正交分頻多工系統之通道估計與I/Q失衡補償

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(1)國立交通大學電信工程學系碩士論文正交分頻多工系統之通道估計與 I/Q 失衡補償. Joint Maximum Likelihood Estimation of Channel and I/Q Imbalance for OFDM Systems. 研究生：許宸睿指導教授：沈文和. 中. 華. 民. 國. 九. 十. 五. 博士. 年. 七. 月.

(2) 正交分頻多工系統之通道估計與 I/Q 失衡補償 Joint Maximum Likelihood Estimation of Channel and I/Q Imbalance for OFDM Systems 研究生：許宸睿. Student：Chen-Jui Hsu. 指導教授：沈文和博士. Advisor：Dr. Wern-Ho Sheen. 國立交通大學電信工程學系碩士班碩士論文. A Thesis Submitted to Institute of Communication Engineering College of Electrical Engineering and Computer Science National Chiao Tung University in Partial Fulfillment of the Requirements for the Degree of Master of Science in Communication Engineering July 2006 Hsinchu, Taiwan, Republic of China.. 中. 華. 民. 國. 九. 十. 五. 年. 七. 月.

(3) 正交分頻多工系統之通道估計與 I/Q 失衡補償研究生：許宸睿. 指導教授：沈文和博士國立交通大學電信工程學系碩士班. 摘要現今許多的應用需要具備高傳輸速率的無線通訊系統，而正交分頻多工 (OFDM)系統正是滿足此特性的主流技術之ㄧ。在接收機架構方面，無論是在業界或是學術界，直接轉換接收機(Direct Conversion Receiver)皆被廣泛地採用，因為其具備有小體積，低成本和高整合度等優勢。然而，此架構因為射頻元件不完美的緣故，無可避免地會產生一些射頻損傷，像是直流偏移(DC Offset)、頻率偏移(Frequency Offset)和 I/Q 失衡(I/Q Imbalance)等。此論文中將對正交分頻多工系統同時考慮通道估測以及 I/Q 失衡估測補償兩項議題，並在假設頻率偏移可被預先被估測出的情況下結合頻率偏移補償。I/Q 失衡有兩種類型，第一種為不隨頻率變動，第二種為隨頻率變動，兩者皆在本研究範圍之內。不同於傳統方法，我們的可能性函數(Likelihood Function)是由頻率域上所載的接收資料所組成，利用最大可能性(Maximum Likelihood)來估測 I/Q 失衡及通道，這個方法能使我們利用到正交分頻多工系統之特性及不同載波間的高相關性優點。論文內容也包含了最大可能性估計的效能分析與電腦模擬的結果，以驗證演算法的正確性和可行性，在訊號雜訊能量比(SNR)高到一定程度下其效能可達到 Cramer-Rao 下限(Lower Bounds)。. i.

(4) Joint Maximum Likelihood Estimation of Channel and I/Q Imbalance for OFDM Systems Student: Chen-Jui Hsu. Advisor: Dr. Wern-Ho Sheen. Department of Communication Engineering National Chiao Tung University. Abstract High data rate wireless communication is required by a variety of applications, and OFDM is one of the promising technologies too meet such a demand. On the other hand, direct-conversion RF receiver (DCR) architecture has become a trend in industry and academic world nowadays because it is small, cheap and low power-consuming. This kind of architecture, however, accompanies with some radio frequency (RF) imperfections such as the direct current (DC) offset, frequency offset, in-phase/quadrature (I/Q) imbalance etc. In this thesis, for OFDM system, we propose an algorithm for joint estimation of delay-spreading channel and IQ imbalance effect combined with frequency offset compensation assuming that frequency offset has been prior estimated. Both of frequency-independent and frequency-dependent IQ imbalance are covered. Different from traditional approach, our likelihood function is constructed by using frequency-domain data to estimate time-domain channel response and IQ imbalance based on ML criterion. This approach enables us to take advantage of characteristics of OFDM and smoothing property. Simulation results show that the estimated values approach their CRLBs once SNR is above certain level ii.

(5) 誌. 謝. 本篇論文得以順利完成，要特別感謝三個人，我的指導教授沈文和博士、徐進發學長以及程瑞曦博士的不斷指教。本篇論文是徐進發學長先前未完成研究的工作及延伸，在此特別感激他找到這麼好的研究題目並盡其所能的傳承給我，包括了指導我對其中不解的疑惑問題讓我能很快投入其中步上此研究軌道、並把實驗所需之模擬程式一一介紹給我讓我能進入狀況，最後並把此論文需要繼續研究或加以改善的結果指示給我讓我能繼續往此方向鑽研其中並做出不錯的結果。在兩年研究過程中，指導教授沈老師給予非常多的指導與建議，讓我對通訊系統有更深一層的思考與認識，尤其對研究謹慎小心的態度，讓我獲益匪淺。而在本篇論文研究中，沈老師及程瑞曦教授給了我很大的指引方向，教我如何把一個研究做完善的思考及呈現，鼓勵我不斷突破才得以做出更佳的研究成果，從中也學到了許多自己不足的專業知識觀念。另外，要感謝口試委員資通所經理賴癸江的指正與建議，使我的碩士論文更加完善。最後感謝實驗室全體同仁大家互相幫忙照顧，讓我能順利完成碩士研究、以及郭志成學長的不吝指教，很有耐心的為我解答不解的問題。也感謝家人給我的照顧與鼓勵，讓我無後顧之憂的進行碩士班研讀及研究，再次甚重感謝大家。. 民國九十五年八月研究生許宸睿謹識於交通大學. iii.

