Double-Directional Estimation for MIMO Channels

Transcription

1 Master Thesis Double-Directional Estimation for MIMO Channels Vincent Chareyre July 2002 IR-SB-EX-0214

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3 Abstract Space-time processing based on antenna arrays is considered to significantly enhance the performance of the third and fourth generation of mobile radio systems. The expected advantages are increased capacity and better quality of supplied services. igh data rates are required and the highest rates per user are expected if multiple antennas are used at both receive and transmit sites. Such a radio propagation channel constitutes a Multiple- Input Multiple-Output (MIMO) system. For the design and performance evaluation of space-time adaptive processors, it becomes important to understand the characteristics of the spatial radio channel. Notably the knowledge of the angular information at both link ends could be used to properly design systems such as smart antennas. This thesis presents and evaluates five methods to estimate jointly the Direction of Arrival (DOA) at the receive site and the Direction of Departure (DOD) at the transmit site in a multipath environment. These methods have been proposed in the sensor array processing literature for the estimation of the DOA in a Single-Input Multiple-Output (SIMO) channel. Two of them, called Beamforming and MUSIC (Multiple Signal Classification) provide a spectrum-like function. Two parametric methods are based on the Maximum Likelihood (ML) principle. Two versions are presented, the stochastic approach and the deterministic approach, referred as to SML and DML respectively. Another algorithm used is a popular high-resolution parameter estimation scheme called Unitary ESPRIT (Estimation of Signals Parameters via Rotational Invariance Technique). The estimation of the angular parameters is based on measurements of the MIMO channel transfer matrix. These methods have been tested on simulated and actual data. In both cases, a Uniform Linear Array (ULA) is employed at each link site. An investigation of the resolution capability and the accuracy is carried out. The performance is also compared to the Cramér-Rao Lower Bound (CRLB). The results show that the ML techniques have the best asymptotic properties, i.e. they attain the CRLB. The 2-D Unitary ESPRIT is often slightly above the CRLB but its low computational cost makes it the preferred method. The performance of the MUSIC algorithm depends on several parameters as the number of realisations, the path Signal-to- Noise Ratio (SNR) and the correlation of the paths. The MUSIC estimators only attain the CRLB if the paths are uncorrelated, and both the number of samples and the path SNR are large. The Beamforming technique is often limited due to its very low resolution. Moreover, we apply parameter estimation schemes to indoor MIMO measurements conducted by the University of Bristol. The algorithms show similar results for different scenarios and tend to fail to resolve the DODs at the transmit side whereas the DOAs are resolved at the receive site. Parameters of the main ray can be approximately resolved in 1

4 a line-of-sight scenario. For multipath components, results are not reliable. It is shown that we deal with highly correlated signals and the failure appears at the transmitter if we estimate the direction of propagation separately at each side. Other tests based on actual measurements data are necessary to verify and validate the double-directional estimation schemes presented in this thesis. 2

5 Acknowledgements This master thesis project has been conducted within the Signal Processing Group at the Department of Sensors, Signals and Systems (S3) at the Royal Institute of Technology (KT) in Stockholm, between January 2002 and July First of all, I would like to thank my advisors Mats Bengtsson (Ph. D.) and Kai Yu (Ph. D. student) for their support and sharing of interesting ideas about sensor array processing field. I would like also to thank my fellow exchange and M. Sc. students who studied at KT during this year for contributing to a good atmosphere. 3

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7 Contents 1 Introduction Previous work Problem formulation Assumptions Performance Criteria Thesis outline Background Channel Modelling in Sensor Array Processing Channel and data modelling MIMO channel model Assumptions Transfer matrix estimation Data model Properties Double-Directional Estimation Spectral Based Methods Beamforming technique MUSIC algorithm Parametric methods Deterministic Maximum Likelihood Stochastic Maximum Likelihood D Unitary ESPRIT Optimisation search Derivation of the CRLB Simulation Results Resolution Spectral-based algorithms Method Results Parametric methods Method

8 Results Accuracy Method Results Comparison of performance Comments Results MIMO measurements Data analysis Results Comments Discussion Conclusions Future work...52 A Some Matrix Algebra...53 A.1 The vec operator...53 A.2 The Kronecker product...53 A.3 Theorem...54 B Direction Estimation at transmitter and receiver sides...55 B.1 Method...55 B.2 Beamforming...56 B.3 MUSIC...56 References

9 Chapter 1 Introduction With the evolution of wireless communication, the growing demand for mobile applications is constantly increasing the need of better coverage and higher quality service. Moreover, to facilitate the implementation of high-speed data services, huge data rates are required for future generations of mobile radio systems. The increase in traffic puts a demand on both manufacturers and operators to provide enough capacity in the networks. One of the means to provide this improvement is to use array antennas. Devices using multiple antennas, combined with particular coding techniques like space time coding [1], have the potential to improve the mobile radio performance in terms of capacity and data rates. The capacity enhancement can be obtained if multiple antennas are used at both the receiver and the transmitter in an environment presenting sufficient scattering. This concept, which has been recently a matter of great interest, constitutes a Multiple-Input Multiple-Output (MIMO) system. Modelling the so-called MIMO radio channel is essential to understand and analyse the performance and then to design such systems. Most of performance investigations have been based on non-physical models, which correspond to a simple stochastic matrix. owever, wireless transmissions are based on multipath wave propagation, therefore, a physical model of the MIMO channel is an alternative model. The angles of departure and arrival of a signal propagating over a MIMO channel are the most important and useful parameters in this kind of models. Smart antennas, also called adaptive antennas, are designed to adapt the antenna patterns to the current radio conditions. The knowledge of the direction of arrival at the transmitting side should be enough to track the user. owever, in an obstructed line-ofsight scenario presenting reflected waves, the angle of arrival becomes insufficient and we need to know the direction of departure. Methods to find these parameters could be implemented in transmit and receive algorithms to improve performance. This thesis aims at working out some algorithms for the estimation of the physical parameters of MIMO channels. The parameters of interest are the Direction of Departure (DOD), the Direction of Arrival (DOA) and the Path Gain of each path component. The different algorithms and models have been extracted from the sensor array processing literature for single-input multiple-output (SIMO) channels. The second purpose of this project is to evaluate the performance of these methods based on both simulation and 7

