Combined Transmitterand Receiver Optimization for Multiple-Antenna Frequency-Selective Channels

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1 Combined Transmitterand Receiver Optimization for Multiple-Antenna Frequency-Selective Channels Ralf Irmer and Gerhard Fettweis Chair for Mobile Communications Systems Dresden University of Technology, Mommsenstrasse, D- Dresden, Germany Abstract Using multiple antennas in the transmitter and/or in the receiver and exploiting all available knowledge about the channel in both the receiver and transmitter, the link capacity can be increased or the transmit power can be decreased. CDMA signals in frequency-selective channels can be preprocessed in the transmitter and postprocessed in the receiver with filters to maximize the SNR at the receiver. The Generalized Selection Combining (GSC) Multiple Input - Multiple Output (MIMO) Eigenprecoder optimizes the SNR for reduced complexity receivers and transmitters. The capacity gains for different antenna configurations and frequency-selective channels are given. Keywords MIMO, MISO, Pre-RAKE, Tx diversity, Rx diversity, GPP- TDD. Introduction There is a high demand for high data rates in limited spectrum bandwidth on the one hand and power efficient transmission on the other hand. These goals can be achieved using multiple antennas in the transmitter and /or the receiver. This is studied in numerous papers for frequency-flat channels or for OFDM systems in frequency-selective channels. The focus of this paper are spread-spectrum systems, like WCDMA, in frequency-selective channels with multiple antennas. Spatio-temporal filters are considered for preprocessing and postprocessing. It is assumed that the channel impulse response (CIR) is known to the receiver and transmitter.the CIR can be estimated in the receiver using pilot symbols and could be conveyed back to the transmitter using a feedback channel. Another option is offered by time division duplex (TDD) systems: since the channel is the same for both links, the channel estimation in the uplink can be used for transmission in the downlink, provided that the channel change is slow between both slots. One well-known concept to exploit multipath is the RAKE. In [] it was shown that the same performance can be achieved using a Pre-RAKE in the transmitter, thus allowing very simple receivers, i.e. code matched filters. This was extended to Multiple Input - Single Output (MISO) transmission in [] [] []. The Post-RAKE [] and Eigenprecoder [] [] [] [9] use filters in the transmitter and receiver, increasing the complexity but enhancing the performance. In this paper, a generalized concept of MIMO transmission of spread-spectrum signals with filters in the transmitter and receiver is derived, including the decision function, the SNR and the SNR maximizing Tx and Rx filters. Using masking matrices, the filters can be optimized data symbols d Spreader c Tx Tx Tx K h Tx h Ch, h Ch, h Ch, QK h Ch n n Q RAKE MRC RAKE Q MRC h Rx Despreader data symbols Figure : MIMO System model for frequency selective channels and transmitter and receiver also for reduced complexity receivers and transmitters, allowing a flexible performance-complexity trade-off. Special cases of this Generalized Selection Combining (GSC) MIMO Eigenprecoder are the RAKE, Pre-RAKE, transmit diversity (MISO) and receive diversity (SIMO) systems. The outage capacity for different antenna configurations and different channels are calculated. In this paper, ideal spreading codes are assumed. Appropriate methods for mitigation of self- and multiuser interference remains a field of further studies. The paper is organized as follows. In section, the frequency selective channel and system model is introduced. Section presents MIMO pre-and postprocessing including the decision variable and the analytical SNR performance, whereas section focuses on the GSC MIMO concepts. Special cases of the GSC MIMO system are considered in section. Performance results in terms of outage capacities are given in section.. Frequency-selective MIMO system model.. Notation In this paper, lower case bold letters are used for vectors, capital bold letters for matrices, T,, H, F for transposed, conjugate and Hermitian, Frobenius norm respectively. Convolution is noted by. The indices T x, Ch and Rx stand for transmitter, channel and receiver. K and Q are the the number of Tx and Rx antennas, respectively and M is the number of chips per burst... Transmit Signal In the MIMO system, the transmitter has K Tx antennas and the receiver has Q Rx antennas, as shown in Fig.. In contrast to space-time block coding (STBC) and other concepts, in this paper the same spread signal is transmitted from each antenna, which is however individually precoded by a filter for each antenna. Since in TDD-Systems the signals are transmitted burst-byburst all signal processing is described by vectors and matrices. ---//$. IEEE

