THE demand for enhancing the data rate of existing mobile

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1 MISO Concepts for frequency-selective channels Ralf Irmer and Gerhard Fettweis Dresden University of Technology, Institut für Nachrichtentechnik, D-0106 Dresden, Germany Abstract In order to limit interference and increase system capacity in CDMA systems, transmitted power required to achieve a certain performance should be as low as possible. Usually, receivers are designed to overcome the problems caused by the frequency-selective wireless channel. However, if the channel impulse response (CIR) is a priori known at the transmitter, the signal we want to transmit can be preprocessed. The CIR for the downlink can be estimated in the uplink in time-division duplex (TDD) systems or conveyed in a feedback channel from the receiver to the transmitter. By using multiple transmit antennas in multiple-input singleoutput(miso) systems, performance improvement potentials are created. In this paper, several MISO concepts using filters both in the transmitter and receiver are compared, their decision variables are given and the SNRs at the receivers are calculated analytically. The MISO generalized selection Eigenprecoder is derived which offers the possibility to trade off performance and complexity. Keywords CDMA, space-time signal processing, MISO, transmit diversity, Pre-RAKE, Eigenprecoder I. INTRODUCTION THE demand for enhancing the data rate of existing mobile communications systems like UMTS and and of systems beyond 3G is still growing. In CDMA systems, the transmit power should be as low as possible for a given data rate to avoid inter-cell and intra-cell interference. The application of multiple antennas in the transmitter, e.g. a base station in the downlink, offers a considerable potential of performance improvement while keeping the receiver (i.e. user equipment) simple. This concept is called multiple-input single-output (MISO) or transmit (Tx) diversity system. In all MISO transmission concepts, it is assumed that the total transmitted power of all antennas is limited. In wideband spread-spectrum systems, the wireless channel is subject to frequency selective fading, or multipath. Therefore, the MISO concepts have to take into account the multipath, which is done by filters in the transmitter and receiver in this paper. The MISO concepts assume, that both the transmitter and receiver have knowledge of the channel impulse responses (CIRs). The CIR could be estimated in the receiver and conveyed back to the transmitter in separate 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 basic concept to exploit multipath is the RAKE receiver. Using a Pre-RAKE [1] [] [3], the multipath combiner is moved from the receiver to the transmitter. Thus, the receiver can be a very simple code-matched filter. The Pre-RAKE using multiple Tx antennas (MISO Pre-RAKE) was proposed in [4]. The Post- RAKE [5] and Pre-Eigenfilter [6] [7] concepts use additionally This work was partly supported by the Deutsche Forschungsgemeinschaft (DFG), contract number Fe 43/4-1. a filter in the receiver, increasing its complexity but enhancing the performance. A good overview of the concepts can be found in [8] [9]. In this paper, a model incorporating all these concepts is proposed. One contribution of this paper is the usage of masking matrices. This allows to optimize the SNR at the receiver by calculating filter weights for transmitters and receivers with reduced complexity. In this paper, the SNR at the receiver is the figure of merit. However, since spreading codes with ideal auto- and crosscorrelation properties are impossible, self interference and multiuser interference limit the performance of the MISO concepts. Therefore, MISO systems maximizing the Signal to Interference and Noise (SINR) at the receiver remain a field of study. The paper is organized as follows. In section II, the system model including the spread signal, channel model, transmit filter and receiver is introduced. Section III presents different MISO concepts. In section IV, the MISO generalized selection Eigenprecoder is introduced and section V analyzes the self interference. Performance results are presented in section VI. A. Notation II. SYSTEM MODEL In this paper, lower case bold letters are