正交分頻多工通信系統於時變與多重路徑衰減通道之通道估測與訊號偵測設計

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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. 正交分頻多工通信系統於時變與多重路徑衰減通道之通道估測與訊號偵測設計 Design of Channel Estimation and Data Detection for OFDM Systems in Time-varying and Multipath Fading Channels. 研究生：蔡金融指導教授：陳紹基博士. 中華民國九十五年七月.

(2) 正交分頻多工通信系統於時變與多重路徑衰減通道之通道估測與訊號偵測設計 Design of Channel Estimation and Data Detection for OFDM Systems in Time-varying and Multipath Fading Channels 研究生：蔡金融. Student：Chin-Jung Tsai. 指導教授：陳紹基博士. Advisor：Sau-Gee Chen. 國立交通大學電子工程學系. 電子研究所所碩士班. 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 in Electronics Engineering. July 2006 Hsinchu, Taiwan, Republic of China. 中華民國九十五年七月.

(3) 正交分頻多工通信系統於時變與多重路徑衰減通道之通道估測與訊號偵測設計. 學生: 蔡金融. 指導教授: 陳紹基博士國立交通大學電子工程學系電子研究所碩士班. 摘. 要. 在本篇論文中，探討了數種正交多工分頻系統於時變與多重路徑衰減通道的通道估測與訊號偵測技術。在單個正交多工分頻系統符元內的通道變化，將會破壞次載波彼此間的正交性，而其所造成次載波間的干擾會使系統效能下降。此效能誤差量將會隨著移動的速度、載波頻率或是符元的持續時間增加而變的嚴重。基於現有的通道估測與訊號偵測方法，我們提出了一些改進的方式。所提出的訊號偵測方法利用了時變通道內隱含的時間多樣性，而不是只將其視為次載波間的干擾。它的效能較一般次載波干擾消去法優異，並且擁有可接受的複雜度。而所提出的通道估測方法基於適當的數學模型假設。此數學模型可以有效減少所需估計的參數，使估測可行並且更為精準。. i.

(4) Design of Channel Estimation and Data Detection for OFDM Systems in Time-varying and Multipath Fading Channels Student: Chin-Jung Tsai. Advisor: Sau-Gee Chen. Department of Electronics Engineering & Institute of Electronics National Chiao Tung University. Abstract In this thesis, several methods for channel estimation and data detection of OFDM systems in time-varying and multipath fading channels are investigated. Channel variations over an OFDM symbol destroy the orthogonality between subcarriers. The corresponding intercarrier-interference (ICI) degrades the system performance. This error floor becomes more severe as mobile speed, carrier frequency, or symbol duration increases. Based on the existing channel estimation methods and the data detection methods, several modifications to improve the system performances or to reduce the system computational complexities are proposed. The proposed data detection methods exploit time diversity involved in the nature of the time-varying channels. Thus, the proposed methods outperform the conventional methods, like ICI cancellation method and least square (LS) detection. For implementation, the proposed methods are modified to have acceptable complexities. The proposed channel estimation methods are based on appropriate mathematical models. The estimated parameters can be reduced by using these mathematical models. Therefore, channel estimation in fast fading channels becomes feasible and trackable. ii.

(5) 誌首先要感謝我的指導教授. 謝. 陳紹基博士。當我遇到研究上的疑惑與困. 難時，能適時的給予指導以及提供思考方向，使我能順利的完成本篇論文，而在生活上老師也提供了很多幫助與寶貴的經驗，在此致上由衷的感謝。在碩士生涯的兩年中，我也要感謝實驗室的夥伴文威、敏杰、譽桀、昀震、彥欽和聖國在生活上的幫助與關心，而跟你們研究上的討論，使我獲益良多。和你們同窗的這段時間也讓我了解到需要學習的事情還很多。另外要特別感謝曲建全學長在研究上與生活上的幫助。能夠在這間實驗室和各位一同奮鬥、歡笑真的覺得很幸運，祝各位將來一切順利。最後感謝我的父母與家人，從小以來對我的呵護與栽培，讓我能順利完成學業，並且擁有寶貴的人生經歷，在此獻上無限的感激與敬意。. iii.

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(7) Contents 中文摘要. i. Abstract. ii. 誌謝. iii. Content. v. List of Tables. ix. List of Figures. x. Chapter 1 Introduction. 1. 1.1 Background and Motivation. 2. 1.2 Organization of the Thesis. 4. Chapter 2 Fundamentals of OFDM Systems 2.1 OFDM System Model. 6 7. 2.1.1 Continuous-time Model. 7. 2.1.2 Discrete-time Model. 9. 2.2 Channel Characteristics in Wireless Communication Environments 10 2.3 Analysis of Intercarrier Interference. 13. Chapter 3 Investigation of Existing Data Detection and Channel Estimation Methods. 17. 3.1 Existing Data Detection Methods. 18. 3.1.1 Single-tap Equalization. 18. 3.1.2 Least-square Detection (Decorrelating Detection). 18. 3.1.3 Jeon’s Method of Least-square Detection. 19. 3.1.4 Intercarrier Interference Cancellation. 21. 3.1.5 Choi’s Method of Successive Detection. 22. 3.1.6 Simulation Results. 25. v.

(8) 3.2 Existing Channel Estimation Methods. 29. 3.2.1 Least-square Channel Estimation. 29. 3.2.2 DFT-based Channel Estimation. 30. 3.2.3 Chen’s Method of Least-square Channel Estimation. 31. 3.2.4 Yeh’s Algorithm of ICI-reduction Method. 34. 3.2.5 Simulation Results. 38. Chapter 4 The Proposed Data Detection and Channel Estimation Methods. 44. 4.1 The Proposed Frequency-domain Successive Detection Methods. 44. 4.1.1 Algorithms of the Frequency-domain Successive Detection Methods. 45. 4.1.2 Complexity Analysis 4.2 The Proposed Channel Estimation Methods 4.2.1 Modification of Yeh’s ICI-reduction Method. 48 55 55. 4.2.2 The Hybrid LS/ICI-reduction Channel Estimation Method 56. Chapter 5 Simulation Results. 58. 5.1 The Performances of the Proposed Frequency-domain Successive Detection methods. 59. 5.2 The Effect of Channel Estimation Error of the Proposed Frequency-domain Successive Detection Methods 5.3 The Performances of the Proposed Channel Estimation Methods. 65 69. Chapter 6 Conclusion. 74. Bibliography. 76 vi.

(9) Autobiography. 81. vii.

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(11) List of Tables Table 3.1. Simulated OFDM system parameters. 25. Table 3.2. Simulated OFDM system parameters. 39. Table 3.3. ETSI Vehicular A channel environment. 39. Table 4.1. Complexity statistics of the proposed and Choi’s successive detection methods in the first detection layer. Table 4.2. Complexity statistics of the proposed and Choi’s successive detection methods in the detection layer before the last one. Table 4.3. 50. Complexity statistics of the proposed and Choi’s successive detection methods in the first detection layer with N = 64, and q = 8, 4. Table 4.4. 50. 50. Complexity statistics of the proposed and Choi’s successive detection methods in the detection layer before the last one with N = 64, and q = 8, 4. 51. Table 4.5. Complexity statistics of the other mentioned methods. 52. Table 4.6. Complexity ratio % of the detection methods N = 64, q = 8, 4 and, I = 3 52. ix.

(12) List of Figures Figure 2.1. Cyclic prefix of an OFDM symbol. 7. Figure 2.2. Continuous-time OFDM baseband modulator. 7. Figure 2.3. Spectrum of an OFDM symbol. 8. Figure 2.4. Continuous-time OFDM baseband demodulator. 9. Figure 2.5. Discrete-time OFDM system model. 10. Figure 2.6. Illustrating the calculation of Doppler frequency. 12. Figure 2.7. SIR for OFDM systems in time-varying channels. 15. Figure 2.8. Normalized magnitude responses for different f nd ’s. 15. Figure 2.9. Channel impulse responses within an OFDM symbol for different f nd ’s 16. Figure 3.1. Transformation of ICI matrix G to block diagonal G . q = 2. 20. Figure 3.2. Choi’s methods of successive detection. 24. Figure 3.3. BER performances of the existing data detection methods in the two-ray equal power channel with f nd = 0.040. Figure 3.4. 26. BER performances of the existing data detection methods in the two-ray equal power channel with f nd = 0.083. 26. Figure 3.5. Regular pilot placement. 30. Figure 3.6. Linear model between two consecutive OFDM symbols. 36. Figure 3.7. Yeh’s algorithm of ICI-reduction method. 37. Figure 3.8. ANMSE performances of Chen’s and Yeh’s methods of channel estimation in the “Vehicular A” channel with f nd = 0.040. Figure 3.9. 40. ANMSE performances of Chen’s and Yeh’s methods of channel estimation in the “Vehicular A” channel with f nd = 0.083. 40. Figure 3.10 BER performances of Chen’s and Yeh’s methods of channel estimation x.