2 Without loss of generality QPSK is assumed as digital modulation scheme. The burst of symbols d {,, j, j} is organized in a vector d of length N. The short spreading code c with G spreading gain G and normalized power i= c (i) = is assumed to have nearly-ideal even and odd autocorrelation behavior. The spread signal of one user is represented by the size M = N G vector s = Cd with the (M N) spreading matrix C = blockdiag{c}. Then, this spread signal is filtered by filters for each Tx antenna K with the impulse response h T x,k of length L T x. Each filter has the Toeplitz-structured size (M M) filter matrix H T x,k = h T x,k () h T x,k (L T x ) h T x,k () () Note, that for simplicity reasons the vector s k has the same length as the transmit vector s. This means that the multipath of the last symbol can not be fully exploited in the receiver. Therefore, the last symbol is not included in the performance calculations. This approach is only possible if M L T x. The length (L T x K) MISO transmitter precoding vector and size (M K M) matrix are h T x = h T T x,,.., h T T T x,k, HT x = H T T x,,.., H T T x,k The (M K) spread signal vector T. () s = H T xs = H T xcd () is the concatenation of the signals transmitted at the K antennas. For all systems the total transmitted power for all antennas is normalized h H T xh T x =. ().. Channel Model Each single branch channel (SISO) is modelled as a chip-spaced tapped-delay line. It is assumed to be constant for the period of one burst. The channel impulse response (CIR) of one channel branch from antenna k to antenna q is expressed by h Ch,q,k = [h Ch,q,k (), h Ch,q,k (),.., h Ch,q,k (L )] T, () with the maximum delay L. The (M M) block channel matrix H q,k is formed according to Eq.(). Hq,k is the upper left (L L) submatrix of H q,k. The (L L) channel correlation matrix is The length M signal at the receiver is R hh,q,k = H H q,k H q,k, () r q,k = H q,k s k + n q, () where n q denotes the vector of additive white Gaussian noise with variance σ. Following, the MIMO channel is composed of SISO channels. In contrast to (), the channel vector has now K Q diversity branches h Ch = h T Ch,,,.., h T Ch,K,,.., hch,k,q T T () and the (M Q M K) and (( L ) Q L K) channel matrices are H H =,.. H,K H, H =,.. H,K H Q,.. H Q,K H Q,.. HQ,K The size (L K L K) correlation matrix is (9) R hh = H H H. () The length M signal at the receiver antenna q, r q = [H q,,.., H q,k]s + n q is used to stack the received signal for all Q Rx antennas,.. Receiver r = [r T,.., r T Q] T = Hs + n. () In this paper, a filter (e.g. RAKE) is used in each single antenna receiver. The impulse response is h Rx,q with the maximum length L Rx,q. The stacked impulse response for all Rx antennas is h Rx = [h Rx,,.., h Rx,Q]. The decision variable in the receiver after the Rx filter and despreader is ˆd = C H H Rxr () with the (M Q M) filter matrix H Rx formed from h Rx according to (). The effective channel is the convolution of the TX filter and the CIR, h q = K k= h T x,q,k h Ch,q,k, h = [h T,.., h T Q] T, () or alternatively expressed with () and (9) h = Hh T x, with the assumption that L T x = L. With that, generality is not lost because masking matrices can be used to handle unequal filter lengths, as will be shown in section.. In a maximum ratio combining (MRC) RAKE receiver, h Rx,q is a matched filter to the effective channel (), i.e its conjugate complex time inverse, h Rx,q = Ah q, h Rx = [h T Rx,,.., h T Rx,Q] T, () with the (L Rx L Rx) swapping matrix A q and the size (L Rx L Rx Q) matrix A A q =......, A = [A ,.., A q]. ().... MIMO Eigenprecoder Following, the SNR maximizing transmit and receive filter coefficients are derived. With the assumption of spreading codes with ideal correlation properties, as mentioned in section., the self interference is neglected. Without loss of generality it is assumed that L Rx = L T x + L = L. This assumption can be made because these vectors can be filled with zero entries and masking matrices can be used. The decision variable of a general MIMO system is ˆd = C H H RxHH T xcd + C H H Rxn. () For an arbitrary transmit filter and the resulting effective channel, the SNR is maximized in the receiver with a filter matched

