra time series: Application to sinusoidal frequencies estimation,” IEEE Trans. Inform. Theory, vol. 47, pp. 1681–1687, May 2001. [17] G. Golub and C. Van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins, 1996. [18] G. Jöngren, M. Skoglund, and B. Ottersten, “Combining beamforming and orthogonal space-time block coding,” IEEE Trans. Inform. Theory, vol. 48, pp. 611–627, Mar. 2002.

Antonio Pascual-Iserte (S’01) was born in Barcelona, Spain, in 1977. He received the degree in electrical engineering from the Universitat Politècnica de Catalunya (UPC), Barcelona, in 2000. Currently, he is working toward the Ph.D. degree in electrical engineering and the degree in mathematics. From September 1998 to June 1999, he worked on microprocessor programming with the Electronic Engineering Department, UPC. From June 1999 to December 2000, he was with Retevision R&D, Barcelona, Spain, where he worked on the implantation of the DVB-T and T-DAB networks in Spain. In January 2001, he joined the Department of Signal Theory and Communications, UPC, where he worked as Research Assistant until September 2003, under a grant from the Catalan Government. Since, September 2003, he is an Assistant Professor at UPC, Barcelona, Spain. Currently, he is involved in several national and European projects. Mr. Pascual-Iserte was awarded the First National Prize of 2001/2002 University Education by the Spanish Ministry of Education and Science and the First National Prize of 2000/2001 University Education by the Spanish Ministry of Education and Culture, in 2000 and 2001, respectively.

REFERENCES [1] T. K. Y. Lo, “Maximum ratio transmission,” IEEE Trans. Commun., vol. 47, pp. 1458–1461, Oct. 1999. [2] Z. Wang and G. B. Giannakis, “Wireless multicarrier communications,” IEEE Signal Processing Mag., vol. 17, pp. 29–48, May 2000. [3] “ETSI TS 101 475 v1.1.1: Broadband radio access networks (BRAN); HIPERLAN Type 2; Physical (PHY) layer,” ETSI, Apr. 2000. [4] K. K. Wong, R. S. K. Cheng, K. B. Letaief, and R. D. Murch, “Adaptive antennas at the mobile and base stations in an OFDM/TDMA system,” IEEE Trans. Commun., vol. 49, pp. 195–206, Jan. 2001. [5] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: A convenient framework for time-frequency processing in wireless communications,” Proc. IEEE, vol. 88, pp. 611–640, May 2000. [6] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [7] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol. 46, pp. 357–366, Mar. 1998. [8] S. Boyd and L. Vandenberghe. (2000) Introduction to convex optimization with engineering applications. Stanford University, Stanford, CA. [Online] Course notes available at http://www.stanford.edu/class/ee364

Ana I. Pérez-Neira (S’92–M’95–SM’01) was born in Zaragoza, Spain, in 1967. She received the degee in telecommunication engineering and the Ph.D. degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain in 1991 and 1995, respectively. In 1991, she joined the Department of Signal Theory and Communication, UPC, where she carried on research activities in the field of higher order statistics and statistical array processing. In 1992, she became Lecturer, and since 1996, she has been an Associate Professor with UPC, where she teaches and coordinates graduate and undergraduate courses in statistical signal processing, analog and digital communications, mathematical methods for communications and nonlinear signal processing. She is the author of nine journal and more than 50 conference papers in the area of statistical signal processing and fuzzy processing, with applications to mobile/satellite communication systems. She has coordinated several private, national public, and european founded projects.

820


Miguel Angel Lagunas (S’73–M’78–SM’89–F’97) was born in Madrid, Spain, in 1951. He received the Telecommunications Engineer degree from the Universitat Politècnica de Madrid (UPM), Madrid, in 1973 and the Ph.D. degree in telecommunications from the Universitat Politècnica de Barcelona (UPB), Barcelona, Spain. From 1971 to 1973, he was a Research Assistant with the Semiconductor Lab ETSIT, Madrid. From 1973 to 1979, he was a Teacher Assistant in Network Synthesis and Semiconductor Electronics. From 1979 to 1982, he was an Associate Professor of digital signal processing, Barcelona, Spain. Since 1983, he has been a Full Professor at the Universitat Politècnica de Catalunya (UPC), Barcelona, where he teaches courses in signal processing, array processing, and digital communications. He was Project Leader of high-speed SCMA (1987–1989) and ATM (1994–1995) cable network. He is also Codirector of the first projects for the European Spatial Agency and the European Union, providing engineering demonstration models on smart antennas for satellite communications using DS and FH systems (1986) and mobile communications GSM (Tsunami, 1994). Currently, he is Director of the Telecommunications Technological Center of Catalonia (CTTC) in Barcelona. His research interests include spectral estimation, adaptive systems, and array processing. His technical activities are in advanced front-ends for digital communications combining spatial with frequency-time and coding diversity. Dr. Lagunas was Vice-President for Research of UPC from 1986 to 1989 and Vice-Secretary General for Research, CICYT, Spain, from 1995 to 1996. He is a member-at-large of Eurasip, and an Elected Member of the Academy of Engineers of Spain and of the Academy of Science and Art of Barcelona. He was a Fullbright Scholar at the University of Boulder, Boulder, CO.