(13) combined with data detection methods in the “Vehicular A” channel with f nd = 0.040. 41. Figure 3.11 BER performances of Chen’s and Yeh’s methods of channel estimation combined with data detection methods in the “Vehicular A” channel with f nd = 0.083. 41. Figure 4.1. The proposed frequency-domain successive detection methods. 47. Figure 4.2. Modified linear model between two consecutive OFDM symbols. 55. Figure 4.3. Procedure of the proposed channel estimation and data detection. 57. Figure 5.1. BER performances of the data detection methods in the two-ray equal power channel with f nd = 0.040. Figure 5.2. BER performances of the data detection methods in the two-ray equal power channel with f nd = 0.083. Figure 5.3. 61. BER performances of the data detection methods in the “Vehicular A” channel with f nd = 0.083. Figure 5.5. 61. BER performances of the proposed method in different detection layers in the “Vehicular A” channel with f nd = 0.083. Figure 5.6. 64. BER performances of the detection methods for f nd = 0.040 , versus normalized mean square error η. Figure 5.8. 64. BER performances of the proposed method in different subcarriers in the “Vehicular A” channel with f nd = 0.083. Figure 5.7. 60. BER performances of the data detection methods in the “Vehicular A” channel with f nd = 0.040. Figure 5.4. 60. 67. BER performances of the detection methods for f nd = 0.083 , versus normalized mean square error η xi. 67.

(14) Figure 5.9. BER performances of the proposed detection methods for f nd = 0.040 , versus normalized mean square error η. 68. Figure 5.10 BER performances of the proposed detection methods for f nd = 0.083 , versus normalized mean square error η. 68. Figure 5.11 BER performances of the proposed detection methods for different normalized Doppler frequencies f nd ’s, versus normalized mean square error η in SNR = 35. 69. Figure 5.12 ANMSE performances of the channel estimation methods in the “Vehicular A” channel with f nd = 0.040. 70. Figure 5.13 ANMSE performances of the channel estimation methods in the “Vehicular A” channel with f nd = 0.083. 70. Figure 5.14 BER performances of the channel estimation methods combined with data detection methods in the “Vehicular A” channel with f nd = 0.040 71 Figure 5.15 BER performances of the channel estimation methods combined with data detection methods in the “Vehicular A” channel with f nd = 0.083 71. xii.

(15) Chapter 1 Introduction. In recent years, the demand for multimedia services increases rapidly, such as high quality video and audio data. Thus, the next generation communication systems are expected to provide high data rate transmissions. To satisfy these requirements, several transmission schemes must efficiently use transmission bandwidth. Multicarrier techniques are such transmission schemes widely adopted in commercial communication systems. When applied in a wireless environment, it is usually referred to as Orthogonal Frequency Division Multiplexing (OFDM). This modulation scheme has been adopted in many wireless transmission standards, such as wireless LAN (Local Area Network) IEEE 802.11a, wireless MAN (Metropolitan Area Network) IEEE 802.16, European Digital Audio Broadcasting (DAB) and Digital Video Broadcasting Terrestrial (DVB-T).. 1.

(16) 1.1 Background and Motivation In OFDM systems, a high data-rate serial data stream is split into numerous low data-rate parallel data streams. They are modulated by several orthogonal subcarriers. For practical implementation, the oscillators generating subcarriers are replaced by inverse fast Fourier transform (IFFT) [1] [2] [3]. The spectrums of these modulated signals overlap each other over the frequency domain only except the frequency points where subcarriers allocate. This is why the transmission bandwidth is efficiently used in OFDM systems. The symbol duration is lengthened by the parallel transmission of the low data-rate data stream. This long symbol duration of the OFDM signal facilitate to mitigate the channel delay spread in wireless communication channels. By adding a cyclic prefix (CP) to the beginning of each OFDM symbol, the intersymbol interference (ISI), and intercarrier interference (ICI) caused by delay spread would be overcome. The corresponding channel response appears as a multiplication in the frequency domain. Therefore, a flat fading channel model can be assumed for each subchannel. A single tap equalizer is good enough for coherent detection. To perfectly combat the ICI, and ISI, the length of the CP must be longer than the maximum channel delay spread. However, the system throughput descends as the length of the CP increases. Usually, it is assumed that the channel response is quasi-static within an OFDM symbol. With such assumption, the effect of the multipath channel equals a complex multiplication to the modulation at the subcarrier. This assumption is reasonable when the symbol duration is short or the transceivers are fixed. However, in some standards, such as DVB-T IEEE 802.16e, and IEEE 802.20, the symbol durations are very long and the systems are not only designed for the fixed users but also for mobile users. The channel response is not quasi-static within an OFDM symbol but time-varying.. 2.

(17) Channel variations within an OFDM symbol destroy the orthogonality between subcarriers and result in ICI. Conventional channel estimation techniques and 1-tap frequency-domain equalizer are subject to the ICI effect. The ICI effect degrades the system performance. The corresponding error floor becomes more severe as mobile speed, carrier frequency, or symbol duration increase, especially for higher-order modulation schemes [4] [5]. To mitigate the ICI due to channel variation, some detection and equalization methods, e.g., iterative ICI cancellation [6] [11] [27], least square (LS) detection [8] [18] [20], iterative LS detection [18], frequency-domain linear filter [21], and frequency-domain decision feed-back filter [22] [23] have been proposed. However, these approaches only treat ICI just as interference obstructing data detection and intend to suppress as much ICI as possible. Thus, the performances of these methods are bounded by the performance of the parallel detection in time-invariant channels. Choi et al. [7] showed a new idea that the time-varying nature of a channel can be exploited as a provider of time diversity and proposed a time-domain successive detection method. Therefore, their method can be applied to rapidly time-varying fading channels and outperform the other mentioned methods. However, the computational complexity of this method is too high to be realized. Some channel estimation methods for time-varying fading channels, e.g., iterative LS estimation [24], frequency-domain pilot-symbol-aided MMSE estimation [7] [21], frequency-domain pilot-symbol-aided estimation [25], and time-domain pilot-symbol-aided estimation [8] [22] also have been proposed. However, time-domain low-pass filter applied in [24] can not suppress ICI perfectly when channel variation increases. MMSE estimation in [7] needs pilot symbols for long time duration. Thus, the approach suffers long system latency. Besides, MMSE estimation in [7] [21] requires the characteristics of the fading channels so that it is hard to be realized. The 3.

(18) approach in [25] can only be used in flat fading channels. The structures of the time-domain pilot symbols used in [8] [22] are different from that of the frequency-domain pilot patterns applied in commercial systems. Besides, the approaches in [8] [22] can only be used in slowly time-varying channels. In [6] [18], the authors have proposed a mathematical model used in the channel estimation assuming linear channel variation. Based on the existing channel estimation methods [6] [18] and the data detection methods [7] [8], several modified methods to improve the system performances or to reduce the system computational complexities are proposed in this thesis. The proposed data detection methods exploit the time diversity involved in the nature of the time-varying channels, which is introduced in [7]. Thus, the proposed methods outperform the conventional methods [6] [8] [11] [20] [21] [22] [23] [27]. For implementation, the proposed methods are modified to have acceptable complexities by utilizing the technique introduced in [8]. The proposed channel estimation method is based on appropriate mathematical models modified from [6] [11] [18]. As a result, the estimated parameters can be reduced. Therefore, channel estimation under fast fading conditions becomes feasible and trackable.. 1.2 Organization of the Thesis This thesis is organized as follows. In Chapter 2, the fundamentals of OFDM systems and the characteristics of wireless multipath Rayleigh fading channels are described. Then, ICI effect of OFDM systems in time-varying channels is analysis. In Chapter 3, five existing data detection methods, single tap equalization, least square (LS) detection, Jeon’s method of LS detection, ICI cancellation, and Choi’s method of successive detection are introduced. Performances of these data detection methods are. 4.

(19) evaluated. Then, two conventional channel estimation methods, LS channel estimation and discrete-time Fourier transform (DFT)-based channel estimation, are presented. Two channel estimation methods for OFDM systems in time-varying channels are introduced. In the end of this chapter, performances of these channel estimation methods are evaluated. In Chapter 4, the proposed data detection methods are introduced. Also, computational complexities of the proposed data detection methods are evaluated. Then, the proposed channel estimation methods are introduced. In Chapter 5, performances of the proposed data detection methods and the channel estimation methods are evaluated. Also, the effect of channel estimation error is discussed. In Chapter 6, conclusion is given.. 5.

(20) Chapter 2 Fundamentals of OFDM Systems. The basic concept of OFDM technique is to increase the symbol duration by splitting the high-rate data stream into numerous lower rate streams and equally to divide the available transmission spectrum into several narrowband subchannels. These low rate streams are transmitted simultaneously over these subchannels. The overlapping and orthogonal transmission spectrums of modulated signals are designed for high spectral efficiency. Due to the property of narrowband, each modulated signal transmitted in the subchannel almost experiences flat fading. Therefore, the channel equalization is simplified as a single-tap equalization. On the other hand, the lengthened OFDM symbol appended with a proceeding CP helps mitigate the ICI and ISI effects. A CP is the extension of an OFDM symbol that equals the tail part of the symbol, which is shown in Figure 2.1, and must be larger than the maximum channel delay spread to avoid ISI. Thus, the signals and their own delayed-version signals resulting from the multipath channel are still whole periodic waves within the DFT in6.

