發展軟體無線電技術---智慧型天線系統/技術之研發---子計劃Ⅴ:低複雜度多使用者解調器之設計與應用(I)

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行政院國家科學委員會專題研究計畫成果報告

子計劃五:低複雜度多使用者解調器之設計與應用(I)

計畫類別：整合型計畫

計畫編號： NSC91-2219-E-011-002-

執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日執行單位：國立臺灣科技大學電子工程系

計畫主持人：方文賢

報告類型：完整報告

處理方式：本計畫可公開查詢

中華民國 92 年 10 月 31 日

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摘要

在無線通信系統中，通道估測具有相當重要性並引起廣大的研究興趣。若能準確估測信號到達方向及信號延遲時間等通道參數，將可提供有效的通道估測並顯著改善整體無線通信系統的效能。許多研究探討如何估測信號到達方向及信號延遲時間等通道參數：例如，以最大可能性(maximum likelihood)為基礎的演算法。這個方法雖然可提高準確的估測，然而在其多維度且非線性的最佳化過程中，複雜度太高而不切實際。另一種方法是基於子空間的方法，此方法在估測參數的過程中，有兩個地方運算量很大：高維度的特徵分解以及在 DOA-delay 平面上進行二維的掃描，因此在即時實現上有困難。

本計畫發展一種基於 MUSIC(MUltiple SIgnal Classification)的樹狀結構演算法則，相對於傳統之用來估測方位角之空間域演算法(S-MUSIC)，我們增加了一個時間域演算法(T-MUSIC)，可同時估測 CDMA 系統中無線通道信號到達方向 (DOAs)以及信號延遲時間(TOAs)。此演算法利用一個空間域上(S-MUSIC)及兩個時間域上(T-MUSIC)的演算法輪流估測信號到達方向及信號到達時間，在過程中，信號將被分群、隔離、進而估測及配對，並結合限制性時間濾波及限制性空間波束形成有效將信號分群並抑制傳遞誤差，因而提高整體效能。所發展之演算法能自動對所估測的信號到達方向及信號延遲時間進行配對，而所使用的天線數目亦遠小於通道之路徑數目，並可解析出在角度或延遲時間中非常靠近之信號。

模擬結果驗證此低複雜度演算法的優異效能。

關鍵詞︰信號到達方向、信號延遲時間、MUSIC、DOA-Delay 估測、分碼多工、

T-MUSIC、S-MUSIC、通道參數估測

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ABSTRACT

In wireless communication systems, channel estimation is of importance and has received a considerable amount of attention. It has been observed that in a multi-ray, multi-channel system an improved channel estimate can be obtained by jointly exploring the channel parameter such as the ray DOAs and ray propagation delays, and thus can significantly improve the system performance. Various algorithms for jointly estimating these two parameters in CDMA system have been proposed. For example, a maximum likelihood-based algorithm was advocated recently, which requires a computationally demanding higher-dimensional nonlinear optimization process. Another approach is the subspace-based algorithm which estimates the parameters via carrying out high dimensional eigendecompositions of the covariance matrices and, in addition, requires a 2-D search on the DOA- delay plane. Despite its effectiveness, the computations required, however, make this algorithm unfavorable for real time implementations.

In the project, we propose a tree-structured MUltiple SIgnal Classification (MUSIC) algorithm to jointly estimate the directions of arrival(DOAs) and propagation delays in CDMA system. In constrast to the traditional MUSIC algorithm, which is referred to as the Spatial-MUSIC (S-MUSIC), for the DOA estimation, we also develop the Temporal MUSIC (T-MUSIC) algorithm for the time delay estimation in a wireless channel. The proposed algorithm makes use of one spatial (S)-MUSIC and two temporal (T)-MUSIC algorithms alternatively to estimate the DOAs and the group delays, respectively. As such, the incoming rays are grouped, isolated, and then estimated, and the pairing of the estimated DOAs and delays are automatically determined. In addition, a constrained temporal filtering process and a constrained spatial beamforming process are conducted. Such a constrained filtering approach can effectively partition the incoming rays and sup- press the propagation error in the tree-structured scheme, thus in turn enhancing the overall performance. The pairing of the estimated DOAs and delays is also automatically determined. The number of antennas required by the algorithm can be much less than that of the incoming rays. The proposed algorithm can either provide superior performance or call for substantially lower computational complexity. Simulations are conducted to verify the relevant results compared with previous works.

