# 臺灣博碩士論文加值系統

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 本論文提出兩種簡單準確的通用公式，主要是用來鑑定正交分頻多工系統中的非線性通道。正交分頻多工系統是一種被廣泛採用的寬頻通訊應用技術。正交分頻多工系統訊號很容易受到通訊網路中非線性通道的影響。使得正確估計OFDM 系統非線性通道的任務變得很重要。非線性通道通常是由VolterraSeries 來表示，非線性通道鑑定的任務就是確定Volterra Kernel。當非線性系統階數增加時，這項任務會變得非常困難。本論文，論述OFDM 信號的高階統計特性和多層分解特性，以克服遇到的瓶頸並推導出解決Volterra Kernel 鑑定的通用方法。這兩種方法各有優缺點。OFDM 信號的高階統計特性法是利用高階自動矩為頻域Volterra Kernel 提出閉式表達式。使用高階統計特性這個方法可以推導出上至5 階非線性通道之簡單公式。多層分解的方法是透過設計具有時域或頻域頻譜凹陷的不同輸入信號來激發非線性系統，這種方法理論上可適用於任意階之非線性通道，且不限於OFDM 系統，但會產生較大的計算複雜度。基於這兩種方法，本論文推導準確簡單的方法來鑑定高階非線性正交分頻多工系統系統Volterra Kernel 的通用公式。理論分析和模擬結果顯示，所得到的解決方法都達到了最小均方誤差。
 The dissertation proposes two general formulas that are simple to derive and applicable to the identification of nonlinear channels in orthogonal frequency division multiplexing (OFDM) systems. OFDM is a widely used wideband communication application technology. As nonlinearities in the communication network can easily impact OFDM signals, the task of correctly estimating the nonlinear channel of an OFDM system is of critical importance. Since the nonlinear channel is usually represented by the Volterra series, the task in nonlinear channel identification is to find and solve for the Volterra kernels. However, this task becomes very difficult when the order of the nonlinear system increases. This dissertation discusses the high-order statistical properties of OFDM signals and multilayer decomposition properties, to overcome related bottlenecks and derive a general method for Volterra kernel identification, respectively. Both methods have their own advantages and disadvantages. The high-order statistics method of OFDM use the high-order moments to propose a closed-form expression for the frequency domain Volterra kernels. This method of high-order statistics can enable the derivation of simple formulas for nonlinear channels up to 5th order. In contrast, the multilayer decomposition method excites the nonlinear system by designing different input signals with time-domain or frequency-domain spectral notches, which theoretically can be applied to nonlinear channels of any order. This method is not limited to ODFM systems. However, it does produce greater computational complexity. By drawing upon these two methods, we derive an accurate and simple method to identify Volterra kernels of high-order nonlinear OFDM systems. The theoretical and simulated results show that the proposed solutions achieve the minimum mean square error (MMSE).
 Contents1 Introduction 11.1 Research motivation and purpose . . . . . . . . . . . . . . . . . 11.2 Literature Discussion . . . . . . . . . . . . . . . . . . . . . . 21.3 Solution . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 41.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . .. . . 52 Concept Discussion 72.1 Baseband Signal and Passband Signal . . . . . . . . . . . . . . . 72.1.1 Baseband signal . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Passband signal . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Nonlinear Channel and Volterra Series . . . . . . . . . . . . . . 82.2.1 Volterra series for baseband channel . . . . . . . . . . . . . .92.2.2 Volterra series for bandpass channel . . . . . . . . . . . . . .112.2.3 Symmetry Properties of Volterra Kernels . . . . . . . . . . . . 142.3 The MMSE Approach for Volterra Kernel Estimation . . . . . . . . .162.3.1 Time-domain kernel estimation for real nonlinear systems . . . .162.3.2 Time-domain kernel estimation for complex nonlinear systems . . 192.3.3 Frequency-domain kernel estimation for real nonlinear systems . 222.3.4 Frequency-domain kernel estimation for complex nonlinear. . . . 243 Identification of Nonlinear Channels with OFDM Inputs . . . . . . . 273.1 OFDM Signal and Its Spectral Properties . . . . . . . . . . . . . 273.1.1 Higher-order auto-moment spectra . . . . . . . . . . . . . . . .303.2 Identification of Volterra Kernels for Nonlinear Bandpass Channels 313.3 Derivation of the Formula for 5th-Order Nonlinear OFDM Channels. . 383.4 Design of Pseudo Random Test Sequences . . . . . . . . . . . . . . 534 Identification by Decomposition . . . . . . . . . . . . . . . . . . .564.1 Time-Domain Decomposition . . . . . . . . . . . . . . . . . . . . .564.2 Frequency-Domain Decomposition . . . . . . . . . . . . . . . . . . 615 Multilayer Decomposition . . . . . . . . . . . . . . . . . . .. . . 