(21) terval. The orthogonality between subcarriers can be maintained after the modulated signals passes through the multipath channel, i.e. no ICI.. Cyclic prefix. Figure 2.1 Cyclic prefix of an OFDM symbol [11]. However, when symbol durations are very long or users have mobility, the channel response is not quasi-static within an OFDM symbol but time-varying. Channel variations within an OFDM symbol destroy the orthogonality between subcarriers and result in ICI. This phenomenon will be depicted in Section 2.3.. 2.1 OFDM System Model. 2.1.1 Continuous-time Model In this section, a continuous-time model of OFDM system is investigated. A typical continuous-time OFDM baseband modulator is shown in Figure 2.2. The input data stream is split into N parallel streams which are modulated by N different subcarriers and then transmitted simultaneously. X i (0) i. Data. X (1). S/P. #. X ( N − 1) i. φ0 (t ) φ1 (t ). xi (t ). φN −1 (t ). Figure 2.2 Continuous-time OFDM baseband modulator. 7.

(22) The k-th modulating subcarrier is ⎧ j 2π k (t −Tg ) T ⎪ , φk (t ) = ⎨ e ⎪⎩ 0 ,. 0 ≤ t ≤ Ts otherwise. (2.1). where T is the symbol duration exclusive of CP, Tg is the length of CP and Ts is the total symbol duration, i.e. Ts = T + Tg. X i (k ) is defined as the transmitted data which is a complex number from a set of signal constellation points at the k-th subcarrier for the i-th symbol. The modulated baseband signal of the i-th OFDM symbol is N −1. xi (t ) = ∑ X i (k )φk (t − iTs ). (2.2). k =0. When an infinite sequence of OFDM symbols is transmitted, the output of the transmitter can be represented as ∞. x(t ) =. ∞. N −1. ∑ x (t ) = ∑ ∑ X i. i =−∞. i =−∞ k = 0. i. (k )φk (t − iTs ). (2.3). The overlapping and orthogonal transmission spectrums of the modulated signals are shown in Figure 2.3. Because of the sinc shape of the spectrums of these modulated signals, the spectrums of them overlap with each other over the frequency domain only except the frequency points where subcarriers allocate. The subcarrier spacing between any two neighbor subcarriers is. 1 . Thus, OFDM systems have high T. bandwidth efficiency. 1 T. N subcarriers Figure 2.3 Spectrum of an OFDM symbol [11] 8.

(23) The received signal y (t ) can be expressed as ∞. y (t ) = ∫ h(t ,τ ) x(t − τ )dτ + w(t ). (2.4). −∞. where time-variant channel impulse response at time t is h(t ,τ ) , and w(t ) is an additive white complex Gaussian noise.. y (t ). Y i (0). ψ 0 (t ). Y i (1). ψ 1 (t ). P/S. #. Y i ( N − 1). ψ N −1 (t ). Figure 2.4 Continuous-time OFDM baseband demodulator. An typical continuous-time OFDM baseband demodulator is drawn in Figure 2.4.. ψ k (t ) is the matched filter for the k-th subcarrier. ⎧ 1 j 2Tπ kt , ⎪ e ψ k (t ) = ⎨ T ⎪0 , ⎩. 0 ≤ t ≤ Ts. (2.5). otherwise. Y i (k ) is the demodulated signal at the k-th subcarrier for the i-th symbol.. 2.1.2 Discrete-time Model In [3], Weinstein suggested that the modulators in the transmitter and the matched filters in the receiver for the OFDM systems can be implemented by IDFT and DFT, respectively. Also, the fast DFT/IDFT algorithm, FFT/IFFT, can be applied to the OFDM systems. Figure 2.5 shows the discrete-time OFDM system model.. 9.

(24) X i (0) i. X (1). # X i ( N − 1). y i (0). xi (0). I xi (1) D P/S F # T xi (N − 1). AWGN. y i (1). w(n). S/P. h ( n, l ). # y (N − 1) i. Y i (0). D F T. Y i (1). # Y i ( N − 1). Figure 2.5 Discrete-time OFDM system model Without loss of generality, the symbol index i is dropped in the following discussion. The modulated signal can be written as N −1. x ( n ) = ∑ X ( k )e. j 2π kn N. (2.6). k =0. where X (k ) is the transmitted data at the k-th subcarrier, and 0 ≤ n ≤ N − 1 . Then, the modulated signal appended with a proceeding CP is delivered over the air and experiences a time-varying and multipath fading channel. Therefore, the received signal can be represented as L −1. y (n) = ∑ h(n, l ) x(n − l ) N + w(n). (2.7). l =0. where h(n, l ) is the l -th channel path at time instant t = n × ts , ts =. T is sampling N. period, L is the number of channel taps, ( ) N represents a cyclic shift in the base of N, and w(n) is sampled additive white complex Gaussian noise with variance σ 2 . The received signal after DFT at the k-th subcarrier is N −1. Y ( k ) = ∑ y ( n)e. −. j 2π kn N. (2.8). n =0. 2.2 Channel Characteristics in Wireless Communication Environments The transmitted signal propagates along several different paths through the wireless channel and arrives at the receiver. Such environment is referred to as a multipath 10.

(25) channel. These paths arise from scattering, reflection, and diffraction of the radiated energy by objects in the environment or refraction in the medium. These duplicated signals add together at the receiver. Thus, the corresponding received signal is distorted. The amplitude of the received signal fluctuates with location and time. This phenomenon is referred to as fading. The impulse response of the fading channel can be expressed as L −1. h(t ,τ ) = ∑ α l (t )δ (τ − τ l ). (2.9). l =0. where L is the number of resolvable channel paths, and α l (t ) and τ l are the path gain and the path delay time of the l-th resolvable channel path, respectively. The path gain α l (t ) is modeled as [9] [10] for all l. M. α l (t ) = ∑ Cn ,l e j (2π f t cos γ d. n ,l +φn ,l. ). (2.10). n =1. where M is the number of the unresolvable paths, Cn ,l , γ n ,l , and φn ,l are respectively the random path gain, the angle of the incoming wave, and the initial phase associated with the n -th unresolvable of the l -th resolvable path, and f d is the maximum Doppler frequency. The central limited theorem justifies that α l (t ) can be approximated as complex Gaussian random process for large M [10]. γ n ,l , and φn ,l are assumed to be mutually independent and to be uniformly distributed over [−π , π ) for all n and l . Further, if path delay time of the resolvable path, τ l , is sample-spaced, i.e. τ l = l × ts and l is an integer, for all l. The impulse response of the fading channel can be simplified as h(n, l ) = α l (nts ). (2.11). for l = 0," , L − 1 . To specify the relationship between the variation of the fading channel and the. 11.

(26) motion of the receiver in (2.10), consider the situation illustrated in Figure 2.6.. S ∆l. γ n ,l. Direction of motion. A′. φn ,l + ∆φn ,l d. A. φn ,l Figure 2.6 Illustrating the calculation of Doppler frequency It is assumed that the transmitted signal propagating along the l -th path is modeled as the source S which is far away from the mobile station A . The mobile station A moves from the point A to the point A′ in the time duration from 0 to t . Thus, the change in the phase angle of the received signal between the point A and the point A′ is given by. ∆φn ,l =. 2π. λ. ∆l. fc d cos γ n ,l C fυ = 2π c t cos γ n ,l C = 2π f d t cos γ n ,l = 2π. (2.12). where λ , f c , C , and υ are the radio wavelength, radio center frequency, velocity of light, and velocity of the mobile station, respectively, and f d . f cυ is the maxiC. mum Doppler frequency occurring at γ n ,l = 0 . In conclusion, the variation of the fading channel is proportional to the ratio center frequency and velocity of the mobile receiver.. 12.

(27) 2.3 Analysis of Intercarrier Interference According to the system model depicted in Section 2.1.2 and the channel model described in Section 2.2, the received signal after DFT at the k-th subcarrier is expressed as N −1. Y ( k ) = ∑ y ( n)e. −. j 2π kn N. (2.8). n =0. Substitute equations (2.6) and (2.7) into (2.8) Y (k ) = G (k , k ) X (k ) +. N −1. ∑. G ( k , m) X ( m) + W ( k ) . (2.13). m = 0, m ≠ k. ICI. where. G ( k , m) =. 1 N. N −1 L −1. ∑∑ h(r , l )e. j 2π r ( m − k ) N. e. − j 2π lm N. N −1. ,. r =0 l =0. W (k ) = ∑ w(n)e. −. j 2π kn N. ,. n =0. 0 ≤ m ≤ N − 1 , and 0 ≤ k ≤ N − 1 . Since the second term on the right hand side of. equation (2.13) is not zero, the desired signal suffers ICI when the channel is time-varying. On the contrary, ICI terms will vanish if f d in (2.10) equals zero, i.e. time-invariant channel. For further ICI analysis when f d ≠ 0 , let’s first define the time average of channel impulse response, h(n, l ) , as have (l ) =. 1 N. N −1. ∑ h(n, l ). (2.14). n=0. and the variation ∆h(n, l ) of h(n, l ) as ∆h(n, l ) = h(n, l ) − have (l ). (2.15). Therefore, L −1. G (k , k ) = ∑ have (l )e. − j 2π lk N. (2.16). l =0. and G ( k , m) =. 1 N. N −1 L −1. ∑∑ ∆h(r , l )e. j 2π r ( m − k ) N. e. − j 2π lm N. , m≠k. (2.17). r =0 l =0. If the channel impulse response is time-invariant, ∆h(n, l ) equals zero and G ( k , m) = 0 , m ≠ k .. The received signal after DFT can be rewritten as the following matrix form 13.