Keywords:Direction Of Arrival, Delay time, MUSIC, DOA-Delay estimation, CDMA, T-MUSIC, S-MUSIC, Channel parameter estimation.

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Chapter 1 INTRODUCTION

In early days, signal processing for wireless communications was confined to the processing of the temporal samples of the signals received by a single antenna. The performance of the systems is therefore limited in capacity, coverage, and interfer- ence cancelation. Due to the mobility of the users and the territorial features in the environment nowadays, radio signals not only experience attenuation fading, but also propagate in a multipath channel. To overcome this severe propagation channel, we may use an antenna array, treating the received signals in both the spatial and temporal domains, as a more flexible space-time signal processing to enhance the system performance. Various techniques exploiting such space diversity along with the time diversity have been developed to increase the system capacity, mitigate the co-channel interferences resulting from the frequency reused in the cellular system, simplify the hand-off scheme, and etc.

Radio signals propagating through a multipath channel arrive at the receivers with different DOAs and delays, as illustrated in Fig. 1.1. The multipath channel can therefore be described by a function of these spatio-temporal parameters. In wireless communications, joint estimation of the DOAs and delays plays an im- portant role in providing prior information for the equalizers to use the channel diversities to estimate data sequences. For example, the Space-Time Minimum Mean Square Error (ST-MMSE) receiver [1] uses the estimated DOAs and the relative delays of the channel to estimate the information sequence. It has also been shown that if the DOAs and delays of the multipath channel are precisely estimated, the system performance can be significantly improved [2, 3]. In addition, the DOA-Delay estimation is a classical problem encountered in radar, sonar, and geophysics. It also finds applications in source localization, accident reporting, cargo tracking, and intelligent transportation [4]. This project thus aims at devel- oping high resolution algorithms for estimating the path DOAs and propagation delays simultaneously.

Several algorithms for joint estimation of the DOAs and multipath propagation delays were addressed recently. For example, Swindlehurst [5] estimated the delays of a multipath channel by the Iterative Quadratic Maximum Likelihood (IQML) algorithms, the spatial signatures (or DOAs) were then solved as a least

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Base Station

Space-Time Proceoing

Subscriber

Figure 1.1: A typical multipath channel

squares problem. Clark et al. [6] proposed a 2-D IQML algorithm which can be extended to jointly estimate the channel parameters. Both [5] and [6] are maximum likelihood (ML)-based algorithms which take advantage of the Vandermonde structure of the estimated channel pulse response in the frequency domain. However, if two or more rays are with close time delays, the data covariance matrix turns ill-conditioned and both [5] and [6] may not work properly in such a situation.

Also, the number of antennas used by both algorithms must be more than that of the incoming rays. In addition, the IQML-based algorithms have initialization problems.

Other alternatives are the subspace-based algorithms. Ogawa et al. [7]

presented a channel sounding method using unmodulated carriers. The parameter pairs are then extracted by invoking a 2-D windowed MUSIC algorithm.

Vanderveen et al. proposed the JADE-MUSIC [8] algorithm which first stacks an observed data matrix into a high dimensional vector, it then use the fact that the covariance matrix for the stacked vector shares the same column space with the space-time steering vectors. After performing a high dimensional eigen- decomposition on the covariance matrix, the channel parameters can be estimated by a 2-D search on the DOA-delay plane. The computations incurred, however, makes the JADE-MUSIC unfavorable for real time implementations.

In the project, we would like to develop a joint DOA-Delay estimation algorithm which possess the following advantages:

• Robustness.

• High accuracy, even when only a small number of data bursts is available.

• Low computational complexity.

• Capability of resolving clustering rays scenarios.