3 to the effective channel. The MRC RAKE () is this channel matched filter. The decision variable in () becomes ˆd = C H H H T xh H HH T xcd + C H H H T xh H n. () The SNR at the decision device is with () and () SNR = hh h σ = hh T x H H Hh T x σ = hh T xr hh h T x σ, () In () and (), the transmit filter h T x is still not defined. The optimum Tx filter for a given channel with a Tx power constraint () is h T x,opt = arg max hh T xr hh h T x σ s.t. h H T xh T x = (9) This was shown to be an Eigenvalue problem for SISO [] [] [9] and MISO [] systems and is now extended to MIMO systems. The SNR maximizing transmit filter h T x,opt subject to a fixed transmit power is the Eigenvector belonging to the largest Eigenvalue of R hh, which is also known as an Eigenfilter. The SNR at the receiver becomes SNR = λmax(r hh) σ. () The Tx filter, maximizing the SNR with a Eigenfilter as a transmit filter and a filter matched to the effective channel as the receive filter is termed in the following MIMO Eigenprecoder []. The largest Eigenvalue and the corresponding Eigenvector can be computed fast and efficiently using the power algorithm [].. Generalized Selection Combining (GSC) MIMO Pre- and Postprocessing.. Generalized Selection (GSC) Receiver In practical implementations of RAKE receivers, only a limited number of RAKE fingers are available due to complexity and cost constraints. In most cases, no significant performance loss must be taken into account in comparison to the full MRC RAKE. Generalized selection combining (GSC) [] [] is a combination of selection combining (SC) and maximum ratio combining (MRC): Only a subset of the strongest P taps is selected out of all L Rx available multipath taps at the receiver. These taps are combined according to the MRC principle. Now, the size (L Rx Q L Rx Q) receiver masking matrix M Rx = blockdiag {M Rx,,.., M Rx,Q}, with () M Rx,q = diag {[a,q,.., a LRx,q]}, a l {, } () is introduced, where entries a l = indicate the paths selected in the receiver. The receiver filter impulse response of the GSC RAKE h Rx = M RxAh is the conjugate complex time inverse of the effective channel impulse response () multiplied with the masking matrix () and the the (L Rx L Rx Q) swapping matrix A (). Two special cases of GSC are important receiver structures: the single tap code matched filter (P =, M Rx,q( LRx, LRx ) = ) and the full MRC RAKE (P = L Rx, M Rx = I)... Generalized Selection Combining (GSC) Transmitter In the generalized selection MIMO Pre-RAKE, the strongest K out of K branches (Tx antennas) are used for transmission. For example, the transmitter with K = uses only one Tx antenna in the downlink, but measures K channels in the uplink. In such a transmitter, only one Tx signal processing and RF unit is needed, but K antennas must be available for transmission. Furthermore, the number and position of activated Tx filter taps can be varied. For antenna selection, the size (K L T x K L T x) Tx antenna selection matrix M T x = blockdiag {M T x,,.., M T x,k} () with M T x,k = I for antenna k selected for antenna k not selected diag{[a,k,.., a,k ]} for any tap sel. similar to () is introduced. The strength indicator for each Tx antenna k is α k = Q q= h Ch,q,k h Ch,q,k. () In the receiver, only the channel branches are seen which where stimulated by the transmitter, leading to the modified effective channel impulse response () h = HM T xh T x. () With this concept, the actual length of the Tx filter can be adjusted without violating L T x = L... GSC MIMO Eigenprecoder As described in the previous sections, the transmitter and receiver can be reduced in their complexity and can be described analytically using masking matrices. The questions arise, how do the SNR-optimizing transmit and receive filters for such a reduced complexity system look like and what performance loss compared to a full system has to be taken into account. The