(28) Y=. 1 QHQ H X + W N. (2.18). where Y = [Y (0)," , Y ( N − 1)]T , X = [ X (0)," , X ( N − 1)]T , W = [W (0)," , W ( N − 1)]T , H (n, m) = h(n, (n − m) N ) , and Q is the N-point DFT matrix with its elements Q(n, m) = e − j 2π mn N . For convenience, G is defined as G . 1 QHQ H . Thus, the N. received signal after DFT can be written as Y = GX + W. (2.19). The effects of ICI have been analyzed in some papers. Li et al. [4] stated that in order to achieve signal to intercarrier interference ratio (SIR) > 20dB, the OFDM symbol duration must be less than 8% of the channel coherence time ( Tcoherence = 1 f d ) when a Wide-Sense Stationary and Uncorrelated Scattering (WSSUS) channel is considered. This effect is shown in Figure 2.7. In [8], Jeon et al. concluded that when the multipath fading channel is slowly time-varying (Normalized Doppler frequency< 0.1), ICI terms which do not significantly affect Y (k ) can be ignored and time variation of the channel impulse response can be approximated by a straight line. These ideas are illustrated in Figure 2.8 and Figure 2.9, respectively. The normalized Doppler frequency, f nd , is defined as f nd f d Tob. (2.20). where Tob is the observation duration. It is used to quantify the variation of the fading channel within the observation duration. Thus, when considering the variation of the fading channel within an OFDM symbol, the observation duration is set to the symbol duration. In this thesis, the investigation focuses on the effect of ICI within an OFDM symbol, i.e. Tob = T . In conclusion, OFDM systems suffer more ICI as normalized Doppler frequency increases.. 14.

(29) 40. SIR(dB). 35. 30. 25. 20. 15 0. 0.02. 0.04. 0.06. 0.08. 0.1. 0.12. f nd. Figure 2.7 SIR for OFDM systems in time-varying channels [4]. 10. 0. Normalized Magnitude. fnd = 0.1 fnd = 0.083 fnd = 0.04. 10. 10. -1. -2. -3. 10 -40. -30. -20. -10 0 10 Subcarrier Index. 20. 30. 40. Figure 2.8 Normalized magnitude responses for different f nd ’s [8]. 15.

(30) 1 0.9. fnd = 0.1 fnd = 0.083 fnd = 0.04. 0.8. Magnitude. 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0. 0.5. 1. 1.5 Time (sec). 2. 2.5 x 10. -5. Figure 2.9 Channel impulse responses within an OFDM symbol for different f nd ’s [8]. 16.

(31) Chapter 3 Investigation of Existing Data Detection and Channel Estimation Methods. A signal propagating through wireless channels will be distorted by it own duplications. The received signal would have a random phase rotation, and the magnitude of the received signal would be attenuated. This phenomenon has been depicted in Section 2.2. In order to coherently detect the transmitted data, the random phase rotation and the magnitude attenuation must be obtained and compensated before demapping the symbols according to the modulation constellation. In this chapter, existing common data detection methods for compensation of the received signal are first introduced. Then, several channel estimation methods for obtaining channel state information are described. These methods are all evaluated by computer simulations. The advantages and the disadvantages of these methods are indicated according to the simulation results.. 17.

(32) 3.1 Existing Data Detection Methods Several existing data detection methods are introduced in this section. The simulation results show that performances of the first four methods are not good enough in the assumed common channel environments. Only, Choi’s method of successive detection performs well in the assumed environments. In the following discussion, the system model introduced in Section 2.3 is considered.. 3.1.1 Single-tap Equalization The impulse response of a slow-fading channel is generally quasi-static within an OFDM symbol. Therefore, there is approximately no ICI between subcarriers. According to (2.13), the compensation for each subcarrier is equivalent to single-tap frequency-domain equalization. Z (k ) =. Y (k ) G (k , k ). (3.1). Then, the equalized signal Z (k ) is passed through a signal demapper. The demodulated data are obtained. This method is simple and widely used in indoor OFDM systems. However, in time-varying fading channels, single-tap equalization will be distorted by ICI. As such, there would be an irreducible error floor.. 3.1.2 Least-square Detection (Decorrelating Detection) Increasing the taps of an equalization process is a straightforward method to suppress the ICI introduced by time-varying fading channels. Given by the linear model (2.18) and basic linear algebra theory, the least-square (LS) detection is expressed by Z = G+Y. (3.2). where G + = (G H G ) −1 G H is the pseudo inversion of G . LS detection can suppress ICI perfectly. However, noise enhancement would be induced when the correlations 18.

(33) of the elements in G are high. Besides, the computational complexity of the LS detection is proportional to O( N 3 ) [16] when G is a square matrix with dimension of N. Thus, this detection method is hard to be implemented for the OFDM systems with numerous subcarriers, such as DVB-T and IEEE 802.16e.. 3.1.3 Jeon’s Method of Least-square Detection The LS detection described in the previous section is widely used for interference suppression. However, its computational complexity is proportional to O( N 3 ) . N is the number of mixed signals. For OFDM systems, N is the number of subcarriers due to the loss of the orthogonality between subcarriers in time-varying fading channels. Jeon et al. [8] proposed a method for computational complexity reduction of LS detection. After ICI analysis, they concluded that when the multipath fading channel is slowly time-varying (Normalized Doppler frequency f nd < 0.1), most power of channel frequency response concentrates on the neighborhood of the subcarriers. Thus, those ICI terms which do not significantly affect Y (k ) can be ignored, i.e. G (m, k ) = 0 , m − k > q 2. (3.3). where q denotes the number of dominant ICI terms. Figure 2.8 depicts this property, obviously. In physical meaning, the diagonal terms of G can be viewed as the D.C. components of the time variation of the channel response, and the neighbor terms of them are the corresponding lowpass components according to equations (2.14)-(2.17). By Jeon’s conclusion, the power of the channel response concentrates on the lowpass band in Doppler spectrum domain. Thus, high frequency components of channel variation could be ignored when the fading channel is slowly time-varying. Consequently, the matrix G becomes sparse and can be transformed as a block diagonal matrix G = diag ([G blk ,0 , G blk ,1 ," G blk , N −1 ]) as shown in Figure 3.1. Note that 19.

(34) there are some revisions compared to Jeon’s method.. G blk ,1 =. G=. 1. G=. 2. 0. G blk ,1 = 1. N −2. N −1. Figure 3.1 Transformation of ICI matrix G to block diagonal G . q = 2 [8]. The system model given in (2.19) is transformed as overlapped block-diagonal Y = GX + W. (3.4). where Y = [Yblk ,0 Yblk ,1 " Yblk , N −1 ]T X = [ Xblk ,0 Xblk ,1 " Xblk , N −1 ]T W = [ Wblk ,0 Wblk ,1 " Wblk , N −1 ]T Xblk ,k = [ X (k − q / 2) N " X (k )" X (k + q / 2) N ]T. (3.5). Yblk ,k = [Y (k − q / 2) N "Y (k )"Y (k + q / 2) N ]T Wblk ,k = [W (k − q / 2) N "W (k )"W (k + q / 2) N ]T. The D.C. components of channel variation and the corresponding dominated ICI terms between subcarriers are separated into a number of sub blocks which do not interfere with each other. Therefore, LS detection of the original system is modified as LS detections of the blocked systems + Zk = G blk , k Yblk , k. (3.6). where k = 0,1," , N − 1 . The middle element of the equalized vector Zk is an approximation to the k-th element of the vector Z in (3.2). Therefore, instead of computing the matrix inversion of the entire G , N matrix inversions of smaller G blk ,k ’s. 20.

(35) are needed in the simplified technique. The computational complexity of LS detection is reduced from O( N 3 ) to O( N (q + 1)3 ) . Usually, q N . Thus, the reduction is large.. 3.1.4 Intercarrier Interference Cancellation In order to avoid the noise enhancement problem and further reduce the computational complexity, the intercarrier interference (ICI) cancellation methods have been proposed [6] [11]. It is a kind of nonlinear detection methods with the decision-directed technique. The demodulated symbols are reconstructed and fed back to cancel ICI. The significant idea is to exploit the discreteness property of the modulated symbols in digital communication systems. Based on the discreteness property, most ICI would be cancelled after several iterations when the initial demodulated symbols are with acceptable correctness. Therefore, the demodulated symbols become increasingly correct during the iterative detection procedure. The ICI cancellation method proceeds as follows: Step 1: Ignoring the ICI effect, the received signal Y is divided by the single tap equalization, i.e. Z (k ) =. Y (k ) . Then, the equalized signal Z is passed through a G (k , k ). ˆ , is obtained. Since hard decision device or a demapper. The demodulated symbol, X. the result is affected by ICI, some demodulated data may be errorneous. Step 2: Performing ICI cancellation on the received signal. Y , i.e.. ˆ . Thus, the ICI effect in the received signal Y is reduced. I YI = Y − G ici X I −1 I. represents the iteration number. G ici denotes the channel response obtained from G only with the ICI elements. Step 3: The signal YI after ICI cancellation is divided by the single tap equaliza-. 21.

