以偵測循環字首類型為重點之LTE初始下行同步

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(1)國. 立交. 通. 大. 學. 電子工程學系電子研究所碩士班碩. 士. 論. 文. 以偵測循環字首類型為重點之 LTE 初始下行同步 Initial Downlink Synchronization for LTE Systems with Emphasis on Detection of Cyclic Prefix Type. 研究生：謝男鑫指導教授：林大衛教授. 中華民國一○二年八月.

(2) 以偵測循環字首類型為重點之 LTE 初始下行同步 Initial Downlink Synchronization for LTE Systems with Emphasis on Detection of Cyclic Prefix Type. 研究生: 謝男鑫. Student: Nan-Shin Hsieh. 指導教授: 林大衛. Advisor: Dr. David W. Lin. 國立交通大學電子工程學系. 電子研究所碩士班. 碩士論文. A Thesis Submitted to Department of Electronics Engineering & Institute of Electronics College of Electrical and Computer Engineering National Chiao Tung University in Partial Fulfillment of the Requirements for the Degree of Master of Science in Electronics Engineering August 2013 Hsinchu, Taiwan, Republic of China. 中華民國一○二年八月.

(3) 以偵測循環字首類型為重點之 LTE 初始下行同步研究生：謝男鑫. 指導教授：林大衛博士. 國立交通大學電子工程學系電子研究所碩士班. 摘要本篇論文介紹LTE-A系統裡關於初始下行同步的問題，包括問題陳述、演算法推導、以及程式模擬。在初始下行同步中，我們發展了一套同步估測演算法，包括了符元時間偏移、載波偏移和訊號傳輸所使用的循環字首類型。首先，我們建立了LTE-A系統下的傳送端和接收端的訊號模型，然後基於最大可能性估測 (maximum likelihood)法推導其解。此解不僅於可加性白色高斯雜訊(additive white Gaussian noise, AWGN)、單一路徑瑞利衰減(single-path Rayleigh fading)通道下在具有普遍正確性，且也適用於多路徑衰減(multipath fading)通道。為了大幅減低計算複雜度，我們在處理載波偏移過程中做一近似，雖然在多路徑衰減通道下此解為次優解，但因為此近似在可加性白色高斯雜訊和單一路徑瑞利衰減通道下不影響估測程序，所以在這些通道下亦為最佳解。在模擬部分，我們設計出一個合理的LTE-A系統接收訊號模型，在可加性白色高斯雜訊通道下得到的數據驗證此演算法，然後在多路徑衰減通道傳輸中使用數種不同多路徑通道、相對移動車速以及雜訊比來測試此演算法對這些環境下所產生的結果。 i.

(4) Initial Downlink Synchronization for LTE Systems with Emphasis on Detection of Cyclic Prefix Type Student: Nan-Shin Hsieh. Advisor: Dr. David W. Lin. Department of Electronics Engineering & Institute of Electronics National Chiao Tung University. Abstract This thesis introduces the topic about initial downlink synchronization, including problem formulation, algorithm derivation and program simulation, in LTE-A system. We develop an algorithm about the joint estimation of symbol timing offset (STO), carrier frequency offset (CFO), and detection of the cyclic prefix (CP) type used in signal transmission. We first establish the transmitted and received signal models of LTE-A system and then derive the solution based on the maximum likelihood (ML) criterion. It is a general solution over not only additive white Gaussian noise (AWGN) and single-path Rayleigh fading channels but also multipath fading channels. In estimating CFO, we consider an approximation so as to reduce computational complexity largely. Though it is the suboptimal solution under multipath fading channels, it is an optimal solution under AWGN and single-path Rayleigh fading channels because the approximate term has no effect on estimation process. In simulation, we establish a reasonable received signal model in LTE-A system, and. then simulate our proposed estimator under AWGN channel to verify its. performance. Moreover, we test it under single-path Rayleigh fading and different multipath fading channels, including Standford University Interim (SUI) and Pedestrian B (PB), at different mobility and different signal to noise ratio (SNR). ii.

(5) 誌謝這篇論文能夠順利完成，首先要感謝的是我的指導教授林大衛老師，在這兩年的研究生涯裡，老師總是很細心、用心的指導我，使得我學習到不少研究的精神與方法。此外，感謝王柏森學長、柯俊言學長、李政憲學長、梁晉源學長給予我在研究過程上的指導與建議。還有感謝葆崧、夏銘、信宏，很幸運可以跟你們在同一實驗室，能和你們共同討論、分享生活上的點點滴滴，讓這兩年的研究生涯充滿許多的歡樂與寶貴的回憶。謝謝通訊電子與訊號處理實驗室所有的成員，志堯、鈞凱、哲瑋、長廷、琬瑜、暐翔、子傑、明孝、中威、家駿以及學弟妹們。謝謝碩二的室友們，家麟、夏銘、執中，每晚回到寢室總是可以開心的閒聊，忘記一天下來的疲憊。最後，我要感謝我的家人，感謝他們一直都在背後支持我，在求學過程中總是不斷的鼓勵我，讓我有幹勁的繼續向前邁進。在此，將此篇論文獻給所有陪伴我走過這一段歲月、幫助過我的貴人們。. 謝男鑫民國一百零二年於新竹 iii.

(6) Contents 1 Introduction. 1. 2 Overview of the LTE-A Downlink Standard. 3. 2.1. 2.2. OFDM and OFDMA Basics . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.1.1. Discrete-time baseband OFDM system model . . . . . . . . . . . . .. 3. 2.1.2. Cyclic prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.1.3. OFDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. Overview of LTE and LTE-A Downlink Specifications . . . . . . . . . . . . .. 6. 2.2.1. Frame structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2.2. Slot structure and physical resources . . . . . . . . . . . . . . . . . .. 7. 2.2.3. Random access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.2.4. Cell search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 2.2.5. Zadoff-Chu (ZC) sequences . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.2.6. Primary synchronization signal (PSS) . . . . . . . . . . . . . . . . . .. 12. 2.2.7. Secondary synchronization signal (SSS) . . . . . . . . . . . . . . . . .. 12. 2.2.8. Broadcast channel (BCH) . . . . . . . . . . . . . . . . . . . . . . . .. 15. 3 Initial Downlink Synchronization 3.1. 17. The Initial Downlink Synchronization Problem . . . . . . . . . . . . . . . . .. 17. 3.1.1. Transmission system model . . . . . . . . . . . . . . . . . . . . . . .. 17. 3.1.2. The ML approach to synchronization . . . . . . . . . . . . . . . . . .. 21. iv.

(7) 3.2 Solution of the ML Estimation Problem. . . . . . . . . . . . . . . . . . . . .. 21. 3.2.1. Derivation of the likelihood function for extended CP . . . . . . . . .. 21. 3.2.2. Derivation of the likelihood function for normal CP . . . . . . . . . .. 31. 3.2.3. Joint estimation of STO, CFO, and CP type . . . . . . . . . . . . . .. 32. 4 Simulation Results and Analysis. 36. 4.1. Simulation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 4.2. Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. 4.3. CP Type Estimation Performance with A Small Observation Window . . . .. 40. 4.3.1. AWGN and Single-Path Rayleigh Fading Channels . . . . . . . . . .. 40. 4.3.2. Multipath Fading Channels . . . . . . . . . . . . . . . . . . . . . . .. 43. CP Type Estimation Performance Under Different Observation Window Sizes. 43. 4.4. 5 Conclusion and Future Work. 56. 5.1. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56. 5.2. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. Bibliography. 58. v.

(8) List of Figures 2.1. Baseband OFDM transmitter and receiver. . . . . . . . . . . . . . . . . . . .. 4. 2.2. OFDM symbols with CP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.3. Types of resource allocation in OFDMA system (modified from [3, Fig. 4.29]).. 5. 2.4. Frame structure type 1 [1, Fig.4.1-1]. . . . . . . . . . . . . . . . . . . . . . .. 7. 2.5. Frame structure type 2 [1, Fig. 4.2-1]. . . . . . . . . . . . . . . . . . . . . . .. 7. 2.6. Slot structure for normal and extended CP [2, Fig. 8.3]. . . . . . . . . . . . .. 8. 2.7. Detailed signal structure for normal CP [1, Figs. 5.2.1-1 and 6.2.2-1], [2, Fig. 8.2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.8. Random access procedure [2, Figure 10.1]. . . . . . . . . . . . . . . . . . . .. 10. 2.9. downlink frame structure for the type of normal CP. . . . . . . . . . . . . . .. 11. 2.10 PSS mapping in the frequency domain (modified from [2, Figure 9.4]). . . . .. 12. 2.11 SSS mapping in the frequency domain (modified from [2, Figure 9.5]). . . . .. 15. 2.12 BCH transmission chain processing [2, Figure 9.6]. . . . . . . . . . . . . . . .. 15. 2.13 PBCH mapping (modified from [2, Figure 9.7]). . . . . . . . . . . . . . . . .. 16. 3.1. Transmission system structure. . . . . . . . . . . . . . . . . . . . . . . . . .. 18. 3.2. Slot structure in the time domain. . . . . . . . . . . . . . . . . . . . . . . . .. 18. 3.3. Structure of the autocorrelation matrix Cr . . . . . . . . . . . . . . . . . . . .. 24. 3.4. Moving average. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. 3.5. Autocorrelation matrix example. . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 3.6. Three-region data correlation. . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. vi.