used for vectors, capital bold letters for matrices, T, and H for transposed, conjugate and Hermitian, respectively. Convolution is noted for by. The indices Tx, Ch and Rx stand for transmitter, channel and receiver. K is the number of antennas and M the number of chips per block. B. Spread Signal In a Multiple-Input Single-Output (MISO) system, the transmitter has K Tx antennas and the receiver has one Rx antenna, as shown in Fig. 1. In contrast to space-time block coding (STBC) and other concepts, 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-by-burst all signal processing is described by vectors and matrices. Without loss of generality QPSK is assumed as digital modulation scheme. The burst of symbols d { 1, 1, j, j} is organized in a vector d of length N. The short spreading code c with spreading gain G and normalized power G 1 i=0 c (i) =1is assumed to have nearly-ideal even and odd autocorrelation behavior. The spread signal of one user is represented by vector s 0 of size M = N G chips: s 0 = Cd, (1) /0/$ IEEE. 40-1

2 Tx 1 Channel 1 n data symbols Spreader Tx k Channel k RAKE MRC Despreader data symbols d c Tx K Channel K h rx h Tx h Ch Fig. 1. Channel Model with Precoder, Channel and RAKE Receiver with the (M N) spreading matrix C. Channel Model C = blockdiag{c} () Each single branch channel (SISO) is modeled as a tappeddelay line. It is assumed to be constant for the period of one burst. The channel impulse response (CIR) of one channel branch L 1 h Ch,k (t) = h Ch,k (l)δ(t τ l ) (3) l=0 can be expressed for equidistant chip-spaced taps by the channel vector h Ch,k =[h Ch,k (0),h Ch,k (1),.., h Ch,k (L 1)] T (4) of length L. The (M M) block channel matrix is h Ch,k (0) h Ch,k (1) h Ch,k (0)... 0 H k =... h Ch,k (1)... 0 h Ch,k (L 1) h Ch,k (L 1) h Ch,k (0) (5) and H k is the upper left (L 1 L) submatrix of H k. The (L L) channel correlation matrix is R hh,k = H H k H k, (6) The power gain of the channel is L 1 l=0 h(l) = h H Ch,kh Ch,k. The length M signal at the receiver is r k = H k s 0 n (7) where n denotes the vector of additive white Gaussian noise with variance σ. Note, that for simplicity reasons the receive vector r k has the same length as the transmit vector s 0. This means that the multipath of the last symbol is not fully exploited in the receiver. The last symbol is not included in the performance calculations. This approach is only possible if M L. Following, the MISO channel is composed of SISO channels. In contrast to (7), the channel vector has now K diversity branches [ T h Ch = h T Ch,1,.., hch,k] T (8) and the (M M K) and ( L 1 L K) channel matrices are H =[H 1,.., H K ] and H [ = H1,.., H ] K (9) and the correlation matrix is R hh = H H H (10) The multiple Tx antenna transmit signal vector s in (11) of size (M K) will be defined in (13). The length M signal at the receiver is now D. Transmit Filter r = Hs n. (11) Each of the K transmit antennas has its own transmit filter with impulse response h Tx,k of length L Tx with the filter matrix H Tx,k, which is defined according to (5). Without loss of generality it is assumed that L Tx = L. This assumption can be made because h Tx,k can be filled with zeros and masking matrices can be used. The length (L Tx K) MISO transmitter precoding vector and size (M K M) matrix are h Tx = [ T [ T h T Tx,1 Tx,K],.., ht, HTx = H T Tx,1 Tx,K],.., HT. The (M K) spread signal vector (1) s = H Tx s 0 = H Tx Cd (13) is the concatenation of the signals transmitted at the K antennas. The length (L 1) effective impulse response of the channel and precoder is h k = h Tx,k h Ch,k, h = h k. (14) 40-

3 E. Receiver In this paper, a filter is used in the single antenna receiver, e.g. a RAKE receiver. The impulse response is h Rx with the maximum length L Rx. The size (L Rx L Rx ) receiver masking matrix M Rx = diag {[a 1,.., a LRx ]}, a l {0, 1} (15) is introduced, where entries a l =1indicate the paths selected in the receiver. The decision variable in the receiver after the Rx filter and despreader is ˆd = C H H Rx r (16) with the (M M) filter matrix H Rx formed by putting h Rx into (5). E.1 Generalized Selection (GSC) RAKE A RAKE receiver is a filter matched to the effective channel h in (14) followed by a despreader. 