Reduced Complexity Space-Time Optimum Processing

Transcription

1 Reduced Complexity Space-Time Optimum Processing Jens Jelitto, Marcus Bronzel, Gerhard Fettweis Dresden University of Technology, Germany Abstract New emerging space-time processing technologies promise a significant performance increase of wireless communication systems. The particular application and scenario strongly influences the amount of possible performance and capacity increase if antenna arrays are deployed at the basestation (BS) and/or at the mobile terminal (MT). The achievable gain is mainly determined by the spatial correlation properties of the underlying physical transmission channel. This paper analyzes the spatial correlation properties for various scenarios and investigates the processing requirements for designing space-time optimum receivers. One aim is to reduce the spatial signal dimension to the information bearing components applying orthogonal transformation techniques. It will be shown that even for virtually uncorrelated spatial channels which are characterized by high delay and angular spread, the spatial dimension can be reduced significantly. This enables less complex receiver structures and more robust channel estimation techniques. 1 Introduction Multiple antenna concepts are commonly regarded as a promising technology to increase the performance of wireless communication systems. The concept of Space Division Multiple Access (SDMA) enables higher user capacity within a cell if the users can be separated spatially. Furthermore, spatial filtering and cancellation of undesired users results in reduced interference. Other multiple antenna concepts include the steadily growing research fields of beamforming, spatial diversity combining and space-time processing. This paper investigates the performance of a single link between a MT and a BS and does not target on system capacity issues. Depending on the given wireless channel conditions the different concepts such as beamforming, space-diversity combining or fully armed space-time processing will be more or less suitable. The multipath situation has to be considered in order to select the appropriate algorithm. If for instance a strong Line-of-Sight (LOS) component is present with only weak temporal and spatial spreading, pure beamforming provides an efficient approach. Space diversity combining would not be appropriate in this scenario since the antenna signals are essentially phase shifted copies of each other and therefore highly correlated. In a scenario with considerable multipath delay and angular spread space-time processing promises optimum results. The degree of signal correlation between different antenna elements influences the possible gain which can be achieved with such a receiver concept. We will investigate the spatial correlation properties at the

2 receiving antennas for various scenarios, which determine the additional information that can be gained by every additional spatial dimension (or antenna) and can be used as a measure of the required spatial receiver complexity. We will introduce a linear transformation based on singular value decomposition (SVD) [1] in order to reduce the potentially correlated M antenna signal streams to an uncorrelated signal stream of lower order D, which is then fed into a reduced complexity space-time receiver (Figure 1). An example will show the potential gains of this approach. ST x Trafo Equal. y MxD DxL M D 1 Figure 1: Reduced Complexity Space Time Receiver Concept 2 Signal Model This section introduces a signal model for a single input multiple output (SIMO) system [2] using one transmit antenna and M receive antennas including the channel characteristics. The basic system model is shown in Figure 2. A binary data stream is mapped onto complex symbols s 0 (k). In order to simplify the discussion linear modulation will be assumed. The continuous-time representation of the symbol stream can be written as s 0 (t) =T sym s 0 (k) δ(t kt sym ), (1) where T sym defines the symbol duration. At the transmitter (Tx), the signal will be band-limited by a pulse shaping filter g Tx (τ). The complex baseband representation of the filtered transmit signal is given by s(t) = s 0 (t) g Tx (t) = s 0 (u) g Tx (t u) du = T sym s 0 (k) g Tx (t kt sym ). (2) The band-limited signal s(t) is transmitted over a linear time-variant channel. Wireless communication channels are affected by multipath propagation with a possible LOS component and indirect paths resulting from reflection, scattering and diffraction at several objects in the propagation TX- Filter Symbol mapping s 0(k) δ(t) s 0(t) RX- Filter s(t) x(t) x(nt s) SIMO Channel i(t)+n(t) Figure 2: System model of a digital transmission system over a wireless channel with multiple receive antennas M Rx