665.1 An Additional Layer of Decomposition . . . . . . . . . . . . . . . 665.2 Multiple Layers of Decomposition . . . . . . . . . . . . . . . . . 716 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . 736.1 Verification of the method derived in Chapter 3 . . . . . . . . . 736.2 Verification of the method derived in Chapter 4 . . . . . . .. . . 787 Conclusion 86List of Figures2.1 The K-th order frequency-domain Volterra model of a nonlinear basebandsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Frequency-domain Volterra model of a nonlinear bandpass channel. . . 142.3 The estimated area of the quadratic Volterra kernel in the time-domain. . 172.4 The estimated area of the cubic Volterra kernel in the time-domain. . . . 182.5 (a) The input x(n) and its output y(n) with a maximal amplitude A.(b) The notched input x(¯p)(n) and its output y(¯p)(n). (c) The differencesignal y(p)(n) between y(n) and y(¯p)(n). . . . . . . . . . . . . . . . . . 202.6 Baseband OFDM system block diagram. . . . . . . . . . . . . . . . . . 232.7 Block diagram of passband OFDM system. . . . . . . . . . . . . . . . 253.1 The 16-QAM constellation is divided into 4 subsets of 4-QAMs withdifferent graphic symbols. 16-QAM is used for the OFDM subcarrierin the simulation. Each data symbol is marked with its 4-digit representationand the symbol number in parentheses. . . . . . . . . . . . . . . 293.2 The feedback shift register corresponds to the original polynomial p(x)in (151) and is used to generate the test sequence. . . . . . . . . . . . . 554.1 (a) A hypothetical spectrum of x(n). (b) The spectrum of x(¯p)(n). . . . 624.2 (a) A hypothetical output spectrum, Y (m), for the inputX(m) in Fig. 4.1(a).(b) A hypothetical output spectrum Y (¯p)(m) for the input X(¯p)(m) inFig. 4.1(b). (c) The difference spectrum Y (p)(m) between Y (m) andY (¯p)(m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Multilayer decomposition diagram. . . . . . . . . . . . . . . . . . . . . 706.1 The shift register corresponds to the original polynomial p(x) = x20 +x3 + 1 in (189). The 20-bit sequence in the sliding window representsthe initial state of the shift register. . . . . . . . . . . . . . . . . . . . . 756.2 Simulation of the power spectrum of the input and output signals of anonlinear satellite channel. . . . . . . . . . . . . . . . . . . . . . . . . 776.3 Estimated NMSE and SNR of linear, cubic, 5th-order, and total frequencydomainVolterra kernels. . . . . . . . . . . . . . . . . . . . . . . . . . 796.4 Estimation of the NMSEs of the time-domain Volterra kernel obtainedby the T-FEX (denoted by the symbol ‘2’) and T-NEX (denoted by thesymbol ‘◦’) methods at SNRs of 20, 30, and 40 dB with different numbersof data frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.5 Actual time-domain linear Volterra kernel (T-ACT) and its estimatedvalues obtained by the T-FEX and T-NEX methods. . . . . . . . . . . . 816.6 (a) Actual time-domain quadratic Volterra kernel and its estimated valueobtained by (b) T-FEX and (c) T-NEX methods. (d) Actual time-domaincubic Volterra kernel and its estimated values obtained by (e) T-FEX and(f) T-NEX methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.7 F-FEX (denoted by the symbol ‘2’) and F-NEX (denoted by the symbol‘◦’) methods use different numbers of data frames to estimate theNMSEs of the Volterra kernels in the frequency domain at an SNR of20, 30, and 40 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.8 The amplitude of the actual frequency domain linear Volterra kernel (FACT)and its estimated value obtained by the F-FEX and F-NEX methods. 846.9 (a) The amplitude of the actual frequency-domain quadratic Volterrakernel and its estimated values obtained by (b) F-FEX and (c) F-NEXmethods. (d) The actual frequency-domain cubic Volterra kernel amplitudeobtained by (e) F-FEX and (f) F-NEX methods. . . . . . . . . . . 85List of Tables6.1 Time-domain Volterra kernels of a satellite communication system. . . . 746.2 States of the shift register in Fig. 6.1. . . . . . . . . . . . . . . . . . . . 756.3 Corresponding 16-QAM symbols for the output sequence in Table 6.2. . 766.4 NMSEs of the linear, cubic, 5th-order, and total frequency-domain Volterrakernels as realized by applying the proposed method under various SNRs. 78
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