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) [10] [11] [1] [13] 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. Two special cases of GSC are important receiver structures: the single tap code matched filter (P =1, M Rx ( 1 L Rx 1, 1 L Rx 1) = 1) and the full MRC RAKE (P = L Rx, M Rx = I). The receiver filter impulse response of the GSC RAKE is the conjugate complex time inverse of the effective channel impulse response, h Rx = M Rx A h Tx,k (17) with the (L Rx L Rx ) swapping matrix A = (18) A. General MISO system III. MISO CONCEPTS Without loss of generality it is assumed that L Rx = L Tx L 1=L 1. 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 MISO system is ˆd = C H H Rx HH Tx Cd C H H Rx n. (19) For all MISO systems the total transmitted power for all antennas is normalized h H Txh Tx =1. Following, the self interference is neglected. The following sections deal with different transmitter and receiver concepts. They differ mostly in the complexity involved in the transmitter and receiver and in the achievable SNR or BER at the receiver. B. MISO Pre-RAKE In TDD systems and in systems with feedback channel the CIR is known a priori in the transmitter. Therefore it is possible to move the RAKE multipath combining technique from the receiver to the transmitter, which is known as the Pre-RAKE concept []. In the receiver, only an integrate-and-dump receiver (code matched filter) has to be applied. Thus most of the signal processing can be concentrated in the base station, which allows simpler mobile receivers. The impulse response of the Pre-RAKE filter precoder is the conjugate complex time inverse of the CIR. In a MISO system, a Pre-RAKE is applied for each branch. In the equal gain MISO Pre-RAKE, the total Tx power is equally distributed to all antennas. In the MRC Pre-RAKE, the signal of each Tx antenna is weighted by the power of the corresponding channel branch under the constraint of fixed total Tx power. The transmit filter impulse response is h Tx,k = α k Ah Ch,k, (0) with the power normalization factor α k for the equal gain Pre- Rake and MRC Pre-RAKE, respectively: 1 α k,eg = and (1) h H Ch,kh Ch,k K α k,mrc = 1 h HChh Ch. () The tap with the highest energy of the effective channel impulse response h k in (14) is in the center. The code matched filter at the receiver has to match to that tap. The receiver impulse response (17) becomes h Rx = M Rx A α k h k, (3) with only one entry in M Rx (L 1,L 1) = 1. The decision variable for the MISO Pre-RAKE is where ˆd = C H HH Tx Cd C H n. (4) H Tx,k = α k H H k (5) The SNR at the receiver of the MISO Pre-RAKE is SNR MRC Pre-RAKE = SNR EGC Pre-RAKE = h H Ch,kh Ch,k σ = hh Ch h Ch σ. (6) ( ) 1 K h H Ch,k h Ch,k σ. (7) 40-3

4 C. Generalized Selection MISO Pre-RAKE In the generalized selection MISO Pre-RAKE, the strongest K out of K branches are used for transmission. The transmitter with K = 1 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. For antenna selection, the size (K L Tx K L Tx ) Tx antenna selection matrix M Tx = blockdiag {M Tx,1,.., M Tx,K } (8) { I for antenna k selected with M Tx,k = (9) 0 for antenna k not selected is introduced. Then, the concept of section III-B for Tx filter design can be used, with the modified CIR, h Ch = M Tx h Ch, H = HM Tx. (30) The Tx filter is now obtained by using Eq. (30) in Eqn. (6), (0) and (1). Depending on α in Eq. (1), the EG selection MISO Pre-RAKE or the MRC selection Pre-RAKE is obtained. The decision variable is (4) and the SNR is (6) and (7) for the MRC and EGC generalized selection MISO Pre-RAKE. D. MISO Pre- and Post-RAKE The MISO Pre-RAKE uses a bank of channel matched filters (i.e. RAKE) in the transmitter and thus concentrates all received power in one peak at the receiver. In the receiver, this signal is demodulated by a simple code matched filter. The proposal of A. N. Barreto and G. Fettweis [5] is to collect the remaining information in the other received signal peaks as well, which is called Post-RAKE. The Post-RAKE is a filter, matched to the combination of Pre-RAKE and multipath channel. It has a significant performance gain over the Pre-RAKE only system, for the cost of added complexity in the receiver. The Pre-RAKE coefficients remain the time-inverted and scaled conjugate complex of the CIR in Eq. (0). The Post-RAKE coefficients (3) are matched to the effective impulse responses h k in (14) with the masking matrix M Rx = I for the full MRC Post-RAKE or different M Rx for the GSC Post-RAKE as introduced in section II-E.1, h Rx = M Rx A α k h k. (31) The decision variable of the MISO Pre- and Post-RAKE is ˆd = C H H H Tx HH HH Tx Cd C H H H Tx HH n. (3) The Signal-to-Noise Ratio becomes E. MISO Eigenprecoder SNR Pre-/Post-RAKE = hh TxR hh h Tx σ. (33) The Pre- and Post-RAKE system has a higher SNR than the Pre-RAKE only, since it collects more energy in the receiver, but is it already optimum? In other words, which set of transmit filters h Tx and receive filter h Rx maximizes the SNR at the receiver under the constraint of equal transmit power for given CIRs, known perfectly to the transmitter and receiver. This question was shown to be an Eigenvalue problem in [6] [14] [7]. Let us first consider the receiver filter. To maximize the SNR, its impulse (3) response has to match to the effective channel (14): h Rx = A The decision variable is The SNR is (h Tx,k h Ch,k ). (34) ˆd = C H H H Tx HH HH Tx Cd C H H H Tx HH n. (35) SNR = hh h σ = hh TxR hh h Tx σ, (36) The precoding gain (the ratio of signal amplification by the effective channel) for the given precoder vector h Tx can be written as the Rayleigh quotient [14] R a (h Tx )= hh Tx R hhh Tx h H Tx h Tx (37) The optimum Tx-filter maximizing R a (h Tx ) and hence the SNR subject to the normalized input power constraint h H Txh Tx =1is the Eigenvector of the largest Eigenvalue λ max, which is also known as Eigenfilter. The SNR at the receiver becomes SNR = λ max σ. (38) The largest Eigenvalue and the corresponding Eigenvector can be computed using the Singular Value Decomposition (SVD), or more efficiently by the power algorithm [15]. IV. MISO GENERALIZED SELECTION EIGENPRECODER In section II-E.1, the generalized selection RAKE receiver was introduced and described using the masking matrix M Rx. The maximized SNR for a MISO system with a GSC RAKE receiver is SNR = (M Rxh) H h σ with the modified correlation matrix = hh TxR hh,rx h Tx σ (39) R hh,rx = H H M H Rx H = H H M Rx H (40) 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 40-4

5 solved so far only by extensive combinatorial search of all constellations [13], but suboptimal settings of M Rx deliver satisfactory performance results. The MISO Eigenprecoder using a generalized selection Tx filter of section III-C can be obtained by using (30) in the Eigenanalysis in (36). The SNR becomes ) H htx( H HMTx HMTx h Tx SNR = = hh Tx σ R hh,t x {}}{ with the modified correlation matrix M H Tx H H HMTx h Tx σ (41) R hh,t x = M H Tx H H HMTx (4) Here again, the optimum M Rx 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 generalized selection MISO Tx filter and a generalized selection RAKE receiver, Eqn. (40) and (4) can be combined to form R hh,t x,rx = M H Tx H H M Rx HMTx, (43) which can be used to calculated the optimum Tx and Rx filter coefficients in the sense of SNR maximization by Eigenanalysis of Eq. (43). Now, for a given antenna and filter structure, the optimum coefficients maximizing the SNR can be obtained. V. SELF INTERFERENCE ANALYSIS By the preprocessing of the signal in the transmitter using an filter the effective channel impulse response h k seen at the receiver is longer than the real CIR. It was shown in this paper, that the receiver does not necessarily have to be more complex. However, a problem could arise for codes with non-ideal autoand crosscorrelation properties, where the self- and multiuser interference is increased. Following, the mean self interference power for short codes is derived. The self interference depends on the data, codes and channel coefficients. For estimation of the data-independent selfinterference, equal propability and independence of the data symbols is assumed (E {d(k)d(k 1)} =0, E { d(k) } =1 ). It is assumed that the channel impulse response is shorter than the symbol length. Therefore, one data symbol can be affected by the following and previous data symbol. The interference on one data symbol d(k) is I(k) = u=1 l=1,l =u u=1 l=1,l<u u=1 l=1,l>u h uh l ϕ ap ( u l ) d(k) h u h lϕ ap (G u l ) d(k 1) h uh l ϕ ap (G u l ) d(k 1), (44) where ϕ ap (l) is the aperiodic autocorrelation of the spreading code the for the relative delays l of the multipaths. If independent random data symbols d are assumed and all other terms in (44) are deterministic, the mean interference power can be calculated to be σsi = E { I } =E{ I 0 } E{ I 1 } E{ I 1 } = h u h lϕ ap ( u l ) u=1 l=1,l =u h u h lϕ ap (G u l ) u=1 l=1,l<u h u h lϕ ap (G u l ). An alternative expression is σ SI = u=1 l=1,l>u (r hh r hh ) T ϕ a r H hh ϕ b r T hh ϕ b =4 Re { r T hh } ϕa r T hh ϕ b (45) (46) with the length (L 1) off-diagonal channel correlation vector r hh = [R hh (, 1),.., R hh (L, 1)] T and the vectors ϕ a = [ϕ ap (1),.., ϕ ap (L 1)] T and ϕ b = [ϕ ap (G 1),.., ϕ ap (G L 1)] T, containing the aperiodic autocorrelation terms for the relative delays of the multipaths. Now, the mean self-interference can be set into relation to the noise variance to evaluate its relevance on the performance. It should be mentioned, that the self interference is not necessarily Gaussian. VI. PERFORMANCE EVALUATION For performance evaluation, the achievable SNR gain at the receiver under a Tx power constraint for different Tx and Rx designs is compared. The self interference is neglected in the simulations since all algorithms are designed to maximize the SNR. Maximizing the SINR is not in the scope of this paper, but is a current field of research. The frequency-selective channel model is based on the 3GPP multipath propagation case 3 [16]. The channel taps are chip-spaced with uncorrelated fading between the taps and the antennas and power delay profile 40-5

6 [0, 3, 6, 9]dB. The channel is constant for one burst. The mean channel power for each branch is E{h H k h k } =1. The Tx power at all Tx antennas is always fixed h H Tx h Tx =1.For each MISO system, 1000 channels are calculated. As a reference, a SISO system (always branch k =1selected) with a conventional Pre-RAKE or RAKE in the same frequency-selective channel is used. Its SNR at the decision device is SNR 0. In figure, the SNR at the receiver normalized to SNR 0 is shown for different number of Tx antennas and different MISO systems. Each additional Tx antenna results in a significant improvement for all MISO concepts. The MISO Eigenprecoder shows the best performance. Already for one antenna, the benefit of systems with Tx and Rx processing can be seen compared to a system with just a RAKE receiver or a Pre-RAKE in the transmitter. In figure 3, the SNR gain at the receiver of a GSC MISO system with different number of Rx filter coefficients is shown. Two different channels are considered. Channel I has 4 taps and channel II 8 taps, and the Tx filter length is the same as the channel length. Hence, the signal at the receiver has 7 or 15 paths, respectively. However, already a small number of Rx RAKE fingers is sufficient to achieve most of the SNR gain, especially for many antennas. Using the GSC MISO concept to optimize the Tx filter and Rx filter coefficients for a reduced complexity RAKE receiver, considerably simpler receivers can be designed without a significant loss of performance. SNR/SNR 0 in db Mean Coherency and Diversity Gain 4 Full Eigenprecoder Pre Post-RAKE PreRAKE MRC PreRAKE EGC Selection PreRAKE # of Tx antennas, K Fig.. SNR gain at receiver for different MISO concepts VII. CONCLUSIONS In this paper, MISO concepts which improve the performance of spread-spectrum signals in frequency-selective fading channels are analyzed. By adapting the transmitted signal to the channel using filters at the transmitter and using an appropriate receiver, the SNR and BER at the receiver can be enhanced satisfying the constraint of a limited transmit power. The discussed concepts include the MISO Pre-RAKE, MISO Pre- and Post-RAKE and MISO Eigenprecoder. Using selection matrices, the performance of transmitters and receivers with reduced complexity can be optimized. For that, the MISO generalized selection Eigenprecoder is a useful framework which includes the conventional Pre-RAKE. Possible performance im- SNR/SNR 0 in db Generalized MISO Eigenprecoder 1 Tx Antenna, 8 channel taps 8 Tx Antennas, 8 channel taps 1 Tx Antenna, 4 channel taps 8 Tx Antennas, 4 channel taps # of RAKE Fingers Fig. 3. SNR gain at receiver for different No. of fingers in GSC MISO RAKE nreceiver provements are discussed for a system similar to the 3GPP TDD specification. REFERENCES [1] R. Esmailzadeh and M. 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