3 environment. This channel can be modeled by a physical channel impulse response (CIR) [3] c(t, τ) = P 1 p=0 c p (t) δ(τ τ p ) (3) with P as the number of propagation paths, each of which is characterized by a delay τ p and a complex weight c p (t). Both parameters are generally time-variant, but changes in the path delays are usually slow compared with the symbol duration. The received signal x(t) can be obtained from the convolution of the transmitted signal (2) with the physical CIR (3) as x(t) = c(t, τ) s(t)+n(t) P 1 = T sym s 0 (k) c p (t) g Tx (t kt sym τ p ) +n(t), (4) p=0 } {{ } h(t,τ=t kt sym) where n(t) is the additional noise term. Interference i(t) from other users as indicated in Figure 2 will not be considered in this paper. From (4) an effective CIR h(t, τ) can be derived, including transmitter filter and physical channel characteristics with fractional delays τ p [3]. Applying multiple antennas with M elements at the receiver effectively introduces M separate channels. Here, it is assumed that the multipath components at the antenna elements differ only in the path length relative to a reference antenna element resulting in additional path delays Rx p,m for every receive antenna m. The overall path delay at antenna m is given by τ p,m = τ p + Rx p,m, (5) which results in an effective CIR h m (t, τ) at antenna m h m (t, τ) = P 1 p=0 c p,m (t) g Tx (t kt sym τ p Rx p,m). (6) If the rate of change of the received signal envelope is slow compared with the propagation time across the array (narrowband assumption), which applies for most wireless systems, as long as the signal bandwidth is small relative to the carrier frequency f, the additional delays Rx p,m can be regarded as pure phase shifts, a p,m = e jω Rx p,m (7) where ω =2πf. The effective CIR h m (t, τ) atantennam can then be written as h m (t, τ) = P 1 p=0 a p,m c p (t) g Tx (t kt sym τ p ), (8) where the path weights c p (t) ofthep elementary rays are assumed to be identical for all antennas. Combining the M impulse responses h m (t, τ) of the SIMO channel h(t, τ) =[h 0 (t, τ) h 1 (t, τ)...h M 1 (t, τ)] T (9)

4 and collecting the phase shifts a p,m in an array response vector a p = [ e jω Rx p,0 e jω Rx p,1...e jω Rx p,m 1] T (10) leads to the reformulation of the SIMO CIR h(t, τ) in matrix notation, h(t, τ) = P 1 p=0 a p g Tx (t kt sym τ p ) c p (t) =AG(τ)c(t). (11) A is the array response matrix of dimension M P containing the P vectors a p as columns. The matrix G(τ) isap P diagonal matrix containing the values of the pulse shaping filter for all pathsaselements,[g(τ)] p = g Tx (t kt sym τ p ). Finally, c(t) containsthep path weights at time t. Using (4) and (11) the resulting M-dimensional received signal vector can now be written as x(t) =T sym s 0 (k) h(t, τ)+n(t) =T sym 3 Spatial Correlation of the Received Signal s 0 (k) AG(τ)c(t)+n(t). (12) The spatial correlation properties of the received signal are strongly influenced by the parameters of the effective CIR h(t, τ). This includes the effects of the pulse shaping filter as well as the multipath characteristics of the physical channel. Additionally, the spatial correlation of the received signal at the antenna array is influenced by the temporal correlation properties of the transmitted symbol sequence s 0 (k). The spatial correlation matrix C S x of the received signal x(t) can with (12) be defined as C S x = Cov(x(t)) = E{x(t)x(t) H } (13) ( )( = E T ) H sym 2 s 0 (k)ag(τ)c(t)+n(t) s 0 (l)ag(τ)c(t)+n(t) (14) The transmitted sequence is i.i.d, l= E{s 0 (k)s 0 (l)} = E{ s 0(k) 2 } δ(l k) (15) and zero-mean. Furthermore, we assume the noise terms n m (t) at the antenna elements m to be zero mean, uncorrelated with s 0 (k) and temporally and spatially white. With these assumptions the spatial correlation matrix as defined in (14) can be simplified to { } C S x = P 0 E AG(τ)c(t)c H (t)g T (τ)a H + C S n (16) with the noise covariance matrix C S n and a constant scaling factor P 0 determined by T sym and the mean symbol energy. This factor can be neglected, since it doesn t influence the matrix properties. With the assumption of constant path delays and AOA s the array response matrix and pulse