(6) Contents 摘要.................................................................................................................................i Abstract ..........................................................................................................................ii 誌. 謝.......................................................................................................................... iii. Contents ........................................................................................................................iv List of Figures ................................................................................................................v Chapter 1 Introduction ...................................................................................................1 Chapter 2 System Model................................................................................................4 2.1 Time Domain I/Q Imbalance Model ....................................................................4 2.2 Proposed I/Q Imbalance Compensation Scheme .................................................7 Chapter 3 Joint ML Estimation of I/Q Imbalance and Channel................................... 11 3.1 Smoothing Property in Time Domain ................................................................12 3.2 ML Estimation of Channel.................................................................................13 3.3 ML Estimation of I/Q Imbalance .......................................................................14 3.4 Algorithm with Frequency Offset Compensation ..............................................17 Chapter 4 Performance Analysis..................................................................................19 4.1 Mean and MSE of I/Q Imbalance Φ .................................................................19 4.2 Mean and MSE of Channel H............................................................................21 Chapter 5 Derivation of Cramer-Rao Lower Bounds ..................................................24 Chapter 6 Simulation Results.......................................................................................33 6.1 MSE of ML Estimates........................................................................................34 6.2 Sensitivities of Proposed Estimators..................................................................36 6.3 Uncoded BERs Performance .............................................................................40 Chapter 7 Conclusions .................................................................................................42 References....................................................................................................................43 iv.

(7) List of Figures Fig.1-1 Spectra of the received signal in DCR with I/Q imbalance ..............................2 Fig.2-1 General architecture of a direct-conversion receiver.........................................4 Fig.2-2 Mathematical model of a direct-conversion receiver with I/Q imbalance. .......4 Fig.3-1 The average magnitude of. Φ. per subcarrier.................................................16. Fig.6-1 MSE of IQ imbalance estimation ....................................................................35 Fig.6-2 MSE of delay-spreading channel estimation...................................................35 Fig.6-3 Channel estimation with various Lh ................................................................36 Fig.6-4 I/Q imbalance Φ estimation with various Lh .................................................37 Fig.6-5 I/Q imbalance Φ estimation with various Lϕ .................................................37 Fig.6-6 I/Q imbalance Φ estimation with various CFO effect....................................38 Fig.6-7 Uncoded BERs with 64 QAM.........................................................................40 Fig.6-8 CFO sensitivities on 64 QAM Uncoded BERs ...............................................41. v.

(8) Chapter 1 Introduction Demands for high data rate wireless communication services and cost-effective devices are getting stronger for past years. In response, OFDM has been adopted in several communication standards due to its higher spectral efficiency and effectiveness in dealing with multi-path delay spreading [1-3].. To combat multi-path. delay spreading, a cyclic-prefix is inserted in front of each OFDM symbols so that OFDM symbols can be compensated in frequency domain using one-tap equalizer [4-6].. On the other hand, to reduce device cost, RF transceivers adopt. direct-conversion architecture have been in favor due to fewer external components required in this approach [7-8].. However, direct-conversion architecture can easily. introduce IQ imbalance effect.. Typically, two types of IQ imbalance exist. For one thing, the imperfect local oscillator (LO) results in frequency-independent I/Q imbalance. The complex carrier generated by imperfect LO is not a true quadrature signal, and the amplitudes of carrier signal for I/Q branches are not equal in practice. For another, I/Q imbalance results from the unavoidable mismatch among all the analog elements on I/Q branches. The signals of I and Q branches are processed by individual analog devices, such as amplifiers, low-pass filter (LPF), A/D converters, and etc. Nevertheless, it is difficult to produce two analog devices with exactly identical responses or properties, especially when the bandwidth (BW) of the system is large, e.g. 40 MHz mode of IEEE 802.11n. This kind of I/Q imbalance is frequency-dependent; in other words, the I/Q imbalance effects tend to vary with frequency. The frequency-independent and frequency-dependent I/Q imbalance have the received signal interfered by its image signal as Fig.1-1 shows. Distorted OFDM symbols can result in higher error rate, if no effective IQ imbalance compensations are applied [9-11]. 1.