To overcome the deficiency of the number of data bursts, one possible solu- tion is to use the 1-D parameter estimation algorithms to estimate the 2-D parameters. The algorithms for the DOA estimation have been well developed in the past

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few years. By properly preprocessing the temporal samples, similar algorithms for the delay estimation can also be extended from those for the DOA estimation [5].

All these delay estimation algorithms use the samples of the delayed pulse shaping function as the temporal vector. In this project, we use the T-MUSIC algorithm to estimate the path delays. Although the T-MUSIC algorithm can precisely estimate the path delays, it also calls for substantially computational overhead to search over the time axis. To alleviate this, the T-ESPRIT algorithm is also developed, which, as the traditional ESPRIT algorithm for the DOA estimation, can estimate the path delays in closed form.

Based on the above discussion, in this project we present a low complexity, yet high accuracy, MUSIC-based algorithm [9] [10] — the TST-MUSIC algorithm

— which combines the temporal filtering and the spatial beamforming techniques in conjunction with three 1-D MUSICs, i.e. one S-MUSIC and two T-MUSIC algorithms to jointly estimate the DOAs-delays of interest from the samples received by an antenna array. The basic ideas behind the proposed approach are to group then isolate the signal of each incoming rays using the space-time characteristics of the multipath wireless channel. To achieve this, the T-MUSIC and the S-MUSIC algorithms are employed to estimate the group delays (or delays) and the DOAs of the incoming rays, which are required by the succeding temporal filtering and spatial beamforming processes, respectively. Thereafter, the other T-MUSIC algorithm is employed to estimate the propagation delays. It is noteworthy that to make the filtering process to be more robust against the propagation error, a constrained approach for the design of the spatial/temporal filters are also addressed.

The proposed approach possesses some distinctive features. First, compared to the ML-based algorithms [5, 6], the tree-structured TST-MUSIC algorithm not only inherently resolves incoming rays either with very close DOAs or with very close propagation delays, but also renders automatic pairing of the estimated DOAs and delays. In addition, the number of antennas required by the TST-MUSIC can be less than that of the incoming rays due to the employment of the temporal filtering process. Second, in contrast to the JADE-MUSIC algorithm [8], the TST- MUSIC algorithm needs only 1-D searches and the associated eigen-decomposition for smaller-sized covariance matrices, thus calling for substantially lower computational complexity.

In Chapter 2, we first introduce the phenomena encountered in a wireless propagation channel. We then describe a mathematical model for the multipath channel, which assumes the propagation paths to be specular rather than dispersed.

By incorporating the antenna array with the temporal sampling, the signal model for the space-time signal processing is addressed.

Chapter 3 reviews the TST method and then proposes the Constrained TST method for the DOA-Delay joint estimation. To simplify the description, a three- ray scenario is used to illustrate the overall procedures. In order to reduce the computational complexity, the ESPRIT version called Constrained TST-ESPRIT is also addressed.

Chapter 4 provides concluding remarks which summarize the whole project and provide some future perspectives of this work.

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Chapter 2 A MATHEMATICAL MODEL FOR THE WIRELESS

COMMUNICATION CHANNELS

As a radio signal propagates through a wireless channel, it will physically be af- fected by the channel in several ways. In this chapter, we describe a mathematical model for the wireless propagating signals received by the base station(uplink model) with the consideration of these physical effects.

2.1 Wireless Communication Channels

A signal propagating through the wireless channel is usually reflected, refracted, and scattered by the objects in the environment. As such, a transmitted signal will arrive at the destination along several paths which are referred to as multipaths.

Multipath and the mobility of the mobile cause the signals to spread in space, time, and frequency, when they are received by the receivers. In addition, the path loss and the fading effects existing in the wireless channel attenuate the power level of the transmitted signals. To follow, we address these effects in more details.

The multipath phenomenon is caused by objects (scatters) in the environment such as buildings, tree, and terrain features, which reflect, diffract, and scatter a radio signal. As a result, the received signals exhibit delay spreads and angle spreads. Furthermore, they also exhibit the Doppler spread if the source is moving. In typical outdoor cellular systems, the delay spread is on the order of 0-10 microseconds, and the angle spread ranges from 2 to 60 degrees and the Doppler spread from 5 to 200 Hz [1, 11].