5 shaping matrices are deterministic, which simplifies (16) to C S x = P 0 A [ G(τ)E{c(t)c H (t)}g T (τ) ] A H + C S n = APA H + C S n. (17) } {{ } P The signal part of the correlation matrix is determined by the matrix form APA H, where P represents the inter-signal coherence matrix of dimension P P corresponding to the number of multipath components. The properties of the inter-signal coherence matrix are influenced by the complex path weights c(t), which are basically determined by the attenuation of the multipath components, and the characteristics of the pulse shaping filter g Tx (τ). Following the P-matrix will be analyzed in more detail for different multipath scenarios with respect to mobility and delay spread to gain some insight into its structure. The reason is that, besides the influence of angular path distribution represented in the array response matrix A, the inter-signal coherence matrix determines the rank of the spatial correlation matrix and therefore is important for determining the possible dimension reduction. One limiting case of the P-matrix occurs when the signal copies arriving over P multipaths are coherent which implies identical path delays τ p τ 0. P will in this case be a rank one matrix. The other limiting case occurs with P mutually uncorrelated signals, where P will have full rank. In realistic scenarios the rank of P may vary between these values. Scenario 1: no Mobility, no Delay Spread A static environment with multipath propagation but negligible delay spread (τ p τ 0 ) results in space-selective fading. The complex path weights c p (t) can be considered as constant due to the stationary environment. For every delay τ = t kt sym τ 0 the entries of [G(τ)] p for all paths are identical. Therefore, the pulse shaping matrix G(τ) can be replaced by a scalar value g Tx 0 (τ) for every k, whichleadstoap-matrix of the form P = g Tx 0 (τ) c(t)ch (t) g Tx 0 (τ) = [ ( ) ] 2 g Tx 0 (τ) c(t)c H (t). (18) Clearly, the inter-signal coherence matrix P has rank one independent from the length of the pulse shaping filter, since the outer product cc H has rank one and k (gtx 0 (τ))2 is a scalar. In this case, the received signal vector x(t) defined in (12) can be written as x(t) =T sym [ ] s 0 (k) g Tx 0 (τ) Ac(t) +n(t), (19) }{{} a(t) where a(t) defines the spatial signature, which is the weighted sum of the array response vectors, a(t) = P 1 p=0 a p c p (t). The spatial correlation matrix C S x for space-selective channels is then given as [ ( ) ] 2 C S x = APA H + C S n = g Tx 0 (τ) a(t)a H (t)+c S n. (21) (20)

6 Relative Path Delay τ/t sym Relative Path Delay τ/t sym Figure 3: P-matrix for 100 path delays uniformly distributed in the interval [0 τ p 5T sym ]and root-raised cosine pulse shaping filter with roll-off factor 0.5, linear attenuation of the path weights with τ; the gray level indicates the inter-signal coherence strength In the noise-free case this matrix will be of rank one according to P, independent of the angular multipath distribution. This suggests a beamforming or optimum combining approach to weight the antenna signals according to the spatial signature, which reduces the signal dimension from M to 1. Scenario 2: no Mobility, Delay Spread For scenarios with considerable delay spread, the channel is considered to be space-frequency-selective. This is typically the case for excess delays of τ max > 0.1T sym [3]. The different path delays τ p cause the pulse shaping matrix G(τ) = diag[g Tx (τ τ 0 ) g Tx (τ τ 1 )...g Tx (τ τ P 1 )] to contain independent entries, which results in the inter-signal-coherence matrix P = G(τ)c(t)c H (t)g T (τ). (22) The rank of this matrix is now determined by the number of multipath components and their respective delays as well as by the length of the pulse shaping filter. Every sub-matrix G(τ)c(t)c H (t)g T (τ) in (22) has rank one. However, the summation of k rank one matrices leads to a matrix with a rank not higher than k. For filters of finite length LT sym the rank of P is limited to min(p, L). Figure 3 shows the magnitudes of a typical P-matrix. Here, the path delays are uniformly distributed in [0 τ p 5T sym ]. The weights c p (t) have random phases and decay linearly with increasing path delay. The pulse shaping filter was modeled as a root-raised cosine filter with roll-off factor 0.5. The gray levels indicate the amount of inter-signal coherence between the delayed paths, where black indicates total coherence and white no coherence. In most cases it has been observed, that this matrix has full rank or some rank deficiency. The degree of the