(9) (a) Received signal in passband. (b) Received signal which is interfered by image signal in baseband Fig.1-1 Spectra of the received signal in DCR with I/Q imbalance. Both of channel equalization and IQ compensation requires channel and IQ imbalance estimation beforehand. these two estimations [12-15].. Previous research has offered several methods for. However, most of them treated these two estimations. separately. Consequently, estimators they obtained are sub-optimal only.. Recently. researchers started to consider joint estimation for these two effects [16-18]. Gil et al [16] started to develop joint ML estimation of carrier offset, channel, IQ imbalance and DC offset using time-domain data.. Their method did not take characteristics of. OFDM into consideration and focus on frequency-independent IQ imbalance only. Tarighat et al [17] considered their joint estimation of IQ imbalance and channel specifically for OFDM systems.. Their estimation is performed completely in. frequency-domain and treats all carriers independently.. As a result, it requires. several training symbols to converge the estimation in their approach. Also, their 2.

(10) proposed method in frequency domain can not deal with frequency offset which may arising inter carrier interferences (ICI). Xing et al [18] considered joint effect of frequency offset, IQ imbalance (both types of IQ imbalance) in OFDM system. However, their method requests special design of training symbols to achieve expected performance.. We develop joint channel and IQ imbalance estimation using ML criterion. The IQ imbalance estimation covers both of frequency-independent and dependent cases. The likelihood function is constructed using frequency-domain data to estimate time-domain channel effect and IQ imbalance.. This approach enables us to make. use of characteristics of OFDM as well as correlation between subcarriers.. We also. consider carrier frequency offset (CFO) effect and propose a compensation method before our joint estimation of I/Q imbalance and channel, but we assume that CFO has been previously estimated within acceptable level.. This assumption can be. justified as seeing most communication standards using different types of training symbols for estimation.. This paper is organized as follows. Chapter 2 describes the model used for IQ imbalance and formulates its equivalent effect both in time domain and frequency domain to develop I/Q imbalance compensation scheme.. In chapter 3, algorithms. using ML estimation are derived, as well as take into account frequency offset compensation before applying the joint estimation.. Performance of proposed joint. estimators are analyzed in chapter 4, and the Cramer-Rao Lower Bounds (CRLBs) of the estimators are derived in chapter 5. Computer simulation based on IEEE 802.11a as well as results compared with analysis and CRLBs are discussed in chapter 6. Finally, Chapter 7 concludes the thesis. 3.

(11) Chapter 2 System Model 2.1 Time Domain I/Q Imbalance Model Fig.2-1 shows a typical architecture of a direct-conversion receiver, and its mathematical model with I/Q imbalance effect is represented in Fig.2-2. The I/Q imbalance due to imperfect LO is frequency-independent. After down conversion, the I and Q branch signals are generated by individual analog components, such as local oscillator (LO), amplifiers, low pass filters (LPF) and A/D converters which in general cause the frequency dependent I/Q imbalance. We use hI [n] and hQ [n] model the cascade effects of analog branch component, and frequency-dependent IQ imbalance occurs once hI [n] and hQ [n] show different responses.. Fig.2-1 General architecture of a direct-conversion receiver.. Fig.2-2 Mathematical model of a direct-conversion receiver with I/Q imbalance. 4.

(12) The frequency-independent I/Q imbalance caused by imperfect LO is expressed as CLO (t ) = cos(2π f ct ) − jg sin(2π f c t + θ ) , where g and θ denote amplitude and phase imbalance. After some arrangement, we could represent it in a complex form. If the LO is perfect, g , θ and ϕ 0 should be zero, and γ 0 should be equal to one. C LO (t ) = cos(2π f c t ) − jg sin(2π f ct + θ ). (2.1). = γ 0 e − j 2π f c t + ϕ 0 e j 2π f c t where γ 0. 1 (1 + ge− jθ ) and ϕ0 2. (2.2) 1 (1 − ge jθ ) 2. (. γ 0 ≈ 1 and. P.S .. ϕ0 ≈ 0. ⎛ proof : ⎜ ⎜ cos(2π f c t ) − jg sin(2π f c t + θ ) = 1 ( e − j 2π f ct + e j 2π fct ) + 1 g e − j( 2π fct +θ ) − e j( 2π fct +θ ) 2 2 ⎜ ⎜ 1 1 = (1 + ge − jθ ) e − j 2π fct + (1 − ge jθ ) e j 2π fct ⎜ 2 2 ⎜ − j 2π f c t j 2π f c t ⎜ = γ 0e + ϕ 0e ⎝. (. ). ). ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠. We define the received signal as. y (t ). {. }. 2 Re y (t )e j 2π fct = y (t )e j 2π fct + y* (t )e− j 2π fct. (2.3). y (t ) = s(t ) ⊗ hCH (t ) + n0 (t ) , ⊗ denotes convolution y (t ), s(t ), hCH (t ) is the baseband representation of the received signal, the transmitted signal, and the channel impulse response, and n0 (t ) is an AWGN.. The received signal y (t ) is multiplied by the output of LO CLO (t ) , which is not orthogonal perfectly. v(t ) = y (t ) × CLO (t ) = ⎣⎡ y (t )e j 2π fct + y* (t )e − j 2π fct ⎦⎤ × ⎣⎡γ 0 e − j 2π fct + ϕ0 e j 2π fct ⎤⎦. = γ 0 y (t ) + ϕ0 y* (t ) + γ 0 y* (t )e− j 4π fct + ϕ0 y (t )e j 4π fct will be removed by the following LPFs. 5. (2.4).