(9) 3.7 Relative positioning of observation window with respect to STO, as well as the correlation structure in the received signal. . . . . . . . . . . . . . . . . .. 27. 4.1. Tap location adjustment by rounding to integer sample spacing. . . . . . . .. 39. 4.2. Error performance of CP type detection in AWGN channel with window size 2Nf f t + 1N512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3. 41. Error performance of CP type detection in single-path Rayleigh fading channel at different speeds with window size 2Nf f t + 1N512 . . . . . . . . . . . . . . .. 41. 4.4. Window sliding structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. 4.5. Pairs of data correlated with window size 2Nf f t + 1N512 . . . . . . . . . . . .. 42. 4.6. Error performance of CP type detection in SUI1 channel at different speeds with window size 2Nf f t + 1N512 . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.7. Error performance of CP type detection in SUI2 channel at different speeds with window size 2Nf f t + 1N512 . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.8. 44. Error performance of CP type detection in SUI3 channel at different speeds with window size 2Nf f t + 1N512 . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.9. 43. 44. Error performance of CP type detection in SUI4 channel at different speeds with window size 2Nf f t + 1N512 . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 4.10 Error performance of CP type detection in Pedestrian-B channel at different speeds with window size 2Nf f t + 1N512 . . . . . . . . . . . . . . . . . . . . . .. 45. 4.11 Pairs of data correlated with window size 6Nf f t + 1N512 . . . . . . . . . . . .. 46. 4.12 Error performance of CP type detection in single-path channel at speed 3 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 47. 4.13 Error performance of CP type detection in single-path channel at speed 120 km with several different window size. . . . . . . . . . . . . . . . . . . . . . .. 47. 4.14 Error performance of CP type detection in single-path channel at speed 360 km with several different window size. . . . . . . . . . . . . . . . . . . . . . .. 48. 4.15 Error performance of CP type detection in SUI1 channel at speed 3 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . . . . .. vii. 48.

(10) 4.16 Error performance of CP type detection in SUI1 channel at speed 120 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 49. 4.17 Error performance of CP type detection in SUI1 channel at speed 360 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 49. 4.18 Error performance of CP type detection in SUI2 channel at speed 3 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. 4.19 Error performance of CP type detection in SUI2 channel at speed 120 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 50. 4.20 Error performance of CP type detection in SUI2 channel at speed 360 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 51. 4.21 Error performance of CP type detection in SUI3 channel at speed 3 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51. 4.22 Error performance of CP type detection in SUI3 channel at speed 120 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 52. 4.23 Error performance of CP type detection in SUI3 channel at speed 360 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 52. 4.24 Error performance of CP type detection in SUI4 channel at speed 3 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53. 4.25 Error performance of CP type detection in SUI4 channel at speed 120 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 53. 4.26 Error performance of CP type detection in SUI4 channel at speed 360 km with several different window size. . . . . . . . . . . . . . . . . . . . . . . . .. 54. 4.27 Error performance of CP type detection in Pedestrian-B channel at speed 3 km with several different window size. . . . . . . . . . . . . . . . . . . . . . .. 54. 4.28 Error performance of CP type detection in Pedestrian-B channel at speed 120 km with several different window size. . . . . . . . . . . . . . . . . . . . . . .. 55. 4.29 Error performance of CP type detection in Pedestrian-B channel at speed 360 km with several different window size. . . . . . . . . . . . . . . . . . . . . . .. viii. 55.

(11) List of Tables 2.1. LTE System Attributes [2, Table 1.1] . . . . . . . . . . . . . . . . . . . . . .. 6. 2.2. Resource Block Parameters (modified from [1, Table 5.2.3-1]) . . . . . . . . .. 9. 2.3. Physical Layer Cell Identities [2, Table 9.1] . . . . . . . . . . . . . . . . . . .. 13. 2.4. Mapping Between NID and Indices m0 and m1 [2, Table 9.2] . . . . . . . . .. 14. 3.1. Joint Estimation of STO, CFO for Extended CP . . . . . . . . . . . . . . . .. 34. 3.2. Joint Estimation of STO, CFO for Normal CP . . . . . . . . . . . . . . . . .. 35. 4.1. Downlink System Parameters Used in Simulation . . . . . . . . . . . . . . .. 37. 4.2. SUI Channel Models for Three Terrain Types . . . . . . . . . . . . . . . . .. 37. 4.3. SUI-1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 4.4. SUI-2 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 4.5. SUI-3 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 4.6. SUI-4 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 4.7. PB Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. (1). ix.

(12) Chapter 1 Introduction Orthogonal frequency division multiple access (OFDMA) is the scheme chosen for the downlink of the 3rd Generation Partnership Project (3GPP) Long Term Evolution (LTE) and LTE-Advanced (LTE-A) standards for 4G cellular mobile communication [2]. When an user equipment (UE) is turned on or performs handoff from one cell to another, a first thing to do is synchronization, which includes OFDM symbol timing recovery (necessitated by the unknown propagation delay), carrier frequency recovery (necessitated by the mismatch of oscillators between the base station and the UE), detection of the cyclic prefix (CP) type used in signal transmission, and detection of the cell identity (cell ID) carried on the synchronization signals. Various algorithms have been proposed to estimate symbol timing offset and carrier frequency offset and they can be classified into two main categories: data-aided estimation and nondata-aided (blind) estimation. In [6] and [9], the authors derive algorithms based on the maximum likelihood (ML) criterion for blind synchronization over an additive white Gaussian noise (AWGN) channel and single-path Rayleigh channel, respectively. In [10], the authors propose a technique to jointly estimate the symbol timing offset and carrier frequency offset over fast time-varying multipath channels. A pilot-aided method (using primary synchronization sequence) based on the ML approach that takes the unknown channel response into account is proposed in [11]. In our study, we consider blind synchronization and focus on LTE and LTE-A initial downlink synchronization estimation schemes based on the 3GPP TS 36.211 release 11 [1], including problem formulation, algorithm derivation based on the ML criterion and computer simulation in MATLAB.. 1.

(13) The contribution of this work is as follows. • We establish the transmitted and received signal models of LTE-A system and derive the quasi-maximum-likelihood estimator for joint estimation of the cyclic prefix (CP) type, symbol timing offset (STO) and carrier frequency offset (CFO). • It is a generalized solution compared to that proposed in [6], [9] and [10]. The first and the second are considered in AWGN and single-path Rayleigh fading channels, respectively. The last is not an optimal ML estimator due to improper derivation. Our solution is derived based on the multipath fading channels. • In estimating CFO, we consider an approximation so that it can reduce complexity largely. Though, by doing so, the solution is not optimal under multipath fading channels. It is an optimal solution under AWGN and single-path Rayleigh fading channels because the approximate term has no effect on estimation process. This thesis is organized as follows. • In chapter 2, we introduce the basic concepts of OFDMA and the LTE (LTE-A) downlink transmission standard based on the 3GPP TS 36.211 release 11 [1]. • In chapter 3, we first formulate our problem into mathematical model and then use the ML criterion to jointly estimate STO, CFO, and CP type. • In chapter 4, we show some simulation results under different environments, and discuss the simulation results. • In chapter 5, we give the conclusion about our proposed estimator and indicate some items of potential future work.. 2.

(14) Chapter 2 Overview of the LTE-A Downlink Standard The LTE-A downlink transmission standard is based on orthogonal frequency-division multiple access (OFDMA), a variant of orthogonal frequency-division multiplexing (OFDM). In this chapter, we first introduce the basic concepts of OFDM and OFDMA, then we give an overview of the LTE-A downlink specifications [1].. 2.1. OFDM and OFDMA Basics. The material in this section is mainly taken from [2] and [3]. OFDM is a most popular scheme for current wireless systems, including 4G broadband wireless networks. The basic principle of OFDM is to derive the available spectrum into narrow-band parallel channels referred to as subcarriers so as to simplify channel equalization.. 2.1.1. Discrete-time baseband OFDM system model. Fig. 2.1 shows the basic concept of baseband OFDM transmitter and receiver. In the transmitter side, the complex modulations symbols X(k), k = 0, 1, ..., N − 1, are mapped to the input of inverse fast fourier transform (IFFT), and there is no information carried on the guard subcarriers. A cyclic prefix (CP) is added after IFFT operation and the resulting sequence is up-converted to RF for transmitting. In the receiver side, the received signal is filtered, amplified and down-converted from RF. Then the CP samples are removed and the received sample sequence are sent into the FFT. The discrete-time baseband channel is 3.

(15) ^. X (0). ^. X (N −1). Figure 2.1: Baseband OFDM transmitter and receiver. assumed to consist of L multipaths with tap coefficients h0 , h1 , · · · , hL−1 , and the received signal after CP removal is given in the frequency domain by Y = HX + z,. (2.1). where H = diag(H(0), H(1), · · · , H(N − 1)) is the diagonal matrix of channel frequency response, and z is a vector of the additive white Gaussian noise (AWGN) with each element distributed according to N (0, σz2 ).. 2.1.2. Cyclic prefix. In OFDM systems, CP is used to overcome the inter-symbol interference (ISI) problem. We usually assume that the CP is longer than the length of the channel impulse response (CIR). Under this condition, successive OFDM symbols do not interfere with each other in the frequency domain if the receiver has proper time and frequency synchronization. CP extends the OFDM symbol length by copying the last samples of the OFDM symbol into its front. In Fig. 2.2, TG denotes the length of CP in terms of samples. After adding 4.

(16) CP. CP ith OFDM symbol. TG. (i+1)th OFDM symbol. T sym = T + TG. T. Figure 2.2: OFDM symbols with CP.. Figure 2.3: Types of resource allocation in OFDMA system (modified from [3, Fig. 4.29]). CP, the OFDM symbol now has the duration Tsym = T + TG .. 2.1.3. OFDMA. In principle, in OFDM, all subcarriers in a symbol are used for transmitting the signal of a single device. OFDMA may share the subcarriers of a symbol among multiple devices or users. In practice, the multiple devices/users are allocated mutually exclusive sets of orthogonal subcarriers, so that the data of different devices/users can be separated in the frequency domain. One typical structure is subchannel OFDMA. A subset of subcarriers is allocated to each user by different types such as block type, comb type or random type. In the first type, each subchannel consists of a set of adjacent subcarriers. In the second type, each subchannel is composed of equi-spaced subcarriers. And in the last, each subchannel is composed of a set of subcarriers distributed randomly over the whole frequency band. Fig. 2.3 shows the three types of resource allocation in an OFDMA system. 5.