SNR at the decision device () for a MIMO GSC RAKE receiver with h Rx from sec.. is SNR = (MRxh)H h σ = hh T x R hh,rx H H M H Rx H h T x σ () with the modified correlation matrix () R hh,rx = H H M Rx H. The maximum Eigenvalue and corresponding Eigenfilter of R hh,rx for a given M Rx can be calculated, and hence the optimum Tx filter vector for the corresponding receiver is obtained. However, the selection matrix M Rx, which maximizes the SNR at the receiver must be estimated. This problem can be solved so far only by extensive combinatorial search of all constellations []. However, suboptimal settings of M Rx by selecting the strongest effective paths of a full MIMO Eigenprecoder deliver satisfactory performance results. The MIMO Eigenprecoder using a GSC Tx filter () can be obtained by using () in the Eigenanalysis in (). The SNR becomes SNR = hh T x R hh,t x M H T x H H HMT x h T x σ ()

4 with the modified correlation matrix R hh,t x = M H T x H H HMT x. Here again, the optimum M T x can only be found by extensive combinatorial search of all constellations, but the selection of the K branches with the highest channel power is a satisfactory solution. In a system using a GSC MISO Tx filter and a GSC MIMO RAKE receiver, R hh,t x and R hh,rx can be combined to form C % MISO Eigenprecoder No. of Tx antennas, K channel taps R hh,t x,rx = M H T x H H M Rx HMT x, () which can be used to calculate the optimum Tx and Rx filter coefficients in the sense of SNR maximization by Eigenanalysis of (). Now, for a given antenna and filter structure, the optimum coefficients maximizing the SNR can be obtained.. Special cases of the GSC MIMO system The MIMO system for frequency-selective channels with filters in the transmitter and receiver described in the previous sections includes some well-known concepts, which will be discussed in this section... Pre-RAKE SISO Systems with a filter in the transmitter and a simple single tap receiver (code matched filter) are known as Pre- RAKE [] []. They have the same performance as the RAKE receiver, but concentrate most of the computational complexity in the transmitter (base station in downlink transmission), which allows simpler mobile receivers. The impulse response of the Pre-RAKE is the conjugate complex time inverse of the CIR, which is as well the Eigenvector corresponding to the maximum Eigenvalue of () when the single nonzero entry receiver masking matrix M Rx,q( LRx, LRx ) = is used... Transmit diversity (MISO) MISO systems with multiple Tx antennas and one Rx antenna (Q = ) are known as transmit diversity systems as well. Usually, no Pre-RAKE is used, but a single weight per Tx antenna. The only nonzero Tx masking matrix () entry for antenna k is now M T x,k (, ) =, and the transmitter and receiver filters can be calculated with (). An example for such a system is the GPP transmit diversity proposal []. A MISO transmit diversity system with Pre-RAKE transmitters is described in [] [] [] []... Receive Diversity (SIMO) A system with one transmit antenna (K = ) and multiple antennas in the receiver is also called Space-Time-RAKE, SIMO, or receive diversity system. Very often, no Pre-RAKE is used, i.e. no preprocessing unit at all is applied. For the Space-Time RAKE with a limited number of fingers, () has to be used additionally. A common framework was found now to calculate and maximize the SNR and to obtain the transmitter and receiver coefficients of different spread-spectrum transceiver concepts. SNR is a valid optimization criterion if ideal spreading codes or spreading codes with a high spreading factor are used, otherwise self- and multiuser interference has to be taken into account. K= channel tap E /N in db S,Tx Figure : Outage capacity of MISO system, channel taps. Performance Evaluation The capacity for one realization of a frequency-selective MIMO channel considering only one transmission layer is with () C = ld + E b,t xλ max N E b,t x ld () log +ld (λ max) N (9) for E b λ max/n >>. The channel is assumed to be constant during the transmission of one burst. The channel capacity (or better its probability distribution) can be calculated by evaluating a large number of channel realizations by Monte Carlo Simulations. The mean channel power for each branch is E{h H q,kh q,k } =, the instantaneous Tx power at all Tx antennas is fixed h H h =. In