(36) tion, i.e. Z I (k ) =. YI (k ) . Then, the equalized signal Z I is passed through a hard G (k , k ). ˆ , is obtained. decision device or a demapper. The demodulated symbol, X I. Step 4: End if I = I max , otherwise repeat Steps 2 and 3 with I = I + 1 . The complexity of the ICI cancellation is proportional to O( N 2 ) due to ICI cancellation procedure in Step 2. For further complexity reduction, Chen et al. [6] suggested that ICI cancellation procedure can be perform in the time domain. However, it requires two more FFT operations.. 3.1.5 Choi’s Method of Successive Detection All the mentioned data detection methods treat the ICI induced by time-varying fading channels just as the interference obstructing data detection. Choi et al. [7] suggested that the time-varying nature of the channel can be exploited as a provider of time diversity. Since the channel response is time-selective, the signal samples received at different time instants have low correlation and provide the diversity. In order to utilize the time diversity provided by time-selective channels, the successive layered detection from [15] performs in the time domain instead of detecting all the data in the frequency domain simultaneously. The data demodulated in the later layers gain more time diversity. Recall the system model described in Section 2.1.2, the received signal is represented as L −1. y (n) = ∑ h(n, l ) x(n − l ) N + w(n). (2.7). l =0. Substitute equation (2.6) into (2.7), the received signal can be represented as y ( n) = L −1. where V (k , n) = ∑ h(n, l )e. 1 N. − j 2π lk N. N −1. ∑ X (k )V (k , n)e. j 2π kn N. + w(n). (3.7). k =0. . Further, the received signal can be rewritten as the. l =0. 22.

(37) matrix form y = VX + w. where y = [ y (0)," , y ( N − 1)]T , w = [ w(0)," , w( N − 1)]T , and V =. (3.8) 1 HQ H . N. The Choi’s method of successive layered detection proceeds in Figure 3.2. In the method, u k and v k are the k-th column vectors of the matrices U and V , respectively. σ 2 is the variance of the additive white complex Gaussian noise. The symbol “slice( )” is the hard decision device, i.e. demapper. In each detection layer, it is necessary to recalculate the matrix inversion in Step P-5. Unfortunately, the number of detection layers and the dimension of the matrix V increase as the number of subcarriers, N, gets larger. According to Section 3.1.2,. the computational complexity of LS detection is proportional to O( N 3 ) . Thus, the computational complexity of this detection method is unacceptable for realization when N is too large.. 23.

(38) Least Square (LS) Version. j =1. ( P − 1). UH = V+. i1 = arg min u k. ( P − 2). 2. k. Loop z (i j ) = uiHj y. ( P − 3). Xˆ (i j ) = slice( z (i j )). y = y − v i j Xˆ (i j ). ( P − 4). V = [ v 0 ," , v i j −1 , 0, v i j+1 ," , v N −1 ]. ( P − 5). UH = V+. i j +1 = arg min u k. ( P − 6). 2. k∉{i1"i j }. j = j +1 Minimum Mean Square Error (MMSE) Version. j =1. ( P − 1). U H = (V H V + σ 2 I ) −1 V H i1 = arg max k. uk , vk. ∑. m,m≠ k. uk , vm. 2. 2. + σ 2 uk. ( P − 2). 2. Loop z (i j ) = uiHj y. ( P − 3). Xˆ (i j ) = slice( z (i j )). y = y − v i j Xˆ (i j ). ( P − 4). V = [ v 0 ," , v i j −1 , 0, v i j+1 ," , v N −1 ]. U H = (V H V + σ 2 I ) −1 V H i j +1 = arg max k∉{i1 ,",i j }. ( P − 5) uk , vk. ∑. m , m ≠ k , m∉{i1 ,",i j }. uk , vm. 2 2. + σ 2 uk. 2. ( P − 6). j = j +1 Figure 3.2 Choi’s methods of successive detection 24.

(39) 3.1.6 Simulation Results Several existing data detection methods are introduced above. In this section, performances of these methods are evaluated by computer simulations. For next-generation communication systems, wireless terminals are expected to operate at high radio frequencies, at high levels of mobility, and with high bandwidth efficiency. Thus, the targeted radio frequency, and the signal bandwidth specified in IEEE 802.20 TDD mode [12] are considered. The simulated OFDM system parameters are listed in Table 3.1. The typical two-ray equal power channel model is considered here. For the simulations, the channel power is normalized to 1. The multipath Rayleigh fading channels in the simulations are generated by the modified Jakes’ model [9] [10]. It is assumed that the perfect channel information is known at the receiver and that synchronization is perfect.. Table 3.1 Simulated OFDM system parameters Operating frequency. 3.5GHz. Signal bandwidth. 2.5MHz. FFT length. 64. Number of piot subcarriers. 16. Number of data subcarriers. 48. Symbol duration. 25.6us. Subcarrier spacing. 39.1kHz. Modulation. QPSK. Channel coding. No. Power delay profile. Two-ray equal power. Normalized Doppler frequency. 0.083 and 0.040. 25.

(40) 10. 10. BER. 10. 10. 10. 10. 10. 0. -1. -2. -3. -4. -5. Single-tap EQ LS Jeon's LS, q=4 ICI cancellation 1st ICI cancellation 3rd ICI cancellation 6th ICI-free Choi's SDMMSE. -6. 0. 5. 10. 15. 20 25 SNR (dB). 30. 35. 40. 45. Figure 3.3 BER performances of the existing data detection methods in the two-ray equal power channel with f nd = 0.040. 10. 10. BER. 10. 10. 10. 10. 10. 0. -1. -2. -3. -4. -5. Single-tap EQ LS Jeon's LS, q=4 ICI cancellation 1st ICI cancellation 3rd ICI cancellation 6th ICI-free Choi's SDMMSE. -6. 0. 5. 10. 15. 20 25 SNR (dB). 30. 35. 40. 45. Figure 3.4 BER performances of the existing data detection methods in the two-ray equal power channel with f nd = 0.083. 26.

(41) The BER performances of the mentioned methods in the two-ray equal power channel with different f nd ’s are shown in Figure 3.3 and Figure 3.4, respectively. The term “ICI-free” indicates the theoretical BER performance of the coherent detection in the time-invariant flat Rayleigh fading channel for the reference of the ICI-free case. It is known that the BER performances of the first four mentioned detection methods are bounded by the ICI-free curve in the figure. The terms “Single tap EQ” and “LS” indicate the single-tap equalization and LS detection, respectively. The ordinal numbers appended to the term “ICI cancellation” indicate the numbers of the iterations of the ICI cancellation method. Jeon’s method of LS detection [8] is denoted as Jeon’s LS. The terms “q=number” appended to “Jeon’s LS” indicate the number of the dominant ICI terms adjacent to the modulated signal in each subchannel, i.e. q. Choi’s method of successive detection with MMSE detection [7] is denoted as Choi’s SDMMSE. Figure 3.3 shows the BER performances of the mentioned data detection methods when f nd = 0.040 . Due to the assumption of neglecting ICI, the single tap equalization suffers from severe ICI effects and has the worst performance of all. The LS detection has a better performance than single tap equalization. However, the performance degradation of the LS detection increases in high SNR environments. Jeon’s method of LS detection suffers from less noise enhancement than LS detection but more ICI effect than LS detection. It also can be found that Jeon’s method of LS detection has worse performance than that of the ICI cancellation with 3 detection iterations when considering the 4 most dominant ICI terms closest to the modulated signal in each subchannel, q = 4, in high SNR case. Comparatively, after 3 iterations, the performance of ICI cancellation method is better than the first four methods when SNR is lower than 40dB. However, when SNR is up to 40 dB, some ICI terms cannot be perfectly cancelled due to the errors of the demodulated symbols. Thus, the error 27.

(42) propagation dominates the performance of the ICI cancellation method and an error floor appears. Due to utilizing the time diversity provided by time-selective channels, Choi’s method of successive detection with MMSE detection outperforms the other mentioned methods and the reference of the ICI-free. This benefit is more apparent in high SNR case. Figure 3.4 illustrates the BER performances of the mentioned data detection methods when f nd = 0.083 . The single tap equalization suffers from severe ICI effects and still has the worst performance of all. LS detection suffers from noise enhancement which increases as f nd gets higher because the minimum nonzero singular value of G becomes smaller. Jeon’s method of LS detection also suffers from severe ICI effect since too many ICI terms are ignored in the detection. Compared to the LS detection, ICI cancellation method is without the noise enhancement problem. However, the error propagation becomes a serious problem for the ICI cancellation method and cannot be ignored even after six iterations. Choi’s method of successive detection with MMSE detection still outperforms the other mentioned methods. Because time selectivity of the fading channel with f nd = 0.083 becomes more apparent than that of the channel with f nd = 0.040 , Choi’s method obtains more benefits by utilizing time diversity. In conclusion, those ICI cancellation methods are not good enough to reduce the ICI effect under the targeted channel environments. An irreducible error floor appears in the high SNR environments regardless of the iteration numbers. Although LS detection outperforms the other detection methods, it still suffers severe noise enhancement problem. The performance degradation is too large. Choi’s method is a good choice for detection of OFDM systems in the targeted time-varying fading channels. However, the computational complexity of this method is too high.. 28.