(13) = γ 0 y (t ) + ϕ0 y* (t ). (2.5). If the last two terms in (2.4) will be completely removed by the following LPF, we would find out that the desired signal is attenuated by a factor γ 0 and interfered by its image signal. This kind of I/Q imbalance effect due to the imperfect LO introduce the two factors γ 0 and ϕ0 constant over all frequencies. Thus, that is why it is called frequency-independent I/Q imbalance.. Now, we rewrite (2.5) into (2.6) to express I branch and Q branch signal of v (t ) more clearly. vI (t ) + j ⋅ vQ (t ). v(t ). = γ 0 y (t ) + ϕ0 y * (t ) 1 1 1 + ge − jθ ) y (t ) + (1 − ge jθ ) y* (t ) ( 2 2 1 1 = ⎡⎣ y (t )+y* (t ) ⎤⎦ + g ⎡⎣e − jθ y (t ) − e jθ y * (t ) ⎤⎦ 2 2 =. 1 1 ⇒ vI (t )= ⎡⎣ y (t )+y* (t ) ⎤⎦ , jvQ (t ) = g ⎡⎣e− jθ y (t ) − e jθ y* (t ) ⎤⎦ 2 2. (2.6). To analyze the effect of branch mismatches, the I part and Q part of signal r (t ) are expressed as following : r (t ) = rI (t ) + jrQ (t ) = vI (t ) ⊗ hI (t ) + jvQ (t ) ⊗ hQ (t ) 1 1 ⎡⎣ y (t )+y* (t ) ⎤⎦ ⊗ hI (t ) + g ⎡⎣e − jθ y (t ) − e jθ y* (t ) ⎤⎦ ⊗ hQ (t ) 2 2 1 1 = y (t ) ⊗ ⎣⎡ hI (t )+hQ (t ) ge − jθ ⎤⎦ + y * (t ) ⊗ ⎡⎣ hI (t ) − hQ (t ) ge jθ ⎤⎦ 2 2 =. h+ ( t ). h− ( t ). = y (t ) ⊗ h+ (t ) + y* (t ) ⊗ h− (t ) = [ s (t ) ⊗ hCH (t ) + n0 (t )] ⊗ h+ (t ) + [ s (t ) ⊗ hCH (t ) + n0 (t ) ] ⊗ h− (t ) *. After r (t ) is digitized at a rate that satisfies the Nyquist sampling theorem, the 6.

(14) resulting baseband discrete-time signal model can be represented as r[ n] = ( s[ n] ⊗ hCH [ n ] + n0 [ n]) ⊗ h+ [ n] + ( s[ n] ⊗ hCH [ n ] + n0 [ n]) ⊗ h− [ n ] *. where h+ [n] h− [n]. (2.7). 1 ⎡⎣ hI [n]+hQ [n]ge − jθ ⎤⎦ , 2 1 ⎡⎣ hI [n] − hQ [n]ge jθ ⎤⎦ 2. (2.8). Equation (2.7) says the joint frequency-independent and frequency-dependent I/Q imbalance effect equivalent to introducing a conjugate interference in time-domain and an interfering image signal in frequency-domain. Furthermore, observing the interference. power. due. to. h− [n]. and. attenuation. due. to. h+ [ n] ,. the. frequency-dependent I/Q imbalance is indeed caused by the degree of different response effect between hI [n] and hQ [n] in time domain.. 2.2 Proposed I/Q Imbalance Compensation Scheme. The received analog baseband OFDM signal r (t ) after digitalized and removed CP (Cyclic Prefix) can be expressed as Equation (2.7). Here we use small letters to denote time-domain signal and system response, their corresponding large letters to denote their representation in frequency domain. Based on equation (2.7), we can remove conjugate interference by introducing. ϕ[n] and applying conjugate cancellation (see Equation (2.9)). After conjugate cancellation, the remaining term of Equation (2.10) can be modeled as convolution between signal and overall channel effect plus noise. r[n] − ϕ[n] ⊗ r *[n] = ( h+ [n] − ϕ[n] ⊗ h−* [n]) ⊗ ( s[n] ⊗ hCH [n] + n0 [n]) + ( h− [n] − ϕ[n] ⊗ h+* [n]) ⊗ ( s[n] ⊗ hCH [n] + n0 [n]). *. This term must be 0 to cancel the conjugate interference. 7. (2.9).