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In a multipath propagation environment, several delayed and scaled version of the transmitted signals, from several angles, arrive at the receiver. These physical spreadings make the wireless channel a selective one. Corresponding to the delay spread, the wireless channel is thus frequency selective, which implies that fading is frequency dependent. Corresponding to the angle spread, the wireless channel can be regarded as a space-selective channel, which means that the amplitude of the signal depends on the spatial location of the antenna. Corresponding to the Doppler spread, the channel is time selective, which means that the amplitude of the received signal varies with time. However, such selective properties of the wireless channels can provide the channel diversities for the receivers, which can be employed to improve the receiver performance [1].

In addition, in a wireless communication channel, the objects, which cause the multipath, can be classified as local to base scatters, local to mobile scatters, and remote scatters. An object is called a local scatter if its size is greater than the wavelength of the propagating signals and its location higher than the receiving antennas. The local to base scatters, which are the objects in the vicinity of the base station (for example, buildings, trees, and etc.), cause significant angle spreads of the incoming signals. However, due to the short traveling distances experienced by the scattered signals, the delay spread caused by the local to base scatters is small and no additional Doppler spread is induced. Similarly, the local to mobile scatters are objects around the mobile. To the base station, the local to mobile scatters cause significant angle spreads and Doppler spreads if the mobile is moving. However, as mentioned above, the local scatters induce small delay spreads. The terrain features and buildings which are far away from both of the mobile and the base station are categorized as remote scatters. The remote scatters cause large delay spreads and angle spreads but no extra Doppler spreads. Fig.

2.1 illustrates the large cell multipath propagation and various scatters.

2.2 Discrete Space-Time Signal Model

In a DS-SS CDMA system of K users with spreading gain N , the transmitted signal can be represented as the convolution of the data bits, the spreading codes and the pulse-shaping function. If we assume that the number of multipaths associated with each user is at most L, P antennas are used, and the length of the pulse-shaping function equals to QTc, where Tc is the chip period, the baseband signals received at the antenna array during the m^th symbol, after sampling, can be expressed as

X_m =

K

X

k=1 L

X

l=1

a(θkl)β_kl^mg(τkl)^TS_kD^m_k + N (2.1) where Xm and N are, respectively, the received signal of dimension P × N and the additive Gaussian noise of dimension P × N , a(θkl) is the P × 1 steering vector of the l^th ray of the k^th user, β_kl^m denotes the fading amplitude of the m^th symbol, g(τkl) is the pulse-shaping function of dimension Q × 1, and the superscript ^T denotes the matrix transposition. Also, the fading rays are assumed to be mutually uncorrelated and their fading amplitudes are zero-mean complex Gaussian distributed The Q×(N +Q−1) code matrix Skis a Toeplitz matrix with

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Figure 2.1: Large cell multipath propagation and scattering

[s_k,1, 0_1×(Q−1)]^T as its first column, and [s_k,1, s_k,2, ..., s_k,N, 0_1×(Q−1)] as its first row, where sk,1, sk,2, ... , sk,N is the normalized spreading code of the k^thuser. The (N + Q− 1) × N data matrix D^m_k, constituted by the present and previous transmitted data bits, is also a Toeplitz matrix with [dk,m, 0_{1×(N −1)}, d_k,m−1, 0_1×(Q−2)]^T as its first column, and [dk,m,01×(N −1)] as its first row, where dk,m is the m^th data bit of the k^th user. The data bits are also assumed to be mutually uncorrelated.

2.3 Data Analysis

In this subsection, we investigate the structure of the temporal and spatial covariance matrices. To achieve this, first note that based on the structure of D^m_k in (2.1), we can divide it into two components as

D^m_k =

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where IN is an N × N identity matrix, and 01 and 02 are (Q − 1) × N and (Q − 1) × (N − Q + 1) zero matrices, respectively. ¯D is a Toeplitz matrix with [1, 0_{1×(N +Q−2)}]^T as its first column and [1, 0_{1×(N −1)}] as its first row, and ˆD is a

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Toeplitz matrix with [01×(N ),1, 01×(Q−2)]^T as its first column and 01×N as its first row. Note that the first and the second components of (2.2) are caused by the present and previous bits, respectively.