(43) 3.2 Existing Channel Estimation Methods In this section, two conventional channel estimation methods, LS channel estimation and DFT-based channel estimation, for OFDM systems in time-invariant channel are introduced. Then, two existing channel estimation methods, Chen’s method of LS channel estimation [6] and Yeh’s algorithm of ICI-reduction method [11], based on channel model with linear variation are described. The simulation results show that the performances of these methods are poor when the normalized Doppler frequency, f nd , is up to 0.08 and that the estimation errors of these methods are too high for the. proposed data detection methods.. 3.2.1 Least-square Channel Estimation LS channel estimation is widely used for OFDM systems and is the foundation of other advanced channel estimation methods. The advantage of LS channel estimation is that the pilot patterns are the only required information to accomplish the estimation. In the following analysis of this section, the number of pilot subcarriers in an OFDM symbol is set to M, and the channel is assumed to be time-invariant. Thus, the received signal after DFT at the k-th pilot subcarrier is represented as Y ( p (k )) = G (k , k )P (k ) + N (p (k )). k = 0,1, 2," , M − 1. (3.9). where p(k) is the position function of the pilot subcarriers and P(k) is the known pattern transmitted at the k-th pilot subcarrier. The channel frequency response at the k-th pilot subcarrier can be roughly estimated by Y ( p (k )) N (p (k )) Gˆ ( p (k ), p(k )) = = G ( p(k ), p(k )) + P(k ) P(k ). (3.10). Then, the channel frequency responses at the other data subcarriers can be estimated by interpolation techniques. Also, the following DFT-based channel estimation is one 29.

(44) kind of the interpolation techniques.. 3.2.2 DFT-based Channel Estimation DFT-based channel estimations have been proposed in [13] [14]. These estimations are based on the technique of interpolation in transform domain to accomplish the estimation and with good performances in the case of sample-spaced channel impulse response. FFT algorithms can be utilized to reduce the computational complexity of the transformation. However, there is a restriction on placement of pilot subcarriers for the DFT-based channel estimation. M pilot subcarriers must be equispaced along frequency direction as shown in Figure 3.5. pilot. data. Df. Df Frequency Time. Figure 3.5 Regular pilot placement [11] For convenience, N/M is assumed to be an integer, the first pilot is placed at Subcarrier 0, i.e. D.C, and the channel is assumed to be time-invariant. Therefore, the received signal at the k-th pilot subcarriers can be expressed as Y ( p (k )) = Y (. N k ) = Y ( D f k ), M. k = 0,1, 2," , M − 1. (3.11). First, the channel frequency response at pilot subcarriers is estimated by LS estimation, i.e. equation (3.10). Observably, the channel frequency response at pilot subcarriers can be viewed as the down-sample of complete frequency response. Then, 30.

(45) M –point IDFT is performed on Gˆ ( p (k ), p (k )) . 1 hˆp (n) = M. M −1. −j ∑ Gˆ ( p(k ), p(k ))e. 2π nk M. ,. n = 0,1, 2," , M − 1. (3.12). k =0. According to the down sampling theorem, the channel impulse response obtained by M–point inverse discrete Fourier transform of {G ( p (k ), p (k )) | k = 0,1, 2," , M − 1} is the overlapped version of the channel impulse response with N points. For analysis, the maximum path delay time is defined as τ max ts and ∆ is the minimum integer which is larger than τ max . To avoid the aliasing effect, the number of the pilot subcarriers must be larger than the sampled maximum path delay, i.e. M > ∆ . According to (2.15) and (2.16), the estimation of channel frequency response can be achieved by padding N-M zeros in the tail of {hˆp (n) | n = 0,1, 2," , M − 1} , ⎪⎧ hˆ (n), hˆ(n) = ⎨ p ⎪⎩0,. 0 ≤ n ≤ M −1 M −1 ≤ n ≤ N −1. (3.13). and then by performing N-point DFT on hˆ(n) N −1. −j Gˆ (k , k ) = ∑ hˆ(n)e. 2π nk N. k = 0,1, 2," , N − 1. (3.14). n =0. 3.2.3 Chen’s Method of Least-square Channel Estimation Recall the system model described in Section 2.3. The number of the parameters required for estimation in the matrix G is N 2 when frequency-domain channel estimation is considered. Apparently, it is impossible to achieve the estimation even if full N training symbols are transmitted at all N subcarriers. To make the problem solvable, the number of parameters required for estimation must be reduced to be less than N. Thus, instead of estimating the channel frequency response, estimating the channel impulse response h(n, l ) is adopted, because the number of channel taps is usually smaller than the number of the subcarriers, i.e. L N . 31.

(46) Without any assumption, there are still LN parameters needed for the estimation within an OFDM symbol when the number of channel taps is L. It is still impossible to achieve the estimation. Fortunately, Jeon et al. [8] stated that the channel variation can be approximated by a straight line when the channel is slowly time-varying (Normalized Doppler frequency, f nd <0.1). This property has been introduced in Section 2.3. Thus, the variation of each channel path can be approximated as [6] [11] [18] ∆h(n, l ) = (n −. N −1 )α l 2. (3.15). where 0 ≤ n ≤ N − 1 , 0 ≤ l ≤ L − 1 , and α l is the slope of the l-th channel path. Given by (2.15), the response of the l-th channel path at n-th time sample can be written as h(n, l ) = have (l ) + (n −. N −1 )α l 2. (3.16). Therefore, the number of parameters required for estimation is reduced to 2L. The channel estimation can be achieved by applying least-square (LS) estimation when pilot subcarriers inserted within an OFDM symbol are enough. Substituting (3.15) into (2.17), the ICI terms of G is given by G ( k , m) = L −1. =−. where. 1 N. ∑αl e l =0. N −1 L −1. ∑∑ (r − r =0 l =0. j 2π r ( m − k ) − j 2π lm N −1 )α l e N e N 2. − j 2π ml N. 1 − e − j 2π ( k − m ) N. (3.17) = Ck − mbTmα, m ≠ k. b m = [1, e − j 2π m N ," , e − j 2π mL N ]T. ,. Ck − m = −(1 − e − j 2π ( k − m ) L N ) −1. ,. and. α = [α 0 ," , α L −1 ]T . Equation (2.16) can be rewritten as G (k , k ) = bTk h ave. (3.18). where h ave = [have (0)," , have ( L − 1)]T . According to (2.13), the received signal after. 32.

(47) DFT at the k-th subcarrier is given by Y (k ) = G (k , k ) X (k ) +. N −1. ∑. G ( k , m) X ( m) + W ( k ). m = 0, m ≠ k. = bTk h ave X (k ) +. N −1. Ck − mbTmαX (m) + W (k ) ∑ m = 0, m ≠ k . (3.19). ICI. By collecting the received signals at all pilot subcarriers, the system model can be expressed as the linear equations ⎡ Y ( p (1)) ⎤ ⎥ = Ah + Bα + e # Y = ⎢⎢ ave ⎥ ⎢⎣Y ( p ( M )) ⎥⎦ ⎡h ⎤ = [ A B ] ⎢ ave ⎥ + e ⎣ α ⎦ = Dh + e. (3.20). where ⎡ X ( p (1))bTp (1) ⎤ ⎢ ⎥ A=⎢ # ⎥ ⎢ X ( p ( M ))bTp ( M ) ⎥ ⎣ ⎦. (3.21). ⎡ ⎤ C p (1) − m X (m)bTm ⎥ ∑ ⎢ ⎢ m∈ pilot ,m ≠ p (1) ⎥ ⎥ # B=⎢ ⎢ ⎥ T ⎥ ⎢ C p ( M ) − m X (m)b m ⎢⎣ m∈ pilot∑ ⎥⎦ ,m≠ p ( M ). (3.22). D = [ A B]. (3.23). ⎡h ⎤ h = ⎢ ave ⎥ ⎣ α ⎦. (3.24). 33.