(17) Table 2.1: LTE System Attributes [2, Table 1.1] Bandwidth 1.25–20 MHz Duplexing FDD, TDD, half-duplex FDD Mobility 350km/h Multiple access downlink OFDMA Uplink Single-carrier-FDMA Multi-input multi-output downlink 2 × 2, 4 × 2, 4 × 4 (MIMO) modes Uplink 1 × 2, 1 × 4 MIMO data rates downlink 173 and 326 Mb/s for 2 × 2 and 4 × 4 MIMO, respectively Uplink 86 Mb/s with 1 × 2 antenna configuration Modulation QPSK, 16-QAM and 64-QAM Channel coding Turbo code Other techniques Channel sensitive scheduling, link adaptation, power control, inter-cell interference coordination (ICIC) and hybrid ARQ. 2.2. Overview of LTE and LTE-A Downlink Specifications. The contents in this section are mainly taken from [1] and [2]. The goal of LTE is to provide a high-data-rate, low-latency and packet-optimized radio access technology supporting flexible bandwidth deployments. In parallel, new network architecture is designed with the goal to support packet-switched traffic with seamless mobility, quality of service and minimal latency. The air-interface related attributes of the LTE system are summarized in Table 2.1. The system supports flexible bandwidths. In addition to frequency division duplexing (FDD) and time division duplexing (TDD), half-duplex FDD is allowed to support low cost user equipment (UE). Unlike FDD, in half-duplex FDD operation a UE is not required to transmit and receive at the same time. This avoids the need for a costly duplexer in the UE. The system is primarily optimized for low speeds up to 15 km/h. However, the system specifications allow mobility support in excess of 350 km/h with some performance degradation.. 6.

(18) Figure 2.4: Frame structure type 1 [1, Fig.4.1-1].. Figure 2.5: Frame structure type 2 [1, Fig. 4.2-1].. 2.2.1. Frame structure. In the specification of the frame structure in LTE, the size in the time domain is normally expressed in time units of Ts = 1/(15000 × 2048) seconds. Both downlink and uplink transmissions are organized into radio frames with frame duration Tf = 307200 × Ts = 10 ms. Fig. 2.4 and 2.5 show frame structure types 1 and 2, respectively. Frame structure type 1 applies to both full duplex and half duplex FDD, and frame structure type 2 applies to TDD. In our study, we focus on frame structure type 1. In this type, each radio frame consists of 20 slots of length Tslot = 15360 × Ts = 0.5 ms, which are numbered from 0 to 19. A subframe consists of two consecutive slots where subframe i consists of slots 2i and 2i + 1. For FDD, there are 10 subframes for downlink and uplink transmissions in each 10 ms interval.. 2.2.2. Slot structure and physical resources. In LTE-A specifications, two CP lengths, called normal CP and extended CP, are defined to support small and large cells deployments, respectively. Fig. 2.6 shows the high-level slot structures for normal and extended CP and Fig. 2.7 shows the detailed slot structure for the 7.

(19) Figure 2.6: Slot structure for normal and extended CP [2, Fig. 8.3].. Figure 2.7: Detailed signal structure for normal CP [1, Figs. 5.2.1-1 and 6.2.2-1], [2, Fig. 8.2]. 8.

(20) Table 2.2: Resource Block Parameters (modified from [1, Table 5.2.3-1]) RB DL UL Configuration Nsc Nsymb Nsymb Normal CP 12 7 7 Extended CP 12 6 6 24 3 NA. case of normal CP. The normal CP length is 5.2 µs (160 × Ts ) in the first OFDMA symbol and 4.7 µs (144 × Ts ) in the remaining six symbols. In the type of extended CP, the CP length is 16.6 µs (512×Ts ) is the same in all the six symbols. The same applies to SC-FDMA symbols in the uplink. RB A resource block (RB) is defined as Nsc consecutive subcarriers in the frequency domain DL UL and Nsymb OFDMA symbols for the downlink or Nsymb SC-FDMA symbols for the uplink. An DL RB UL RB RB therefore consists of Nsymb ×Nsc (Nsymb ×Nsc ) resource elements (REs) in the downlink. (uplink), which corresponds to one slot in the time domain and 180 kHz bandwidth in the RB frequency domain. The number of subcarriers within an RB NSC is 12 or 24 for the case. of 15 kHz or 7.5 kHz subcarrier spacing, respectively. Table 2.2 summarizes the parameter sets. The minimum and maximum number of RBs in a slot are 6 (1.08 MHz transmission bandwidth) and 110 (19.8 MHz transmission bandwidth), respectively. The relation between the RB number (or index) nRB in the frequency domain and RE index (k, l) in a slot is given by nRB = b. 2.2.3. k c. RB Nsc. Random access. Fig. 2.8 shows the random access procedure. When a UE is turned on or performs handoff from one cell to another, the first thing to do is to acquire timing and frequency synchronization. This procedure is achieved by using the received primary synchronization sequences (PSS), secondary synchronization sequences (SSS) and the broadcast channel (BCH). After synchronization and getting system information including random access (RA) parameters, the UE can then transmit the RA preamble. The purpose of RA is to let the base station (eNB in LTE terminology) estimate and adjust the UE uplink transmission timing to within a fraction of the CP. When the eNB receives an RA preamble successfully, it sends the 9.

(21) Figure 2.8: Random access procedure [2, Figure 10.1]. response sequence indicating the successfully received preamble, timing advance (TA) and information about uplink resource allocation to the UE. The UE determines if its RA preamble has been successfully received by matching the preamble number with that received from the eNB. If they match, the UE uses the TA information to adjust its uplink timing, sends uplink scheduling or a resource request indicated in the RA response message.. 2.2.4. Cell search. Besides timing and frequency synchronization, when a UE turns on or performs handoff from one cell to another, it also should detect the cell identity (cell ID), transmission bandwidth of the system, CP type, and numbers of transmit antenna ports, etc. This procedure is achieved by using the PSS, SSS, and BCH. To facilitate such work, the PSS, SSS and BCH are transmitted using the same minimum bandwidth of 1.08 MHz in the central part of the bandwidth. PSS and SSS are carried in the last and second last OFDM symbols respectively in slot numbers 0 and 10, and in the frequency domain over the middle 62 subcarriers out of a total of 72 subcarriers (1.08 MHz). Fig. 2.9 shows the position of PSS and SSS under normal CP.. 10.

(22) . .

(23). . . . . . . . . . . . . . .

(24). . . . . . . Figure 2.9: downlink frame structure for the type of normal CP.. 2.2.5. Zadoff-Chu (ZC) sequences. A ZC sequence is defined as ( xu (m) =. 2. −j πum N. e. ZC. ,. when NZC is even,. πum(m+1) −j N ZC. e. , when NZC is odd,. (2.2). with m = 0, 1, ..., NZC − 1, and the sequence index u is relatively prime to the length of ZC sequence NZC . ZC sequences are used in many sequences in the LTE system such as PSS, uplink reference signals, uplink physical control channel (PUCCH) and RA channel. Some properties of the ZC sequences are as follows. First, the periodic autocorrelation of a ZC sequence is zero for all time shifts other than zero. Second, ZC sequences for different u exhibit low cross-correlation, though are not orthogonal. And third, if the sequence length NZC is a prime number, then √ there are (NZC − 1) different sequences with periodic cross-correlation of 1/ NZC between any two sequences regardless of time shift.. 11.

(25) Figure 2.10: PSS mapping in the frequency domain (modified from [2, Figure 9.4]).. 2.2.6. Primary synchronization signal (PSS). In the LTE system, a total of 504 unique physical layer cell IDs are provided. It is derived (1). (2). from a physical layer cell ID index NID ∈ [0, 167] and another physical layer ID NID ∈ [0, 2], (1). (2). (cell). where NID and NID are integers. The cell ID is defined to be NID. (1). (2). = 3NID + NID .. The PSS in the frequency domain is defined as ( πun(n+1) e−j 63 , n = 0, 1, ..., 30, du (n) = πun(n+1)(n+2) −j 63 , n = 31, 32, ..., 61, e. (2.3). (2). where the ZC root sequence index u is 25, 29, and 34 for NID = 0, 1, and 2, respectively. The mapping of the PSS sequence du (n) to the REs is as shown in Fig. 2.10. The mapping for the non-reserved REs is as follows: ak,l = du (n), k = n − 31 + b. DL RB NRB Nsc DL c, l = Nsymb − 1, n = 0, ..., 61. 2. The signal values of the reserved REs are set to null, i.e., for k = n − 31 + b. (2.4). DL N RB NRB sc c, 2. l=. DL Nsymb − 1, n = −5, −4, ..., −1, 62, 63, ..., 66.. 2.2.7. Secondary synchronization signal (SSS). The sequence of SSS is an interleaved concatenation of the two length-31 binary sequences. The concatenated sequence is scrambled with the sequence of PSS. The combination of two length-31 sequences defining the SSS differs between subframes 0 and 5, and is defined as. ( d(2n) = ( d(2n + 1) =. (m ). s0 0 (n)c0 (n), in subframe 0, (m ) s1 1 (n)c0 (n), in subframe 5, (m ). (2.5). (m ). s1 1 (n)c1 (n)z1 0 (n), in subframe 0, (m ) (m ) s0 0 (n)c1 (n)z1 1 (n), in subframe 5,. 12. (2.6).