Substituting (2.2) into (2.1) leads to Xm =

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k=1 L

X

l=1

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where ¯g(τkl)^T=g(τkl)^TS_kD¯ and ˆg(τkl)^T=g(τkl)^TS_kDˆ are the temporal vectors caused by the present and previous bits, respectively.

As such, the temporal covariance matrix R^t becomes R^t = E[X^H_mX_m]

= E[

K

X

k=1 K

X

i=1 L

X

l=1 L

X

j=1

β_kl^m∗β_ij^m{dk,md_i,m¯g(τkl)a(θkl)^Ha(θij)¯g(τij)^T +

d_k,md_i,m−1¯g(τkl)a(θkl)^Ha(θij)ˆg(τij)^T + dk,m−1d_i,mˆg(τkl)a(θkl)^Ha(θij)¯g(τij)^T + d_k,m−1d_i,m−1ˆg(τkl)a(θkl)^Ha(θij)ˆg(τij)^T}] (2.3) where E[·] represents the statistical average operation, and (·)^H and (·)^∗ denote the Hermitian operation and the complex conjugate operation, respectively. Note that since both of the data bits between users and the present and previous bits are mutually uncorrelated, the first and the fourth terms of (2.3) equal to zero when k 6= i, and the second and the third terms equal to zeros. Also, E[β_kl^m∗β_ij^m] = σkl if k = i and l = j, Using these facts, (2.3) can be readily shown to be

R^t =

K

X

k=1 L

X

l=1

σ_kl{¯g(τkl)¯g(τij)^T + ˆg(τkl)ˆg(τij)^T}

= GE ˜˜ G^H (2.4)

where ˜G = [˜g₁,g˜₂,· · · , ˜g_2KL], in which ˜g(τkl) = ¯g(τkl) for l = 1 to L and = ˆ

g(τk(l−L)) for l = 1 + L to 2L, k = 1, · · · , K, and E = diag[σ1, σ₂,· · · , σ2KL]. Based on the R^t determined in (2.4), we can note that R^tand ˜Gpossess the same column space.

Similarly, the spatial covariance matrix R^s, after some manipulations, can be shown to be

R^s= E[XmX^H_m] = ACA^H (2.5)

where A = [a(θ1), a(θ2), · · · , a(θKL)] and C = diag[σ1c₁, σ₂c₂,· · · , σKLc_KL] with c_kl = ¯g(τkl)^T¯g(τkl) + ˆg(τkl)^Tˆg(τkl). We can note again that R^s and A possess the same column space.

In this chapter, the wireless propagation channel is described and the data model for the space-time signal processing in such a channel is also developed. The model uses appropriate parameters to represent the angle spread, the delay spread, and the fading effect for a specular multipath channel. In subsequent chapters, the algorithm addressed will be based on this model.

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Chapter 3 PROPOSED CONSTRAINED TST ALGORITHM

In this chapter, we review the tree-structured algorithm (TST-MUSIC) and propose the Constrained tree-structured algorithm (Constrained TST-MUSIC). In order to alleviate the computational complexity, the constrained TST-ESPRIT algorithm is also addressed. At first, we introduce the DOA or delay estimation by MUSIC and ESPRIT method.

3.1 DOA or Delay Estimation by MUSIC and ESPRIT

T-MUSIC and S-MUSIC

Based on the temporal covariance matrix R^t and the spatial covariance matrix R^s determined above, we can then employ the T-MUSIC and S-MUSIC algorithms to estimate the ray delays and the ray DOAs, respectively. Note that the temporal samples in the T-MUSIC can be regarded as a kind of temporal antenna array. Compared to the commonly used antenna array for spatial sampling, adding extra antennas in the temporal array is cost free, and the known pulse shaping function provides a perfect temporal array manifold in which no calibration is required.