(15) ⇒ r[n] − ϕ[n] ⊗ r *[n] = ( h+ [ n] − ϕ[n] ⊗ h−* [n]) ⊗ ( s[n] ⊗ hCH [n] + n0 [n]) = s[n] ⊗ h[n] + n[n]. (2.10). where h[n] = ( h+ [n] − ϕ[n] ⊗ h−* [n]) ⊗ hCH [n] overall channel effect , n[n] = ( h+ [n] − ϕ[n] ⊗ h−* [n]) ⊗ n0 [n]. and h− [n] − ϕ[n] ⊗ h+* [n] = 0 ⇒ ϕ[n] = ( h+* [n]) ⊗ h− [n] −1. (2.11). We suppose that the effective length of ϕ[n] plus effective channel length after I/Q imbalance effect ( see (2.10), i.e. ( h+ [n] − ϕ[n] ⊗ h−*[n]) ⊗ hCH [n] ) is smaller than GI (guard interval) to maintain the orthogonality of OFDM symbol. Then, we can take FFT (Fast Fourier Transform) on Equation (2.10) and (2.11) and assume total l subcarriers per OFDM symbol. We can obtain k ' th subcarrier :. R[k ] − Φ[k ]R*[l − k ] = S[ k ]H [k ] + N [k ]. for k = 1 ~ l − 1. (2.12). We discard the DC tone k = 0 in our algorithm analysis since it does not carry any information due to implementation issues, such as in 802.11a standardized OFDM systems.. Note that from (2.8) h+ [n] h− [n]. 1 1 ⎡⎣ hI [n]+hQ [n]ge − jθ ⎤⎦ ⇒ FFT ( h+ [n]) = ( H I [k ] + ge jθ H Q [k ]) 2 2 1 1 ⎡⎣ hI [n] − hQ [n]ge jθ ⎤⎦ ⇒ FFT ( h− [n]) = ( H I [k ] − ge jθ H Q [k ]) 2 2. Φ[k ] = FFT (ϕ[n]) = FFT. ( ( h [ n]) * +. −1. ⊗ h− [n]. 8. ).

(16) 1 H I [k ] − ge jθ H Q [k ]) ( H − [k ] 2 = * = * 1 H+[N − k] H I [ N − k ] + ge− jθ H Q [ N − k ]) ( 2 =. Ψ 0 [k ] Γ [N − k]. (2.13). * 0. H Q [k ] ⎞ 1⎛ where Ψ 0 [k ] = ⎜ 1 − ge jθ ⎟ , Γ 0 [k ] = 2⎝ H I [k ] ⎠. H Q [k ] ⎞ 1⎛ − jθ ⎜ 1 + ge ⎟ 2⎝ H I [k ] ⎠. Note that Ψ 0 [k ] 0 and Γ 0 [k ] 1 .. Here, we define some notation and operation for simplicity. For a vector X of size l − 1 , we write its vector form as. ⎡ X [1] ⎤ ⎢ X [2] ⎥ ⎢ ⎥ ⎢ ⎥ X =⎢ ⎥ ⎢ X [k ] ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ X [l − 1]⎦. ⎡ ⎢ ⎢ ⎢ and its mirror vector by the ︵ , X = ⎢ ⎢ ⎢ ⎢ ⎣. X [l − 1] ⎤ X [l − 2]⎥⎥ ⎥ ⎥ X [k ] ⎥ ⎥ ⎥ X [1] ⎦. Matrix D X with a vector X will denote a diagonal matrix with X on its main diagonal, and. denotes elementwise product. Then, the subcarrier index k can be. dropped and thus Equation (2.12) can be written in its equivalent vector form Equation (2.14).. R−Φ. R * = R - DΦ R * = D S H + N. ( ). where DΦ = DΨ 0 DΓ*. (2.14). −1. 0. Some characteristics of proposed compensation scheme in (2.14) can be stated as follows. The perfect compensation coefficients Φ in (2.14) can be physically viewed as being constructed from attenuation factor Ψ 0 [k ] and Γ 0 [k ] which 9.

(17) quantifies the severity of I/Q imbalance effect on each subcarier. For more detail, the variation of Φ between subcarriers reflect on H Q [k ] ≠ H I [k ] caused by severity of frequency dependent I/Q imbalance effect, and the DC constant term of Φ on each subcarrier reflect on g and θ which represent the frequency independent I/Q imbalance effect. Therefore, if only frequency independent I/Q imbalance exists, Φ will be its constant term of frequency-independent contribution and reduced to single tap coefficient. Moreover, if there is no I/Q imbalance effect, i.e., g = 1 , θ = 00 and. H Q [k ] = H I [k ] , then Ψ 0 [k ] = 0 and Γ 0 [k ] = 1 , thus Φ is reduced to 0 indicating no need for I/Q imbalance compensation.. The difference between time domain compensation method (2.10) and frequency domain compensation method (2.14) can be compared as follows. For time domain compensation method, it needs convolution operation, but for frequency domain compensation, it only needs elementwise product operation. Thus, if we perform Lϕ taps to compensate in time domain, the complexity of time domain method is about. Lϕ multiplications and. (L. ϕ. − 1) additions per subcarrier, whereas the complexity of. frequency domain method is always only 1 multiplication and 1 addition per subcarrier, which is independent of Lϕ . In conclusion, the compensation complexity of frequency domain method is much lower than time domain method.. 10.