(26) Table 2.3: Physical Layer Cell Identities [2, Table 9.1] (2) PHY layer cell ID PHY layer ID NID PHY layer cell ID (1) (cell) (1) (2) group NID NID = 3NID + NID 0 0 0 1 1 2 2 0 3 1 1 4 2 5 .. .. .. . . . 0 1 2. 167. 501 502 503. (1). where 0 ≤ n ≤ 30. The indices m0 and m1 are derived from NID according to m0 = m0. mod 31,. m1 = (m0 + bm0 /31c + 1) (1). mod 31,. m0 = NID + q(q + 1)/2, q = b. (1) NID. (2.7). + q 0 (q 0 + 1)/2 (1) c, q 0 = bNID /30c. 30. (1). The mapping between NID and indices m0 and m1 is illustrated in Table 2.4. (m0 ). The two sequences s0. (m1 ). (n) and s1. (n) are defined as two different cyclic shifts of the. m-sequence s˜(n) according to (m0 ). (n) = s˜((n + m0 ). mod 31),. (m1 ). (n) = s˜((n + m1 ). mod 31),. s0 s1. (2.8). where s˜(i) = 1 − 2x(i), 0 ≤ i ≤ 30, with x(i) given as x(j + 5) = (x(j + 2) + x(j)). mod 2, 0 ≤ j ≤ 25,. (2.9). with initial conditions x(0) = x(1) = x(2) = x(3) = 0, x(4) = 1. The two length-31 scrambling sequences c0 (n) and c1 (n) which are dependent on PSS are defined by two different cyclic shifts of the m-sequence c˜(n) as (2). c0 (n) = c˜((n + NID ) (2). mod 31),. c1 (n) = c˜((n + NID + 3) 13. mod 31),. (2.10).

(27) (1). Table 2.4: Mapping Between NID and Indices m0 and m1 [2, Table 9.2] (1) (1) NID m0 m1 NID m0 m1 .. .. .. 0 0 1 . . . 1 1 2 113 26 30 .. .. .. 114 0 5 . . . 29 29 30 115 1 6 .. .. .. 30 0 2 . . . 31 .. .. 1 .. .. 3 .. .. 58. 28. 30. 59 60 .. .. 0 1 .. .. 3 4 .. .. 86 87 88. 27 0 1. 30 4 5. 139. 25. 30. 140 141 .. .. 0 1 .. .. 6 7 .. .. 164. 24. 30. 165 166 167. 0 1 2. 7 8 9. where c˜(i) = 1 − 2x(i), 0 ≤ i ≤ 30, with x(i) given as x(j + 5) = (x(j + 3) + x(j)). mod 2, 0 ≤ j ≤ 25,. (2.11). with initial conditions x(0) = x(1) = x(2) = x(3) = 0, x(4) = 1. (m0 ). The two length-31 scrambling sequences z1. (m1 ). (n) and z1. (n) are defined by a cyclic. shift of the m-sequence z˜(n) according to (m0 ). (n) = z˜((n + (m0. mod 8)). mod 31),. (m1 ). (n) = z˜((n + (m1. mod 8)). mod 31),. z1 z1. (2.12). where z˜(i) = 1 − 2x(i), 0 ≤ i ≤ 30, with x(i) given as x(j + 5) = (x(j + 4) + x(j + 2) + x(j + 1) + x(j)) mod 2, 0 ≤ j ≤ 25,. (2.13). with initial conditions x(0) = x(1) = x(2) = x(3) = 0, x(4) = 1. The SSS mapping for the non-reserved REs has the following relationship: ak,l = du (n), k = n − 31 + b. DL RB NRB Nsc DL c, l = Nsymb − 2, n = 0, ..., 61. 2. The signal values of the reserved REs are set to null, i.e., k = n − 31 + b. (2.14). DL N RB NRB sc c, 2. l =. DL Nsymb −2, n = −5, −4, ..., −1, 62, 63, ..., 66. Fig. 2.11 shows the SSS mapping in the frequency. domain. 14.

(28) Figure 2.11: SSS mapping in the frequency domain (modified from [2, Figure 9.5]).. Figure 2.12: BCH transmission chain processing [2, Figure 9.6].. 2.2.8. Broadcast channel (BCH). Fig. 2.12 shows the transmission processing chain for the BCH. First, error detection capability is provided on BCH transport blocks through cyclic-redundancy-check (CRC) coding. After CRC attachment, the bit sequence is coded using a rate 1/3 tail-biting convolutional code. Then the bits are rate matched using a certain circular buffer approach to obtain the rate-matched sequence b(0), b(1), · · · , b(Mbit − 1), where Mbit is the number of transmitted bits. And then the rate-matched sequence is scrambled with a cell-specific sequence as ˜b(i) = (b(i) + c(i)). mod 2. (2.15). with frame number nf fulfilling the condition nf. mod 4 = 0. (2.16). and c(n) = (x1 (n + Nc ) + x2 (n + Nc )). mod 2,. x1 (n + 31) = (x1 (n + 3) + x1 (n)) mod 2,. (2.17). x2 (n + 31) = (x2 (n + 3) + x2 (n + 2) + x2 (n + 1) + x2 (n)). mod 2,. where Nc = 1600, the m-sequence x1 (n) is initialized with x1 (0) = 1, x1 (n) = 0, n = 30 X 1, 2, ..., 30, and the m-sequence x2 (n) is initialized by cinit = x2 (i) × 2i with the value depending on the application of the sequence.. i=0. Fig. 2.13 shows the mapping of the physical BCH. It is transmitted over 4 subframes with a 40 ms timing interval. The number of subcarriers used for PBCH is 72 in the third 15.

(29) Figure 2.13: PBCH mapping (modified from [2, Figure 9.7]). and fourth OFDMA symbols in the slot without reference signals and is 48 in the first and second OFDMA symbols with reference signals.. 16.

(30) Chapter 3 Initial Downlink Synchronization In this chapter, we first formulate the the initial downlink synchronization problem for the 4G LTE-A FDD system. Then we derive a solution algorithm by taking a maximum likelihood (ML) approach.. 3.1. The Initial Downlink Synchronization Problem. When an user equipment (UE) tries to gain connection with a base station (eNB), the UE first needs to synchronize to the eNB. In so doing, it first has to estimate the symbol timing offset (STO), carrier frequency offset (CFO), and the cyclic prefix (CP) type. The STO can arise from the unknown propagation delay between the transmitter and receiver, and the CFO from the mismatch between transmitter and receiver oscillators and the Doppler spread due to mobility. As to the CP type, recall that the LTE-A system defines two CP lengths, namely normal CP and extended CP, to support small and large cells deployments respectively. Hence the UE also has to detect the CP type for signal reception as well as transmission.. 3.1.1. Transmission system model. The overall system structure is shown in Figure 3.1. Figure 3.2 depicts the slot structure in the time domain that captures the essence. Let m = 1 and m = 2 represent extended CP and normal CP, respectively. Let dim (j) donote the frequency domain signal vector transmitted in the jth OFDMA symbol of the ith slot under CP type m. The size of dim (j) is N , the FFT size used by the system. Let sim 17.

(31) Figure 3.1: Transmission system structure.. k = 0. k = 0. 512 × TS. k =1. k =1. 160 × TS. k=2. k =3. k = 2. 144 × T S. k=4. k = 3. k =5. k =4. k =6. k =5. TS = 1 / f s , where f s = 30 .72 M samples / sec. Figure 3.2: Slot structure in the time domain.. 18.

(32) be a collection of the signal vectors in each slot as ½ [di1 (0) di1 (1) · · · di1 (5)]T , m = 1, i sm = [di2 (0) di2 (1) · · · di2 (6)]T , m = 2.. (3.1). The baseband transmitted signal can be written as i xim = Gm · FH m · sm. (3.2). where Fm is as Nm × Nm block diagonal matrix given by ½ diag(F, F, F, F, F, F), m = 1, Fm = diag(F, F, F, F, F, F, F), m = 2,. (3.3). with N1 = 6N , N2 = 7N and F being the normalized N × N DFT matrix given by  0·0  0·(N −1) 0·1 ωN ωN · · · ωN 1·(N −1)  1·1 1·0 ωN · · · ωN 1   ωN  F= √  .  . . . . . . .  . N . . . (N −1)·0 (N −1)·1 (N −1)·(N −1) ωN ωN · · · ωN 1. where ωN = e−j2π N ; Gm is the CP insert  GCP 1 0   0 GCP 1   0  0 G1 =  .  0 ..   ..  0 . 0 ··· with. (3.4). matrix given by ···. ··· .... ··· .... GCP 1. 0. .... 0. 0 .. .. GCP 1. 0. 0. 0. 0. 0 GCP 1 0 GCP 1. 0 ···. ···. . 0.          . (3.5). . GCP 1        G2 =       with.  0N512 ×(N −N512 ) IN512 0 , =  I(N −N512 ) 0 IN512. GCP 2. 0. ···. 0. GCP 3. 0. ··· .. .. 0. 0 .. . .... GCP 3. 0. ··· .. . .. .. 0 .... GCP 3. 0. ··· ···. ··· ···. 0 0 0 0. 0 ··· ···. ··· .. . .. . .. .. (3.6) 0 0 0 0. GCP 3 0 0 0 GCP 3 0 ··· 0 GCP 3.             . . GCP 2.    0N160 ×(N −N160 ) IN160 0512×(N −N512 ) IN144 0  , GCP 3 =  I(N −N144 ) 0 , =  I(N −N160 ) 0 IN160 0 IN144. (3.7). 19. (3.8).