(48) ⎡ e( p (1)) ⎤ ⎥ e = ⎢⎢ # ⎥ ⎢⎣e( p ( M )) ⎥⎦ N −1 ⎡ ⎤ C p (1) − m X (m)bTm ⎥ ∑ ⎢ ⎡ W ( p (1)) ⎤ ⎢ m =0,m∉ pilot ,m ≠ p (1) ⎥ ⎥ ⎥α + ⎢ =⎢ # # ⎢ ⎥ ⎢ ⎥ N −1 ⎢ ⎥⎦ ( ( )) W p M ⎣ ⎢ C p ( M ) − m X (m)bTm ⎥ ∑ ⎢⎣ m =0,m∉ pilot ,m ≠ p ( M ) ⎥⎦. (3.25). where p(k) is the position function of the pilot subcarriers, M is the number of the pilot subcarriers, and e is composed of noise at the pilot subcarriers and ICI contribution from non-pilot subcarriers. Thus, the estimated channel is obtained by the LS solution of the linear equations from (3.20) when M ≥ 2 L hˆ = D+ Y. (3.26). However, the simulation results (see Section 3.2.5) show that an irreducible error floor still appears in the BER performance when only pilot subcarriers are used for the estimation in the fading channel with f nd = 0.083 , and that this method only takes effect in the all-pilot preamble case.. 3.2.4 Yeh’s Algorithm of ICI-reduction Method Chen’s method of Channel estimation is based on the LS estimation. According to the conclusion in the previous section, this method only takes effect in the all-pilot preamble case. Thus, it is necessary to find out a channel tracking method in the data transmission region where only pilot subcarriers are utilized for the estimation. Yeh [11] proposed a channel tracking method with ICI-reduction technique in linearly-variant fading channels. Recall the system model described in Section 2.3. The received signal after DFT is given by. 34.

(49) N −1. Y (k ) = G (k , k ) X (k ) +. ∑. G ( k , m) X ( m) + W ( k ) . (3.27). m = 0, m ≠ k. ICI. Figure 2.8 shows that the power of G (k , k ) is much larger than the power of G (k , m) when k ≠ m . Therefore, the channel frequency response at k-th pilot sub-. carrier can be obtained by LS estimation introduced in Section 3.2.1 N −1. ∑. Y ( p (k )) = G ( p (k ), p (k )) + m =0,m ≠ k Gˆ ( p (k ), p(k )) = P(k ). G ( k , m) X ( m) + W ( k ) P(k ). (3.28). The second term on the right hand side of (3.28) represents the estimation error and increases as normalized Doppler frequency, f nd , becomes large. According to (2.16), the estimation of have (l ) can be obtained by DFT-based channel estimation introduced in Section 3.2.2 1 hâve (l ) = M. L −1. ∑ Gˆ ( p(k ), p(k ))e. j 2π lk M. (3.29). k =0. Besides, (3.15) shows that have (l ) is the channel impulse response at the midpoint of the symbol interval when channel variation is approximated to linear fashion, i.e. h( n =. N −1 , l ) = have (l ) . In order to estimate the slopes, α l ’s, of the channel paths 2. within single OFDM symbol by using the two have (l ) ’s of the two consecutive OFDM symbols, it is assumed that channel variation within the two consecutive OFDM symbols is linear, i.e. f nd ≤ 0.05 . Therefore, the slopes of the channel paths can be obtained by. αˆli =. i i −1 hâve (l ) − hâve (l ) N + N CP. (3.30). i where α li denotes the slope of the l-th channel path of the i-th symbol, have (l ) de-. notes the average response of the l-th channel path of the i-th symbol, and N CP denotes the length of the cyclic prefix (CP). This concept is shown in Figure 3.6. The solid line and the dashed line represent the response of one channel path and the cor35.

(50) responding estimated response, respectively. The estimation error is the region between the solid line and the dashed line.. previous symbol. current symbol. CP. CP. i have (l ). i −1 have (l ). mid-point. mid-point. time. Figure 3.6 Linear model between two consecutive OFDM symbols [18] Thus, the ICI terms in (3.19) can be cancelled by using the estimation of the slope,. αˆli , and the demodulated data, Xˆ (k ) . The whole procedure of Yeh’s method is shown in Figure 3.7 modified from [11]. I represent the iteration number. I max is the number of the iterations. In the first iti i eration, the estimation hâve (l ) of have (l ) suffers from whole ICI and has large error.. Thus, there are still many erroneous data after demodulation. After several iterations, the ICI terms at the pilot subcarriers would be mitigated by the ICI cancellation proi (l ) becomes small. A more reliable slope of the cedure. The estimation error of hâve. channel path αˆli is obtained. The ICI terms at the data subcarriers would be cancelled more clearly. Therefore, the error floor is reduced. Although Yeh’s method operates well in the aspect of the performance and the computational complexity, it is only feasible in the assumption circumstance with f nd ≤ 0.05 . Therefore, this method needs some modification to operate in the targeted. fast fading channels. Besides, the performance of this method is bounded by the ICI cancellation method with ideal channel state information, i.e. ICI cancellation in Figure 3.3 and Figure 3.4.. 36.

(51) I =0 Y ( p (k )). Gˆ ( p (k ), p (k )). i hâve (l ). i,I hâve (l ). i,I hâve (l ). i −1, I max hâve (l ). i,I hâve (l ). Gˆ (k , k ). αˆli. Y i , I (k ). Gˆ (k , k ). Xˆ (k ). αˆli. Xˆ (k ). Y i , I +1 (k ). i , I +1 hâve (l ). I = I +1. I = I max. Figure 3.7 Yeh’s algorithm of ICI-reduction method [11] 37. Xˆ (k ) i , I max hâve (l ).

(52) 3.2.5 Simulation Results In this section, the performances of the last two mentioned channel estimation methods are evaluated by computer simulations. Multipath Rayleigh fading channels in the simulations are generated by the modified Jakes’ model [9] [10]. The power delay profile is chosen as the “Vehicular A” channel model defined by ETSI for the evaluation of UMTS radio interface proposals [17]. The simulated OFDM system parameters and the channel parameters are listed in Table 3.2 and Table 3.3, respectively. The 16 pilot subcarriers grouped into 4 groups are equispaced onto the DFT grid, i.e. the indices of the pilot subcarriers are {0, 1, 2, 3, 16, 17, 18, 19, 32, 33, 34, 35, 48, 49, 50, 51} within 64 subcarriers. These pilot patterns are used for Chen’s method of LS channel estimation. For Yeh’s algorithm of ICI reduction method, the 16 pilot subcarriers are equispaced along frequency direction, i.e. the indices of the pilot subcarriers are {0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60} within 64 subcarriers, due to DFT-based channel estimation. Beside, an all-pilot preamble is attached in front of the data symbols for the initial channel estimation. It is assumed that the maximum path delay time is known at the receiver and that synchronization is perfect.. 38.

(53) Table 3.2 Simulated OFDM system parameters Operating frequency. 3.5GHz. Signal bandwidth. 2.5MHz. FFT length. 64. Number of piot subcarriers. 16. Number of data subcarriers. 48. Symbol duration. 25.6us. Subcarrier spacing. 39.1kHz. Modulation. QPSK. Channel coding. No. Power delay profile. ETSI Vehicular A. Normalized Doppler frequency. 0.083 and 0.040. Table 3.3 ETSI Vehicular A channel environment Tap Relative delay (ns) Average power (dB) 1. 0. 0.0. 2. 310. -1.0. 3. 710. -9.0. 4. 1090. -10.0. 5. 1730. -15.0. 6. 2510. -20.0. 39.

(54) 20 Chen's LS, preamble Chen's LS, pilot Yeh's ICI reduction, pilot. 10. ANMSE (dB). 0 -10 -20 -30 -40 -50 0. 5. 10. 15. 20 25 SNR (dB). 30. 35. 40. 45. Figure 3.8 ANMSE performances of Chen’s and Yeh’s methods of channel estimation in the “Vehicular A” channel with f nd = 0.040. 20 Chen's LS, preamble Chen's LS, pilot Yeh's ICI reduction, pilot. 10. ANMSE (dB). 0 -10 -20 -30 -40 -50 0. 5. 10. 15. 20 25 SNR (dB). 30. 35. 40. 45. Figure 3.9 ANMSE performances of Chen’s and Yeh’s methods of channel estimation in the “Vehicular A” channel with f nd = 0.083. 40.

(55) 10. 10. BER. 10. 10. 10. 10. 10. 0. -1. -2. -3. -4. -5. Yeh's ICI reduction 3rd ICI cancellation 3rd, Ideal CSI ICI-free FSDMMSE, Chen's LS, pilot FSDMMSE, Ideal CSI Choi's SDMMSE, Ideal CSI. -6. 0. 5. 10. 15. 20 25 SNR (dB). 30. 35. 40. 45. Figure 3.10 BER performances of Chen’s and Yeh’s methods of channel estimation combined with data detection methods in the “Vehicular A” channel with f nd = 0.040. 10. 10. BER. 10. 10. 10. 10. 10. 0. -1. -2. -3. -4. -5. Yeh's ICI reduction 3rd ICI cancellation 3rd, Ideal CSI ICI-free FSDMMSE, Chen's LS, pilot FSDMMSE, Ideal CSI Choi's SDMMSE, Ideal CSI. -6. 0. 5. 10. 15. 20 25 SNR (dB). 30. 35. 40. 45. Figure 3.11 BER performances of Chen’s and Yeh’s methods of channel estimation combined with data detection methods in the “Vehicular A” channel with f nd = 0.083. 41.