(18) Chapter 3 Joint ML Estimation of I/Q Imbalance and Channel In chapter 2, we have proposed an easier frequency domain method to compensate I/Q imbalance effect and demodulate signal in (2.14), once Φ and H are estimated. In this chapter, Φ and H can be estimated in an ML sense using (2.14) when the training symbols s[n] are transmitted. We now analyze the statistical property of the noise N after compensation of the received signal in (2.14).. h+ [n] h− [n]. 1 1 ⎡ hI [n]+hQ [n]ge− jθ ⎦⎤ ⇒ FFT ( h+ [n]) = ( H I [k ] + ge jθ H Q [k ]) = Γ 0 [k ]H I [k ] ⎣ 2 2 1 1 ⎡ hI [n] − hQ [n]ge jθ ⎦⎤ ⇒ FFT ( h− [n]) = ( H I [k ] − ge jθ H Q [k ]) = Ψ 0 [k ]H I [k ] ⎣ 2 2. H Q [k ] ⎞ 1⎛ where Ψ 0 [k ] = ⎜ 1 − ge jθ ⎟ , Γ 0 [k ] = 2⎝ H I [k ] ⎠. H Q [k ] ⎞ 1⎛ − jθ ⎜ 1 + ge ⎟ 2⎝ H I [k ] ⎠. Note that n0 [n] is AWGN and its frequency domain N 0 is also AWGN. Therefore, N [k ] = FFT. {( h [n] − ϕ[n] ⊗ h [n]) ⊗ n [n]} * −. +. 0. ( Γ [k ] − Φ[k ]Ψ [l − k ]) H [k ]N [k ] (1 − Φ[k ]Φ [l − k ]) Γ [k ]H [k ]N [k ] * 0. 0. 0. I. *. 0. ⇒N. (I. N. I. − DΦ D*Φ ) DΓ0 D H I N 0. (3.1). (3.2). 0. (3.3). The AWGN n0 [n] is not a periodic signal, thus the convolution of AWGN n0 [n] with. ( h [ n] − ϕ [ n] ⊗ h [ n]) +. * −. in (3.1) is not equivalent to circular convolution. Then,. N [ k ] is indeed a linear combination of N 0 [ k ] to become a color noise in (3.1). However, the correlation between N [k ] is very low, hence can be approximated to be independent in (3.2). Therefore, vector N is approximated as equation (3.3) and 11.

(19) its. Σ. covariance. matrix. is. also. expressed. as. E { NN H } = σ 02 ( I N − DΦ D*Φ ) DΓ0 D H I D HH I DΓH0 ( I N − DΦ D*Φ ). H. For given Φ and H, the vector R - DΦ R* in (2.14) is Gaussian with mean. DS H , and covariance matrix Σ derived and approximated from above. Thus, the conditional. probability. {. p R − D R* Φ Φ , H. }. density. function. for. R - DΦ R *. is. {. written. }. H 1 exp − ( R − D R* Φ − DS H ) Σ −1 ( R − D R* Φ − DS H ) π det ( Σ ) n. as. (3.3). The ML estimates of Φ and H can be obtained by maximizing the following likelihood. function. (3.4). over. different. trial. value. of. Φ. and. H. :. Λ (Φ , H ) = − ( R − D R* Φ − DS H ) ∑ −1 ( R − D R* Φ − DS H ) H. H H = − ( R − D R* Φ − DS H ) ⎡⎢( I N − DΦ D*Φ ) DΓ0 D H I D HH I DΓH0 ( I N − DΦ D*Φ ) ⎤⎥ ⎣ ⎦. −1. (R − D. R*. Φ − DS H ). (3.4). 3.1 Smoothing Property in Time Domain Although direct estimation for Φ and H based on Equation (3.4) is possible, it does not make use of frequency correlation properties since it treats every carrier independently. It has been reported that due to the structure of OFDM symbol, time and frequency correlation can attain significant gain for channel estimation [20]. Actually, time domain I/Q imbalance coefficients ϕ and impulse response channel h have the part of “Fourier matrix” relation with their frequency domain Φ and H shown below :. 12.