(33) where N512 , N160 , N144 are the CP lengths of different OFDMA symbols under different CP type as shown in Figure 3.2. The above gives the signal structure in a slot, we can concatenate the slots together to obtain the overall transmitted signal. For this, we define the rectangular window matrix Wi ∈ R∞×M for the ith slot as W i = e i ⊗ IM ,. (3.9). where ⊗ is the Kronecker product operation. Then the infinite-length transmitted signal can be written as y=. ∞ X. Wi · xim .. (3.10). i=−∞. At the receiver, let the observation window size be K samples. Then the observed received signal r ∈ CK×1 suffers from the influence of the multi-path channel and the effect of the carrier frequency offset and symbol timing offset. It can be modeled as r = D(θ) · A(²) · H · y + w,. (3.11). where w is the additive white Gaussian noise (AWGN) vector with zero mean and variance σw2 , A(²) ∈ C∞×∞ is the CFO matrix with the (p, q)th element given by [A(²)]p,q = ej2πp²/N δ(p − q),. (3.12). D(θ) ∈ RK×∞ is the STO matrix, which means collecting K samples from the overall signal, and its (p, q)th element is given by [D(θ)]p,q = δ(q − p − θ),. (3.13). H ∈ C∞×∞ is the channel impulse response matrix given by     H=   . ... .. ... .. ... .. .. ... . .. ... .. 0 .. . .. . .. .. ... .. ... h(Ng − 1, 0). ···. 0 .. . .. .. .. h(Ng − 1, 1) 0 .. .. ... .. ... .. ... .. ···. h(0, 0). 0. ···. ···. h(0, 1). h(Ng − 1, 2) .. .. ··· .. .. ··· .. .. . ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. 0 h(0, 2) .. .. 0 .. ..    ,   . (3.14). where the (p, q)th element is given by [H]p,q = h(l, n)|0≤l=p−q≤Ng −1,n=p ,. (3.15). with l and n being the delay and time indices. We have assumed that the channel impulse response (CIR) length is smaller than the CP length in order to avoid the inter-symbolinterference (ISI) problem, so h(l, n) is nonzero only when 0 ≤ l ≤ Ng − 1 and Ng = N144 . 20.

(34) 3.1.2. The ML approach to synchronization. Eq. (3.15) gives the received signals. We take the ML approach to estimate the parameters θ, ² and m, i.e., the STO, CFO and CP type, which give the ML estimation as ˆ ²ˆ, m} {θ, ˆ = max f (r|θ, ², m) θ,²,m. = max max f (r|θ, ², m) m. θ,²|m. (3.16). where f (r|θ, ², m) is the likelihood function. Therefore, the two key issues are the derivation of the likelihood function and the solution of the ML estimation problem. We first derive the likelihood function for extended CP. Then we consider the normal CP case. And finally we derive the solution.. 3.2 3.2.1. Solution of the ML Estimation Problem Derivation of the likelihood function for extended CP. The likelihood function for extended CP is given by p(r|θ1 , ²1 , m = 1) =. exp(−rH · C−1 r · r) K π · det(Cr ). (3.17). where Cr is the autocorrelation matrix of the received signals r given by (3.11). To derive Cr , consider first the autocorrelation matrix of the transmitted signal y given by Cy = E{y · yH } ∞ ∞ X X i = E{( Wi · x1 ) · ( Wj · xj1 )H } i=−∞. = =. ∞ ∞ X X. (3.19). j=−∞ j H i H H Wi · G1 · FH 1 · E{s1 · (s1 ) } · F1 · G1 · Wj. i=−∞ j=−∞ ∞ σd2 X. N. (3.18). H W i · G1 · GH 1 · Wi. (3.20) (3.21). i=−∞. σd2 I∞ ⊗ G1 GH = 1 N. (3.22). T where G1 GH 1 = G1 G1 is a tri-diagonal matrix given by   IN512 0 IN512 IN −N512 0 . G1 GT1 =  0 IN512 0 IN512. 21. (3.23).

(35) Because of the shape of the matrix G1 GT1 , we consider partitioning the matrix H as H = [ · · · H1 H2 H3 · · · ] = [ · · · Hl1 Hm1 Hr1 Hl2 Hm2 Hr2 · · · ]. (3.24) (3.25). where Hi is the ith N + N512 columns of H, Hli and Hri consist of the left most and right most N512 columns of Hi , respectively, and Hmi is composed of the middle N − N512 columns of Hi . The autocorrelation matrix of received signals r is then given by © ª Cr = E r · rH (3.26) © ª = E (D(θ)A(²)Hy + w)(yH HH AH (²)DH (θ) + wH ) (3.27) ¡ ¢ H H ª σ2 © H H A (²)D (θ) + σw2 I (3.28) = d E D(θ)A(²)H I∞ ⊗ G1 GH 1 N Ã ! ∞ ∞ X X σd2 H H D(θ)A(²)E HH + Hri Hli + = Hli HH AH (²)DH (θ) + σw2 I. ri N i=−∞ i=−∞ (3.29) As mentioned, we assume that the length of the channel impulse response (CIR) is smaller than the CP length. Here we also assume the channel to be wide sense stationary uncorrelated scattering (WSSUS), that is, E{h(l, n)h∗ (l + ∆l, n + ∆n)} = ϕh (l, ∆n)δ(∆l). (3.30). where ϕh (l, ∆n), by the above assumption on CIR length, can be nonzero only when 0 ≤ l ≤ N512 − 1. The first term in the parenthesis of (3.29) leads to ª © E [HHH ]p,q =. ∞ X. E {h(p − k, p)h∗ (q − k, q)}. (3.31). k=−∞. = = =. −∞ X l=∞ −∞ X. E {h(l, p)h∗ (l + (q − p), p + (q − p))}. (3.32). ϕh (l, q − p)δ(q − p). (3.33). l=∞ NX 512 −1. ϕh (l, q − p)δ(q − p).. l=0. 22. (3.34).

(36) Therefore, E{. ∞ X. HHH } is a scaled identity matrix given by. i=−∞ 512 −1 © ª NX H E HH = ϕh (l, 0)I ≡ σh2 I.. (3.35). l=0. The second term in the parenthesis of (3.29) leads to " #  ∞ ∞ NX 512 −1  X  X © ª H E Hri Hli = E [Hri ]p,k [HH ] q,k li   i=−∞. i=−∞. p,q. =. ∞ NX 512 −1 X i=−∞. (3.36). k=0. E{h(p − iM512 − N − k, p)h∗ (q − iM512 − k, q)}. k=0. (3.37) Ng −1. =. ∞ X X. ϕh (p − iM512 − N − k, q − p)δ(q − p + N ). (3.38). i=−∞ k=0. where M512 = N + N512 . Similarly, the third term leads to " #  ∞ ∞ NX 512 −1  X  X © ª H E Hli Hri E [Hli ]p,k [HH ] = q,k ri   i=−∞. i=−∞. p,q. =. ∞ NX 512 −1 X i=−∞. (3.39). k=0. E{h(p − iM512 − k, p)h∗ (q − iM512 − N − k, q)}. k=0. (3.40) Ng −1. =. ∞ X X. ϕh (p − iM512 − k, q − p)δ(q − p − N ).. (3.41). i=−∞ k=0. From (3.31) to (3.41), we see that E{. ∞ X. Hri HH li } is a matrix with all its nonzero. i=−∞ ∞ X. elements located on a certain subdiagonal, E{. Hli HH ri } is a matrix with all its nonzero. i=−∞. elements located on a certain superdiagonal, and E{HHH } is a matrix with all its nonzero elements located on the diagonal. The matrix Cr is then given by  N512 −1 X  σd2   ϕh (k, 0) + σw2 ≡ σr2 ,  N    k=0   ∞ N512 −1   2  σd +j2π² X X e ϕh (θ1 + p − iM512 − N − k, −N ), N [Cr ]p,q = i=−∞ k=0   ∞ NX  512 −1 X  2  σd −j2π²   Ne ϕh (θ1 + p − iM512 − k, +N ),     i=−∞ k=0  0, 23. q = p, q = p − N, q = p + N, else.. (3.42).

(37) N+ (N512) N.

(38) . K.

(39) . . K. Figure 3.3: Structure of the autocorrelation matrix Cr .. …. Figure 3.4: Moving average. Figure 3.3 depicts the structure of autocorrelation matrix Cr . First, the bold line on the diagonal indicates where the nonzero elements of E{HHH } are located and each of them has the same value σr2 . The two thinner lines along a subdiagonal and a superdiagonal, which ∞ X appear like a long dashed line each, indicate where the nonzero elements of E{ Hri HH li } and E{. ∞ X. i=−∞. Hli HH ri } may lie. These elements are not all of the same value along each. i=−∞. diagonal, but each segment is given by a scaled version of the N512 -point moving average of the channel PDP. The xy-plot located near the first sequent of the subdiagonal in Figure 3.3 illustrates the concept, where for illustration purpose we assume that PDP has an exponential shape. Figure 3.4 illustrates the essence of the moving-average operation that results in the above property. Finally, the dotted square represents the matrix Cr ∈ C K×K when the 24.

(40) Figure 3.5: Autocorrelation matrix example. observation window has a finite size K and located at an STO θ1 . Note from (3.17) that we have to compute the inversion of Cr . Due to its tridiagonal structure, a closed-form expression can be obtained that expresses the elements of C−1 in r terms of that of Cr . Before presenting the general form, we first give a simple example for illustration purpose. Consider a case with FFT size N = 6, CP length Ng = 2, observation window size K = 18, and STO θ1 = 0. The corresponding Cr is shown in Figure 3.5. From this figure, we can see that Cr can be divided into 3-by-3, 2-by-2 and 1-by-1 submatrix whose inverse can be computed separately to make up the full C−1 r . Thus the inversion of Cr can be obtained rather simply. For this example, the joint PDF of received signal r is given by f (r|²1 , θ1 ) =. Y n∈{0,1}. Y n∈{2,3}. f (r(n), r(n + N )|²1 , θ1 ) f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ) f (r(n), r(n + N ), r(n + 2N )|²1 , θ1 ) f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ) · f (r(n + 2N )|²1 , θ1 ). Y n∈{10,11}. Y f (r(n), r(n + N )|²1 , θ1 ) f (r(n)|²1 , θ1 ) f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ) n 25. (3.43).