(56) In Figure 3.8 and Figure 3.9, ANMSE indicates the average normalized mean square error, which is defined as 2 ⎧ N −1 N −1 i ˆ i ( k , m) ⎫ G ( k , m ) − G ∑∑ ⎪⎪ ⎪⎪ 1 k =0 m=0 ANMSE = ⎨ ⎬ ∑ N −1 N −1 2 N sym i =1 ⎪ i ⎪ G ( k , m) ∑∑ ⎪⎩ ⎪⎭ k =0 m=0 N sym. (3.31). where N sym is the number of the transmitted OFDM symbols, Gˆ i (k , m) is the estimation of G i (k , m) , and G i (k , m) is the channel frequency response within the i-th OFDM symbol defined in Section 2.3. The terms “Chen’s LS, preamble” and “Chen’s LS, pilot” indicate that Chen’s methods of LS channel estimation, introduced in Section 3.2.3, are applied to the all-pilot preamble and the pilot subcarriers, respectively. The term “Yeh’s ICI reduction, pilot” indicates that Yeh’s algorithm of ICI reduction method [11] with 3 detection iterations, introduced in Section 3.2.4, is applied to the pilot subcarriers. In Figure 3.10 and Figure 3.11, the term “ICI-free” indicates the theoretical BER performance of the coherent detection in the time-invariant flat Rayleigh fading channel for the reference of the ICI-free case. The ordinal numbers appended to “ICI cancellation” and “Yeh’s ICI reduction” indicate the numbers of the iterations performed in the two methods. The terms “FSDMMSE” and “Choi’s SDMMSE” indicate that the proposed frequency-domain successive detection method with MMSE detection (see Section 4.1) and Choi’s method of successive detection with MMSE detection [7] are adopted in data detection, respectively. The term “Ideal CSI” indicates that ideal channel state information is applied to data detection. Figure 3.8 and Figure 3.9 illustrate ANMSE performances of Chen’s and Yeh’s methods of channel estimation in the “Vehicular A” channel with f nd = 0.040 , and f nd = 0.083 , respectively. Chen’s method applied to the all-pilot preamble has the. least channel estimation error of all. It can be viewed as the performance boundary of 42.

(57) Chen’s method applied to any other pilot pattern when the number of pilot subcarriers is less than N. Apparently, the performance degradation of Chen’s method applied to the pilot subcarriers is very large. Under the same ANMSE condition, Yeh’s method with 3 detection iterations applied to the pilot subcarriers has about 4~5dB performance loss compared to Chen’s method applied to the all-pilot preamble when f nd = 0.040 ~ 0.083 . Besides, the channel estimation errors of these methods increase. as f nd becomes large, especially in the high SNR case. Figure 3.10 and Figure 3.11 illustrate the BER performances of Chen’s and Yeh’s methods of channel estimation combined with data detection methods in the “Vehicular A” channel with f nd = 0.040 , and f nd = 0.083 , respectively. Because of the large channel estimation error, Chen’s method applied to the pilot subcarriers combined with the proposed detection method has the worst performance of all. There is a large performance degradation compared to the proposed detection method with ideal channel state information. As known in Section 3.2.4, the performance of Yeh’s method is bounded by the ICI cancellation method with ideal channel state information. In conclusion, the Yeh’s method is not good enough to reduce the ICI effect under the targeted channel environments. The channel estimation error of the Chen’s method applied to the pilot subcarriers is too large for the proposed detection methods. Thus, it is necessary to develop more reliable channel estimation.. 43.

(58) Chapter 4 The Proposed Data Detection and Channel Estimation Methods. In the previous chapter, some existing data detection and channel estimation methods are described, briefly. However, the simulation results show that these methods except Choi’s method of successive detection are not effective in the time-varying environments, where normalized Doppler frequency, f nd , is up to 0.08. Based on the concepts of the mentioned methods [6] [7] [8] [11], modified data detection and channel estimation methods are proposed for the targeted channel environments. The computational complexities of the proposed methods and their introduced additional costs are also discussed.. 4.1 The Proposed Frequency-domain Successive Detection Methods 44.

(59) As mentioned in the previous chapter, the computational complexity of Choi’s method of successive detection is too high to be realized. Fortunately, some properties of the time-varying fading channels described in Section 2.3 combined with the technique introduced in Section 3.1.3 help to reduce the computational complexity of the successive detection method and make the method feasible.. 4.1.1 Algorithms of the Frequency-domain Successive Detection Methods In order to exploit time diversity involved in time-selective fading channels, Choi et al. proposed their successive detection method performing in the time domain, intuitively. However, in most-considered application environments, the correlation of the channel impulse response within OFDM symbol duration is still high. It is hard to predict which time samples suffer less fading effect and only to process these time samples for better efficiency. Besides, since the modulated signals transmitted in different subchannels are mixed in the time domain, the whole N received time samples must be used for the separation of modulated signals from the received signal without additional ICI induced by the non-periodic DFT windows. Thus, no special properties of the system model in the time domain can be utilized to reduce the computational complexity. Fortunately, some properties of the channel frequency response can help reduce the computational complexity of the successive detection method. First, Parseval’s Theorem states that average power of the signal in time equals sum of average power of the signal in the k-harmonics, i.e. power conservation property. 1 N N −1. where T (k , l ) = ∑ h(n, l )e. − j 2π kn N. N −1. N −1. ∑ h ( n, l ) = ∑ T ( k , l ) 2. n =0. 2. (4.1). k =0. . Therefore, the power gain of time diversity obtained. n =0. 45.

(60) in the frequency domain is the same as that obtained in the time domain. Second, recall the system model shown in Section 2.3. The received signal after DFT is represented as Y = GX + W. (2.19). The successive detection is modified to perform in the frequency domain. According to the properties mentioned in Section 3.1.3, the power of the time-varying fading channel is compressed by DFT, i.e. most power of the time-varying fading channel concentrates in the neighborhood of subcarriers and does not spread over the spectrum. Consequently, the computation of linear detection in Step P-1 in Figure 3.2 is changed from pseudo inversion of the N × N matrix G to N pseudo inversions of the (q + 1) × (q + 1) matrices. G blk ,k . Hence, the computational complexity of the linear. detection in Step P-1 is reduced from O( N 3 ) to O( N (q + 1)3 ) . Thus, by utilizing these two properties, the successive detection performing in the frequency domain not only gains time diversity but also has lower computational complexity. There are additional benefits when the successive detection performs in the frequency domain. Only the pseudo inversions and the norms of the corresponding equalization vectors of the subcarriers adjacent to the demodulated subcarrier in the current detection layer are necessarily updated for the proposed methods with LS detection, which is shown in Step P-5 and Step P-6 in Figure 4.1. This property can also be applied tor the signal to interference and noise ratio (SINR) calculation of the proposed methods with MMSE detection. Therefore, the computational complexity can be further reduced. The complexity analysis is detailed in Section 4.1.2. The structures of G blk ,k ’s are depicted in Figure 3.1. The proposed detection methods proceed as follows:. 46.

(61) Least Square (LS) Version. j =1 H + U blk , k = G blk , k. ( P − 1). 0 ≤ k ≤ N −1. i1 = arg min ublk ,k ,q 2+1. 2. ( P − 2). k. Loop H z (i j ) = ublk ,i j , q 2 +1Yblk ,i j. ( P − 3). Xˆ (i j ) = slice( z (i j )). Y = Y − g i j Xˆ (i j ). ( P − 4). G = [g 0 ," , g i j −1 , 0, g i j+1 ," , g N −1 ] H + U blk , k = G blk , k. ( P − 5). k ∈ {i j − q / 2," , i j − 1, i j + 1," , i j + q / 2}. i j +1 = arg min ublk ,k ,q 2+1. 2. ( P − 6). k∉{i1 ,",i j }. j = j +1 Minimum Mean Square Error (MMSE) Version. j =1 H H 2 −1 H U blk , k = (G blk , k G blk , k + σ I ) G blk , k , 0 ≤ k ≤ N −1. ublk ,k ,q 2+1 , g blk , k , q 2+1. i1 = arg max SINRk = arg max k. k. ∑. m , m ≠ q 2 +1. Loop. ( P − 1) 2. ublk ,k ,q 2+1 , g blk , k ,m. 2. + σ 2 u blk , k , q 2+1. 2. ( P − 2). ( P − 3). H z (i j ) = ublk ,i j , q 2 +1Yblk ,i j. Xˆ (i j ) = slice( z (i j )). Y = Y − g i j Xˆ (i j ). ( P − 4). G = [g 0 ," , g i j −1 , 0, g i j+1 ," , g N −1 ] H H 2 −1 H U blk , k = (G blk , k G blk , k + σ I ) G blk , k. ( P − 5). k ∈ {i j − q / 2," , i j − 1, i j + 1," , i j + q / 2}. i j +1 = arg max k∉{i1 ,",i j }. j = j +1. u blk ,k ,q 2+1 , g blk ,k ,q 2+1. ∑. m , m ≠ q 2 +1. u blk ,k ,q 2+1 , g blk ,k ,m. 2. 2. + σ 2 u blk ,k ,q 2+1. 2. ( P − 6). Figure 4.1 The proposed frequency-domain successive detection methods 47.