(20) Φ = Fϕϕ. ,. H = Fh h. ⎛ 2π ⋅ i ⋅ n ⎞ exp ⎜ − ⎟ , 1 ≤ i ≤ l − 1, 0 ≤ n ≤ Lh − 1 N ⎠ ⎝ {Fϕ }i,n exp ⎜⎝⎛ − 2π ⋅Nj ⋅ m ⎟⎠⎞ , 1 ≤ j ≤ l − 1, 0 ≤ m ≤ Lϕ − 1 Lh is effective length of channel impulse response.. where {Fh }i , n and. Lϕ is degree of I/Q imbalance effect.. That means Φ and H just have Lϕ and Lϕ degrees of freedom instead of l − 1 degrees of freedom in l − 1 dimension vector; therefore, only much fewer. independent parameters in time domain instead of frequency domain need to be estimated, which can improve estimation performance drastically. Equation (3.5) enables us to express Φ and H in terms of their time-domain forms and use frequency-domain received data to do time domain estimation.. R - DR* Fϕϕ = DS Fh h + N. (3.5). 3.2 ML Estimation of Channel Due to the smoothing criterion indicated in section 3.1, Φ is replaced with. Fϕϕ 、 H is replaced with Fh h in (3.4), then the modified likelihood function become Λ (ϕ , h) H H = − ( R − D R* Fϕϕ − DS Fh h ) ⎡( I N − DΦ D*Φ ) DΓ0 D H I D HH I DΓH0 ( I N − DΦ D*Φ ) ⎤ ⎢⎣ ⎥⎦. = − ⎣⎡( I N − DΦ D*Φ ) DΓ0 DH I ⎦⎤. −1. (. R − DR* Fϕϕ − DS Fh h. = − W (ϕ ) ( R − DR* Fϕϕ − DS Fh h ). ). −1. (R − D. R*. Fϕϕ − D S Fh h ). 2. 2. where W (ϕ ) = ⎡⎣( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦. (3.6) −1. 13.

(21) The I/Q imbalance coefficients ϕ are fixed first in (3.6) and the estimated hˆML can be treated as maximum of. time-domain response channel response. Equation (3.6) in least square method. It can be expressed as. † hˆML (ϕ ) = ( W (ϕ ) DS Fh ) W (ϕ ) ( R − D R Fϕϕ *. where. ( ). †. ). (3.7). denotes Psudo Inverse. Then, the frequency–domain channel Hˆ ML can be computed through Fourier matrix.. Hˆ ML (ϕ ) = Fh hˆML (ϕ ) = Fh ( W (ϕ ) DS Fh ) ( R − D R Fϕϕ †. *. ). (3.8). 3.3 ML Estimation of I/Q Imbalance Equation (3.8) expresses the solution of Hˆ ML as function of ϕ . To obtain optimal ϕ , we substitute Hˆ ML (ϕ ) back to Equation (3.6) Λ (ϕ ). Λ(ϕ , h) H = Hˆ. ML. (ϕ ). † = − W (ϕ ) ⎡ R − D R* Fϕϕ − DS Fh ( W (ϕ ) DS Fh ) W (ϕ ) ( R − D R Fϕϕ ) ⎤ ⎣ ⎦. 2. *. (. ). = − I N − PW (ϕ ) DS Fh W (ϕ ) ( R − D R* Fϕϕ ). 2. H W (ϕ )DS Fh ⎡( W (ϕ )DS Fh ) W (ϕ )DS Fh ⎤ ⎣ ⎦ is a projection matrix. where PW (ϕ ) DS Fh. W (ϕ ) = ⎡⎣( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦. −1. −1. ( W(ϕ )DS Fh ). H. is functions of ϕ , so it is difficult to maximize. 14.

(22) Λ (ϕ ). Λ(ϕ , h) H = Hˆ. ML. over ϕ . Some approximation on W (ϕ ) is needed to derive. (ϕ ). the ML estimate of I/Q imbalance Φ in a close form. 1.. First, express DΓ0 as function of DΦ H [k ] ⎞ 1⎛ Γ 0 [k ] = ⎜1 + ge − jθ Q ⎟ 2⎝ H I [k ] ⎠. H [k ] ⎞ 1⎛ ∵ Ψ 0 [k ] = ⎜1 − ge jθ Q ⎟ , 2⎝ H I [k ] ⎠ ∴Ψ 0 [k ] = 1 − Γ*0 [l − k ]. ⇒ DΨ 0 = I − DΓ*. 0. ( ) (. ⇒ DΦ = DΨ0 DΓ*. −1. 0. = I − D*Γ0. )( ) ( ) DΓ*. 0. −1. = DΓ*. −1. −I. 0. ⇒ DΓ−10 = I + DΦ*. 2.. (3.13). Because the magnitude of Φ[k ] is around 0.1 in moderate I/Q imbalance level shown in fig.3-1, the second order of Φ[k ] can be eliminated.. (I − D 3.. Φ. D Φ* ). −1. I. From 1 and 2, we can approximate W (ϕ ) as W (ϕ ) = ⎡⎣( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦. (D D ) = (I + D ) D. -1. −1. Γ0. HI. Φ*. 4.. −1 HI. However, this approximation is not enough, approximate again W (ϕ ). D−H1I. 15.