More specifically, if we carry out the eigen-decomposition of the covariance matrices, R^t and R^s can be, respectively, expressed as

R^t = V^t_sΛ^t_sV^t_s^H + V^t_nΛ^t_nV_n^t^H (3.1) and

R^s= V_s^sΛ^s_sV_s^s^H + V^s_nΛ^s_nV_n^s^H (3.2)

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The column(signal) subspace of R^t and R^s is, respectively, equal to the column space of V_s^t and V^s_s, which are constituted by the eighevectors corresponding to the KL largest eigenvalues R^t and R^s. Menwhile, the null(noise) subspace of R^t and R^s is, respectively, equal to the column space of V^t_n and V^s_n, which are constituted by the remaining N − KL and P − KL eighevectors of R^t and R^s. Both (Λ^s_s, Λ^s_n) and (Λ^t_s, Λ^t_n) are diagonal matrix pairs with the associated eigenvalues as their diagonal elements. Also, V_s^t are orthogonal complements to V_s^sand V^t_n and V_n^s, respectively.

Using the orthogonality property between the signal and the noise subspaces and (2.4), the T-MUSIC estimates the KL ray delays by searching τ over the range of interest using the pseudospectrum of the T-MUSIC given by

P_music^t (τ ) = 1

˜g(τ )^H(I − V^t_sV^t_s^H)˜g(τ ) (3.3)

Similarly, the S-MUSIC algorithm estimates the DOAs by searching and θ over the range of interest using the pseudospectrum of the S-MUSIC given by

P_music^s (θ) = 1

a(θ)^H(I − V^s_sV^s_s^H)a(θ) (3.4)

T-ESPRIT and S-ESPRIT

For the computationally less demanding reason, we can adopt the T-ESPRIT and S-ESPRIT algorithm. The closed-form algorithms are described below via the flow chart in Fig 3.1, Fig 3.2 and Fig 3.3.

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3.2 TST MUSIC Algorithm

Note that the MUSIC algorithms can not distinguish the delays or DOA if they are very close. In lieu of this, we can make use of the space-time characteristics of the multi-ray channel by arranging the MUSIC algorithms employed in a tree- structured manner. We can then utilize the estimated estimated delays or DOAs to help us separate the incoming rays into some groups so that the DOAs and delays can be more precisely determined in the next stage. To achieve this, this section addresses a temporal filtering process and a spatial beamforming process, which are a refinement of those addressed in [12], to separate the incoming rays into several groups according to the estimated delays and DOAs in the previous stages.

First, we consider a temporal filtering process, which can separate the rays into several groups based on the estimated delays. To explain this, we assume that the received data only contains one user with J rays with delays τl, l = 1 · · · J . For simplicity, if we neglect the effect caused by the previous bit, (2.1) can be written as

X_m =

J

X

l=1

a(θl)β_l^mg(τ˜ l)^Td_m

If we want to remove the ray with delay θi, we can consider the following filtering matrices {U^t_i} given by

U^t_i = I − ˜g⁰(τi)˜g⁰(τ )^T_i , l= 1 · · · J (3.5) where ˜g⁰(τi) denotes the normalization of ˜g(τi). Postmultiplying Xm by U^t_i then results in

X⁰_m = Xm· U^t_i

=

J

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where we have used (2.1) and the fact that ˜g(τi)^TU^t_i = 0. As such, the component corresponding to the ray with delay τi is removed. Note that the components of the temporal vectors of the other rays now become ˜g(τl)^T − ˜g(τl)^T˜g⁰(τi)˜g⁰(τi)^T rather than ˜g(τl)^T, as all rays are not orthogonal to each other. Therefore, if we intend to filter out more than one rays from the data, we need to use the Gram-Schmidt orthogonalization procedure to obtain a new set of temporal vectors given by ˜g⁰(τl) as:

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g⁰(τl) = g(τ˜ l) −^P^l−1_j=1c_ljg˜⁰(τj)

k˜g(τl) −^P^l−1_j=1c_ljg˜⁰(τj)k (3.7) where k · k denotes the Frobenius norm, and clj = ˜g(τl)^T · ˜g⁰(τj). It then follows that if we want to filter out all rays except the j^th ray in the received data, the data after filtering can be readily shown to be

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where U^s_l is constructed in the same way as (3.5) and (3.7), but with ˜g⁰(τl) being replaced by a⁰(θl) , the normalization of a(θl) .