fig., the % outage capacity is shown for a MISO Eigenprecoder with K =.. Tx antennas in a tap frequency selective channel. The difference C % to the reference capacity of a single tap SISO channel is plotted in the subsequent figures. In fig., C % is shown as a function of the total number of antennas K + Q. All configurations have a growing capacity, where the MIMO capacity for K = Q has a higher capacity than MISO or SIMO. For L >, the SIMO capacity is slightly higher than the MISO capacity. The capacity gain of L = shrinks from bit/channel use at K = Q = down to. bits per channel use at K + Q =. In fig., C % is shown for different channel lengths L =... The additional capacity gain is falling with L. Fig. shows the capacity for different antenna configurations and a fixed number of antennas, K + Q =. The number of spatial diversity branches is K Q, explaining the maximum at about K = Q. The GSC MIMO Eigenprecoder performance vs. number of Rx space-time RAKE fingers is shown in fig.. A Tx precoder, which is adapted to the available number of Rx fingers, shows a performance degradation compared to the full Eigenprecoder. This degradation is, however, negligible for a sufficient number of fingers. This offers complexity reduction potentials in the receiver.. Conclusions For transmission of spread-spectrum signals in frequencyselective channels multiple antennas at the transmitter and receiver can be used. With filters in both the transmitter and receiver, the SNR at the decision device and hence the capacity can be maximized using the MIMO Eigenprecoder. The Generalized Selection Combining (GSC) MIMO Eigenprecoder op-

5 Eigenprecoder Eigenprecoder C % Channel Taps, L= Channel Tap, L= MISO Q= SIMO K= MIMO K=Q # of Antennas, K+Q C %..... tap channel tap channel K+Q= MISO SIMO. # of Rx Antennas, Q Figure : C % for different number of antennas, K + Q Figure : C % for antennas, L= and L= C % MIMO Eigenprecoder L= L= C % GSC MIMO Eigenprecoder K=, Q= K=, Q= K=, Q= K=, Q= # of Antennas K+Q total # of Rx Fingers Figure : C % for different number of taps, L, K = Q (MIMO) Figure : GSC MIMO Eigenprecoder: C % as a function of of used Rx fingers timizes the SNR for reduced complexity hardware configurations.. References [] R. Esmailzadeh and M. Nakagawa, Pre-RAKE Diversity Combination for Direct Sequence Spread Spectrum Mobile Communications Systems, IEICE Trans. Commun., vol. E-B, no., pp., Aug. 99. [] I. Jeong and M. Nakagawa, A Novel Transmission Diversity System in TDD-CDMA, IEICE Trans. Commun., vol. E-B, no., pp. 9, July 99. [] R. Choi, K. Letaief, and R. Murch, CDMA Pre-RAKE Diversity System with Base Station Transmit Diversity, in Proc. IEEE VTC,, pp [] R. Choi, K. Letaief, and R. Murch, MISO CDMA Transmission with Simplified Receiver for Wireless Communication Handsets, IEEE Trans. Commun., vol. 9, no., pp. 9, May. [] A. N. Barreto and G. P. Fettweis, Performance Improvement in DS-Spread Spectrum CDMA Systems Using Preand Post-Rake, in Proc. of Int. Zurich Seminar on Broadband Communications, Zurich (Switzerland),.-. Feb., pp. 9. [] J.-B. Wang, M. Zhao, S.-D. Zhou, and Y. Yao, A Novel Multipath Transmission Diversity Scheme in TDD- CDMA Systems, IEICE Trans. Commun., vol. E-B, no., pp. 9, Oct [] R. Irmer, A. N. Barreto, and G. Fettweis, Transmitter Precoding for Spread-Spectrum Signals in Frequency- Selective Channels, in Proc. G Wireless, San Francisco,, pp [] R. Irmer and G. Fettweis, MISO concepts for frequencyselective channels, in Proc. International Zurich Seminar on Broadband Communications (IZS ), Zurich, Switzerland, Feb., pp. / /. [9] J.-K. Han, M.-W. Lee, and H.-K. Park, Principal ratio combining for pre/post-rake diversity, IEEE commun. letters, vol., no., pp., June. [] G. Golub and C. Van Loan, Matrix Computations, John Hopkins University Press, 99. [] T. Eng, N. Kong, and L. Milstein, Comparison of diversity combining techniques for Rayleigh-fading channels, IEEE Transactions on Communications, vol., pp. 9, 99. [] J. Winters and M. Win, Hybrid-Selection/Optimum Combining, in Proc. IEEE VTC Spring, Rhodes, Greece, May. [] R. Esmailzadeh, E. Sourour, and M. Nakagawa, Prerake Diversity Combining in Time-Division Duplex CDMA Mobile Communications, IEEE Transactions on Vehicular Technology, vol., no., pp. 9, May 999. [] GPP, Physical layer procedures (TDD). Technical Specification GPP TS.,