(41) multipath. CP. CP. CP. CP Figure 3.6: Three-region data correlation. where f (r(n), r(n + N )|²1 , θ1 ) = f (r(n), r(n + N ), r(n + 2N )|²1 , θ1 ) = f (r(n)|²1 , θ1 ) = f (r(n + N )|²1 , θ1 ) = f (r(n + 2N )|²1 , θ1 ) =. ¡ ¢ −1 exp −rH 1 C1 r1 , π 2 det(C1 ) ¡ ¢ −1 exp −rH 2 C2 r2 , π 2 det(C2 ) ´ ³ |r(n)|2 exp − σ2 r , 2 πσr ´ ³ )|2 exp − |r(n+N 2 σr , 2 πσr ´ ³ )|2 exp − |r(n+2N σ2 r. πσr2. (3.44) (3.45) (3.46) (3.47) ,. (3.48). with r1 = [r(n), r(n+N )], r2 = [r(n), r(n+N ), r(n+2N )], C1 and C2 are the autocorrelation matrixes of vectors r1 and r2 , respectively. From (3.43) we can see that there are three kinds of data correlation. The first is because of the CP structure. The second is due not only to the CP structure but also the multi-path channel spread, as illustrated in Figure 3.6. And the third is due to the multi-path channel spread alone. Now consider a general FFT size N , CP length N512 , and STO θ1 . For illustration, consider an observation window size K = 3M512 . Figure 3.7 depicts the various conditions of data correlation. We divide an OFDM symbol into 4 sections, RI = {m|N ≤ m ≤ M512 −1}, RII = {m|0 ≤ m ≤ +N512 − 1}, RIII = {m|N512 ≤ m ≤ 2N512 − 1}, and RIV = {m|2N512 ≤ m ≤ N − 1}, where m = [(n − θ1 ) mod M512 ] is the distance between timing index n and STO θ1 . In Figure 3.7, Rji denotes the ith region in section j under symbol timing offset 26.

(42) Case 1 1 ≤ θ1 ≤ N 512 CP . . . CP. . . CP . . . . R 1I. CP . . . . . . . RI2 RI3. n θ. R I4. 1. Case 2 N512 +1 ≤ θ1 ≤ N − N512. CP . . . CP . . . . . CP . . . . CP . . . RIV1. . RIV2 3 n RIV. θ. 1. Case 3 N − N512 +1≤θ1 ≤ N. CP . . . CP. . . . . . CP . . . . CP . 1. . . RIII. . 2. R III 3. R III. n θ. 4. RIII. 1. Case 4 N +1≤θ1 ≤ N + N512 −1,0 CP CP . . . . . 1. . . . CP . . . . CP . R II . . . 2. R II 3. R II 4. n θ. R II. 1. Figure 3.7: Relative positioning of observation window with respect to STO, as well as the correlation structure in the received signal.. 27.

(43) θ1 , and there are four cases due to the different sections where in θ1 is located. Noted that if the observation window is infinite, then data correlation is always over the three regions RIII , RII and RI . Two-region data correlation occurs only at the boundaries of the observation window which cuts one of three regions, RIII or RI , out of the picture. The data that do not correlate with others are those that situate in the middle of an OFDM symbol and are not repeated in the CP or affected by the ring-down of the multipath spread caused to the signal in the CP. From the previous discussion, we see that likelihood function is in general of the form Y. f (r|²1 , θ1 ) =. n∈R0. f (r(n), r(n + N )|²1 , θ1 ) f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ). Y. f (r(n), r(n + N ), r(n + 2N )|², θ1 ) f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ) · f (r(n + 2N )|²1 , θ1 ) n∈R1 Y Y f (r(n), r(n + N )|²1 , θ1 ) f (r(n)|²1 , θ1 ), f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ) n n∈R. (3.49). 2. where R0 = {m|0 ≤ m ≤ N512 − 1} ∩ {n < N } ∩ {n + N ≤ K}, R1 = {m|N512 ≤ m ≤ 2N512 − 1} ∩ {n + 2N ≤ K}, R2 = {m|N512 ≤ m ≤ 2N512 − 1} ∩ {n + 2N > K} ∩ {n + N ≤ K},. (3.50). with m = (n − θ1 ) mod M512 . According to (3.49) and (3.50), the log-likelihood function of the received signals r can be written as " Λ1 (θ1 , ²1 ) = log. Y. n∈R0. ". f (r(n), r(n + N )|²1 , θ1 ) f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ). #. # f (r(n), r(n + N ), r(n + 2N )|²1 , θ1 ) + log f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ) · f (r(n + 2N )|²1 , θ1 ) n∈R " # # " 1 Y Y f (r(n), r(n + N )|²1 , θ1 ) + log f (r(n)|²1 , θ1 ) + log f (r(n)|² 1 , θ1 ) · f (r(n + N )|²1 , θ1 ) n n∈R Y. 2. =. X n∈R0. S. Φ1 (n, ²1 , θ1 ) + R2. X. " Φ2 (n, ²1 , θ1 ) + log. Y n. n∈R1. 28. # f (r(n)|²1 , θ1 ). (3.51) (3.52).

(44) where Φ1 (n, ²1 , θ1 ) ≡ log. f (r(n), r(n + N )|²1 , θ1 ) f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ). −|ρ(m)|2 (|r(n)|2 + |r(n + N )|2 ) = + 2< σr2 (1 − |ρ(m)|2 ). (3.53) ½. r(n)r∗ (n + N )ρ(m) +j2π²1 e σr2 (1 − |ρ(m)|2 ). − log(1 − |ρ(m)|2 ). ¾. (3.54). and Φ2 (n, ²1 , θ1 ) ≡ log. =. f (r(n), r(n + N ), r(n + 2N )|²1 , θ1 ) f (r(n)|²1 , θ1 ) · f (r(n + N )|²1 , θ1 ) · f (r(n + 2N )|²1 , θ1 ). (3.55). −|ρ(m)|2 (|r(n)|2 + |r(n + N )|2 ) − |ρ(m − N512 )|2 (|r(n + N )|2 + |r(n + 2N )|2 ) σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ) ½ +2< ½ −2<. r(n)r∗ (n + N )ρ(m) + r(n + N )r∗ (n + 2N )ρ(m − N512 ) +j2π²1 e σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ) r(n)r∗ (n + 2N )ρ(m)ρ(m − N512 ) +j4π²1 e σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ). ¾. ¾. − log(1 − |ρ(m)|2 − |ρ(m − N512 )|2 ). (3.56). NX 512 −1 σ2 β ϕh (m − k, N ) and β = d . with ρ(m) = 2 · σr N k=0. Assume that the Doppler spectrum follows the classical Jakes model. Then P 512 −1 10SN R/10 J0 (2πfd T ) · N ϕh (m − k, 0) k=0 ρ(m) = SN R/10 10 J0 (2πfd T ) · σh2 + 1. (3.57). where SN R ≡ σd2 /N , J0 (·) is the zero-order Bessel function of first kind, fd is the maximum Doppler shift and T is the useful symbol period. Q First, from (3.46), we can see the product n f (r(n)|²1 , θ1 ) is independent of θ1 and ²1 ; so we can drop this term. Second, according to (3.54) and (3.56), we can group these terms into two types, one including the effect of the CFO ²1 , the other not. Thus the log-likelihood. 29.

(45) function can be rewritten as     X X   Λ1 (θ1 , ²1 ) = λ1 (n, θ1 ) + λ2 (n, θ1 ) + 2<   S n∈R0. n∈R1. R2. (Ã +2<. X. !. ). T2 (n, θ1 ) e+j2π²1. (Ã − 2<. n∈R1. X.  X n∈R0. S. T1 (n, θ1 ) e+j2π²1 R2. !.   . ). T3 (n, θ1 ) e+j4π²1. (3.58). n∈R1. = A1 (n, θ1 ) + 2 |A2 (n, θ1 )| cos (2π²1 + ∠A2 (n, θ1 )). −2 |A3 (n, θ1 )| cos (4π²1 + ∠A3 (n, θ1 )) ,. (3.59). where A1 (n, θ1 ) =. X n∈R0. A2 (n, θ1 ) =. X. n∈R0. A3 (n, θ1 ) =. S. X. S. λ1 (n, θ1 ) +. X. λ2 (n, θ1 ),. (3.60). T2 (n, θ1 ),. (3.61). n∈R1. R2. T1 (n, θ1 ) +. X. n∈R1. R2. T3 (n, θ1 ),. (3.62). n∈R1. with λ1 (n, θ1 ) =. −|ρ(m)|2 (|r(n)|2 + |r(n + N )|2 ) − log(1 − |ρ(m)|2 ), σr2 (1 − |ρ(m)|2 ). λ2 (n, θ1 ) =. −|ρ(m)|2 (|r(n)|2 + |r(n + N )|2 ) − |ρ(m − N512 )|2 (|r(n + N )|2 + |r(n + 2N )|2 ) σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ) − log(1 − |ρ(m)|2 − |ρ(m − N512 )|2 ),. (3.63). (3.64). and T1 (n, θ1 ) =. r(n)r∗ (n + N )ρ(m) , σr2 (1 − |ρ(m)|2 ). (3.65). T2 (n, θ1 ) =. r(n)r∗ (n + N )ρ(m) + r(n + N )r∗ (n + 2N )ρ(m − N512 ) , σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ). (3.66). r(n)r∗ (n + 2N )ρ(m)ρ(m − N512 ) T3 (n, θ1 ) = . σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ). 30. (3.67).