7 rank reduction depends on the given multipath situation. This also holds for the received signal correlation matrix C S x = APAH + C S n = A [ G(τ)c(t)c H (t)g T (τ) ] A H + C S n, (23) where in the noise free case the rank of this matrix is less or equal to min(m,rank(p)). However, since in most cases this information is not sufficient to be applied efficiently as a dimension reduction criterion, analyzing the eigenvalue properties of the spatial correlation matrix will provide a valuable tool for estimating the effective signal dimension, as will be shown later. Scenario 3: Mobility, Delay Spread In this most general case the wireless channel is selective with respect to space, frequency, and time. The space-time correlation function for two antenna elements m 0 and m 1 is defined as R xm0,m 1 ( t) =E{x m 0 (t) x m1 (t + t)}. (24) The analysis of this correlation function can only be simplified for particular scenarios. One important special case occurs when the temporal correlation is decoupled from the spatial correlation. Then the matrix of space-time correlation functions can be written as R( t) =ρ( t)r S x (25) with a separable matrix of spatial correlation functions 1 R S x and a temporal correlation factor ρ( t). The P- andc S x-matrices can be determined from (17). 4 Subspace Methods and Spatial Dimension Reduction As stated earlier, the rank of the spatial signal correlation matrix is often not a sufficient measure to determine the appropriate spatial dimension of the receiver. Therefore, an eigenanalysis of the spatial correlation matrices is conducted using SVD [1]. The C S x-matrix can be decomposed into C S x = UΛV H (26) with U and V as left- and right-hand side eigenvalue matrices of C S x. Since C S x is hermitian, it follows U = V. Λ is a diagonal matrix containing eigenvalues of C S x sorted in descending order, Λ = diag [λ 0 λ 1...λ M 1 ]. An important property of the normalized eigenvalues is, that their magnitudes represent a measure of signal energy contained in the signal components after an orthogonal coordinate transformation (OT). If the spatial correlation matrix has full rank, the signal energy is distributed across all components. However, applying an OT concentrates the signal energy within the first components. 1 The matrix of spatial correlation functions R S is related to the spatial correlation (covariance) matrix C S through C S =(R S ) T due to the differences in the definitions for complex valued autocorrelation functions and complex valued correlation matrices for vector processes [4].

8 Normalized SNR Gain [db] Number of Signal Branches Figure 4: Average SNR improvement for M signal components (dashed line: gain achievable by beamforming or Maximum Ratio Combining; solid line: gain typically achievable after orthogonal transformation). Usually in array signal processing the performance gain in noise-limited environments can be determined by the SNR gain per antenna branch. With the assumptions of identical average SNR s at the M antenna elements and spatially uncorrelated noise, the SNR gain is 10 log M. Figure 4 shows the average SNR performance gain depending on the number of signal branches considered. These results are normalized with respect to the average SNR in each branch. Applying an orthogonal transform to the received signal shows that the first signal components already provide most of the possible SNR gain, although the corresponding spatial correlation matrix has full rank. The maximum possible SNR gain is the same as without transformation, if all signal dimensions are used. Commonly applied criteria for rank estimation in noisy environments such as the Akaike information-theoretic criterion (AIC) or Minimum description length criterion (MDL) [5] will not provide a sufficient estimate for the required signal dimension. The simulation of several scenarios with different numbers of multipath components P,delay spread σ τ and angular spread σ θ for an antenna array with M = 8 elements has shown that using more than 3 signal branches does not provide any significant SNR improvement, as indicated in Figure 4. Basic results from these simulations are: for scenarios with small angular spread σ θ 10, the signal dimension after OT can be reduced to 3 or less with negligible loss of SNR gain independent of the respective delay spread, for scenarios with limited excess delay τ max 4T sym the signal dimension after OT can be reduced to 3 or less with negligible loss of SNR gain independent of the respective angular spread,

9 for arbitrary scenarios with unrestricted angular spread and large excess delay τ max 15T sym more than 82% of the available signal energy after OT is contained in the first 3 components resulting in a loss of SNR gain of less than 0.9 db. The results from simulations using arbitrary parameter combinations have been verified using tap delays and average relative path powers from the GSM and ITU channel models [6, 7] as parameters. The delays have been normalized to the GSM symbol period (T sym = 3700ns) for the GSM models and to a chip period of T C =1/4.096MHz = 244.1ns for the ITU models. The channel parameters delay spread σ τ, maximum excess delay τ max and their normalized values are summarized in Table 1. For each tap of the different channel models, a path angle θ p has been selected which is uniformly distributed within [ max θ p max ], with max varying from 0 to 180. The resulting eigenvalues from 100 trials have been averaged for each value of max.table1 shows the corresponding loss of SNR gain for the smallest sum of the 3 largest eigenvalues, which is considered the worst case (*). The last column of this table shows the required receiver dimension, for a maximum loss of 0.25 db SNR gain. The results obtained for the GSM HT and TU scenarios with a restricted angular spread are even more promising as apparent from Table 1. Channel model σ τ σ τ T C,S τ max τ max T C,S max EV SNR Gain loss Required [ns] [ns] [ ] [%] [db] Dimension ITU I-A * ITU I-B * ITU IO-A * ITU IO-B * ITU HA-A * ITU HA-B * GSM RA * GSM HT * GSM TU * Table 1: Comparison of the worst case performance loss for dimension reduction to 3 components compared with full 8 component case for ITU- and GSM-channel models These results suggest, that it makes sense to remove the partial coherence properties of multipath channels through spatial dimension reduction to simplify space-time receiver structures and to improve channel estimation techniques. 5 Reduced Dimension Space-Time Receiver As an example for the potential benefits of the dimension reduction approach we will discuss channel estimation and space-time MLSE using antenna arrays. An equalizer structure using an space-time Viterbi algorithm (VA) combined with reduced rank channel estimation (RRCE) was presented in

10 [8]. Starting from the received signal vector in (12) we can define a discrete time signal vector x(lt sym ) sampled at symbol rate 1/T sym with unknown timing error t 0 as x(lt sym + t 0 )= s 0 (k) h(lt sym + t 0, (l k)t sym + t 0 )+n(lt sym + t 0 ) (27) assuming that the channel remains constant during one time frame. Using further the assumptions, that s 0 (k) is i.i.d (15) and zero mean and that the noise is spatially and temporally white and uncorrelated with s 0 (k) and that the CIR is of finite duration LT sym, we can reformulate (27) with l := lt sym + t 0 for notational convenience as l x(l) = s 0 (k) h(l k)+n(l) =Hs(l)+n(l). (28) k=l L+1 Here, H =[h(l) h(l 1)...h(l L+1)] is the channel matrix and s =[s 0 (l) s 0 (l 1)...s 0 (l L+1)] T is the input symbol sequence which affects x(l). Considering a block of data of N symbol intervals the received signal in matrix notation can be written as X = HS + N (29) with X = [x(l) x(l +1)...x(l + N 1)] (30) N = [n(l) n(l +1)...n(l + N 1)] (31) s 0 (l) s 0 (l +1)... s 0 (l + N 1) s 0 (l 1) s 0 (l)... s 0 (l + N 2) S =.. (32)..... s 0 (l L +1) s 0 (l L +2)... s 0 (l L + N) As discussed in [8], under the assumption of white noise samples the VA can be implemented as a distance measure with the corresponding branch metric being d(l) = x(l) Hs(l) 2. (33) The performance of the VA depends on the estimate of H. If S defined in (32) contains known training sequence symbols, the least-squares estimate of H is given by Ĥ = XS H ( SS H) 1 = XS +, (34) with S + as the Moore-Penrose pseudo-inverse of S. In [8] it was proposed to replace this estimate by a subspace based approach. Using the decomposition of the spatial correlation matrix (26), the M-dimensional space is divided into a D s -dimensional signal subspace and a noise subspace by splitting the eigenvector matrix as U =[u 0 u 1...u Ds 1 u Ds...u M 1 ]=[U s U n ]. The signal