10 actual measurements data. The investigation is focused on indoor scenarios, using Uniform Linear Arrays (ULA) at both sides. 1.1 Previous work In the wireless communications area, antenna array processing has been an attractive technology. Most of the work has been concentred on the Direction of Arrival estimation. Characterisation of the channel in terms of DOA is an important property and has been used in wireless communications for many reasons. Traditionally, the main application has been source localization in radar and sonar. To make the distinction with a MIMO channel, the term Single-Input Multiple-Output (SIMO) channel is used. A large number of DOA estimation techniques and algorithms are well-known and they are listed in references [2] and [3] of the antenna array literature. In this thesis, we have extended these methods to the MIMO channel case. The first method called Beamforming dates back to the second world war and is based on the spectral analysis of spatio-temporary sampled data. The idea is to filter the output data measured at the receiver array (Wiener filtering). This leads to a spectrum where the matched location, which results in maximum power, yields the DOA estimates. This method, straightforward to implement, does not show a good ability to resolve closely spaced signal sources. This was the main drawback for the foreseen applications. The limitations of the beamformer started a development of others algorithms. By exploiting the physical and statistical data model, so-called parametric methods have been worked out during the last two decades. The two first methods were based on the Maximum Likelihood (ML) principle whose basic idea is that the best model is most consistent with the observations. Two different assumptions on the emitter signals have led to two approaches of the ML principle. The Deterministic Maximum likelihood (DML) technique assumes that the signal waveforms are deterministic but unknown contrary to the noise, which is modelled as white and Gaussian. The second approach is termed Stochastic Maximum Likelihood (SML) method. It is obtained by modelling the source signals and the receiver noise as Gaussian random processes. Another algorithm termed Spacing Alternating Generalized Expectation Maximization (SAGE) is based on the ML principle using the technique of expectation and maximization (EM) [4]. The expectation is with respect to the unknown underlying variables, using the current estimate of the parameters. The maximization step then provides a new estimate of the parameters. These two steps are iterated until convergence. The introduction of subspace-based estimation techniques in 1980 with the MUSIC algorithm (Multiple Signal Classification) was a tremendous step in the antenna array processing field. This algorithm is based on the eigen-analysis and second order statistics of a SIMO channel. It leads to a spectrum-like function of the DOA parameters where the peaks yield the corresponding estimates. The Estimation of Signals Parameters via Rotational Invariance Technique (ESPRIT) has been introduced later in Rapidly, it has become a powerful algorithm due to its simplicity. This algorithm requires a special array configuration, the ULA. Actually it exploits a property of ULAs, the so-called shift invariance. 8

11 Recently, a new version has been presented as the Unitary ESPRIT algorithm [5]. This algorithm has been extended to multidimensional parameters estimation and more generally can be apply for centro-symmetric arrays [6]. Other techniques have been reported in the array antenna literature and are most of the time an extension of the previous ones (Capon s beamformer, JADE, Root-Music, Subspace Fitting algorithm, see references [2] and [3] for a detailed presentation). In this thesis, the most significant methods are worked out to perform a good comparison. Concerning MIMO systems, techniques of identification of the multipath channel parameters are still rare. Very recently, some methods have been presented like the Double-Directional Super Resolution, as defined in [7] and the Joint M-Dimensions Parameter Estimation, as defined in [8]. The former one presents a technique that uses estimation and beamforming alternatively, and that relies on ESPRIT. In the latter one, they use the M-D Unitary ESPRIT algorithm with a circular uniform beam array (CUBA) at the transmitter side and a rectangular array at the receiver. Finally no studies about MIMO channels identification have been done using the parametric methods like DML or SML before. An evaluation of several methods for MIMO parameters estimation has not been done before. 1.2 Problem formulation This Master s Project aims to estimate multipath parameters of a wireless transmission over a MIMO channel. Each sub-signal, propagating over the MIMO channel, has to be identified in terms of departure angle at the transmit array and arrival angle at the receive array. Moreover, the path-gain has to be estimated as well when the method allows it. In order to estimate multipath channel parameters, different techniques proposed from the sensor array processing literature for SIMO will be worked out and extended to the case of MIMO. As defined in [2, 3], the methods subject to evaluation are: Beamforming, Multiple Signal Classification (MUSIC), Deterministic Maximum Likelihood (DML), Stochastic Maximum Likelihood (SML), Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT), The objective is to adapt these methods to estimate the pairs DOD/DOA in a MIMO channel. The evaluation will be based either on simulation data using the model presented in chapter 3 or on actual measurements data. The MIMO measurements have been performed at the University of Bristol in an indoor environment as part of the SATURN project. First the investigation of the performance will be focused on the resolution of these methods. This term generally refers in the antenna array field to the ability to distinguish two close rays, i.e. to find the smallest aperture angle, separating two arriving signals (or 9

12 two leaving signals) for which the corresponding rays can be resolved. This is straightforward to see on a spectrum-like function that exhibits peaks. owever, for parametric methods, the notion of resolution is not intuitive and we will give a specific definition of this notion later in this thesis. Another goal of this thesis is to compare the performance of each method. We will naturally compare them to the Cramér-Rao Lower Bound (CRLB) [9], which is a lower bound on the covariance matrix of any unbiased estimator. 1.3 Assumptions Throughout this thesis, we will consider Uniform Linear Arrays at both the transmitter and the receiver sites. We will assume propagation in the horizontal plane in a multipath environment. The number of multipath rays, denoted K in this report, is assumed to be known. The estimation of K could be done using standard model order estimation techniques, such as the Akaike Information Criterion, but this problem is out of the scope of this thesis. We are dealing with narrowband flat fading communications so the time-delay of arrival will not be taken into account. This term could be another parameter to estimate. 1.4 Performance Criteria As performance criteria, we will essentially study the Root Mean Square (RMS) error of one angular parameter, measured in degrees versus the source separation in degrees if we assume a two-path environment. Moreover, we will also use the Path Signal-to-Noise Ratio (SNR) measured in db, which corresponds to the power of one path gain divided by the power of the noise at the receiver. We will see that the performance depends on several parameters as the number of realizations of the measured channel matrix or the presence of correlated signals. 1.5 Thesis outline This thesis is organised as follows. In Chapter 2, we review the well-known SIMO channel model in the area of sensor array processing, using a ULA as receiver. Then Chapter 3 presents the MIMO channel model commonly used. We then describe the data model used to perform the simulations. Chapter 4 gives a detailed description of the algorithms we have evaluated for the Double-Directional Estimation. These algorithms are introduced in the same way as in the SIMO literature but extended to the MIMO case. Chapter 5 presents the results of the simulations we have performed using simulated data. Then Chapter 6 gives the results obtained with actual measurements data. Finally, Chapter 7 summarizes the entire work and we discuss about future work in this area. 10

13 Chapter 2 Background This section presents the data model commonly used in the field of Sensor Array Processing [2]. This model will be used to define the MIMO propagation transfer matrix provided we deal with the same antenna arrays and signals. On the other hand, the MIMO model will be transformed to match the SIMO case in order to apply the same algorithms. These models will be presented in the next chapter. 2.1 Channel Modelling in Sensor Array Processing Throughout this thesis, we will consider narrow-band signals represented by their baseband equivalents. A real-valued signal bandpass signal with center frequency f c, transmitted over a physical channel, can be written as z ( t) = Re s( t) { e } j 2π f c t where the baseband signal s(t) is the complex envelope of the transmitted signal z(t). The baseband equivalent representation is easier to work with than the bandpass representation. Moreover, an important property of narrowband signals is that time delays shorter than the coherence time, i.e. the inverse of the bandwidth, amount to phase shifts. Then, if the center frequency of the signal is f c, a small time delay will correspond to a complex j2 fc phase shift e in the complex valued baseband representation of the signal. Let us consider a linear array consisting of m antenna elements, as in Figure 2.1. A far field source emits a narrow-band signal modulated at the carrier frequency f c. The distance between the array and the source is assumed large enough to make the wave front planar. 11

14 Direction toward far-field. 0 1 m 1 Figure 2.1: Geometry of the equidistant linear array. For a Uniform Linear Array (ULA), the antenna elements are equally spaced by wavelengths, denoted as, along a line. The antenna elements are assumed omnidirectional and the antenna gain pattern of each antenna, which is a constant scalar, is normalized to 1. Denote by s(t) the signal as it is received at the first element (reference), then the signal received at any antenna element will experience an additional delay i and can be written as x ( t) = e i s( t) = e s( t) j2π fc τi j( i 1)2π fcτ d = a s( t) where d is the time delay between two sensors. It depends on the direction of the incoming wave front, which is denoted as. From the geometry of the problem illustrated in Figure 2.2, we have the following relationship cτ d cos θ = δλ where c is the velocity of the propagation. The distance c. d is referred to as d in Figure 2.2. i d. Figure 2.2: A Uniform Linear Array receiving a planar wave front. 12

15 Collecting the signals received by the individual elements into a vector x(t), we obtain x(t) = a( )s(t) where a( ) is the so-called array response vector or steering vector which is a function of the parameter. For a ULA, we have the following expression 1 j2π δ cosθ e a ( θ ) =. e j( m 1)2π δ cosθ If K signals impinge on the linear array from distinct DOAs, the output vector takes the form x(t) = A( )s(t) using linear superposition. Thus we define a steering matrix that contains the array response vectors of all the K sources and a vector of signals waveforms as A( ) = [ a( θ 1) a( θ 2 ) a( θ K )] [ s ( t),..., s ( t ] T s( t ) = ) 1 K In the presence of an additive noise n(t), we now get the following model x ( t) = A( ) s( t) + n( t), (2.1) which is a model commonly used in sensor array processing employing a ULA at the receiver side. 13

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17 Chapter 3 Channel and data modelling This chapter describes the model used for a MIMO channel employing ULAs at both ends. Statistical assumptions about data are stated and a method to estimate the corresponding transfer matrix of the channel is given. Then we define the actual data model we have chosen for the double-directional estimation and its basic properties are reviewed. 3.1 MIMO channel model The MIMO propagation model is described by the physical properties of the channel as it is illustrated in Figure 3.1. scatterer Transmitter Receiver Figure 3.1: MIMO propagation channel in a 3-path environment. 15

18 Considering the physical environment, we assume the antenna arrays are far enough to get far-field patterns. Therefore, the individual paths are assumed to have planar wave fronts. Moreover, we consider propagation in the horizontal plane and only the azimuth is estimated. owever, the extension to the real 3-D environment is possible with the estimation of the elevation. The MIMO transfer matrix relates the transmit array excitation vector u(n) to the observed receive array element response vector y(n) as ( n) u( n) y = (3.1) where the array output vectors y(n) and u(n) are the complex baseband representations. The transfer matrix is formed as a superposition of array responses and defined as K b k k= 1 = a ( θ ) a ( ϕ ). (3.2) Rx The parameters k and k are the departure and arrival angles of the kth ray respectively. Each path is weighted with a complex path gain b k. The vector a Rx ( k) is the receive array response vector for direction k. It is defined as in the previous section provided we use a ULA with the same assumptions. The vector a Tx ( k) is the corresponding transmit array response vector for a signal emitting from the direction k. We will assume throughout this thesis there are m transmit elements and n receive elements, then the channel matrix here is an n by m matrix. The previous formulation can be expressed in a matrix form k Tx k = A BA (3.3) Rx Tx where B = diag{b} with b = [b 1,, b K ] T. The steering matrices A Tx and A Rx are defined in the same way as the steering matrix in the SIMO case. 3.2 Assumptions In all measurement situations, noise inevitably appears and has to be modelled. At the previous model (3.1), we include an additive term as y ( n) = u( n) + v( n). (3.4) There are many sources of noise [10], like for instance environmental noise, including cosmic noise and atmospheric noise. Another kind of noise is man-made noise, caused by electrical equipment in the vicinity of the receiver. The receiver also generates some noise, mainly thermal noise caused by the thermal motion of particles in all materials. Thermal noise is readily modelled as a stationary Gaussian Stochastic process, with 16

19 essential flat spectral characteristic, so called white noise. The receiver noise is often the dominating noise source then we assume v(n) as an Additive White Gaussian Noise (AWGN). The noise is assumed to be spatially and temporally white and Gaussian distributed. E 2 T { v( n) } = 0 E{ v( n) v( s) } = σ Iδ E{ v( n) v( s) } = 0 n, s v The transmitted source signal is assumed to be Gaussian distributed in a similar manner. It is also assumed that the transmitted signal is uncorrelated with the noise. The complex gain b k of the kth path can be written as ns b k e jψ k = β k. The amplitude k is usually assumed as a Rayleigh distributed random variable as we are dealing with flat fading signals [10]. The phase k represents a random phase associated to each path, uniformly distributed on (0,2 ). This added phase could be seen as a random phase shift when the ray encounters a scatterer. Then it yields that b is complex and the two parts become uncorrelated Gaussian processes. This is the assumption required for the Stochastic Maximum Likelihood algorithm. For the deterministic approach of the ML principle, we will assume b is deterministic but unknown. Moreover, we will assume that the complex path gain is time varying. 3.3 Transfer matrix estimation From Equation (3.4), we can estimate the channel matrix using a Least Squares (LS) method provided we have got an over-determined system. The antenna arrays are appropriately pre-processed and sampled with a period T, normalized to T = 1. If we collect a batch of N samples into a matrix, we will get Y = U + V where Y = [y(1) y(n)], U = [u(1) u(n)] and V = [v(1) v(n)]. If U is already known, because of training for instance, then ˆ = YU + = YU ( UU ) 1 where U + denotes the Moore-Penrose pseudo-inverse of U. Finally we get the following expression ˆ ~ + = A BA + N (3.5) = Rx Tx where N is the estimation error that is still colored since we have N = VU +. owever we make the simplifying assumption that N is white. 17

20 If U is white, the variance of the noise N is related to the noise power of V as 2 v σ =. N 2 σ This relationship has been verified with simulated data. The larger the number of snapshots is, the smaller the variance of the estimation error on is. The final equation, defined in (3.5), is the model of interest although it will be reshaped in order to apply the different direction estimation methods, as we will see in the following section. 3.4 Data model The ultimate goal of this thesis is to estimate the angles of departure and arrival, with knowledge only of the transfer matrix. As we have seen before, several algorithms have been proposed to carry out space-time processing of data sampled at an array of sensors. In the special case of a double directional channel, the problem gets one dimension larger. The provided data become matrices instead of vectors. Then the main idea is to vectorize the matrix by stacking its columns in order to separate the angles of departure and arrival. Using the vec operator, see Appendix A, we get from (3.2) In a matrix form, we get vec = vec = K k = 1 K k k = 1 vec b a ( a ( θ ) a ( ϕ ) ) Rx Rx ( θ ) a k k Tx Tx ( ϕ ) k k b k. vec = where A( ϕ θ ) = vec a Rx ( θ1 ) atx ( ϕ1 ) c = a ( ϕ ) a ( θ ) A( ϕ, θ ) b [ ( ) vec( a Rx ( θ K ) atx ( ϕ K ) )] c [( ) ( a ( ϕ ) a ( θ ))], (3.6) Tx 1 Rx 1 The superscript c denotes the complex conjugate and denotes the Kronecker product. The matrix A here is an M by K matrix where M = m n. Let us denote vec() by h. In presence of additive white Gaussian noise, the vec operation does not change the properties of the noise. If different realisations of are available, we finally obtain Tx ( t) = A( θ ) b( t) n( t) K h ϕ, + (3.7) We could now identify the previous equation with the model (2.1) defined in the SIMO case. The given finite data set {h(t)} corresponds to the data set {x(t)} of the SIMO Rx K 18

21 channel and the vector b(t) containing the complex path gains corresponds to the vector of signals waveforms noted s(t) in the previous chapter. The main difference is that the steering matrix A is a function of two parameters, i.e. two angles per ray. 3.5 Properties This section presents basic properties of the model (3.7). Many of the schemes used to estimate the angles of departure and arrival rely on the properties of the second-order moment of the measured data. Therefore, we can define the covariance matrix of the finite data set {h(t)} as R h 2 { h( t) h( t) } = A( ϕ, θ ) PA( ϕ, θ ) + σ I = E (3.8) where P = E{b(t)b(t) } is the path covariance matrix. In the derivation of the covariance matrix, the cross-terms vanish because the noise n(t) is independent of the path gains b(t) and has zero mean. The spectral factorisation of the data covariance matrix is of great interest and is commonly used in the field of array signal processing [2] thus we can also represent the covariance matrix R h by its eigenvalue decomposition. If K paths are present in the MIMO channel and if they are not fully correlated, the product APA has rank K and R h can be written as a sum of two parts. R h EΛE = Es ses + n n n = E E. One part is related to the signal where E s = [e 1,, e K ] denotes the signal eigenvectors and s is a diagonal matrix containing the K dominant eigenvalues. The second part consists of the eigenvectors, denoted E s = [e K+1,, e M ], corresponding to eigenvalues equal to the noise variance then n = 2 I. Since the subspace spanned by the columns of E s is equal to the subspace spanned by the columns of A, then it is usually called the signal subspace, or path subspace in our case, and the corresponding space spanned by E n is therefore called the noise subspace. These two spaces are orthogonal. A natural estimate of the covariance matrix is based on the sampled data using N 1 R ˆ = h, h N t = 1 ( t) h( t) for which a spectral representation similar to that of R is defined as Rˆ = Eˆ ˆ Eˆ + Eˆ h s s s This representation will be extensively used in the description and implementation of the subspace-based estimation algorithms. n ˆ n Eˆ n 19

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23 Chapter 4 Double-Directional Estimation This section describes the well-known parameter estimation methods we have used to estimate both the angles of departure and arrival of a signal propagating over a MIMO channel. The algorithms are pulled out of the SIMO sensor array processing literature and extended to the case of the MIMO channel. We can classify these techniques into two main categories, namely Spectral-Based and Parametric approaches. In the former, a spectrum-like function can be plotted as a function of the two parameters of interest, the DOD and DOA. The location of the peaks in the spectrum gives the corresponding DODs and DOAs. The latter approach requires a simultaneous search for all parameters and increases the computational cost. 4.1 Spectral Based Methods Beamforming technique The idea of the 2-D beamformer is that the energy propagating over the MIMO channel is focused to one direction of departure and one direction of arrival, at a time. This can be expressed by y(t) = w h(t), where w is a weighting vector that emphasizes one particular path and thus two angles. Given samples y(1), y(2),, y(n), the corresponding power is measured by computing N N P ( w) = y( t) = w h( t) h( t) w = w Rˆ hw (4.1) N N t= 1 t= 1 where Rˆ h has been defined in the previous section. Many different choices of the weighting vector w can be made leading to different properties of beamforming schemes, see [2]. In this thesis, only the conventional beamformer is developed for the MIMO channel. This scheme is a natural extension of classical Fourier-based spectral analysis. 21

24 The choice of the weighting vector to maximize the power for a certain ray path [3] leads to a( ϕ, θ ) w = BF a ϕ, θ a ϕ, θ ( ) ( ) where a(, ) is the MIMO steering vector similar to one column vector of the model (3.6). Inserting the weighting vector into Eq. (4.1), the classical beamforming spectrum is obtained ( ϕ, θ ) Rˆ h a( ϕ, θ ) a( ϕ, θ ) a( ϕ, θ ) a PBF ( ϕ, θ ) =. This spectrum is a function of the DOD and DOA parameters. It shows the peak locations that result in maximum power. In practice the maximization of the cost function is determined through a grid search over the parameters and. The Figure 4.1 shows a 2D beamforming spectrum in the case when two rays are propagating over the MIMO channel. An 8-element antenna array of half-wavelength inter-element spacing is used at both ends. The true pairs (DOD, DOA) are (60º, 90º) and (100º, 50º) and a batch of 100 samples of the transfer matrix are available. Figure 4.1: Cost function of the Beamforming algorithm. Scenario involving two paths of equal power, path SNR = 20dB. 22

25 The step size of the mesh is 5º. The accuracy of peak detection depends on the step size of the grid search, which costs in time and computation. We will at the end of this section see means to optimize the peak detection in a spectrum MUSIC algorithm The multiple signal classification (MUSIC) method is a spectral-based algorithm but relies on the properties of the eigenvalue decomposition of the covariance matrix. As we have seen before, the signal and noise subspaces are orthogonal. This fundamental property means that the noise eigenvectors are perpendicular to the matrix A or the signal subspace spanned by E s. Then we have the following orthogonality condition n ( ϕ θ ) = 0, ( ϕ, θ ) { ( ϕ, θ ),..., ( ϕ, θ )} E a., 1 1 The idea of the algorithm is to find the K vectors a(, ) which are the most orthogonal to the estimate of E n. The signal subspace is estimated from an eigendecomposition of an estimate of the covariance matrix R h. Finally, we can define a MUSIC spectrum as ( ϕ, θ ) P M = a a ( ϕ, θ ) a( ϕ, θ ) ( ϕ, θ ) ˆ ˆ E E a( ϕ, θ ) The scenario presented in the previous section has been processed with the MUSIC algorithm. The results are shown in Figure 4.2. The step size of the mesh in this case is 2º. The cost function P M (, ) shows very sharp peaks in the vicinity of the true couples (DOD,DOA). Very sharp peaks can be missed if the step size of the grid search is too large. Comparing the spectra in Figure 4.1 and 4.2, this parametric method gives a better resolution, i.e. the ability to resolve two close peaks, than the traditional beamforming technique since we obtain very sharp peaks. The resolution and performance of this method will be studied in details further in this report. n n K K 23

26 Figure 4.2: Cost function of the MUSIC algorithm. Scenario involving two paths of equal power, path SNR = 20dB. 4.2 Parametric methods The spectral-based methods presented in the previous section are quite attractive regarding the computational cost. owever for certain bad scenarios, involving for instance high correlated paths, these methods fail to resolve each path. In the array signal processing literature some parametric methods are proposed that exploit the underlying data model. These more powerful methods have been extended for the search of two angles in the MIMO case. The two methods are based on the Maximum Likelihood (ML) technique. They are based on different statistical properties of the ray paths. The Deterministic and Stochastic ML techniques have been defined Deterministic Maximum Likelihood The deterministic approach of the ML technique does not require any assumption about the ray paths. The complex path gains are modelled as arbitrary deterministic sequences although they are unknown (the carrier frequency of the propagating waves over the 24

27 MIMO channel is known). owever the noise term is assumed to be white Gaussian with zero mean as we have seen before. As a consequence of the statistical assumptions on the noise and the path gains, the observed process is distributed as h(t) N(A(, )b(t), 2 I). The unknown parameters of this model are = 1,, K and = 1,, K as well as b(1),, b(n), 2. The probability density function (PDF) of all observations viewed as a function of the unknown parameters is termed the likelihood function. The PDF of one measurement of the vectorized channel matrix is the complex M-variate Gaussian PDF defined by 2 1 e 2 h( t) A( ϕ, ) b( t) σ M 2 ( πσ ) The PDF of all observations is the product of the PDF of each one since the measurements are independent. For the sake of convenience, we take the negative loglikelihood function. Normalized by N and dropping the parameter-independent term, we get l DML N ( ϕ,, b( t), σ ) = M logσ + h( t) A( ϕ, ) b( t). 2 σ N The ML estimates of the unknown parameters are computed as the maximizing arguments of the likelihood function therefore the latter expression has to be minimized. Introduce the pseudo-inverse of A and the orthogonal projector onto the nullspace of A by A P + A ( ϕ, ) = ( A A) 1 A t= 1 + ( ϕ, ) = I AA = I P, The so-called DML estimates are the minimizing arguments of the function l DML and are obtained from [2, 11], { P R ˆ } 1 ˆ σ 2 = Tr A h M, ˆ + b( t) = A h( t), ˆ, ϕ ˆ θ = arg mintr P R. ( ) { } First, the angles of departure and arrival are of interest. If K rays are assumed to propagate over the MIMO channel, the minimization problem leads to a 2 K dimensions search. This is computationally expensive. owever a numerical optimization technique can be used to obtain accurate results, but the numerical optimization must be initialised. ϕ, θ A A. h 25

28 As this remains the main problem of the parametric methods, we will discuss this point further in this thesis. Once the angles have been found, these estimates are inserted into the two first equations above to obtain the estimates of the noise variance and the vector of complex path gains for each time step. Notice that the last equation giving both DOD and DOA parameters reduces to the Beamforming cost function in the case of a single path (K=1) Stochastic Maximum Likelihood This second ML approach is referred as to the Stochastic Maximum Likelihood technique in the literature. It is based on another assumption of the path gain distribution and uses the model initially defined in the third chapter. The complex path gains are modelled as stationary, temporally white Gaussian random processes with covariance P. This model leads to the definition of the channel covariance matrix (3.8) R h = A 2 ( ϕ, θ ) PA ( ϕ, θ ) + σ I The observed process is now distributed as h(t) N(0, APA + 2 I). The unknown parameters in this stochastic model are,, P and 2. By using the model given in (3.7) and the statistical assumptions, the negative log-likelihood function to be minimized becomes l SML 2 1 (,, P, σ ) = log R + Tr{ R Rˆ } ϕ (4.2) where R h is the definition of the covariance matrix and a function of the parameters to estimate whereas Rˆ is the covariance matrix estimated from the observed data set h {h(t)}. This expression can be put in a concentrated form [12]. For fixed parameters 2 and, the estimations of P and becomes σ ˆ 2 h 1 ( ϕ, θ ) = Tr{ P A R ˆ h } M K + +, θ = A ( Rˆ ˆ σ ϕ θ I) A ( ϕ ) (, ) Pˆ 2 h By inserting these expressions into (4.2), the concentrated negative log likelihood function is obtained as ( ϕ ˆ θ ) { APˆ 2 ˆ, = arg min log ( ϕ, θ ) A + ˆ σ ( ϕ, θ ) I } ϕ, θ and both the DOD and DOA estimates are taken as the minimizing arguments. This method requires a numerical search as well, which can be computationally heavy. h h. 26

29 SML estimates are consistent since they converge to the true value when the number of data tends to infinity. With the deterministic model, consistent estimation of all model parameters is impossible since the time-varying path gains are regarded as unknown parameters. It follows that the number of parameters to estimate grows without bound with increasing N. Moreover it has been shown in [13] that the SML algorithm provides more accurate estimates than the deterministic ML technique. The difference is significant for very small number of sensors, low SNR and highly correlated signals. owever the difference in performance is negligible in most scenarios of practical interest as we will see further on in this thesis. This property holds regardless the actual distribution of the path gains and both ML techniques are actually applicable. Under the Gaussian assumption, the SML method is asymptotically efficient as the estimation error attains the CRLB D Unitary ESPRIT In this section, we review the two-dimensional extension of the Unitary ESPRIT algorithm. This method is applied to the MIMO channel in order to estimate the 2-D parameters jointly, i.e. both the DODs and the DOAs. Due to its simplicity and high performance, ESPRIT has become one of the most powerful subspace-based techniques for DOA or frequency estimation schemes. For certain array geometries, namely centrosymmetric arrays, an ESPRIT-type algorithm has been formulated to reduce the computational complexity significantly. The resulting algorithm is called Unitary ESPRIT. This algorithm has been extended to Two-, Three-, and Multidimensional cases [5]. We will here only present one way to apply the algorithm to the MIMO case. Some theoretical elements are presented in [14] and a thorough description of the Multidimensional extension of Unitary ESPRIT is presented in [15]. First we review the model. Considering one element of the channel matrix (t), it can be written as K jµ k ( l 1) jυk ( p 1) ( t) = b p ( t) e e + nl, ( ) l, p p t k= 1 where l denotes the row and p denotes the column. The frequencies µ k and ν k are function of the DOA and the DOD of the kth ray respectively. Using the present algorithm, we can estimate the two frequency vectors = [ µ µ ] T and [ ν ν ] T 1 2 µ K with correct pairing of the estimates. 1 2 ν K =. All the MIMO channels are vectorized and stacked into a matrix as shown in (3.7). These measurements vectors are placed as columns of an M N data matrix denoted X. Let us denote r the dimension, which can take the values 1 or 2. The Unitary ESPRIT Algorithm exploits the centro-symmetric property of sensor arrays. A sensor array is called centro-symmetric if its elements locations are symmetric with respect to the centroid (i.e. generally these elements occurs in pair) and the complex 27

30 response patterns of the paired elements are the same. A MIMO channel matrix with M = M 1 M 2 = n m elements has this property and can be seen as a Uniform Rectangular Array. Such a sensor array has a dual-invariance structure, two identical sub-arrays of m 1 = M 2 (M 1 1) elements can be formed along the first dimension. In the same way, another pair of identical sub-arrays, consisting of m 2 = M 1 (M 2 1), is displaced along the second dimension. Then we can define two pairs of selection matrices that are centro-symmetric with respect to one another, i.e. J ( )1 = J r m ( r)2 M r = r where Π M denotes the M M exchange matrix with ones on its anti-diagonal and zeros elsewhere. The reference [15] gives explicit formulas to obtain the 2-D selection matrices. Figure 4.3 visualizes a possible choice of the selection matrices for a MIMO channel employing a 4-element ULA at both sides. 1,2 J (1)1 J (1)2 J (2)1 J (2)2 Figure 4.3: Sub-array selection for a MIMO channel of M = 4 4 sensor elements. Then forward-backward averaging is achieved by transforming the given measurement data matrix X into the matrix [ X Π M X c Π M ] of size M 2N, which is a complexvalued extended data matrix. It is transformed into a real-valued matrix of the same size and we obtain c ( X ) = Q M [ X M X N ] Q 2N where the matrices Q denotes the left Π-real matrix. For instance, the matrix of odd order is defined as I n 0 1 = T Q n n 0 ji j n n T

31 A unitary left Π-real matrix of size 2n 2n is obtained from the latter one by dropping its central row and central column. Then we compute the K dominant eigenvectors of the covariance matrix of Γ(X) and form the signal subspace E s. Then it is shown in [16] that the spatial frequencies, related to the parameters of interest, can be obtained by solving the over-determined set of equations K ( r ) 1 Es r K ( r ) 2Es r = 1,2. using a (Total) Least Squares technique. The selection matrices K (r)1 and K (r)2 are obtained from the selection matrices in the following way ( J J ) M j ( J r 1 J r ) Q M K r 1 = Q + Q r 2 ( ) m ( r ) 1 ( r ) K ( r ) 2 Q mr ( ) ( ) 2 =. The next step in the algorithm is to estimate the spatial frequencies by computing the eigendecomposition of the complex-valued matrix defined by 1 1 j 2 = T with { } K k k = + T = diag λ. 1 Taking the real and imaginary part of the previous eigenvalues automatically pairs the frequencies. We obtain ( Re{ k }) k 1, 2, K ( Im{ }) k 1, 2, K µ k = 2 arctan λ =..., ν k = 2 arctan λk =..., Finally, we can easily extract the pair of DODs and DOAs from the frequencies. The 2-D Unitary ESPRIT technique turns out to be more powerful than the other spectralbased methods and ML techniques in term of computational cost. Indeed this method does not require any preliminary search of the estimates and can be applied directly. Another important characteristic is that this technique requires only one eigenvalue decomposition and some matrix products. It does not need any optimisation search. owever, we will see further on that the 2-D Unitary ESPRIT estimation is not as asymptotically efficient as the ML methods. 4.3 Optimisation search Observing the description of the different techniques, most of these algorithms require a final optimisation search for the angle estimation of interest. The spectral-based methods give a two-dimensional spectrum-like function. The peak locations yield to the pairs of parameters DOD and DOA. The maximization of the 29

32 function is determined through a grid search over the two angular parameters. Then we apply a numerical optimisation search at each maximum. Given the previous values as a starting point, the problem is to find the parameters that maximize the two-variable function. We have used the Matlab function fminsearch provided in the Optimisation toolbox. This function is based on the Nelder-Mead simplex method presented in [17]. If the step size of the grid search is too large, some peaks can be missed especially if they are too sharp. On the other side, false extra peaks can occur in the scan of the spectrum if it is too noisy. Subspace-based methods like the ML techniques require a multidimensional search to calculate the estimates of the angles. Given the starting values, the previous function can be used to solve the multivariable optimisation problem. The main problem is to estimate the initial estimates first to feed the optimisation algorithm. Obtaining sufficiently accurate initial estimates is generally a computational expensive task. One possibility is often to apply a spectral-based technique like MUSIC to provide the initial values. Another approach is to compute a multidimensional grid of the cost function provided the number of paths is known. In simulations, this technique has been used considering a two-path environment. This leads to a 4-D search. For a model order higher than two rays, the time of computation becomes too enormous. 4.5 Derivation of the CRLB In this section, we derive the Cramér-Rao lower bound (CRLB) on the covariance matrix of any unbiased estimator of and. The CRLB is derived under SML assumptions, i.e. the path gains are stochastic Gaussian processes. In theory the SML method outperforms the DML method [13]. This justifies the stochastic model being appropriate for the CRLB. The expression of the CRLB has been derived in [18] with the point source model presented in (2.1) in the SIMO case. As we have seen, our reshaped MIMO data model (3.7) is based on the same assumptions and the covariance matrix of data set {h(t)} is given by 2 R = A ϕ, θ PA ϕ, θ + σ I = E E + E E. h ( ) ( ) ( ) Let us denote the parameter vector containing both the DOD and DOA parameters such as = [ 1 1 ϕ K θ K ] T. Then the CRLB matrix is given by [18] CRLB [ R ( ) ] h 1 c 1 [ R ] [ R ] s s s vec Rh ( ) ( ) h h n n n [ ] 1 1 vec = Re s n T 2N where [ ] s and [ ] n denote the pseudo-signal and pseudo-noise parts respectively, i.e. 1 1 [ R h ] = E s s s E s 1 1 [ R ] = E E h n In the following sections, the performance of the different algorithms will be compared to the ultimate performance of the CRLB. n n n 30

33 Chapter 5 Simulation Results This chapter gives a collection of results on the performance of techniques to estimate directions of propagation through a MIMO channel. First we investigate the resolution and accuracy of double-directional estimation methods. Then we compare their performance in different scenarios. The simulations carried out along this chapter are based on generated data. 5.1 Resolution In array processing field, the resolution is the ability to resolve two closely spaced sources. For spectral-based methods, this is easy to represent as we obtain a spectrum showing peaks: two sources are not resolved if only one peak appears in the spectrum. The resolution of the conventional beamformer is well known in the SIMO literature and a simple expression for the minimum separation in pulsations, i.e. (πcos θ 2 πcos θ 1 ) for a half-wavelength spacing, is given by = 2 /n, n being the number of the elements of the receive array. For the MUSIC algorithm, the resolution threshold has been derived in [19] and depends on several parameters under certain limited conditions. For parametric methods, the notion of resolution is not trivial and no specific work has been reported. For the MIMO case, the problem gets one dimension larger. We then define two resolutions: one at the transmitter denoted and one at the receiver denoted. It is obvious that one resolution depends on the other one Spectral-based algorithms Method We assume a two-path MIMO channel. We apply a spectral-based technique and analyse the spectrum. When two beams are too close, that results in a merging of the two peaks and a local minimum appears between them, as it is shown in Figure 5.1. Data have been 31

34 generated from model (3.7). This example includes two rays emitted and received by an 8-element ULA. Their parameters are (100, 90 ) and (100, 105 ). The path SNR is 20 db. Figure 5.1: Beamforming spectrum with two close rays. Mesh step size = 3. Given the cost function of the Beamformer, we can compute the ratio between the power of the top and the power of the minimum called the valley. It is getting smaller when the rays get closer. Therefore this ratio depends on both source separations. It also depends on the relative phase of the path gains however it is appropriate to determine the resolution. Let us call it the Peak-to-Valley Ratio. This ratio can be plotted as a function of the source separation at both the transmitter and the receiver. By setting a threshold, we can determine the behaviour of the spectral-based algorithm in resolving two close paths. This method has been tested with both Beamforming and MUSIC algorithm. The results are described in the following section. In practice, we apply one algorithm to find the two peaks for a given source separation at each side. The point corresponding to the valley is on the curve passing by the two summits. This is a parametric curve of the angles and. By using the Lagrange Interpolation, the previous curve can be reduced to a curve function of one parameter. Finally we apply a local min search to find the minimum of the curve. Then we can easily obtain the power difference measured between the lowest peak and the local minimum. The gap is set to 0 db if only one peak is resolved. 32

35 Results First we study the resolution of the Beamforming technique. We simulate a scenario with two uncorrelated rays propagating over a MIMO channel. A ULA of m = 10 sensors of half-wavelength inter-element spacing is used at the transmitter side whereas the receiver array is an 8-element antenna array. Different numbers of sensors are used to better see the implication of this parameter on the resolution. The source separation at each end varies between 0 and 20 by step of 1. Figure 5.2 shows the Peak-to-Valley ratio measured as a function of the DOD and DOA separation. The path Signal-to-Noise ratio for both rays is 30 db. Figure 5.2: Peak-to-Valley Ratio versus the DOD and DOA separation based on measurements of Beamforming spectrums. For a source separation lower than 10 at each side, the peaks are not resolved since the ratio is 0 db. The next step is setting a threshold on the ratio. We have set this limit to 0.8 db, which corresponds to a gap of 15%, according to observations on simulated data. This threshold is an upper bound. Under this limit, rays cannot always be separated. The figure 5.3 below shows the intersection, viewed from above, of a horizontal plane giving the threshold with the previous the parametric surface. 33

36 Figure 5.3: Beamforming resolution limits of a 10 8-element MIMO channel. The grey area corresponds to Peak-to-Valley ratio lower than the threshold, i.e. scenario where only one peak is resolved. According to Figure 5.3, the limit at the transmitter side is 11.5, equivalent to a pulsation separation of = 2 /10. In the same way, the limit at the receiver side is 14.5, giving a separation of = 2 /8. Moreover the shape of the area delimiting the ability to resolve two paths is approximately an ellipse. Based on the previous observations, we obtain the following equation θ 2 π n 2 ϕ + 2 π m where n and m are the number of elements at the receiver and transmitter respectively. The resolution depends only on these parameters and not on the path SNR. This result has been verified with different simulated scenarios. The same method has been tested with MUSIC. We consider a two-path environment with path gains of equal power and a path SNR set to 30 db. The transmitter is a ULA of 10 elements, as well as the receiver. A batch of N = 100 realizations of the transfer matrix is available. Using the method seen before, the Peak-to-Valley ratio has been computed for DOD and DOA separations varying from 0 to 2 by step of 0.1. Figure 5.4 shows the resolution limits for this scenario. The threshold is 0.8 db. 2 = 1 34

37 Figure 5.4: MUSIC resolution limits of a element MIMO channel. Path SNR = 20 db, N = 100 realisations. MUSIC algorithm shows better results than the 2-D Beamformer. Considering the shape, we should obtain a circle as we have identical arrays at both ends in this scenario and if MUSIC follows the same hypothesis we stated for the Beamforming technique. This shape is nearly obtained. A small spot appears in the lower and middle part of the figure. This is due to the measured Peak-to-Valley ratio that depends on the relative phase of the complex path gains. In the SIMO literature, the resolution of MUSIC has been derived [19] under certain conditions. For equal power sources, it depends on a complicated manner on several parameters as the number of realisations, the number of elements, and the SNR. We have the following equation [19] 2880 Nm ( m 2) Nm 1+ 60( m 1) = ξ (5.1) where m is the number of elements of the array, N is the number of samples, is the SNR per source and is the resolution. Figure 5.5 shows the resolution in degrees as a function of the SNR for different number of elements, where N = 100 samples. 35

38 Figure 5.5: MUSIC resolution versus SNR, in a SIMO scenario, for 8 and 10 halfwavelength-spaced antenna elements. 100 samples are taken. For a 10-element array, the resolution is 0.9 with a SNR equal to 30 db. In the MIMO case, see Figure 5.4, the resolution limits are close to 1 at both ends. We use a 10- element array at both ends with 30 db as path SNR. The results are quite similar. We have been testing this method for different scenarios and we have got analogous results provided the path gains have equal power. Under this condition, we can make the hypothesis that the MUSIC resolution is given by d MU θ 2 + ( n, ξ, N ) d ( m, ξ, N ) MU ϕ 2 = 1 where d MU is the function, obtained from Eq. (5.1), that gives the resolution as a function of the number of elements, the path SNR and the number of samples N Parametric methods Method For these techniques, the previous method cannot be applied. We have to define the resolution in another way. Indeed algorithms such as DML, SML or Unitary ESPRIT need the number of rays K. If we set K to 2, they will always give two solutions whatever the scenario is. If two rays are extremely close, parametric methods will give two wrong estimated rays opposed to spectral-based methods where only one ray is found in the spectrum. 36

39 Therefore, we have to take into account the estimated separation at both side as well as the error on each parameter. We define the resolution at the transmitter as the angular separation for which the average distance between the DOD estimates corresponds to the true separation and the RMS error on each DOD is within a certain limit. The resolution at the receiver is defined in the same way on the other side. The threshold on the RMS error of one parameter is set to 25 % of the true separation at the corresponding side. If the RMS error is higher than the threshold, we assume the rays are not always resolved. This is based on practical observations. Both resolutions depend on each other. Then the previous conditions have to be considered simultaneously at each side Results We deal with 2-D Unitary ESPRIT algorithm. We consider a two-path environment. One beam has got its parameters fixed. The second ray has its parameters, both DOD and DOA, varying. We compute the RMS error of each parameter and the average distance between the beams at the receiver and the transmitter, based on 800 trials to get sufficient statistics. The DOD and DOA separations are kept when the conditions are reached. The Figure 5.6 shows the results for a path SNR of 20 and 30 db. Figure 5.6: Unitary ESPRIT resolution for an 8 8-element MIMO channel. The 2-D Unitary ESPRIT shows a high resolution, better than traditional spectral-based methods. owever if the rays separation is getting very large at one side, the minimum angular separation at the other side becomes smaller but does not attain 0 with the Unitary ESPRIT method. For instance, if two rays are separated from 40 at the transmitter, the minimum separation at the receiver has to be in order to resolve the DOAs, under a path SNR of 30 db. For spectral-based methods, the minimum separation is 0 as the function giving both resolutions is an ellipse-like equation. 37

40 As these methods are very computationally heavy, other scenarios involving different array sizes or SNR have not been studied. 5.2 Accuracy Method To investigate the accuracy of different algorithms, we consider a single-path environment. Simulations have been performed using a basic scenario with an 8-element ULA with half wavelength element separation at both sides. DOD and DOA of the single beam are respectively and All data are generated from the physical model (3.7). The angle estimates are computed from a batch of N =100 samples of the channel. For different path SNR, the estimated RMS error of θˆ is studied. Naturally, it is compared with the optimal CRLB on the variance of θˆ Results The plot in Figure 5.7 shows the performance of MUSIC and 2-D Unitary ESPRIT. The estimated RMS value of θˆ is plotted, calculated from 500 trials, versus the path SNR. The CRLB has been included as well. Figure 5.7: RMS error of θˆ using MUSIC and Unitary ESPRIT, versus the path SNR. Estimated with MUSIC, θˆ is asymptotically efficient as it attains the CRLB even for low SNR. owever Unitary ESPRIT attains the CRLB only for very high path SNR, which are complicated to get in practice. This scenario has been reproduced using the Maximum Likelihood technique. We have obtained identical results as MUSIC regardless the 38

41 initialisation of these algorithms, i.e. either with true values or by carrying out a 4- dimension search previously. In the same way, the Beamformer gives the same results. In a single-path environment, Beamforming and DML have the same cost function. For a two-path scenario, we investigate the performance of each method as a function of the source separation at one side. This is the topic of the following section 5.3 Comparison of performance We consider a scenario involving two beams to compare the performance of each method. Two 8-element ULAs are used. One ray is kept fixed at = 89.5 and = The second ray has its departure angle fixed at = 90.5 whereas the arrival angle is varied between 93 and 97. The data are generated from the model (3.7). The different parameters, such as the path SNR, and the number of snapshots are varied one at a time. The performance of MUSIC, SML and Unitary ESPRIT are investigated. For MUSIC, a grid search of step 0.5 is conducted followed by an optimisation search. SML is initialised with true values since similar results are obtained using a 4-D search to find values for initialisation. According to the theory [13], the SML estimates are more accurate than the DML estimates. owever the difference is noticeable in worse situations involving highly correlated rays, path SNR close to 0 db and small number of sensors. In the scenarios presented in this section, both DML and SML techniques have shown the same performance so we do not deal with DML in this section. Finally Beamforming technique is not presented in this section, due to its low resolution. The plot in Figure 5.8 shows the estimated RMS values of θ ˆ, i.e. the estimator of 2 DOA of the moving source, calculated from 600 trials. In this scenario, each estimate was calculated from a burst of N = 50 samples. The path SNR is 20 db and the rays are uncorrelated. Figure 5.8: RMS values of 2 ˆ θ versus the DOA separation. Uncorrelated paths, 20 db SNR, 50 snapshots. 39

42 In this scenario, the SML estimator of 2 attains the CRLB whereas the estimate given by Unitary ESPRIT never reaches the CRLB but it is close. owever MUSIC shows better performance as the DOA separation is getting larger. For this test, MUSIC never failed to resolve the two peaks in the spectrum. In the next scenario, the number of channel snapshots available has been increased to 200. Figure 5.9 shows the corresponding results. Figure 5.9: RMS values of θ ˆ versus the DOA separation. Uncorrelated paths, 20 db 2 SNR, 200 snapshots. As the number of samples increased, the performance of MUSIC is better and the MUSIC estimator attains the CRLB. The SML estimator is still asymptotically efficient whereas the Unitary ESPRIT estimator does not reach the CRLB as in the previous scenario. The following step is to keep 50 samples and to increase the path SNR to 30 db. The performance of MUSIC is improved in the same way. For a worse scenario as correlated signals, the results are shown in Figure The paths have equal power and the path SNR is 20 db. The path covariance matrix is given by 1 P = If we increase the correlation rate between two paths, MUSIC fails to resolve the peaks. 40

43 Figure 5.10: RMS values of ˆ θ versus the DOA separation. Correlated paths, 20 db SNR, snapshots. The performance of SML and Unitary ESPRIT are still the same, the SML estimator is asymptotically efficient while the estimate of ESPRIT is close to the CRLB but never reaches it. We have tried to increase the power of one source but in this case the MUSIC algorithm fails to resolve the two peaks for a source separation of 7, even for a large number of snapshots and a higher SNR. The resolution remains the main problem in this scenario. 5.4 Comments Parametric methods show very high resolution. owever, MUSIC offers a large improvement over rough beamforming techniques. But it is difficult to compare the resolution of parametric methods and spectral-based methods as we do not define the resolution for these methods in the same way. On the other hand, methods based on the Maximum Likelihood principle show very good performance and their estimators are statistically efficient whatever the conditions in the scenario. The Unitary ESPRIT is not as efficient as ML techniques for a One- or Twopath environment. 41

44 Finally MUSIC suffers from a few drawbacks: The performance is degraded for scenarios presenting low path SNR and small number of snapshots. The algorithm presents difficulties to resolve correlated rays over a MIMO channel. In that case the Covariance matrix of the set {h(t)} is not full rank and the separation into signal and noise subspace becomes harder. A summary of the properties of each method is given in Table 5.1. The resolution and the statistical performance compared to the CRLB are presented. The major computational requirements are also included where the term EVD corresponds to an eigenvalue decomposition. Method Resolution Single-path Statistical Performance Multipath Computations Beamforming Low Efficient Poor 2-D Search MUSIC Good Efficient Efficient for large N and SNR, uncorrelated paths EVD, 2-D Search DML igh Efficient Efficient 2 K-D Search SML igh Efficient Efficient 2 K-D Search 2-D Unitary ESPRIT igh Good Good 2 EVD Table 5.1: Summary and comparison of the properties of the five methods. 42

45 Chapter 6 Results This chapter reports the results based on indoor MIMO measurements conducted by the University of Bristol (UoB) as part of the SATURN project (Smart Antenna Technology in Universal Broadband wireless Networks) funded by the EU IST program. Estimation of ray parameters is investigated for two different situations. One situation has a line-of-sight while in the second case, the environment yields to several bounce reflections. 6.1 MIMO measurements The measurements were performed in the Merchant Venture s Building (MVB) at the UoB. The test site is the entrance floor including one big corridor with several office rooms and computer labs, as it is shown on the map 6.1. Different scenarios were carried out including both line-of-sight (LOS) and non-line-of-sight (NLOS) situations. The measurement platform is based on a Medav RUSK BRI vector channel sounder [20] using an 8-element omnidirectional ULA at the transmitter side and an 8-element ULA with 120º beamwidth at the receiver side [21]. Both ULA have a distance between two antenna elements of half-wavelength. The sounder employs a test-signal with a maximum bandwidth of 120 Mz, centred at 5.2 Gz. The measurement equipment provides channel impulse response estimates in the frequency domain. The channel excess delay window was 0.8 µs, corresponding to 97 sub-channels. By using switching and synchronisation control circuitry, a complete MIMO snapshot was obtained in µs. The device allowed recording 3184 MIMO snapshots for each frequency sub-channel. The total time for one measurement was 5.3 s. A more detailed description of the measurement procedure is given in [22]. Figure 6.1 shows the environment and the different possible locations of the antenna arrays. The RUSK receiver array was located at Rx1, Rx2 and Rx3. There were fifteen different positions for the transmitter, from Tx1 to Tx15. In this chapter we have studied one LOS case with the transmitter placed at Tx9 and the receiver at Rx2. The second scenario is NLOS when the transmitter is located at Tx8. 43

46 Rx 2 Tx 9 Tx 8 Tx 1 Rx 1 Figure 6.1: Measurement scenario. 6.2 Data analysis First we examine the provided data. For each scenario, we have got 3184 MIMO snapshots over 97 sub-channels. We have noticed the data are quite stationary in time for one sub-channel. As we need several independent snapshots to compute the covariance matrix, 97 samples are taken over the sub-channel range but they are still correlated. The second step is to examine the covariance matrix of the data set {h(t)} and its eigendecomposition. For a LOS scenario, one eigenvalue is very high comparing to the others. Two scenarios are probable: either the covariance matrix is not full rank and paths are thus correlated, or it is a single-path environment. In the former case, the small eigenvalues should be all close to one value corresponding to the noise variance, which is not the case. For a NLOS, the eigenvalues decrease gradually and the partition into two subspaces is not noticeable. One main difficulty has been the estimation of the number of rays for parametric methods. The second problem encountered has been the separation of eigenvalues/vectors into two subspaces for subspace-based methods. Only one method, the Beamforming technique, requires only an estimate of the covariance matrix of the data and can be applied directly. 44

47 6.3 Results First we investigate a LOS scenario. The transmitter is located at the position Tx9 and the receiver is placed at Rx2. The most straightforward method to apply is the beamforming technique. The corresponding spectrum is plotted in Figure 6.2. Figure 6.2: Beamforming spectrum based on LOS scenario Tx9 Rx2. Step size = 3. One main direction of arrival is found but several directions of departure are resolved. We observe five distinct DODs paired with one unique DOA. This value is estimated to 105. Table 6.1 shows the results obtained from a peak detection. DOD (degrees) DOA (degrees) Table 6.1: Estimation of the pairs DOD/DOA based on a beamforming spectrum. Scenario Tx9 Rx2. 45

48 According to the map, the line of sight has the following features ( = 67, = 113 ). As the map is not accurate enough, some errors can occur. Then the highest peak in the spectrum with parameters (72.87, 105 ) may correspond to the line of sight. On the other hand, another DOA is estimated to 129º and gives two different DODs, ϕˆ = 73.7 or ϕˆ = The first could correspond to a double bounce reflection whereas the latter one could correspond to a single bounce reflection if we count large errors of 5. Therefore, results at the transmitter are not reasonable whereas at the receive side, the true angles seem to be resolved. With MUSIC, the main problem is to separate signal and noise subspaces based on the number of rays K. With K = 1, we obtain a spectrum similar to the Beamforming spectrum but the MUSIC spectrum is heightened by a continuous component. Figure 6.3 shows the MUSIC spectrum obtained with K = 3. Figure 6.3: MUSIC spectrum based on LOS scenario Tx9 Rx2. Mesh step size = 3. Number of rays K = 3. With different values of K, we obtain similar spectra. The continuous component could be due to strong noise. This case has been reproduced in simulations with a bad scenario involving highly correlated rays and a noise power higher than the path power: According to simulations, the presence of a stronger beam does not change the spectrum if the noise 46