(46) 3.2.2. Derivation of the likelihood function for normal CP. The derivation of algorithm for normal CP is similar to that for extended CP except that we have to modify the region size and location and change some parameters. For normal CP, we set the observation window size K is large enough to comprise the slot timing θ2 , and we then modify the likelihood function in (3.49) as Y f (r(n), r(n + N )|²2 , θ2 ) f (r|²2 , θ2 ) = f (r(n)|²2 , θ2 ) · f (r(n + N )|²2 , θ1 ) n∈R0 Y f (r(n), r(n + N ), r(n + 2N )|², θ2 ) f (r(n)|²2 , θ2 ) · f (r(n + N )|²2 , θ2 ) · f (r(n + 2N )|²2 , θ2 ) Y Y f (r(n), r(n + N )|²2 , θ2 ) f (r(n), r(n + N )|²2 , θ2 ). n∈R1. n∈R2. Y. f (r(n)|²2 , θ2 ) · f (r(n + N )|²2 , θ2 ) n∈R f (r(n)|²2 , θ2 ) · f (r(n + N )|²2 , θ2 ) 3. f (r(n)|²2 , θ2 ),. (3.68). n. where R0 = {m|0 ≤ m ≤ N144 − 1} ∩ {n < N } ∩ {n + N ≤ K}, R1 = {m|N144 ≤ m ≤ 2N144 − 1} ∩ {n + 2N ≤ K}, R2 = {m|N144 ≤ m ≤ 2N144 − 1} ∩ {n + 2N > K} ∩ {n + N ≤ K}, R3 = {n|θ2 + N144 ≤ n ≤ θ2 + N160 − 1} ∩ {n + N ≤ K},. (3.69). with.  n < (θ2 + N144 ) ,  (n − θ2 ) mod M144 , N144 − 1, (θ2 + N144 ) ≤ n ≤ (θ2 + N160 − 1) , m=  (n − (N160 − N144 ) − θ2 ) mod M144 , else.. (3.70). From the previous discussion, we can conclude that the log-likelihood function can be written as. .    γ2 (n, θ2 ) + 2<   . X. Λ2 (θ2 , ²2 ) =  n∈R0. S. R2. (Ã +2<. S. γ1 (n, θ2 ) +. n∈R1. R3. X. X. ). ! D2 (n, θ2 ) e+j2π²2. (Ã − 2<. n∈R1. X.  X n∈R0. S. R2. S. D1 (n, θ2 ) e+j2π²2 R3. ! D3 (n, θ2 ) e+j4π²2.   . ) (3.71). n∈R1. = B1 (n, θ2 ) + 2 |B2 (n, θ2 )| cos (2π²2 + ∠B2 (n, θ2 )). −2 |B3 (n, θ2 )| cos (4π²2 + ∠B3 (n, θ2 )) , 31. (3.72).

(47) where X. B1 (n, θ2 ) = n∈R0. B2 (n, θ2 ) = n∈R0. B3 (n, θ2 ) =. X. S. R2. S. X S. R2. S. X. γ1 (n, θ2 ) +. γ2 (n, θ2 ),. (3.73). n∈R1. R3. D1 (n, θ2 ) +. X. D2 (n, θ2 ),. (3.74). n∈R1. R3. D3 (n, θ2 ),. (3.75). n∈R1. with γ1 (n, θ2 ) =. −|ω(m)|2 (|r(n)|2 + |r(n + N )|2 ) − log(1 − |ω(m)|2 ), 2 2 σr (1 − |ω(m)| ). γ2 (n, θ2 ) =. −|ω(m)|2 (|r(n)|2 + |r(n + N )|2 ) − |ω(m − N144 )|2 (|r(n + N )|2 + |r(n + 2N )|2 ) σr2 (1 − |ω(m)|2 − |ω(m − N144 )|2 ) − log(1 − |ω(m)|2 − |ω(m − N144 )|2 ),. (3.76). (3.77). and. with. 3.2.3. D1 (n, θ2 ) =. r(n)r∗ (n + N )ω(m) , σr2 (1 − |ω(m)|2 ). (3.78). D2 (n, θ2 ) =. r(n)r∗ (n + N )ω(m) + r(n + N )r∗ (n + 2N )ω(m − N144 ) , σr2 (1 − |ω(m)|2 − |ω(m − N144 )|2 ). (3.79). D3 (n, θ2 ) =. r(n)r∗ (n + 2N )ω(m)ω(m − N144 ) , σr2 (1 − |ω(m)|2 − |ω(m − N144 )|2 ). (3.80). P 144 −1 10SN R/10 J0 (2πfd T ) · N ϕh (m − k, 0) k=0 ω(m) = . 10SN R/10 J0 (2πfd T ) · σh2 + 1. (3.81). Joint estimation of STO, CFO, and CP type. From (3.59), the joint estimation for θ1 and ²1 can be achieved as follows: max Λ1 (θ1 , ²1 ) = θ1 ,²1. max. max Λ1 (θ1 , ²1 ),. θ1 ∈{0,1,··· ,K−1} ²1 |θ1. (3.82). that is, θ1 takes a value from 0 to K − 1 (the observation window size), and for each θ1 , the corresponding ²1 is computed, which entails a solution of a 4th-order polynomial equation. Because by doing so, the complexity will be very high, we consider an approximation. We 32.

(48) assume that if the set {n|n ∈ R1 } is large enough, then the angle of A3 (n, θ1 ) is close to −4π²1 , and so the last cosine term in (3.59) is close to one. Thus, the log-likelihood function becomes Λ1 (θ1 , ²1 ) ≈ A1 (n, θ1 ) + 2 |A2 (n, θ1 )| cos (2π²1 + ∠ (A2 (n, θ1 ))) − 2 |A3 (n, θ1 )| .. (3.83). Maximizing the approximate log-likelihood function (3.83), we get the joint quasi-ML estimation of θ1 and ²1 in two steps as arg max Λ1 (θ1 , ²1 ) = arg max max Λ1 (θ1 , ²1 ) = arg max Λ1 (θ1 , ²ˆ1 (θ1 )). θ1 ,²1. θ1. θ1. ²1 |θ1. (3.84). For each θ1 , the maximization of Λ1 (θ1 , ²1 ) with respect to ²1 is achieved with the cosine term in (3.83) equal to one. This yields ²ˆ1 (θ1 ) = −. 1 ∠A2 (n, θ1 ) + n 2π. (3.85). where n is an integer. In this work, we assume |²1 | < 0.5; thus n = 0. With the cosine term in (3.83) equal to one, the log-likelihood function becomes Λ1 (θ1 , ²ˆ1 (θ1 )) = A1 (n, θ1 ) + 2 |A2 (n, θ1 )| − 2 |A3 (n, θ1 )| .. (3.86). Finally, the estimation of θ1 for extended CP can be obtained as θˆ1 = arg max {A1 (n, θ1 ) + 2 (|A2 (n, θ1 )| − |A3 (n, θ1 )|)} . θ1. (3.87). Similarly, the joint estimation of θ2 and ²2 for normal CP is given by θˆ2 = arg max {B1 (n, θ2 ) + 2 (|B2 (n, θ2 )| − |B3 (n, θ2 )|)} , θ2. ²ˆ2 (θ2 ) = −. 1 ∠B2 (n, θˆ2 ). 2π. (3.88) (3.89). In summary, the overall joint estimation of STO, CFO, and CP type is given by ˆ ²ˆ, m} {θ, ˆ = max f (r|θ, ², m) = max max max Λ(r|θ, ²). m. θ,²,m. θ|m. ²|θ,m. (3.90). Tables 3.1 and 3.2 give a summary of the solution and various quantities defined in the derivation of the solution.. 33.

(49) Table 3.1: Joint Estimation of STO, CFO for Extended CP θˆ1. Estimator. = arg max {A1 (n, θ1 ) + 2 (|A2 (n, θ1 )| − |A3 (n, θ1 )|)} θ1. ²ˆ1 (θ1 ) =. R0. Regions. R1 R2 m. 1 − ∠A2 (n, θˆ1 ) 2π. = {m|0 ≤ m ≤ N512 − 1} ∩ {n < N } ∩ {n + N ≤ K} = {m|N512 ≤ m ≤ 2N512 − 1} ∩ {n + 2N ≤ K} = {m|N512 ≤ m ≤ 2N512 − 1} ∩ {n + 2N > K} ∩ {n + N ≤ K} = (n − θ1 ) mod M512 X. A1 (n, θ1 ). n∈R0. S. X. A2 (n, θ1 ). n∈R0. S. λ1 (n, θ1 ) +. X. λ2 (n, θ1 ). n∈R1. R2. T1 (n, θ1 ) + R2. A3 (n, θ1 ). X. X. T2 (n, θ1 ). n∈R1. T3 (n, θ1 ). n∈R1. ¡ ¢ |ρ(m)|2 |r(n)|2 + |r(n + N )|2 − − log(1 − |ρ(m)|2 ) σr2 (1 − |ρ(m)|2 ). λ1 (n, θ1 ). λ2 (n, θ1 ). −. ¡ ¢ ¡ ¢ |ρ(m)|2 |r(n)|2 + |r(n + N )|2 + |ρ(m − N512 )|2 |r(n + N )|2 + |r(n + 2N )|2 σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ). −. log(1 − |ρ(m)|2 − |ρ(m − N512 )|2 ). T1 (n, θ1 ). r(n)r∗ (n + N )ρ(m) σr2 (1 − |ρ(m)|2 ). T2 (n, θ1 ). r(n)r∗ (n + N )ρ(m) + r(n + N )r∗ (n + 2N )ρ(m − N512 ) σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ). T3 (n, θ1 ). r(n)r∗ (n + 2N )ρ(m)ρ(m − N512 ) σr2 (1 − |ρ(m)|2 − |ρ(m − N512 )|2 ). ρ(m). PN512 −1 10SN R/10 J0 (2πfd T ) · k=0 ϕh (m − k, N ) 10SN R/10 J0 (2πfd T ) · σh2 + 1. 34.

(50) Table 3.2: Joint Estimation of STO, CFO for Normal CP θˆ2. Estimator. =. ²ˆ2 (θ2 ) =. Regions. R0. =. R1 R2 R3. = = =. m =. arg max {B1 (n, θ2 ) + 2 (|B2 (n, θ2 )| − |B3 (n, θ2 )|)} θ2. 1 − ∠B2 (n, θˆ2 ) 2π. {m|0 ≤ m ≤ N144 − 1} ∩ {n < N } ∩ {n + N ≤ K} {m|N144 ≤ m ≤ 2N144 − 1} ∩ {n + 2N ≤ K} {m|N144 ≤ m ≤ 2N144 − 1} ∩ {n + 2N > K} ∩ {n + N ≤ K} {n|θ2 + N144 ≤ n ≤ θ2 + N160 − 1} ∩ {n + N ≤ K} ( (n − θ2 ) mod M144 , n < (θ2 + N144 ) N144 − 1, (θ2 + N144 ) ≤ n ≤ (θ2 + N160 − 1) (n − (N160 − N144 ) − θ2 ) mod M144 , else X. B1 (n, θ2 ) n∈R0. S. R2. X. B2 (n, θ2 ) n∈R0. S. R2. S. S. γ1 (n, θ2 ) + D1 (n, θ2 ) +. B3 (n, θ1 ). X. γ2 (n, θ2 ). n∈R1. R3. R3. X X. D2 (n, θ2 ). n∈R1. D3 (n, θ2 ). n∈R1. ¡ ¢ |ω(m)|2 |r(n)|2 + |r(n + N )|2 − − log(1 − |ω(m)|2 ) σr2 (1 − |ω(m)|2 ). γ1 (n, θ2 ). γ2 (n, θ2 ). −. ¡ ¢ ¡ ¢ |ω(m)|2 |r(n)|2 + |r(n + N )|2 + |ω(m − N144 )|2 |r(n + N )|2 + |r(n + 2N )|2 σr2 (1 − |ω(m)|2 − |ω(m − N144 )|2 ). −. log(1 − |ω(m)|2 − |ω(m − N144 )|2 ). D1 (n, θ2 ). r(n)r∗ (n + N )ω(m) σr2 (1 − |ω(m)|2 ). D2 (n, θ2 ). r(n)r∗ (n + N )ω(m) + r(n + N )r∗ (n + 2N )ω(m − N144 ) σr2 (1 − |ω(m)|2 − |ω(m − N144 )|2 ). D3 (n, θ2 ). r(n)r∗ (n + 2N )ω(m)ω(m − N144 ) σr2 (1 − |ω(m)|2 − |ω(m − N144 )|2 ). ω(m). PN144 −1 10SN R/10 J0 (2πfd T ) · k=0 ϕh (m − k, N ) 10SN R/10 J0 (2πfd T ) · σh2 + 1. 35.

(51) Chapter 4 Simulation Results and Analysis In this chapter we first give the simulation conditions. Then we present and analyze some simulation results for LTE-A systems with our quasi-ML synchronization method derived previously. Our focus is on the performance in detecting the CP type.. 4.1. Simulation Conditions. We consider the FDD mode of LTE-A system in the SISO operating condition. The system parameters used are listed in Table 4.1. We consider the following channel models: AWGN, single-path Rayleigh, Standford University Interim (SUI) and Pedestrian B (PB), which is one of the ITU-R models. The SUI channel models consist of 6 different radio channel models in three terrain categories [3]. The three terrain types corresponding to the SUI channels in suburban area are shown in Table 4.2. SUI-1 and SUI-2 are Rician multi-path channels, whereas the other four are Rayleigh multi-path channels which exhibit greater root-mean-square (RMS) delay spread. The SUI2 represents a worst-case condition for terrain type C. For the SUI channels, we only consider the modified SUI-1 to SUI-4 models in our simulation such that the CIR lengths are less than the minimum CP length. The channel characteristics of SUI-1 to SUI-4 and Pedestrian B are shown in Tables 4.3– 4.7. Since the power delay profiles (PDPs) of those channel models are based on actual measurements, it may not be equal to integer multiples of the sampling period in the LTEA system. In computing, it will cost a very large amount of memory if we directly call 36.

(52) Table 4.1: Downlink System Parameters Used in Simulation Parameters Setting Carrier frequency 2.5 GHz Bandwidth 1.4 MHz Sampling frequency 1.92 MHz FFT size 128 CP types Extended, Normal CP lengths (N512 , N160 , N144 ) 32, 10, 9 Modulation type 16 QAM Channel models Modified SUI-1, SUI-2, SUI-3, SUI-4 Normalized CFO [−0.5, 0.5] STO [1, M ] Velocity 3 km, 120 km, 360 km Window size 2N + 1N512 6N + 5N512. Table 4.2: SUI Channel Models for Three Terrain Types Terrain type Description SUI channels A Hilly terrain, heavy tree SUI-5, SUI-6 B Between A and C SUI-3, SUI-4 C Flat terrain, light tree SUI-1, SUI-2. Tap 1 2 3. Table 4.3: SUI-1 Channel Model Relative delay Average power (µs or sample number) µs sample numbers dB normalized dB 0 0 0 −0.1771 0.4 1 −15 −15.1771 0.9 2 −20 −20.1771. “f ilter()” function in MATLAB. So we modify the PDPs to solve the problem. The term modif ied means that PDP is adjusted by forcing the CIR taps into integer multiples of the sampling period by rounding delays. This method is to shift the taps into the closest sampling instances, for preserving the path number and the path powers. Figure 4.1 depicts the method of rounding. Because the system bandwidth we used for simulation is 1.4 MHz, some of taps of PB channel in this case will overlap on the same sample, and for them we adjust the PDP path locations in order to avoid overlapping.. 37.

(53) Tap 1 2 3. Table 4.4: SUI-2 Channel Model Relative delay Average power (µs or sample number) µs sample numbers dB normalized dB 0 0 0 −0.3930 0.4 1 −12 −12.3930 1.1 2 −15 −15.3930. Tap 1 2 3. Table 4.5: SUI-3 Channel Model Relative delay Average power (µs or sample number) µs sample numbers dB normalized dB 0 0 0 −1.5113 0.4 1 −5 −6.5113 0.9 2 −10 −11.5113. Tap 1 2 3. Table 4.6: SUI-4 Channel Model Relative delay Average power (µs or sample number) µs sample numbers dB normalized dB 0 0 0 −1.9218 1.5 3 −4 −5.9218 4 8 −8 −9.9218. 38.

(54) Table 4.7: PB Channel Model Relative delay Average power (µs or sample number) µs sample numbers dB normalized dB 0 0 0 −3.9114 0.2→0.3 0→1 −0.9 −4.8114 0.8 2 −4.9 −8.8114 1.2→1.5 2→3 −8 −11.9114 2.3 4 −7.9 −11.8114 3.7 7 −23.9 −27.8114. Tap 1 2 3 4 5 6. Original. 0. Adjusted. ts. 2t s. 3ts. Delay. Figure 4.1: Tap location adjustment by rounding to integer sample spacing.. 4.2. Simulation Method. We describe the simulation setting and computational flow in this section. The simulation result under this environment is based on observing the error rate of the CP type. First, we construct a two-slot long transmitted signal in the time domain with a random CP type. Then we put the transmitting signals into channel by calling “f ilter()” function after setting up channel parameters by calling the “awgn” or “rayleighchan” function in MATLAB. Finally, we estimate the error rate of CP type at the receiver side. In the following sections, all the results shown in the figures are averages over 2 × 105 runs. The estimation process can be devided into several steps as follows: • Step 1: Use (3.87) and (3.88) to find the STOs θ1 and θ2 . • Step 2: Substitute the estimated θ1 and θ2 into (3.85) and (3.89) to find the CFOs ²1 and ²2 . 39.

(55) • Step 3: Compare the log-likelihood function values corresponding to the two different CP types and pick the one having the larger value.. 4.3. CP Type Estimation Performance with A Small Observation Window. We consider the CP type estimation performance at an observation window size of 2Nf f t + N512 in this section. The channels simulated include AWGN, single-path Rayleigh, and multipath fading channels.. 4.3.1. AWGN and Single-Path Rayleigh Fading Channels. Figures 4.2 and 4.3 show the performance of the CP detection error rate in AWGN and single-path Rayleigh channels, respectively. In these figures, the lines with “CP1” legend mean the error rate of that eNB transmits the extended CP type (denoted CP1 for simplify), but the receiver detects to the normal CP type (denoted CP2 for simplify). And the lines with “CP2” legend mean the error rate of that eNB transmits the CP2 type, but the receiver detects to the CP1 type. From these figures, the performance for transmitting CP1 type is much better (having less error) than the other. This can be explained as follows. Because the channel considered in these two cases are AWGN and single-path Rayleigh, the data that are correlated are just in two regions due to the CP structure (with no effect of spread of multi-path channel). And we assume that the log-likelihood function is maximized when the observed window includes the maximum pairs of correlated data. Figure 4.4 depicts the window sliding structure that captures the essence. If CP1 type is transmitted, the numbers of maximum pairs of correlated data are always N512 for the CP1 estimator, and they are changed with maximum value N160 + N144 for the CP2 estimator. The difference of maximum number of pairs between CP1 estimator and CP2 estimator with transmitting CP1 type is N512 − (N160 + N144 ). However, if CP2 type is transmitted, the maximum number of pairs of correlated data is N160 + N144 for the CP2 estimator, while it is N160 + N144 (the same value) for the CP1 estimator. There is no difference of maximum number of pairs between CP1 estimator and CP2 estimator with transmitting CP2 type. Figure 4.5 shows the maximum number of 40.