(23) Averge Magnitude of φ per subcarrier. 0.12 0.11 0.1 0.09 0.08 0.07 0.06 10 1.1 1.08. 5. 1.06 1.04 1.02 0. Phase Imbalance θ (degree). 1. Gain Imbalance g. Fig.3-1 The average magnitude of Φ per subcarrier. Replace W (ϕ ) with D−H1I , hence, the likelihood function is finally approximated as. (. ). Λ(ϕ ) ≈ − I N − PD−1 D F D HI. (R − D. Fϕ. R* ϕ. (. D D S Fh ⎡ D−H1I D S Fh ⎢⎣ −1 HI. where PD−1 D F HI. S h. −1 HI. S h. ). H. ). 2. (3.9). D D S Fh ⎤ ⎥⎦ −1 HI. −1. (D. −1 HI. D S Fh. ). H. Similar to Equation (3.6), the maximum of Equation (3.9) can be expressed as. (. ϕˆML = ⎡ I N − PD ⎣⎢. (. ) F ) (I. −1 H I DS Fh. = ⎡ D−H1I DR* ⎣⎢. †. H. ϕ. ( )D. ). D−H1I DR* Fϕ ⎤ I N − PD−1 D F D−H1I R HI S h ⎦⎥ N. − PD−1 D F HI. S h. −1 HI. DR* Fϕ ⎤ ⎦⎥. −1. (D. −1 HI. DR* Fϕ. ). H. (I. N. ). − PD−1 D F D−H1I R HI. S h. (3.10). The frequency domain of I/Q imbalance coefficients can be obtained through FFT matrix. 16.

(24) ˆ = F ϕˆ Φ ML ϕ ML. 3.4 Algorithm with Frequency Offset Compensation So far, we haven’t considered the carrier frequency offset (CFO) effect at the receiver. In this section, we propose the CFO compensation method combined with our algorithm. The received signal path considering CFO effect can be derived as following. {. y (t ). 2 Re y (t )e. j 2π ( f c +Δf )t. } = y(t )e. j 2π ( f c +Δf )t. + y* (t )e. − j 2π ( f c +Δf )t. where Δf is the CFO. v(t ) = y (t ) × CLO (t ) = ⎡ y (t )e ⎣. j 2π ( f c +Δf )t. + y * (t )e. − j 2π ( f c +Δf )t. ⎤ × ⎡γ 0 e − j 2π fct + ϕ0 e j 2π fct ⎤ ⎦ ⎦ ⎣. = γ 0 y (t )e j 2πΔft + ϕ0 y* (t )e − j 2πΔft + γ 0 y* (t )e. − j ( 4π f c + 2πΔf )t. + ϕ0 y (t )e. j ( 4π f c + 2πΔf )t. will be removed by the following LPFs. = γ 0 y (t )e j 2πΔft + ϕ0 y* (t )e− j 2πΔft. r (t ) = rI (t ) + rQ (t ) = vI (t ) ⊗ hI (t ) + vQ (t ) ⊗ hQ (t ) 1 1 ⎡ y (t )e j 2πΔft +y* (t )e− j 2πΔft ⎦⎤ ⊗ hI (t ) + g ⎣⎡e− jθ y (t )e j 2πΔft − e jθ y* (t )e− j 2πΔft ⎤⎦ ⊗ hQ (t ) ⎣ 2 2 1 1 = y (t )e j 2πΔft ⊗ ⎡⎣ hI (t )+hQ (t ) ge − jθ ⎤⎦ + y* (t )e− j 2πΔft ⊗ ⎡⎣ hI (t ) − hQ (t ) ge jθ ⎤⎦ 2 2 =. h− ( t ). h+ ( t ). = y (t )e j 2πΔft ⊗ h+ (t ) + y* (t )e− j 2πΔft ⊗ h− (t ) = e j 2πΔft [ s (t ) ⊗ hCH (t ) + n0 (t ) ] ⊗ h+ (t ) + e − j 2πΔft [ s (t ) ⊗ hCH (t ) + n0 (t )] ⊗ h− (t ) *. The finally resulting baseband signal model is given by. 17.

(25) r[n] = e j 2πΔn ( s[n] ⊗ hCH [n] + n0 [n]) ⊗ h+ [n] + e − j 2πΔn ( s[n] ⊗ hCH [ n] + n0 [ n]) ⊗ h− [ n] *. After conjugate cancellation and CFO compensation in time domain, the modified algorithm is shown in equation (3.11).. r[n] − ϕ[n] ⊗ r *[n] = ( h+ [n] − ϕ[n] ⊗ h−* [n]) ⊗ ( s[n] ⊗ hCH [n] + n0 [n]) e j = ( s[n] ⊗ h[n] + n[n]) e j. ⇒ e− j. ωn. ωn. ωn. ( r[n] − ϕ[n] ⊗ r [n]) = s[n] ⊗ h[n] + n[n] *. For frequency domain compensation, we must modify the equation as following : ( Note that ϕ[n] is not ϕ[n] at all ). (e. − j ωn. r[ n ] ) − ϕ [ n ] ⊗ ( e − j. r [n]) = s[n] ