Joint Clustering and Tracking for MIMO Radio Channel Modeling RUIYUAN TIAN
Communication Systems Group Department of Signals and Systems Chalmers University of Technology Göteborg, Sweden, 2007
EX050/2007
This work was performed at The Telecommunications Research Center Vienna (ftw.) Supervised by: Dipl.-Ing. Nicolai Czink (ftw.) Prof. Erik Agrell (Chalmers University of Technology) Examiner: Prof. Erik Agrell
Joint Clustering and Tracking for MIMO Radio Channel Modeling
MASTER THESIS
Ruiyuan Tian Supervisor: Nicolai Czink Examiner: Erik Agrell Chalmers University of Technology Department of Signals and Systems Report EX050/2007 2007-05-07
iii
Abstract This thesis investigates clustering and tracking aspects for cluster-based MIMO radio channel modeling. A joint clustering-and-tracking approach for MIMO channel characterization is proposed. MIMO radio channel modeling draws great attention because the channel determines the entire system performance fundamentally. This thesis is focusing on physical channel models, which characterize the propagation environment by describing the multipath components (MPCs), i.e., discrete propagation paths between Tx and Rx. In the double-directional channel description, each multipath is characterized by several parameters, including power, delay, direction of arrival and direction of departure. These MPCs are appearing in clusters which consist of a group of multipaths that have similar parameters. Cluster-based channel modeling requires the accurate parametrization of the cluster characteristics. For cluster identification, we characterize each cluster by a single multivariate Gaussian distribution. This distribution is described by its first-order and second-order statistics. In this way, the multipath channel is characterized by a mixture of Gaussian distributions. In this thesis, the cluster characteristics are extracted by applying the Gaussian Mixture Modeling (GMM) clustering approach using the Expectation-Maximization (EM) algorithm. In order to extract the time-varying channel characteristics, it is also necessary to track the movement of multipath clusters. A tracking approach by a prediction step and an update step using a Kalman filter is applied. Furthermore, the tracking is combined with the clustering, which in turn improves the initialization required by the clustering algorithm. The joint clusteringand-tracking framework for multipath cluster identification and parametrization is developed. The cluster tracking can be easily combined with both the GMM-based clustering approach and other available approaches for multipath clustering. In this work, the double-directional MPCs parameters are estimated using a high resolution method from MIMO channel measurements. Various cluster characteristics are extracted and presented as the results. These parameters significantly contribute to cluster-based stochastic MIMO radio channel models. Index term: MIMO, channel modeling, multipath cluster, Gaussian mixture, tracking
Preface The work presented in this thesis is carried out at the Telecommunications Research Center Vienna (ftw.), during the period between August of 2006 and March of 2007. First of all, I would like to express my gratitude to Nicolai Czink, my supervisor at ftw., for making it possible of me to work at such a great institute, and for guiding me to step into research. “Thanks a lot” to Niki for introducing such an interesting topic to me. I learnt very much from him. ftw. is sincerely acknowledged for offering me this valuable experience. I would also like to thank Christoph Mecklenbrauker from Vienna University of Technology, and Erik Agrell from Chalmers University of Technology, for their valuable discussion and persistent encouragement. Bernard Fleury, Troels Pedersen, Xuefeng Yin and Thomas Zemen also have many valuable ideas input during the discussions. Besides, I enjoy my working at ftw. with all the colleagues there, especially Pavle Belanovic and Erwin Riegler for sharing the office. I appreciate all my friends for their understanding. Last but not the least, I express my most love to my parents from Beijing. “Life is a journey; I’m on the road.”
Ruiyuan Tian Vienna, March 2007
v
vi
PREFACE
:-)
Contents Preface
v
Contents
vii
List of Figures
ix
List of Tables
xi
Acronyms
xiii
1 Introduction
1
2 Overview of MIMO radio channel modeling 2.1 MIMO channel models . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Physical MIMO radio channel models . . . . . . . . . . . . . . . . . 2.3 Cluster-based channel modeling . . . . . . . . . . . . . . . . . . . . .
5 5 6 7
3 Clustering 3.1 Cluster analysis . . . . . . . . . 3.2 Gaussian Mixture Model . . . . 3.3 Likelihood Maximization . . . . 3.4 EM-GMM clustering algorithm 3.5 Clustering simulation . . . . . .
. . . . .
11 11 12 13 14 17
4 Tracking 4.1 Target tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19 19 20
5 Multipath clustering 5.1 Multipath clusters . . . . . . . . 5.2 Power mean . . . . . . . . . . . . 5.3 EM-GMM multipath clustering . 5.3.1 Initialization . . . . . . . 5.3.2 Convergence of EM-GMM 5.4 Clustering real-world data . . . .
25 25 26 27 31 31 35
. . . . .
vii
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . .
. . . . . .
. . . . . .
viii
CONTENTS
6 Tracking multipath clusters 6.1 Tracking multipath clusters . . . . . . . . . . . . . . . . . . . . 6.1.1 Multipath cluster as target . . . . . . . . . . . . . . . . 6.1.2 Cluster association . . . . . . . . . . . . . . . . . . . . . 6.1.3 Multiple multipath clusters . . . . . . . . . . . . . . . . 6.2 Joint clustering and tracking . . . . . . . . . . . . . . . . . . . 6.2.1 Initialization by prediction . . . . . . . . . . . . . . . . 6.2.2 Joint clustering and tracking approach . . . . . . . . . . 6.3 Tracking multipath clusters from MIMO channel measurements 6.3.1 MIMO channel measurements . . . . . . . . . . . . . . . 6.3.2 Geometry-based analysis . . . . . . . . . . . . . . . . . . 6.3.3 Stochastic parameters . . . . . . . . . . . . . . . . . . . 6.3.4 Illustration of a tracked multipath cluster . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
39 39 39 41 43 44 44 45 48 48 49 50 51
7 Summary
55
Bibliography
57
List of Figures 1.1
A multipath fading channel . . . . . . . . . . . . . . . . . . . . . . .
2
2.1 2.2 2.3
Channel of MIMO systems . . . . . . . . . . . . . . . . . . . . . . . Double-directional multipath propagation . . . . . . . . . . . . . . Multipath clusters characteristics Note that delay and delay spread are also parameters of multipath clusters, which are however not shown in this 2-D figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 7
3.1 3.2
5.1
5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
6.1 6.2
Mixture density by 2 Gaussian distributions . . . . . . . . . . . . EM-GMM clustering iteration procedure From left top to right down, clustering results at iteration 4, 15, 30, 50 . . . . . . . . . . . . . Cluster mean with power weighted The two axes indicate two dimensional data. The power weight of each data point is indicated by its color. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MPC clusters in parameter space AoAs and AoDs are between −π and +π; Delays are in 10ns (1 : 10(−8) s) . . . . . . . . . . . . . . . . . . MPC clusters in different dimensions AoAs and AoDs are between −π and +π; Delays are in 10ns (1 : 10(−8) s) . . . . . . . . . . . . . . . . Initialization procedure for MPC clustering strong indicates power within the range of 20 dB below the strongest power. . . . . . . . . . . . Framework of MPC clustering using EM-GMM . . . . . . . . . . Example 1: Clustering result for MPCs from measurements AoAs and AoDs are between −π and +π; Delays are in 10ns (1 : 10(−8) s) Example 1: Mixture of multipath clusters (in AoA and AoD) AoAs and AoDs are between −π and +π . . . . . . . . . . . . . . . . . Example 2: clustering result for MPCs from measurements AoAs and AoDs are between −π and +π; Delays are in 10ns (1 : 10(−8) s) Example 2: Mixture of multipath clusters (in AoA and AoD) AoAs and AoDs are between −π and +π . . . . . . . . . . . . . . . . . Update multiple multipath clusters . . . . . . . . . . . . . . . . . . Initialization procedure using prior prediction information strong indicates power within the range of 20 dB below the strongest power. . ix
9 12 18
27 30 30 32 34 36 36 37 38 45 46
x
6.3 6.4 6.5
6.6 6.7 6.8
List of Figures
Joint clustering-and-tracking framework . . . . . . . . . . . . . . . Route of the channel measurements . . . . . . . . . . . . . . . . . Geometry based multipath cluster analysis left: possible multipaths between Tx and Rx. right: illustration of AoA at Rx and AoD at Tx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example A - F of the channel measurements . . . . . . . . . . . . Example route of tracked multipath cluster . . . . . . . . . . . . Trajectory of a tracked multipath cluster . . . . . . . . . . . . . .
47 48
49 51 52 53
List of Tables EM-GMM clustering algorithm . . . . . . . . . . . . . . . . . . . . Cluster weight by EM-GMM iteration procedure The average cluster weight probability is 1/K = 1/8 = 0.125. . . . . . . . . . . . . .
16
4.1
Kalman filter by prediction and update step . . . . . . . . . . . .
23
5.1 5.2 5.3 5.4
EM-GMM clustering algorithm for MPCs . . . . . . Implementation specifications for MPC parameters Example 1: multipath cluster parameters . . . . . . . Example 2: multipath cluster parameters . . . . . . .
. . . .
28 29 37 38
6.1 6.2 6.3
Kalman prediction step for tracking multipath cluster . . . . . . Kalman update step for tracking multipath cluster . . . . . . . . Multipath cluster parameters of the geometry based analysis example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example statistics of multipath cluster parameters . . . . . . . . Example parameter statistics of one tracked multipath cluster
41 42
3.1 3.2
6.4 6.5
xi
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
18
49 51 53
Acronyms AoA AoD CMD DoA DoD EM GMM KF LoS MCD MIMO ML MPC Rx SAGE Tx
azimuth of arrival azimuth of departure correlation matrix distance direction of arrival direction of departure expectation-maximization Gaussian mixture model Kalman filter line of sight multipath component distance multiple input multiple output maximum likelihood multipath component receiver subspace-alternating generalized expectation-maximization transmitter
xiii
Chapter 1
Introduction Communication systems utilizing multiple antennas at both transmitter (Tx) and receiver (Rx) are so called multiple-input multiple-output (MIMO) systems. MIMO is a promising technique to achieve high capacity in future mobile radio communication systems [1]. However, these MIMO techniques have not been sufficiently tested under realistic propagation conditions. Their performance in real applications is still of question. This fact underlines the importance of physically meaningful, yet easy-to-use methods to understand and model the wireless channel and the underlying radio propagation [2, 3]. From a propagation point of view, the characterization of wireless channels can be described in terms of the double-directional impulse response [4]. This impulse response consists of contributions from all individual propagation paths. These paths are usually referred to as multipath components (MPCs). Each component is characterized by several propagation parameters in a multi-dimensional parameter space [5]. These parameters are typically direction-of-arrival (DoA), direction-ofdeparture (DoD), power and delay. DoA and DoD can be on both azimuth and elevation angle, but only azimuth-of-arrival (AoA) and azimuth-of-departure (AoD) are considered in this thesis. Figure 1.1 shows a typical multipath channel. When a single pulse is transmitted over a multipath channel, the received signal will appear as a pulse train, with each pulse corresponding to either the Line-of-Sight (LoS) component, or a distinct multipath component associated with a distinct scatterer, or cluster of scatterers [6]. Many advanced radio channel models have been developed based on the concept of multipath clusters. A multipath cluster consists of a group of MPCs that have similar multipath parameters such as AoA, AoD and delay [7,8]. The prerequisite of these cluster-based models are accurate parameters of the multipath clusters, which one needs to extract from measurement data. MIMO channel measurements provide numerous snapshots of the impulse response of the (time-varying) radio channel. To identify a number of MPCs from these impulse responses, they are processed by a
1
2
CHAPTER 1. INTRODUCTION
Figure 1.1. A multipath fading channel
high-resolution estimation algorithm, for example SAGE [9]. A group of MPCs that have similar parameters are treated as a multipath cluster. The problem is to automatically identify and track such clusters from MIMO channel measurements [10]. Given a data set at one time instant (snapshot), there are many ways to do clustering. For example, the classic KMeans algorithm can classify the data set into K groups by assigning paths to clusters with different indexes. This method has been extended to the KPowerMeans algorithm [10] which is able to perform clustering for MPCs. In this thesis, the Gaussian Mixture Model (GMM) based clustering approach is applied using the Expectation-Maximization (EM) algorithm. In the GMM-based clustering approach, the distribution of all possible MPCs are modeled by a mixture of Gaussian densities. Each Gaussian density represents a single multipath cluster. This distribution is described by its first-order and second-order statistics. In this way, the multipath channel is described by a number of clusters, where each cluster is individually Gaussian distributed. The accuracy of clusterbased radio channel models strongly depends on the accurate parameterization of these cluster parameters.
3
In a time-varying radio channel, the cluster parameters should evolve smoothly over time. This requires to track the clusters in order to extract the time-variant properties of the cluster parameters. In this thesis, a Kalman filter (KF) is applied to track multipath clusters. Furthermore, cluster tracking is combined together with the clustering procedure which results in a joint clustering-and-tracking framework for MIMO radio channel characterization. This thesis is organized as follows: Chapter 2 provides the overview of MIMO radio channel models and the background of the cluster-based channel modeling. Chapter 3 discusses the GMM-based clustering method by using the EM algorithm. Chapter 4 discusses the target tracking problem by using the Kalman filter. Chapter 5 proposes the MPC clustering approach by using the EM-GMM clustering method. Chapter 6 proposes the joint clustering-and-tracking approach for multipath clustering. Results are also presented in this chapter. Chapter 7 summarizes and concludes the thesis. A prospect of the future work is also provided.
Chapter 2
Overview of MIMO radio channel modeling MIMO is a promising technique for future wireless communications. However, the radio propagation channel plays a fundamental role to determine the characteristics of the entire MIMO communication system. Therefore, accurate modeling of MIMO channels is an important prerequisite for MIMO system design, simulation and deployment [5, 14]. This chapter gives an brief overview of MIMO radio channel modeling and cluster-based propagation modeling.
2.1
MIMO channel models
A communication system can be simply described mathematically by y = Hs + n
(2.1)
where s is the transmitted signal vector, y denotes the received signal vector, H describes the channel transfer function, and n is the system noise. Generally, communication channels describe how the signal is transmitted from the Tx side to the Rx side. On one hand, the channel transfer function can be represented analytically by the channel matrix H. It describes the link between the Tx and the Rx. On the other hand, the channel environment can be also represented physically by describing the electromagnetic wave propagation [2]. In a MIMO system equipped with multiple antennas at both Tx and Rx, the channel can be illustrated as in Figure 2.1. Different MIMO channel models can be classified into analytical models and physical models [2]. Analytical MIMO channel models describe the channel transfer function H statistically. They are widely used in system performance evaluation. The channel of a MIMO system with Nt transmit 5
6
CHAPTER 2. OVERVIEW OF MIMO RADIO CHANNEL MODELING
Figure 2.1. Channel of MIMO systems
antennas and Nr receive antennas can h1,1 h2,1 H= . .. hNr ,1
be specified by a Nr × Nt matrix, h1,2 . . . h1,Nt h2,2 . . . h2,Nt .. .. .. . . . hNr ,2 . . . hNr ,Nt
(2.2)
where hi,j denotes the channel transfer function between the j-th transmit antenna and the i-th receive antenna. In contrast to the analytical models, physical MIMO channel models characterize the channel environment on the basis of electromagnetic wave propagation by a double-directional model [2, 4]. In this thesis, parameters for physically-based MIMO channel models are investigated.
2.2
Physical MIMO radio channel models
In physical MIMO channel models, the environment is described in terms of electromagnetic wave propagation, using the double-directional model [4]. The channel impulse response consists of contributions from all L individual MPCs [2]. h(τ, ϕ, ψ) =
L X
γl δ(τ − τl )δ(ϕ − ϕl )δ(ψ − ψl ).
(2.3)
l=1
Each MPC is characterized by several propagation parameters, which typically include the complex amplitude or power (γ), DoA (ϕ), DoD (ψ) and delay(τ ). Figure 2.2 shows a typical multipath propagation channel by using the doubledirectional description, where (XT x , XRx ) indicates the location of the Tx and the Rx antenna array. Each path is characterized by its parameters θ = [γ, ϕ, ψ, τ ]. In this way, the physical models are able to reproduce a channel by specifying these multipath parameters. This approach is independent of the antenna configuration
2.3. CLUSTER-BASED CHANNEL MODELING
7
Figure 2.2. Double-directional multipath propagation
and system bandwidth [2]. A first approach for channel modeling is using stored measurements. The starting point is the channel sounder campaign. Measurement data can be recorded and reused to generate channels for this specific measurement environment. This deterministic model is perfect in terms of accuracy, but provides only samples for the measured channel. Alternatively, one could obtain the statistical characteristics of channel parameters by a couple of channel measurements and analytical analysis for a typical propagation environment. This leads to the stochastic channel models.
2.3
Cluster-based channel modeling
A multipath cluster is defined as a group of MPCs that have similar propagation parameters. The complete channel impulse response (2.3) is determined by the parameter vector θ, where θl describes the parameters for the l-th MPC, θ = [θ1 , θ2 , . . . , θl , . . . , θL ]
(2.4)
θl = [γl , ϕl , ψl , τl ]T
(2.5)
Given a measured impulse response, the parameter θl can be estimated for every MPC by the SAGE algorithm. These multipath parameters, typically AoA, AoD and delay, were observed to appear in clusters. For example, [15] and [16] observe multipath clusters with similar parameters. Many advanced MIMO channel models
8
CHAPTER 2. OVERVIEW OF MIMO RADIO CHANNEL MODELING
are developed based on the cluster concept, such as in [7, 8, 17–22]. The existence of multipath clusters can be motivated from rough scattering. The paths propagating through the channel are scattered not individually but in groups. We can characterize each cluster by its centroid parameters, i.e., mean AoA (ϕ), ¯ ¯ and mean delay (¯ mean AoD (ψ) τ ), as well as its spreads, i.e., AoA spread (σϕ ), AoD spread (σψ ) and delay spread (στ ). We define these parameters as cluster centroid, µl = [ϕ¯l , ψ¯l , τ¯l ] (2.6) and cluster spread, σl = [σϕ,l , σψ,l , στ,l ]
(2.7)
respectively, for the l-th cluster. An illustration of the meaning of cluster characteristics is given in Figure 2.3. Cluster-based channel models aim to use these multipath clusters to represent the channel characteristics. Therefore, if one can identify clusters properly and obtain cluster parameters from channel measurements accurately, cluster-based models can maintain the accuracy while reducing the complexity. In this thesis, the parameters of multipath clusters are assumed to be parameters of multivariate Gaussian distributions. These distributions are characterized by their centroids and spreads. The centroid and the spread represent the mean and the variance of each Gaussian distribution. One can then use a mixture of multipath clusters to represent the total multipath channel, and use a mixture of Gaussian distributions to describe the total channel characteristics. Moreover, time-varying channels require time-evolving cluster characteristics. These can be identified by cluster tracking. Both clustering and tracking are discussed in detail in the following chapters.
2.3. CLUSTER-BASED CHANNEL MODELING
Figure 2.3. Multipath clusters characteristics Note that delay and delay spread are also parameters of multipath clusters, which are however not shown in this 2-D figure.
9
Chapter 3
Clustering The main idea of cluster-based MIMO channel modeling is to represent the radio propagation channel by using multipath clusters. This requires the accurate parameterization of MPC clusters. An existing automatic multipath clustering is the KPowerMeans algorithm [5, 10, 13]. In this thesis, another approach applying the GMM-based clustering method is discussed. The section is based on the study of [11], on the theoretical aspects of GMM-based clustering.
3.1
Cluster analysis
Clustering is an established research field in statistics, image analysis and data mining. According to the definition of [31], the process of grouping a set of physical or abstract objects into classes of similar objects is called clustering. A cluster is a collection of data objects that are similar to one another within the same cluster, and are dissimilar to the objects in other clusters. A cluster of data objects can be treated collectively as one group in many applications [31]. Given a data set D which consists of N samples, one is interested to classify this data set into K groups, where K N . The problem is that all the data points are unlabeled, and K is even unknown in some cases. This requires an approach to classify the unlabeled data by clustering them into groups with similar characteristics, and to obtain the K most representative characteristics of set D [11]. The cluster features found are useful for the categorization. It is also valuable to perform exploratory data analysis and thereby gain some insight into the nature or structure of the data [11]. Various clustering approaches, such as the KMeans method and Hierarchical methods, all show both benefits and disadvantages. A detailed description of cluster analysis can be found in [31–34]. Apart of them, the clustering approach based on Gaussian mixture model (GMM) is addressed in this thesis. 11
12
CHAPTER 3. CLUSTERING
Figure 3.1. Mixture density by 2 Gaussian distributions
3.2
Gaussian Mixture Model
Gaussian Mixture Models estimate the mixture of probability densities through the weighted sum of multiple Gaussian densities. The GMM not only provides a smooth overall distribution fit, its components can also clearly detail a multimodal density [37]. The use of Gaussian mixture models has a long history, see [38] for a review and thorough introduction [39]. In the GMM-based approach, the data set D = {x1 , . . . , xN } is assumed to be a mixture of K clusters. Each cluster is producing observations with a Gaussian density. This Gaussian density is described by its first-order and second-order statistics, i.e., the mean µi and the variance Σi . In this way, one can describe the characteristics of the data set D by K Gaussian models in a mixture. Figure 3.1 gives an example of a distribution mixture by two Gaussian densities. To estimate clusters from the given data set, the parameter vector θ is defined as θ = {λ1 , . . . , λK ; µ1 , . . . , µK ; Σ1 , . . . , ΣK } ,
(3.1)
where µk and Σk are the mean and the variance of the k-th cluster. The cluster weight is given by λk which describes the cluster’s probability in the whole mixture. If the data is assumed to be a mixture of Gaussian models, the likelihood density p(X|θ) is given as K X p(X|θ) = λk G(X| {µk , Σk }), (3.2) k=1
3.3. LIKELIHOOD MAXIMIZATION
13
where G represents the d dimensional multivariate Gaussian density for each cluster, and it is given by 1
G(X| {µ, Σ}) =
3.3
1/2
(2π)d/2 |Σ|
1 exp(− (X − µ)T Σ−1 (X − µ)). 2
(3.3)
Likelihood Maximization
For most cases of practical interest, the Maximum Likelihood (ML) estimator gives the optimal performance for data records large enough [12]. In this section, the data set X = {x1 , . . . , xN } is assumed to be drawn independently from one unknown distribution p(X|θ) (which is not a mixture here, but a single distribution, and θ is only referred to {µ, Σ} for this single distribution). The likelihood of the parameter θ with respect to the data set X is defined as ∆
`(θ) = p(X|θ) =
N Y
p(xn |θ).
(3.4)
n=1
In the Gaussian model, p(xn |θ) indicates a d dimensional multivariate Gaussian density as p(xn |θ) = G(xn | {µ, Σ}) =
1 (2π)d/2 |Σ|1/2
1 exp(− (xn − µ)T Σ−1 (xn − µ)), 2
(3.5)
where µ and Σ are the mean and the variance of the density. The ML estimate of θ is defined as the value θb that maximizes the likelihood p(X|θ). In other words, this ML estimate gives the optimal estimation result. The likelihood maximization approach can be also illustrated by the Bayesian Philosophy [12]. According to the Bayes theorem, the likelihood in (3.4) can be written as N N Y Y p(xn ) `(θ) = p(xn |θ) = p(θ|xn ) , (3.6) p(θ) n=1
n=1
where the probability p(θ|xn ) is known as the ‘a posteriori probability’. Therefore, the ML estimate maximizing the ‘a posteriori probability’ is given as θb = arg max θ
N Y
p(xn |θ) = arg max θ
n=1
N Y
p(θ|xn )p(xn ).
(3.7)
n=1
In practice, the logarithm likelihood ∆
L(θ) = log`(θ) =
N X n=1
log p(xn |θ)
(3.8)
14
CHAPTER 3. CLUSTERING
is used, where 1 1 log p(xn |θ) = lnG(xn | {µ, Σ}) = − ln((2π)d |Σ|) − (xn − µ)T Σ−1 (xn − µ). (3.9) 2 2 One can find the optimal estimation by setting the gradient of the log-likelihood to zero. The ML estimate of the parameter θb = {µ, Σ} is then obtained as N 1 X µ b= xn , N
(3.10)
N X b = 1 Σ (xn − µ b)(xn − µ b)T . N
(3.11)
n=1
n=1
3.4
EM-GMM clustering algorithm
The problem is to maximize the likelihood with respect to a data set D = {x1 , . . . , xN }. This data set shall be characterized by a mixture consisting of K Gaussian distributions. Each cluster is indicated by a label ωk . In order to estimate the cluster parameters, we recall the definition of the parameter vector θ as θ = {λ1 , . . . , λK ; µ1 , . . . , µK ; Σ1 , . . . , ΣK } .
(3.12)
The membership probability p(ωk |xn ) describes the conditional probability of the data sample xn to belong to the cluster ωk . Then, the ML estimate solution is obtained as [11] N 1 X b p(ωk |xn ), λk = N
(3.13)
n=1
PN
n=1 µ bk = P N
xn p(ωk |xn )
n=1 p(ωk |xn )
PN bk = Σ
n=1 (xn
,
−µ bk )(xn − µ bk )T p(ωk |xn ) . PN n=1 p(ωk |xn )
(3.14)
(3.15)
This membership probability p(ωk |xn ) can be shown to depend on the data sample xn and the parameter vector θ, by using the Bayes theorem as p(ωk |xn ) =
p(ωk , xn ) p(ωk ) = p(xn |ωk ). p(xn ) p(xn )
(3.16)
Note that, • p(ωk ) is the probability of the k-th cluster, which depends on the weight λk of the k-th Gaussian in the mixture.
3.4. EM-GMM CLUSTERING ALGORITHM
15
• p(xn |ωk ) corresponds to the k-th Gaussian density. p(xn |ωk ) = G(xn | {µk , Σk }).
(3.17)
• p(xn ) can be written as p(xn ) =
K X
p(ωk )p(xn |ωk ) =
k=1
K X
λk G(xn | {µk , Σk })
(3.18)
k=1
By substituting (3.17) and (3.18) into (3.16), the probability p(ωk |xn ) is obtained as
λk G(xn | {µk , Σk }) , p(ωk |xn ) = PK j=1 λj G(xn | {µj , Σj })
(3.19)
where G is the Gaussian density from (3.5). To solve this problem, the Expectation-Maximization (EM) algorithm can be used iteratively to maximize the log-likelihood in following way [11, 35]: Expectation The estimate of the parameter θ at iteration t is used to compute the probability p(ωk |xn ) for every of K clusters Maximization The probability p(ωk |xn ) for K clusters is used to compute the ML estimate of the parameter θ at iteration t + 1 The EM-GMM clustering algorithm is summarized in Table 3.1.
16
CHAPTER 3. CLUSTERING
EM-GMM clustering algorithm Initialization at iteration t = 1 • data set D = {x1 , . . . , xN } • set maximum possible K • set iteration time T • give initial value for λ(1) , µ(1) and Σ(1) From iteration t = 1, . . . , T (t+1) (t+1) (t+1) λ ,µ ,Σ =EM-GMM(D, λ(t) , µ(t) , Σ(t) ) Expectation step For every k and n, calculate n o (t) (t) (t) λk G(xn | µk , Σk ) n o p(ωk |xn ) = P (t) (t) (t) K ) j=1 λj G(xn | µk , Σk
(3.20)
p(ωk |xn ) ρkn = PN n=1 p(ωk |xn )
(3.21)
Maximization step For every k, calculate (t+1)
λk
N 1 X ρkn N
=
(3.22)
n=1
(t+1)
µk
=
N X
xn ρkn
(3.23)
n=1
(t+1) Σk
=
N X
(t+1)
(xn − µk
(t+1) T
)(xn − µk
) ρkn
n=1
Next t Table 3.1. EM-GMM clustering algorithm
(3.24)
3.5. CLUSTERING SIMULATION
3.5
17
Clustering simulation
After the theoretical investigation on the EM-GMM clustering algorithm, we discuss clustering simulation in this section. For demonstration, the data set consists of samples in only 2 dimensions . The example is shown iteration by iteration in Figure 3.2. The red dots denote the 2 dimensional data set to be clustered. The first step is to set the initialization. First, the maximum number of clusters K is set to be 8, which is large enough to cluster the data set. Secondly, the weight λ is set to be 1/K for every cluster which indicates the same probability of each cluster initially. The mean vector µ and the variance matrix Σ are chosen randomly from the samples for each one of the K clusters. Then, the EM-GMM algorithm is run iteratively according to Table 3.1. The results at iteration 4, 15, 30 and 50 are shown Figure 3.2. The blue curve indicates the contour of each cluster. As can be seen, the algorithm converges finally at iteration 50 when 5 clusters are established. Their mean vector locations are indicated by the black dot. The 8 initially overestimated clusters converge to 5 final stable estimates. In order to illustrate this convergence, the weight λ for each cluster is provided in Table 3.2. It shows that the EM-GMM algorithm adjusts the estimation according to the data set iteratively. The weights for clusters whose ID are 3, 5 and 6 decrease gradually and finally fall below the average weight probability 1/K = 1/8. Clusters are only regarded when their weight probability λ is greater than 1/K. This is a simple example to show how the EM-GMM clustering algorithm works. It is applied in order to obtain local optimal estimation. The greatest advantage is its ability to decide the number of clusters in a Gaussian mixture. Note that EM-GMM converges to a local maximum of the likelihood, but the convergence is relatively fast. The initially overestimated number of clusters can be eliminated by their weight and likelihood with respect to the data set. Still, the algorithm is possible to get trapped in suboptimal solutions and one must set constraints to make the EM iteration feasible [11]. As to the clustering for MPCs, multipath clusters have parameters in AoA, AoD and delay domain which are different from linear scale. The MPCs’ powers are important so that this requires extra consideration. The application of the EM-GMM approach to MPC clustering is discussed explicitly in Chapter 5.
18
CHAPTER 3. CLUSTERING
Figure 3.2. EM-GMM clustering iteration procedure From left top to right down, clustering results at iteration 4, 15, 30, 50
Cluster ID Iteration 1 λ Iteration 4 λ Iteration 15 λ Iteration 30 λ Iteration 50 λ
1
2
3
4
5
6
7
8
0.1250
0.1250
0.1250
0.1250
0.1250
0.1250
0.1250
0.1250
0.1798
0.1914
0.2642
0.0203
0.0006
0.1694
0.1065
0.0679
0.1969
0.1991
0.2578
0.0210
0.0027
0.1132
0.1661
0.0431
0.1972
0.2003
0.1095
0.1554
0.0040
0.0485
0.1803
0.1047
0.1974
0.2006
0.0280
0.1813
0.0037
0.0174
0.1940
0.1778
Table 3.2. Cluster weight by EM-GMM iteration procedure The average cluster weight probability is 1/K = 1/8 = 0.125.
Chapter 4
Tracking Whenever the channel is of time-varying nature, it is necessary to include the timevariant properties in channel models. When using multipath clusters, a tracking approach is required to model the time-varying channel characteristics. This section introduces the tracking method using the Kalman filter.
4.1
Target tracking
Target tracking can be generally described as the process to determine the state of the moving target. By tracking, one can draw some conclusions on the motion of the target based on available observations. Examples include vehicle tracking, computer visual tracking and radar tracking. A state-space model is usually considered in target tracking problems. The target to be tracked can be an object that is moving in space. The state of a target is usually its position parameters in space. This position state is evolving over time according to the movement of the target. In the state-space model, the motion of the target is described by a target motion model, defined as x(n) = F[x(n − 1)] + w(n),
(4.1)
where n indicates the current time step while n − 1 indicates the previous time step. F represents the motion function which defines the movement of the target. The random variable w(n) represents the process noise. The target motion model can be also viewed as a Markov process. In addition, the state-space model also includes a measurement model which describes the observation of the target. This is defined as z(n) = H[x(n)] + v(n),
(4.2)
where H is the observation function and v(n) is the measurement noise. In this way, the observation z(n) is a function of the target state x(n) that is embedded in noise.
19
20
CHAPTER 4. TRACKING
Using Bayesian statistics is currently the most commonly accepted theoretical framework in the tracking community [23]. The core of the approach is to maximize a posterior distribution which is given by Bayes’ rule fposterior (x|z) =
f (z | x)fprior (x) . f (z)
(4.3)
In the tracking application, the prior distribution fprior (x) and posterior distribution fposterior (x|z) are the probability distribution of the target state x before and after collection of observations z. In this case, the prior and the posterior distribution are defined as fprior (x) = fXn |Z1:n−1 (Xn | Z1:n−1 ),
(4.4)
fposterior (x|z) = fXn |Z1:n (Xn | Z1:n ),
(4.5)
where {1 : n} denotes the time from 1 to n. Furthermore, the target motion model defines the motion density fXn |Xn−1 (Xn | Xn−1 ). The prior distribution can then be obtained as Z fXn |Z1:n−1 (Xn | Z1:n−1 ) =
fXn |Xn−1 (Xn | Xn−1 )fXn−1 |Z1:n−1 (Xn−1 | Z1:n−1 )dXn−1 .
The measurement model defines the observation likelihood fZn |Xn (Zn which can be included into the posterior distribution
fXn |Z1:n (Xn | Z1:n ) = R
fZn |Xn (Zn | Xn )fXn |Z1:n−1 (Xn | Z1:n−1 ) . fZn |Xn (Zn | Xn )fXn |Z1:n−1 (Xn | Z1:n−1 )dXn
(4.6) | Xn ),
(4.7)
The tracking is achieved through the recursive update of the posterior probability. If the motion process is assumed to be a first-order Markov process1 , and if the motion density and the observation likelihood are both Gaussian distributed, the Bayesian statistics reduce to the Kalman filter [23].
4.2
Kalman filter
Kalman filter estimates the state of a process in a way that minimizes the mean squared error. The state process is non stationary and it is characterized by a dynamic motion model. 1
In this case, the current target state is only dependent on the previous one, which means fXn |Z1:n−1 = fXn |Zn−1
4.2. KALMAN FILTER
21
First of all, the state-space model is specified. If the target is moving in a 3 dimensional space, for example, the target state at time n is defined as Θ = [x, ∆x, y, ∆y, z, ∆z]T ,
(4.8)
where [x, y, z] indicates the position (first-order parameters) in space and [∆x, ∆y, ∆z] indicates the velocity (second-order parameters) of the movement. In this case, the motion model can be written as Θn = ΦΘn−1 + wn . The motion function is represented by the state given as xn 1 T 0 0 0 0 ∆xn 0 1 0 0 0 0 yn 0 0 1 T 0 0 = ∆yn 0 0 0 1 0 0 zn 0 0 0 0 1 T ∆zn 0 0 0 0 0 1
(4.9) transition matrix Φ which is xn−1 ∆xn−1 yn−1 ∆yn−1 zn−1 ∆zn−1
+ wn ,
(4.10)
where T is the time interval between time steps, w is the process noise that models the speed noise, which is assumed to be i.i.d. Gaussian distributed with zero mean and variance Q. The measurement model for the observation µ is specified by the matrix H as µn = HΘn + vn , 1 0 0 0 0 0 µn = 0 0 1 0 0 0 0 0 0 0 1 0
(4.11) xn ∆xn yn ∆yn zn ∆zn
+ vn ,
(4.12)
where v models the observation noise, and it is assumed to be i.i.d. Gaussian distributed with zero mean and variance R. b (n|n−1) denote the prior state estimation (the prediction) Furthermore, let Θ b (n|n) denote the estimation given the given the observations up to time n − 1, and Θ observations up to time n, where e(n|n−1) and e(n|n) are the corresponding estimation errors, as b (n|n) . e(n|n) = Θ(n) − Θ Then we denote P(n|n−1) and P(n|n) as the error covariance matrices, as
(4.13)
22
CHAPTER 4. TRACKING
P(n|n) = E{e(n|n) eH (n|n) }.
(4.14)
b (n|n) that miniIn the Kalman filter, one aims to find the best estimation of Θ mizes the mean-square error (n) from [24] (n) = tr{P(n|n) }.
(4.15)
b (n|n) is obtained by a In the Kalman filter, the estimation of the target state Θ prediction step and an update step. In the prediction step, a prior state estimab (n|n−1) and the error covariance P(n|n−1) are calculated by using the motion tion Θ model and the estimation from n − 1. In the update step, the Kalman gain K is first obtained by setting the derivative of the mean-square error (n) to be zero. b (n|n) and P(n|n) are updated by the current available Then the state estimation Θ b (n|n−1) and observation µn using the measurement model and the prior estimation Θ P(n|n−1) . This prediction step and update step of the Kalman filter are summarized in Table 4.1 [24] [25]. The Kalman filter is applicable in various target tracking problems. However, multipath clusters have parameters in AoA, AoD and delay domain which are different from linear scale. Besides, there are multiple clusters in each scenario. Data association is necessary to find the matched observations. Furthermore, multipath clusters do not exist forever. New clusters are also discovered from time to time. This condition requires a flexible tracking approach where the number of targets has to be adjustable. Using the Kalman filter to track multipath clusters is discussed explicitly in Chapter 6.
4.2. KALMAN FILTER
23
Kalman Filter Initialization at time n = 1 • specify the state transition matrix Φ • specify the observation matrix H • set the process noise covariance Q • set the observation noise covariance R b (1|1) and P(1|1) • give initial value for Θ From time n = 2, . . . , T b (n|n−1) and P(n|n−1) Prediction Step the prior estimation of Θ b (n|n−1) = ΦΘ b (n−1|n−1) Θ
(4.16)
P(n|n−1) = ΦP(n−1|n−1) ΦT + Q
(4.17)
b (n|n) and P(n|n) Update Step update the estimation of Θ K(n|n) = P(n|n−1) HT (HP(n|n−1) HT + R)−1
(4.18)
b (n|n) = Θ b (n|n−1) + K(n|n) (µ(n) − HΘ b (n|n−1) ) Θ
(4.19)
P(n|n) = (I − K(n|n) H)P(n|n−1)
(4.20)
Table 4.1. Kalman filter by prediction and update step
Chapter 5
Multipath clustering The impulse response of a radio channel consists of contributions from all individual MPCs. Cluster-based channel models describe the channel characteristics in terms of multipath clusters. A multipath cluster is defined as a group of MPCs that have similar multipath parameters, i.e. AoA, AoD and delay. In this case, the characterization of all possible MPCs are modeled by a number of multipath clusters. Each multipath cluster is characterized by a 3-dimensional multivariate Gaussian distribution (in AoA, AoD and delay). If one can identify and extract the cluster distribution properly, the total multipath channel can then be characterized by a Gaussian mixture model (GMM). This chapter draws a connection between multipath clusters and a Gaussian mixture model by using an improved EM-GMM clustering algorithm.
5.1
Multipath clusters
One approach to automatically identify multipath clusters is available through the KPowerMeans approach [5, 10, 13]. This is an extended KMeans clustering algorithm including paths’ power. The clustering is performed by assigning each path with different cluster indexes. In this thesis, another multipath clustering method is developed using Gaussian mixture models. As discussed in Chapter 3, the EM-GMM approach is capable of clustering a data set into different Gaussian models according to the distributions of its feature parameters. For multipath clusters, the data set D consists of L MPCs. Each MPC is characterized in a parameter space of power (γ), AoA (ϕ), AoD (ψ) and delay (τ ). These are defined by D = {x1 , x2 , . . . , xl , . . . , xL }
(5.1)
xl = [γl , ϕl , ψl , τl ]T
(5.2)
25
26
CHAPTER 5. MULTIPATH CLUSTERING
In order to identify multipath clusters and to extract the cluster distributions, one aims to classify the MPCs by clustering them into groups with similar characteristics, and to obtain the K most representative characteristics of set D. Each multipath cluster is defined to be of a multivariate Gaussian distribution in AoA, AoD and delay1 . This distribution is characterized by its mean µ and variance Σ [27], by µ = [ϕ, ψ, τ ]T , (5.3) 2 σϕ ρϕψ · σϕ · σψ ρϕτ · σϕ · στ ρ · σ · σ σψ2 ρψτ · σψ · στ , Σ= (5.4) ϕ ψϕ ψ ρτ ϕ · σ τ · σ ϕ ρ τ ψ · σ τ · σ ψ στ2 where σϕ denotes the spread of AoA so as σψ and στ to AoD and delay, respectively. ρϕψ denotes the covariance between AoA and AoD, and so on for all other parameters. For each multipath cluster, the mean µ describes the position of the cluster centroid in the parameter space. The variance Σ describes the cluster spread which determines the shape of the cluster.
5.2
Power mean
The MPCs appear to be in clusters within the AoA, AoD and delay domain. However, it is noted that the characterization of each MPC also includes the power component γ which is so important that it requires extra consideration. The authors of [5], [10] and [13] proposed the KPowerMeans clustering algorithm for multipath clusters by considering power components. Each MPC xn is weighted by its path power γn . The cluster centroid should concentrate more on the strong paths other than just the geometric average centroid. The estimation of the power weighted mean µ and variance Σ are not the same as in (3.10) and (3.11). They are defined as µp and Σp by p
PN
n=1 µ = P N
γn xn
n=1 γn
,
(5.5)
PN
− µp )(xn − µp )T . (5.6) PN n=1 γn Figure 5.1 illustrates the difference between the geometric mean and the powerweighted mean. The power weight of each data point is indicated by its color. The mean and the variance for this data set are calculated with and without power consideration, respectively. The red circle indicates the geometric mean and the green curve indicates its contour. The red triangle indicates the power-weighted mean and the black curve indicates its contour. It shows the power mean concentrates the cluster centroid on the paths with stronger power. This is very beneficial for practical multipath clustering of MPCs. p
Σ =
1
n=1 γn (xn
It is noted here that power component is no cluster parameter but it is addressed later in this chapter.
5.3. EM-GMM MULTIPATH CLUSTERING
27
Figure 5.1. Cluster mean with power weighted The two axes indicate two dimensional data. The power weight of each data point is indicated by its color.
5.3
EM-GMM multipath clustering
In GMM, the total multipath channel is characterized by a mixture of K Gaussian distributions. Each Gaussian is described by its first-order and second-order statistics, the mean µ and the variance Σ. As discussed in Chapter 3, each cluster is also characterized by itself a weight λ which describes its probability in the whole mixture. In order to estimate the multipath clusters, the parameter vector θ is defined again as θ = {λ1 , . . . , λK ; µp 1 , . . . , µp K ; Σp 1 , . . . , Σp K } ,
(5.7)
where µp and Σp are from (5.5) and (5.6). This corresponds to the EM-GMM approach from Table 3.1. The goal is to iteratively estimate the parameter θ, but with power weight consideration. By substituting (5.5) and (5.6) into Table 3.1, the EM-GMM clustering algorithm for MPCs is summarized in Table 5.1. Note that, besides the power mean, other specific considerations for multipath clustering are required. Some multipath parameters come in the angular domain,
28
CHAPTER 5. MULTIPATH CLUSTERING
EM-GMM clustering algorithm for MPCs Initialization at iteration t = 1 • data set D = {x1 , . . . , xL }:
L MPCs of the multipath channel
• set maximum possible K • set iteration time T • give initial value for λ(1) , µp,(1) and Σp,(1) From iteration t = 1, . . . , T (t+1) (t+1) (t+1) λ ,µ ,Σ =EM-GMM(D, λ(t) , µ(t) , Σ(t) ) Expectation step For every k and l, calculate n o (t) p,(t) p,(t) λk G(xl | µk , Σk ) n o p(ωk |xl ) = P (t) p,(t) p,(t) K λ G(x | µ , Σ ) l j=1 j k k
(5.8)
p(ωk |xl ) ρkl = PL l=1 p(ωk |xl )
(5.9)
Maximization step For every k, calculate the power mean L
(t+1)
λk
=
1X ρkl L
(5.10)
l=1
p,(t+1) µk
p,(t+1) Σk
PL =
l=1 γl (xl
PL
= Pl=1 L
γl xl p(ωk |xl )
(5.11)
l=1 γl p(ωk |xl )
p,(t+1)
− µk PL
p,(t+1) T ) p(ω
)(xl − µk
k |xl )
l=1 γl p(ωk |xl )
Next t Table 5.1. EM-GMM clustering algorithm for MPCs
(5.12)
5.3. EM-GMM MULTIPATH CLUSTERING
29
which is not of linear scale but spherical. So, the principal value need to be taken into account when calculating the cluster mean and variance in (5.11) and (5.12). The equations are provided in Table 5.2. The function PVang takes the principal value between −π and +π.
µpk
µϕ = µψ = µτ
P
L γl p(ωk |xl ) exp(jϕl ) Pl=1 L PVang l=1 γl p(ωk |xl ) exp(jψl )
PVang
P P
L l=1 γl p(ωk |xl )τl L l=1 γl p(ωk |xl )
PVang (ϕ1...L − µϕ ) (xl − µpk ) = PVang (ψ1...L − µψ ) τ1...L − µτ PL p T p p l=1 γl (xl − µk )(xl − µk ) p(ωk |xl ) Σk = PL l=1 γl p(ωk |xl )
(5.13)
(5.14)
(5.15)
Table 5.2. Implementation specifications for MPC parameters
The following part discusses a simulation example of EM-GMM multipath clustering. The data set is generated synthetically in the 3 multipath dimensions, AoA, AoD and delay. Each path is also weighted by its power component. The EM-GMM procedure is run iteratively as described in Table 5.1. Figure 5.2 shows the centroid and the ellipsoid contour for each multipath cluster in the parameter space. The centroid position is obtained from the cluster mean µp . The ellipsoid contour is calculated by the cluster variance Σp . Figure 5.3 shows the results in different views, in AoD-AoA, delay-AoA, and delay-AoD. Clusters show different shapes because of the different spreads at different dimensions. This example shows that the EMGMM clustering algorithm is able to identify multipath clusters properly. However, real-world data from channel measurements are more challenging.
30
CHAPTER 5. MULTIPATH CLUSTERING
Figure 5.2. MPC clusters in parameter space AoAs and AoDs are between −π and +π; Delays are in 10ns (1 : 10(−8) s)
Figure 5.3. MPC clusters in different dimensions AoAs and AoDs are between −π and +π; Delays are in 10ns (1 : 10(−8) s)
5.3. EM-GMM MULTIPATH CLUSTERING
5.3.1
31
Initialization
The proposed EM-GMM algorithm for clustering MPCs (multipath components) requires an initialization. This gains importance when one is clustering MPCs from channel measurements. As discussed, the multipath power is a crucial factor. Therefore, the initialization approach should also have emphasis on the power component. Various initialization approaches are developed. Its main idea is summarized in Figure 5.42 . This method selects the path with strongest power as the initial cluster centroid. Then the variance for this selected cluster is calculated, according to the EM framework in Table 5.1. The normalization of the paths’ power guarantees the next strongest path to lie out of the already selected clusters. In other words, for the selection of the next clusters, the paths which have been already assigned to belong to a cluster are considered with smaller weights compared to the paths which have not been assigned a cluster yet. This path-disregarding is performed by a power normalization. The powers of the already selected paths are normalized in order to result in smaller weights. In this way, the strong paths lying out of the selected clusters would have bigger weights. This could lead the selection of the next strongest path to lie out of the already selected clusters. The selection process stops if there are no more strong paths that are still left out of all already selected k clusters. For example, no path, whose power is higher than 20 dB below the strongest power of all MPCs in the scenario, is left out of selected clusters .
5.3.2
Convergence of EM-GMM
In order to do clustering for MPCs, one aims to identify and extract the multipath cluster parameters properly to model the multipath channel. Therefore, the convergence of the EM-GMM approach must be guaranteed for stable solutions. The initialization approach proposed above is deterministic as it is decided by the power distribution of all MPCs. However, in Gaussian mixture models, each cluster has a weight λ which describes its probability in the mixture. It is illustrated from Table 3.2 in Chapter 3 that the λs for the initially over estimated clusters would decrease gradually. The initially over estimated clusters should be eliminated if their weights λ are reduced to be very small. However, this change would influence the stability of the clustering results. The outcome clusters after the EM-GMM algorithm are sometimes co-located3 , because of the structure of the MPCs and the initial over estimation. Co-located clusters are defined as clusters whose means are similar and whose variances are relevant to each other. In other words, clusters are treated as co-located if they are close to each other in terms of centroid and if they are of similar shape in terms 2 3
This is different if tracking is included. See Chapter 6 This problem is quite troublesome for tracking
32
CHAPTER 5. MULTIPATH CLUSTERING
Set K_max K=1
1> Select the strongest path p 2> Set mu(K) = p’s parameter 3> Calculate the variance of var(K) 4> Normalize the paths’ power by excluding powers in clusters [1...K]
Still strong path left out of clusters [1, …, K] ? && K
K=K+1
Stop
Initialization with mu(k) and var(k) with k=1,…,K
Figure 5.4. Initialization procedure for MPC clustering strong indicates power within the range of 20 dB below the strongest power.
5.3. EM-GMM MULTIPATH CLUSTERING
33
of spread. For the cluster centroid, the distance between cluster mean vectors are calculated with consideration of principal angle values. For the cluster spread, the distance between cluster variance matrices are calculated by Correlation Matrix Distance (CMD) [28]. If the distance of both mean and variance are relatively small, these co-located clusters must be united to be one cluster. The parameters for the united cluster are obtained by using the paths from the co-located clusters before they are united. However, this change would also influence the stability of the clustering results. Therefore, in order to obtain the stable convergence, any changes from the clustering results must be fed back. The EM-GMM algorithm would employ the feedback information to cluster the MPCs again until the stable results are obtained. Figure 5.5 proposes the framework by using the EM-GMM algorithm to cluster MPCs which helps to guarantee the stability of the clustering results.
34
CHAPTER 5. MULTIPATH CLUSTERING
K_max
Initial guess K_ini mu_ini var_ini lamb_ini
Do EM-GMM Eliminate cluster(s) if its probability weight is too small Unite clusters if they are co-located K_out mu_out var_out lamb_out
K_out == K_ini ?
K_ini = K_out mu_ini = mu_out var_ini = var_out
Stop
Figure 5.5. Framework of MPC clustering using EM-GMM
5.4. CLUSTERING REAL-WORLD DATA
5.4
35
Clustering real-world data
Examples of multipath clustering using the above described EM-GMM approach are given as results. These examples employ multipath estimates from real-world MIMO channel measurements. Example 1 is illustrated in Figure 5.6, in which each multipath cluster is indicated by an index number. The contour of the cluster is obtained from the variance matrix of the corresponding Gaussian density for each cluster. The shape describes the spread and the correlation among multipath parameters. In this example, clusters with ID 3, 5, 6 and 7 are well defined clusters. Clusters 1 and 2 are characterized by relatively strong power. Cluster 4 describes the MPCs that are left around the cluster 1 and 2, which in turn result in a very large spread in AoD domain. The cluster parameters obtained in this example are given in Table 5.3. It shows that clusters 1 and 2 characterize more than half of the total number of MPCs in this scenario. And they are of relatively strong power. Cluster 4 has a very large AoD spread which characterizes many weak MPCs that are left around clusters 1 and 2. Cluster 3 is distinct clusters with relatively strong power, little spread and small number of paths. This can correspond to a cluster of multiple bounced MPCs. Other clusters are of relatively weak power that characterize MPCs which are left out. The total mixture of distributions are illustrated in Figure 5.7. The centroids of cluster 1 and 4 are very close. However, their distributions are of big difference. It is noted that EM-GMM clustering approach models the data set as a mixture of Gaussian densities. Large-spread cluster like cluster 4 is important to model this mixture in a proper way. This is based on the assumption that the distribution of all possible MPCs can be modeled by a mixture of Gaussian distributions. However, it is quite troublesome for tracking these clusters because they have similar cluster centroids as target states. The centroid of large-spread clusters does not have meaningful target state for tracking. More detailed results on this issue are discussed in Chapter 6. Example 2 is illustrated in Figure 5.8 whose mixture distribution is given in Figure 5.9. Obtained cluster parameters are presented in Table 5.4.
36
CHAPTER 5. MULTIPATH CLUSTERING
Figure 5.6. Example 1: Clustering result for MPCs from measurements AoAs and AoDs are between −π and +π; Delays are in 10ns (1 : 10(−8) s)
Figure 5.7. Example 1: Mixture of multipath clusters (in AoA and AoD) AoAs and AoDs are between −π and +π
5.4. CLUSTERING REAL-WORLD DATA
ID
1 2 3 4 5 6 7
cluster Power mean (dBm) -38.7 -43.4 -47.5 -46.9 -52.1 -50.4 -50.6
cluster Power spread (dBm) 4.4 7.5 3.6 5.1 0.1 4.5 1.1
cluster Delay mean (ns) 32.6 43.4 35.1 35.5 103.7 56.2 39.9
cluster Delay spread (ns) 3.9 3.7 4.0 4.6 2.7 28.9 21.1
37
cluster AoA spread (Degree) 3.8 6.4 6.9 9.8 5.9 18.9 11.2
cluster AoD spread (Degree) 2.2 7.9 4.0 65 7.3 18.8 10.4
number of paths (%) 40.5 14.75 3.5 31.75 1 6.75 1.75
Table 5.3. Example 1: multipath cluster parameters
Figure 5.8. Example 2: clustering result for MPCs from measurements AoAs and AoDs are between −π and +π; Delays are in 10ns (1 : 10(−8) s)
38
CHAPTER 5. MULTIPATH CLUSTERING
Figure 5.9. Example 2: Mixture of multipath clusters (in AoA and AoD) AoAs and AoDs are between −π and +π
ID
1 2 3 4 5 6 7 8 9 10 11 12 13 14
cluster Power mean (dBm) -39.1 -50.6 -53.2 -44.8 -48.1 -60.7 -51.9 -45.6 -55.7 -50.5 -61.3 -54.3 -51.3 -59.4
cluster Power spread (dBm) 7.6 8.6 1.6 3.4 0.6 2.8 0.8 3.2 4.4 4.3 1.9 2.2 5.4 1.7
cluster Delay mean (ns) 38.4 38.9 136.5 45.2 39.4 111.2 101.4 40.3 100.7 39.6 47.4 38.6 44.9 80.8
cluster Delay spread (ns) 1.9 1.0 2.4 1.5 0.6 11.6 1.8 0.3 17.9 1.8 1.9 2.1 3.7 34.2
cluster AoA spread (Degree) 2.1 15.8 1.6 1.8 3.0 72.8 6.5 4.0 4.3 8.9 10.0 13.8 11.1 3.3
cluster AoD spread (Degree) 1.1 1.3 2.6 1.6 1.2 77.9 5.9 3.3 4.6 24.3 7.1 1.1 49.4 7.0
Table 5.4. Example 2: multipath cluster parameters
number of paths (%) 11 3.25 3 1 1 4.25 2.25 2 3.5 27.5 1.25 2.5 36 1.5
Chapter 6
Tracking multipath clusters Cluster tracking is necessary to identify the time-varying channel characteristics. Authors in [26] present a channel tracking approach for multipath radio propagation. In cluster-based MIMO radio channel modeling, multipath clusters are used to characterize the channel properties. Therefore, in order to model the channel of time-varying nature, it is necessary to track these multipath clusters and to extract the time-variant cluster parameters. This chapter discusses the tracking of multipath clusters. A joint clustering-and-tracking approach is developed to parameterize multipath clusters for cluster-based channel modeling. This cluster tracking can be combined with both the discussed EM-GMM clustering method or other available methods. Results from clustering and tracking of real-world MIMO channel measurements are finally presented.
6.1
Tracking multipath clusters
Tracking is important to provide the multipath cluster characteristics with timevariant parameters. The Kalman filter from Chapter 4 is applied to track the movement of the multipath clusters. One aims to track the movement of the multipath clusters and to extract the time-variant cluster parameters.
6.1.1
Multipath cluster as target
The tracking approach for a single multipath cluster is addressed first. The Kalman filter is suitable for this target tracking problem. As a target, the multipath cluster position is determined by its cluster centroid µ in the multipath parameter space of AoA (ϕ), AoD (ψ) and delay (τ ). It is written as µ = [ϕ, ψ, τ ].
(6.1)
In this case, the target state Θ from (4.8) is defined as Θ = [ϕ, ∆ϕ, ψ, ∆ψ, τ, ∆τ ]T , 39
(6.2)
40
CHAPTER 6. TRACKING MULTIPATH CLUSTERS
where ∆ indicates the velocity (second-order parameters) of the movement in the corresponding domain. The state-space model, Θn = ΦΘn−1 + wn ,
(6.3)
µn = HΘn + vn ,
(6.4)
is given by
ϕn ∆ϕn ψn ∆ψn τn ∆τn
=
1 T 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 T 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1 T 0 1
1 0 0 0 0 0 µn = 0 0 1 0 0 0 0 0 0 0 1 0
ϕn−1 ∆ϕn−1 ψn−1 ∆ψn−1 τn−1 ∆τn−1
ϕn ∆ϕn ψn ∆ψn τn ∆τn
+ wn ,
(6.5)
+ vn ,
(6.6)
to model the movement of the target, where Φ and H are indicated by the state transition matrix and observation matrix, correspondingly, T is the time interval between time steps which is set to be 1 for convenience, w and v are the process noise and the observation noise, which are i.i.d. Gaussian distributed with variances R and Q, respectively. The Kalman filter, employing a prediction step and an update step, is applied as in Table 4.1. Note that the target as a multipath cluster is in the angular domain which is not of linear scale. One needs to obtain the principal value when doing calculations for AoA and AoD. A target can move form −π to +π which remains to be the same target. The Kalman filter prediction step and update step for tracking a multipath cluster is given in Table 6.1 and Table 6.2. The prediction provides the predicted multipath cluster centroid at time n given n − 1. The observation is the multipath cluster centroid obtained at time n. The update corrects the prediction using the observation. The cluster spreads, for both the predicted and the updated clusters, are the variances calculated for the corresponding means from the current MPCs data at time n.
6.1. TRACKING MULTIPATH CLUSTERS
41
Prediction step for tracking a multipath cluster
b (n|n−1) = ΦΘ b (n−1|n−1) Θ
ϕ b(n|n−1) b ∆ϕ(n|n−1) ψb(n|n−1) b (n|n−1) ∆ψ
τb(n|n−1) b (n|n−1) ∆τ
=
1 T 0 1 PVang 0 0 0 0 1 T 0 1
0 0 1 0
ϕ b(n−1|n−1) 0 ∆ϕ b 0 (n−1|n−1) T ψb(n−1|n−1) b (n−1|n−1) 1 ∆ψ τb(n−1|n−1) b (n−1|n−1) ∆τ
(6.7)
P(n|n−1) = ΦP(n−1|n−1) ΦT + Q
(6.8)
Table 6.1. Kalman prediction step for tracking multipath cluster
There are two major facts that might influence the tracking performance. First, there are multiple MPC clusters at each time snapshot. In this case, the targets to be tracked need to be associated to the observed targets. An introduction for this problem can be found in [36]. Second, the multipath clusters do not exist forever and new clusters are discovered from time to time. This condition requires a flexible tracking approach in which the number of targets has to be adjustable. This corresponds to a multitarget tracking problem [23, 36].
6.1.2
Cluster association
There are multiple clusters to track. For example, at time n, the prior estimation is the prediction of kn−1 clusters which are obtained at time n − 1. The update step takes the observations which are kn clusters obtained at time n. The number of cluster kn and kn−1 can be different because they are decided by the distribution of MPCs from time to time. This problem is discussed in the next paragraphs. The detailed problem addressed here is how to match these multiple observed clusters at time n to those multiple predicted clusters from time n − 1. In order to track the clusters, the observed targets should be matched properly to predicted targets. Each target state can be predicted independently by Table 6.1. The data association procedure is required before updating the predicted target states. Usually, one can process the data association by finding the nearest neighbor in terms of the Euclidean distance, between the predictions and the observations. This approach is suitable for a cartesian multiple target tracking problem. However, the multipath cluster is not only characterized by its centroid but also by
42
CHAPTER 6. TRACKING MULTIPATH CLUSTERS
Update step for tracking a multipath cluster
Γϕ Γ∆ϕ Γψ Γ∆ψ Γτ Γ∆τ
K(n|n) = P(n|n−1) HT (HP(n|n−1) HT + R)−1
(6.9)
b (n|n) = Θ b (n|n−1) + K(n|n) (µ(n) − HΘ b (n|n−1) ) Θ
(6.10)
ϕ b(n|n−1) b (n|n−1) ∆ϕ ψb(n|n−1) b (n|n−1) ∆ψ
µ 1 0 0 0 ϕ − PVang µ 0 0 1 0 ψ = K(n|n) τb(n|n−1) [µτ ] − 1 0 b ∆τ(n|n−1)
ϕ b(n|n) b (n|n) ∆ϕ ψb(n|n) b (n|n) ∆ψ
τb(n|n) b (n|n) ∆τ
=
ϕ b(n|n−1) Γϕ ∆ϕ b Γ∆ϕ PVang b (n|n−1) + ψ(n|n−1) Γψ b (n|n−1) Γ∆ψ ∆ψ τb(n|n−1) Γτ b (n|n−1) + Γ∆τ ∆τ
P(n|n) = (I − K(n|n) H)P(n|n−1)
(6.11)
Table 6.2. Kalman update step for tracking multipath cluster
its spread. The spread is determined by the variance of the cluster distribution. In this case, finding the nearest neighbor considering cluster centroids only is not sufficient for association. To capture both the cluster centroid and the cluster spread, a probability distance metric for cluster association is proposed by using both the mean and the variance of the cluster distributions.
A probability distance metric The proposed probability distance metric is described by this example. One aims to match the predicted cluster i to the observed cluster j . Cluster i is described b (n|n−1,i) and the variance of Σ b (n|n−1,i) . by a Gaussian density with the mean of HΘ Cluster j is described by a Gaussian density with the mean of µ(n,j) and the variance of Σ(n,j) . We recall that H is the observation matrix from the state-space model. The probability distance from cluster i to cluster j is defined as the probability pj|i
6.1. TRACKING MULTIPATH CLUSTERS
43
that cluster j belongs to cluster i. This is given as
b (n|n−1,i) , Σ b (n|n−1,i) }) = pj|i = G(µ(n,j) |{HΘ
b
1
(2π)d/2 |Σ(n|n−1,i) |
1/2
(6.12)
b (n|n−1,i) )T Σ b −1 b exp − 12 (µ(n,j) − HΘ (µ − H Θ ) . (n|n−1,i) (n|n−1,i) (n,j)
The desired matching between predicted and observed clusters is of the maximum pj|i among all possible matchings. One aims to match the observed cluster j to the predicted cluster i if it results in a maximum of all possible pj|i . In the case of multiple clusters, it is necessary to check this metric for all possible associations to find the best matched observations. Matching both mean and spread In order to perform the multipath cluster association to update the tracking, both the mean and the spread of the cluster distribution must be considered. Besides the probability distance metric proposed above, an alternative method is described. The desired matching between two clusters, on one hand, must be relevant to each other in terms of the cluster centroids, but on the other hand, must have comparable cluster shape in term of cluster spreads. The distance between the predicted cluster means and the observed cluster means can be calculated by using the Multipath Component Distance (MCD) [29]. The distance between their spreads can be obtained through the Correlation Matrix Distance (CMD) [28] between cluster variance matrices. These distance metrics are discussed in detail in [28] and [29]. They are not addressed in this thesis.
6.1.3
Multiple multipath clusters
There are multiple multipath clusters and the number of clusters is changing from time to time. This requires the tracking approach to be flexible to capture the appearance and vanishment of clusters. For example, at time n − 1, there are kn−1 observed multipath clusters. Each of these kn−1 clusters is predicted by the Kalman prediction step in parallel. This results in kn−1 predicted clusters. At time n, kn clusters are observed. One aims to track clusters from time n − 1 by using the observations at time n through the update. The observations must be associated to the predictions properly before performing the updates. In this case, a forward and backward data matching is proposed to obtain the association. In the forward association, each predicted cluster from n − 1 is matched to an observed cluster at n. The probability distance of all possible pj|i , that the observed cluster j owns a membership into the predicted cluster i, are calculated for i = {1, . . . , kn−1 } and j = {1, . . . , kn }. The best forward match is to obtain a pair of i and j by searching for the maximum probability distance of pj|i among all possible
44
CHAPTER 6. TRACKING MULTIPATH CLUSTERS
pairs. In the backward association, each observed cluster at n is matched to a predicted cluster from n − 1 in the same way. The probability distance of all possible pi|j , that the predicted cluster i owns a membership into the observed cluster j, i = {1, . . . , kn−1 } and j = {1, . . . , kn }. The best backward match is to obtain a pair of i and j by searching for the maximum probability distance of pi|j among all possible pairs. • If a pair of clusters are matched to each other by both the forward and the backward association, the predicted cluster is updated and tracked by the associated observation. • If the predicted cluster can not be matched to any of the observations, the cluster has vanished. • If the observed cluster is not matched by any of the predictions, a new cluster has appeared. Figure 6.1 illustrates the update procedure for multiple multipath clusters. In this way, the matched multipath clusters are tracked by the update step in parallel. The cluster appearance and vanishment is also addressed. This tracking information is important for the clustering approach to capture the time-varying cluster properties. Note that, the forward probability distance pj|i can be also viewed as the probability fXn |Xn−1 (Xn | Xn−1 ) in (4.6). The backward probability distance pi|j corresponds to the probability fXn |Zn (Xn | Zn ) in (4.7). In this way, finding the maximum of pj|i and pi|j contributes to the maximization of the posterior probability.
6.2
Joint clustering and tracking
It is discussed in Chapter 5 that the initialization of the clustering approach is crucial for the multipath clustering. To extract the time-variant characteristics of the multipath channel, the tracking result is provided as prior information to the initialization step. This proposes a joint clustering-and-tracking approach to identify multipath clusters parameters. The approach is described in the following paragraphs.
6.2.1
Initialization by prediction
Assuming kn−1 multipath clusters are obtained from time n − 1, in order to include the time-variant characteristics, these kn−1 multipath clusters are employed
6.2. JOINT CLUSTERING AND TRACKING
45
EM-GMM clustering for MPCs at time n
K(n-1) prediction multipath clusters
K(n) observation multipath clusters
Cluster association - forward matching - backward matching -
matched clusters
appearing clusters
vanished clusters
tracked clusters
Figure 6.1. Update multiple multipath clusters
as prior information for clustering MPCs at time n. In the proposed joint clusteringand-tracking approach, the clustering outputs from previous time step are used for predictions. This prediction is then employed as the prior information to the initialization for clustering MPCs at the current time step. If there are no strong paths left out of the already selected clusters, for example, all paths left out are less than 20dB below the strongest path, the initialization may stop before all kn−1 predicted clusters are employed. Alternatively, if there are still strong paths left after all kn−1 predicted clusters are selected, the initialization continues until no more strong paths are left. By giving this prior information, the selection for the clusters are performed similarly as in Figure 5.4. Note that after all predicted clusters are regarded, the selection is performed in the same way as in Section 5.3.1. The proposed initialization procedure by using the prior prediction information is illustrated in Figure 6.2.
6.2.2
Joint clustering and tracking approach
Finally, the framework of the joint clustering-and-tracking approach is proposed as follows. • The predicted clusters from previous time step are used as prior information for clustering initialization. • The EM-GMM clustering algorithm is performed for multipath clustering at current time step.
46
CHAPTER 6. TRACKING MULTIPATH CLUSTERS
Set K(max) MPCs at time n Prior information K(n-1) predicted clusters K=1 1> Set mu(K) = K-th prediction 2> Calculate the variance of var(K) 3> Normalize the paths’ power by excluding powers in clusters [1...K]
Strong path left out ?
K<=K(n-1) ?
K=K+1
K=K(n-1)+1 1> Select the strongest path p 2> Set mu(K) = p’s parameter 3> Calculate the variance of var(K) 4> Normalize the paths’ power by excluding powers in clusters [1...K]
Still strong path left out ? K<=K(max) ?
K=K+1
Stop Initialization with mu(k) and var(k) with k=1,…,K Figure 6.2. Initialization procedure using prior prediction information strong indicates power within the range of 20 dB below the strongest power.
6.2. JOINT CLUSTERING AND TRACKING
47
convergence procedure
prediction
initialization
update
Figure 6.3. Joint clustering-and-tracking framework
• The result clusters are then associated to the tracked clusters. A Kalman filter is applied for tracking clusters. Figure 6.3 illustrates the procedure for the joint clustering-and-tracking approach. The EM-GMM clustering algorithm is capable to identify the multipath clusters and to extract cluster parameters at each time snapshot. Then, Kalman tracking by using a prediction step and an update step is able to capture the timevarying nature of the multipath cluster characteristics. The approach is proposed for improved cluster-based modeling of time-varying MIMO radio channels. The joint clustering-and-tracking approach is useful to extract the time-varying cluster characteristics, and the tracking information in turn improves the clustering performance. The framework above employs the GMM clustering approach using the EM algorithm. Note that the cluster tracking can be combined with any other clustering methods proposed for multipath clustering. For example, the framework can be easily combined with KPowerMeans approach [5, 10, 13] by substituting the EM-GMM clustering, see [40]. The joint clustering-and-tracking results by applying this framework are presented in the next section.
48
CHAPTER 6. TRACKING MULTIPATH CLUSTERS
Figure 6.4. Route of the channel measurements
6.3
6.3.1
Tracking multipath clusters from MIMO channel measurements MIMO channel measurements
MIMO channel measurements provide numerous impulse responses of the measured time-varying channel, from time snapshot to time snapshot. These channel impulse responses at each time snapshot are input to a high resolution parameter estimator, for example SAGE [9], to estimate multipath parameters. In this way, all MPCs are described in the parameter space by these multipath parameters, which are typically AoA, AoD, delay and power. The proposed joint clustering-and-tracking framework aims to identify multipath clusters properly and to extract cluster parameters with time-variant characteristics. This information is useful to find stochastic cluster parameters for cluster-based MIMO radio channel models. The channel measurement employed in this analysis was conducted at 2.55 GHz in a Cafeteria. In this example1 , the Rx is fixed, while the Tx moves around the Rx according to the route shown in Figure 6.4. The elevator in the lower part of the figure shadows the LoS propagation paths.
1
Configuration details on MIMO channel measurement campaign are not of emphasis in this thesis. Please see [13] and [30] for more information.
6.3. TRACKING MULTIPATH CLUSTERS FROM MIMO CHANNEL MEASUREMENTS
49
Figure 6.5. Geometry based multipath cluster analysis left: possible multipaths between Tx and Rx. right: illustration of AoA at Rx and AoD at Tx.
6.3.2
Geometry-based analysis
Stochastic MIMO channel models based on physical considerations recently became important because they allow to model the spatial structure of the channel in a convenient way with low complexity [30]. A geometry based analysis is discussed first. One example is shown in Figure 6.5. The multipath parameters are provided in Table 6.3;
cluster power mean (dBm) cluster power spread (dBm) cluster delay mean (ns) cluster delay spread (ns) cluster AoA mean (degree) cluster AoA spread (degree) cluster AoD mean (degree) cluster AoD spread (degree) number of paths (%)
Cluster 1 -52.4 3.2 86.3 6.1 -96.1 23.5 -63.3 3.7 5
Cluster 2 -49.9 4.4 61.8 2.2 136.8 3.7 -57.8 4.2 10.75
Cluster 3 -44.0 6.3 45.4 2.8 -153.7 1.3 21.8 0.9 10.25
Cluster 4 -54.7 1.3 121.1 6.6 -87.3 14.1 73.3 7.5 2
Cluster 5 -56.3 6.1 60.0 26.8 -167.6 67.9 30.7 65.2 72
Table 6.3. Multipath cluster parameters of the geometry based analysis example
In this example, Cluster 3 corresponds to the direct LoS propagation. It is characterized by strong power, short delay, and small spreads (power, delay, AoA, AoD). Clusters 1 and 2 correspond to the single-bounce reflection propagation from
50
CHAPTER 6. TRACKING MULTIPATH CLUSTERS
the wall around. Compared to the direct LoS propagation, these two clusters are characterized by weaker power and longer delay. Their spreads are also larger because of the rough scattering. Note that Cluster 1 has relatively larger AoA spread than it should be. This is due to the special spatial structure around the reflection area on the left wall. Cluster 4 can be viewed to characterize a multiple bounced cluster. It has weaker power and longer delay. Cluster 5 is characterized by very weak power and very large spreads. It models the dispersed propagation paths which do not belong to any established clusters. This is the result of the EM-GMM clustering with which the mixture distribution can model the total scenario in a proper way. However, this kind of large spread clusters are highly data dependent and do not reflect the included multipath characteristics in the desired way. They are quite troublesome for tracking clusters as these clusters do not have meaningful target states. This must be taken into account when applying the joint framework of clustering and tracking.
6.3.3
Stochastic parameters
The advantage of the joint clustering-and-tracking approach is its capability to extract the time-variant multipath cluster parameters. The obtained parameters are useful for cluster-based stochastic channel modeling. Some parameters obtained from the examples are presented here as results. The approximate period for these examples (A - F) are indicated in Figure 6.6. The first-order statistics of clustering parameters obtained from these examples (A - F) are summarized in Table 6.4. These are the average cluster parameters for the corresponding time period. The second-order parameters can be also obtained by finding the variance of parameters over time. Among these examples, A and F have similar parameters as they are similarly located. Example B, C and D have comparable parameters for the number of clusters and the cluster spreads. All of them are modeling the radio propagation channel in the similar environment. Example E is addressed because it is a NLoS scenario. There is no direct propagation paths between Tx and Rx. All multipaths are due to reflection and scattering. In this case, it shows conspicuously more clusters than other examples with LoS propagation. It also has large AoD spread due to the rich scattering environment around Tx.
6.3. TRACKING MULTIPATH CLUSTERS FROM MIMO CHANNEL MEASUREMENTS
51
Figure 6.6. Example A - F of the channel measurements
Firstorder parameters A B C D E F
number of clusters 4.1 7.3 8.9 7.4 11.4 4.6
cluster Power (dBm) -41.4 -45.6 -49.4 -49.3 -58.6 -42.4
cluster Delay (ns) 34.4 44.5 63.9 56.3 82.1 35.9
cluster Delay spread (ns) 5.08 3.6 3.45 3.6 2.8 4.4
cluster AoA spread (Degree) 11.4 8.9 6.7 6.5 7 7.1
cluster AoD spread (Degree) 8.8 5.7 5.8 6.8 10 8.3
Table 6.4. Example statistics of multipath cluster parameters
6.3.4
Illustration of a tracked multipath cluster
Finally, the statistical parameters of a tracked multipath cluster are illustrated. The Example G of the tracked multipath cluster is shown in the measurements route in Figure 6.7. In the parameter space of (AoA, AoD, delay), this multipath cluster moves from (161, −17, 47) to (−145, 28, 49), in degree and ns. It shows that AoA and AoD change smoothly according to the movement of the cluster, while values do not change much in delay domain. The movement’s trajectory of this tracked multipath cluster is illustrated in Figure 6.8. It shows that the multipath cluster is tracked properly when its AoA is moving from +π to −π.
52
CHAPTER 6. TRACKING MULTIPATH CLUSTERS
Figure 6.7. Example route of tracked multipath cluster
The obtained statistics of multipath cluster parameters are given in Table 6.5. The first-order statistics are the parameters’ average over the time period. The second-order statistics are the parameters’ standard deviation over the time period.
6.3. TRACKING MULTIPATH CLUSTERS FROM MIMO CHANNEL MEASUREMENTS
Figure 6.8. Trajectory of a tracked multipath cluster
Example G cluster Power (dBm) cluster Delay (ns) cluster Delay spread (ns) cluster AoA spread (Degree) cluster AoD spread (Degree) number of paths (%) number of total clusters
first-order -43.5 46.4 2.7 2.2 1.5 23.7 8.7
second-order 1.4 0.9 0.5 1.2 0.5 9.4 1.6
Table 6.5. Example parameter statistics of one tracked multipath cluster
53
Chapter 7
Summary Cluster-based MIMO radio channel models describe the radio propagation in terms of multipath clusters. The framework of this modeling is processed as follows. The starting point is the channel measurement for a typical time-variant channel. Then the double-directional multipath parameters are estimated using a high resolution estimator. This thesis investigates cluster parameterization for MIMO channel characterization. It employs the estimation results of MPCs from previous work. Then the multipath clusters are identified and tracked using the proposed joint clustering-and-tracking framework. The obtained cluster parameters would improve cluster-based MIMO radio channel models. A multipath cluster consists of a group of MPCs which are described by the double-directional impulse response. These MPCs show similar parameters including AoA, AoD and delay in a cluster. The characterization of all possible MPCs in the multipath channel is proposed to form a number of multipath clusters, where each cluster is characterized by a multivariate Gaussian distribution. In this way, the total channel characteristics are represented by a mixture of Gaussian distributions, that is, a Gaussian mixture model. Each cluster in the mixture is described by its corresponding Gaussian model via the cluster centroid and the cluster spread. The cluster centroid is obtained from the mean of the Gaussian model which describes its position in the multipath parameter space. The cluster spread is obtained from the variance of the Gaussian model which describes the shape of cluster. The EM-GMM approach is proposed in this thesis for multipath clustering. This approach is extended to perform clustering in the multipath parameter space. The power of MPCs is also taken into account. The EM-GMM multipath clustering approach is capable to identify and parameterize multipath clusters in a proper way. The advantage of the proposed approach is its optimum estimation for parameters of the mixture cluster distributions. In order to extract the time-variant cluster characteristics, a Kalman filter is
55
56
CHAPTER 7. SUMMARY
employed for tracking multipath clusters with a prediction step and an update step. This is furthermore extended to combine with the clustering which in turn proposes a joint clustering-and-tracking framework for identification and parametrization of multipath clusters. In the prediction step, the predicted multipath clusters from the previous time are input as prior information to initialize the clustering for current MPCs. In the update step, the observed multiple clusters are first associated with a probability distance metric. This metric employs both the cluster centroid and the cluster spread information. Then, tracked clusters, vanished clusters and appearing clusters can be identified correspondingly. Note that the output of the EM-GMM approach is the estimation of the mixture Gaussian distribution in a local optimal way. However, this sometimes results in large spread cluster which do not have meaningful target states for tracking. This must be taken into account when doing further modeling. Furthermore, the joint framework for clustering and tracking can be easily combined with both the EMGMM clustering approach and other available approaches for multipath clustering. Finally, exemplary clustering results for MPCs from real-world MIMO channel measurements are obtained via the proposed joint clustering-and-tracking framework. The extracted parameters from the multipath clusters are useful to find the stochastic properties in order to improve cluster-based stochastic MIMO radio channel models. The future work is to model this radio channel from the typical channel measurements by using the stochastic properties extracted from the obtained multipath clusters. This would propose a cluster-based stochastic MIMO radio channel model. The validity of this model would be the major discussion in the future work. Besides, the cluster tracking performance can possibly be improved by applying more advanced tracking methods. Applications from multitarget tracking can be useful for this improvement.
Bibliography [1] A. Paulraj, R. Nabar, D. Gore, “Introduction to Space-Time Wireless Communications”, Cambridge University Press, 2003 [2] P. Almers, E. Bonek, A. Burr, N. Czink, M. Debbah, V. Degli-Espoti, H. Hofstetter, P. Kyosti, D. Laurenson, G. Matz, A.F. Molisch, C. Oestges and H. Ozcelik, “Survey of Channel and Radio Propagation Models for Wireless MIMO Systems”, accepted for publications EURASIP Journal on Wireless Communications and Networking, 2006 [3] A. F. Molisch, “Wireless Communications”, IEEE Press - Wiley, 2005 [4] M. Steinbauer, A. F. Molisch, E. Bonek, “The Double-Directional Radio Channel”, IEEE Magazine in Antennas and Propagation, vol. 43, no. 4, pp. 51-63, Aug. 2001 [5] P. Cera, “A Framework for Automatic Clustering of Parametric MIMO Channel Data”, Master Thesis, Vienna University of Technology, 2005 [6] A. Goldsmith, “Wireless Communications”, Cambridge University Press, 2005 [7] A. F. Molisch, “Modeling the MIMO propagation channel”, Belgian Journal of Electronics and Communications, no. 4, pp. 5-14, 2003 [8] L. Correia, Ed., “Towards Mobile Broadband Multimedia Networks”, COST 273 Final Report, 2006 [9] B. H. Fleury, M. Tschudin, R. Heddergott, D. Dahlhaus, K. I. Pedersen, “Channel parameter estimation in mobile radio environments using the SAGE algorithm”, IEEE JSAC, no. 3, pp. 434-450, 17 1999 [10] N. Czink, P. Cera, J. Salo, E. Bonek, J.P. Nuutinen, J. Ylitalo, “A framework for automatic clustering of parametric MIMO channel data including path powers”, Proceedings of IEEE VTC, September, 2006 [11] L. Weruaga, “Unsupervised Learning and Clustering”, Lecture notes on Learning Theory for Information and Signal Processing, Vienna University of Technology, summer semester, 2006, available at http://www.viskom.oeaw.ac.at/∼luis/ 57
58
BIBLIOGRAPHY
[12] S. M. Kay, “Fundamentals of Statistical Signal Processing: Estimation Theory”, Upper Saddle River, New Jersey, USA, Prentice-Hall, 1993 [13] N. Czink, E. Bonek, L. Hentila, J.P. Nuutinen, J. Ylitalo, “Cluster-based MIMO channel model parameters extracted from indoor time-variant measurements”, in Proc. 49th Annual IEEE GLOBECOM Technical Conference and IEEE Communications EXPO, San Francisco, CA, USA, November, 2006 [14] H. Ozcelik, “Indoor MIMO channel models”, Ph.D Thesis, Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology, 2004 [15] C. C. Chong, C. M. Tan, D. Laurenson, S. McLaughlin, M. Beach, A. Nix, “A new statistical wideband spatio-temporal channel model for 5-GHz band WLAN systems”, IEEE Journal on Selected Areas in Communications, vol. 21, no. 2, pp. 139-150, Feb. 2003 [16] K. Yu, Q. Li, D. Cheung, C. Prettie, “On the tap and cluster angular spreads of indoor WLAN channels”, in Proceedings of IEEE Vehicular Technology Conference Spring 2004, Milano, Italy, May. 2004 [17] J. Fuhl, A. F. Molisch, E. Bonek, “Unified Channel Model for Mobile Radio Systems with Smart Antennas”, IEEE Proc. - Radar, Sonar and Navigation: Special Issue on Antenna Array Processing Techniques, vol. 145, no. 1, pp. 3241, Feb. 1998 [18] A. Saleh, R. Valenzuela, “A Statistical Model for Indoor Multipath Propagation”, IEEE Journal on Selected Areas in Communications, vol. 5, no. 2, pp.128137, Feb. 1987 [19] http://www.lx.it.pt/cost273 [20] 3GPP - 3GPP2 Spatial Channel Model Ad-hoc Group3GPP TR 25.996, “Spatial Channel Model for Multiple Input Multiple Output(MIMO) Simulations”, v6.1.0 (2003-09) [21] Q. Spencer, B. Jeffs, M. Jensen, A. Swindlehurst, “Modeling the Statistical Time and Angle of Arrival Characteristics of an Indoor Multipath Channel”, IEEE Journal on Selected Areas in Communications, vol. 18, no. 3, pp. 347-360, Mar. 2000 [22] T. Zwick, C. Fischer, W. Wiesbeck, “A Stochastic Multipath Channel Model Including Path Directions for Indoor Environments”, IEEE Journal on Selected Areas in Communications, vol. 20, no. 6, pp. 1178-1192, Aug. 2002 [23] R. Mahler, “Engineering Statistics for Multi-Object Tracking”, p. 0053, IEEE Workshop on Multi-Object Tracking (WOMOT’01), 2001
59
[24] M. H. Hayes, “Statistical Digital Signal Processing and Modeling”, New York: Wiley Press, 1996 [25] G. Welch, G. Bishop, “An Introduction to the Kalman Filter”, Dept. of Computer Science, University of North Carolina at Chapel Hill, available at http://www.cs.unc.edu/∼welch/kalman/ [26] J. Salmi, A. Richter, V. Koivunen, “Enhanced Tracking of Radio Propagation Path Parameters Using State-Space Modeling”, 14th European Signal Processing Conference, Florence Italy, Sept. 2006 [27] X. Yin, T. Pedersen, N. Czink, B. H. Fleury, “Parametric characterization and estimation of bi-azimuth and delay dispersion of path components”, in Proceedings of The First European Conference on Antennas and Propagation (EuCAP), Acropolis, Nice, France, November 2006 [28] M. Herdin, N. Czink, H. Ozcelik, E. Bonek, “Correlation matrix distance, a meaningful measure for evaluation of non-stationary MIMO channels”, in Vehicular Technology Conference, 2005, VTC 2005-Spring [29] M. Steinbauer, H. Ozcelik, H. Hofstetter, C. Mecklenbrauker, E. Bonek, “How to quantify multipath separation”, IEICE Transaction on Electronics, vol. E85, no. 3, pp. 552-557, March 2002 [30] N. Czink, E. Bonek, J. Nuutinen, J. Ylitalo, “Parameterizing Geometry-Based Stochastic MIMO Channel Models from Measurements Using Correlated Clusters”, in Proceedings of International ITG/IEEE Workshop on Smart Antennas, Vienna, Austria, February 26-27, 2007 [31] J. Han, M. Kamber, “Data Mining: Concepts and Techniques”, Morgan Kaufmann Publishers, 2001 [32] B. Mirkin, “Mathematical Classification and Clustering”, Kluwer Academic Publishers, 1996 [33] R. O. Duda, P. Hart, D. G. Stork, “Pattern Classification”, John Wiley Sons, Inc., 2001 [34] L. Kaufman, P. J. Rosseeuw, “Finding Groups in data: an introduction to cluster analysis”, John Wiley Sons, 1990 [35] A. W. Moore, “Clustering with Gaussian mixtures”, Tutorial, School of Computer Science, Carnegie Mellon University, available at http://www.cs.cmu.edu/∼awm/tutorials [36] Y. Bar-Shalom, T.E. Fortmann, “Tracking and Data Association”, Academic Press, 1988
60
BIBLIOGRAPHY
[37] D. A. Reynolds, “An Overview of Automatic Speaker Recognition Technology”, in Proceedings of International Conference of Acoustics, Speech, and Signal Processing, vol. 4, pp. 4072-4075, 2002 [38] R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm”, in SIAM Review, 26(2):195-239, 1984 [39] J. Samuelsson, “Towards Asymptotic Vector Quantization”, Ph.D Thesis, School of Electrical and Computer Engineering, Chalmers University of Technology, 2001 [40] N. Czink, R. Tian, S. Wyne, F. Tufvesson, J. Nuutinen, J. Ylitalo, E. Bonek, A. Molisch, “Tracking Time-Variant Cluster Parameters in MIMO Channel Measurements”, submitted to ChinaCom, 2007