11 x(t) 8 Transform U H s X 3 x s(t) 3 3 Viterbi 3 ŝ raw BER U s 10-4 C S x SVD select 1-3 S + ˆQ SNR [db] (a) (b) Figure 5: (a) RD receiver structure; (b) Performance of RD receiver (solid line) using subspace channel estimation ˆQ compared with full 8-dimensional space-time receiver (dashed line) using full channel estimation Ĥ (ITU-IO-B channel, 8 tap channel estimation) subspace dimension D s is determined using the AIC or MDL criterion. Applying this decomposition a subspace estimate of the channel matrix was proposed as Ĥ ˆQ = U s ˆQ (35) with ˆQ = U H s XS +. (36) The main advantage of this estimate is the usage of additional knowledge about the signal subspace in U s, which is estimated from the entire frame and does not rely on the training sequence alone. However, analyzing (35) and (36) of the RRCE receiver suggests a reduced dimension (RD) receiver structure with the same performance at lower receiver complexity. The received signals are first transformed to reduced-dimensional data X s using the signal subspace eigenvector matrix U s, X s = U H s X, (37) where the dimension is now truncated to 3 instead of using the AIC or MDL criteria. All the remaining processing can then be performed with reduced complexity. The channel estimation based on the reduced data set is given by Ĥ s = ˆQ = X s S +, (38) which provides the same estimates as obtained with (36). The resulting receiver structure is shown in Figure 5(a). In Figure 5(b) the performance of a full space-time receiver (dashed line) and the RD receiver (solid line) is compared. The usage of signal subspace information results in a performance advantage of the RD receiver depending on the training sequence length. The performance is

12 identical to the RRCE receiver in [8], but at significantly lower receiver complexity. If interference is considered the RD receiver may even outperform the RRCE receiver since truncation of the non-signal components will also reduce interference. However, this depends on the interference characteristics and further research is needed. 6 Conclusion The spatial correlation properties of the received signal at an antenna array using a signal model which includes the transmit filter have been investigated. The rank of the spatial correlation matrix doesn t provide a sufficient measure for determining the possible dimension reduction in scenarios with partially coherent multipath signals. Here, analyzing the eigenvalue strength associated with the signal energy distribution after orthogonal transformation is more appropriate. The main advantages of the dimension reduction are the reduced spatial receiver complexity and the more robust channel estimation. However, further research needs to be carried out in order to replace the SVD by numerically less complex algorithms to derive the reduced dimension transformation matrix. References [1] Gene H. Golub and Charles F. van Loan, Matrix Computations, The Johns Hopkins University Press, third edition, [2] Arogyaswami Paulraj and Constantinos B. Papadias, Space-time processing for wireless communications, IEEE Signal Processing Magazine, vol. 14, no. 6, pp , Nov [3] Heinrich Meyr, Marc Moeneclaey, and Stefan A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation, And Signal Processing, John Wiley & Sons, Inc., [4] Steven M. Kay, Fundamentals of Statistical Signal Processing, Volume II: Detection Theory, Prentice Hall, [5] Mati Wax and Thomas Kailath, Detection of signals by information theoretic criteria, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 33, no. 2, pp , Apr [6] ETSI EN , Digital cellular telecommunications system (phase 2+); Radio transmission and reception, Tech. Rep. GSM version Release 1998, ETSI, [7] Radio Communications Study Group, Guidelines for evaluation of radio transmission technologies for IMT-2000/FPLMTS, Tech. Rep. 8/29-E, ITU, [8] Ayman F. Naguib, Babak Khalaj, Arogyaswami Paulraj, and Thomas Kailath, Adaptive channel equalization for TDMA digital cellular communications using antenna arrays, in Proceedings of the 1994 IEEE International Conference on Acoustics, Speech, and Signal Processing, 1994, vol. IV, pp