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According to the above discussion, the proposed tree-structured MUSIC is to use two T-MUSIC algorithms and one S-MUSIC algorithm aletrnatively in conjunction with a temporal filtering process and a spatial beamforming process to enhance the estimation accuracy of the MUSIC algorithms. The overall procesures can be summarized as follows[14]:

Step 1: Rough Delay Estimation:

From the received data, we can estimate the temporal covariance matrix by Rˆ^t = 1

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X^T_m· X^∗_m (3.10)

We can then apply the T-MUSIC to ˆR^tto estimate the group delays {ˆτ₁, . . . ,τˆ_q}, where q is the number of the total group delays of all users.

Step 2: Temporal Filtering:

Exploit (3.5) to determine the temporal filtering matrices {U^t_i} for i = 1, 2, . . . , q, and use (3.8) to obtain X^j_m for j = 1, 2, . . . , q, which are the data after the temporal filtering.

Step 3: DOA Estimation:

From each X^j_m, we can estimate the spatial covariance matrices by Rˆ^s_j = 1

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M

X

m=1

X^j_m· X^jH_m (3.11)

for j = 1, 2, . . . , q. We can then apply the S-MUSIC to all ˆR^s_j to estimate DOAs {ˆθ_1,1, . . . , ˆθ_1,p₁, . . . , ˆθ_q,1, . . . , ˆθ_q,p_q}, where pj denotes the number of the estimated DOAs in the j^th group.

Step 4: Spatial filtering:

Employ (3.9) but with Xm being replaced by X^j_m to obtain X^0k_m,j for k = 1, 2, . . . , q, j = 1, 2, . . . , pk, which are the data only containing the ray with DOA θˆ_k,j.

Step 5: Delay Estimation:

From all X^0k_m,j, repeat (3.26) but now based on X^0k_m,j to find the temporal covariance matrices. Then apply the T-MUSIC algorithm again but with different temporal array manifolds, (^Q^q_i=1;i6=jU^t_i)^Tg(τ ), to estimate the delays˜ {ˆτ_1,1⁰ , . . . ,τˆ_1,p⁰

1, . . . ,τˆ_q,1⁰ , . . . ,τˆ_q,p⁰ _q}. Finally, the pair of θi,jand ˆτ_i,j⁰ for i = 1, 2, . . . , q, j = 1, 2, . . . , pq are the resulting DOA-delay estimates.

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Figure 3.4: The evolution of the signal contents for the estimation of the parameters of ray one.

The rationale of the TST-MUSIC is to incorporate three 1-D MUSIC (S- MUSIC and T-MUSIC) algorithms with beamforming techniques and the filtering techniques to group, isolate, and then estimate and pair the 2-D parameters of a fading channel. To simplify the algorithm description, we first assume that there are only three rays present in the system.

As shown in Fig. 3.4(a), three rays are characterized by their temporal- spatial coordinates on the DOA-delay plane. Note that ray 1 and ray 2 possess close time delays (τ1 ≈ τ2), but diverse DOAs (θ1 < θ₂); while ray 1 and ray 3 are close in the DOAs (θ1 ≈ θ3), but with far apart delays (τ1 < τ₃). The tree structure of the TST-MUSIC algorithm for this scenario is illustrated in Fig. 3.5.

In addition, corresponding to Fig. 3.4, Fig. 3.5 shows the evolution of the data contents as the parameters of ray 1 are estimated in the tree structure.

The TST-MUSIC treats those temporally-close rays, which flatten the T- MUSIC spectrum, as a group. Therefore, ray 1 and ray 2 in Fig. 3.4(a) are regarded as one group, while ray 3 is considered another group. By applying the T-MUSIC to the rows of Xt, the resulting group delays are estimated, denoted by ˆt₁ and ˆt₂. Based on the group delay estimates, ˆt₁ and ˆt₂, we define the temporal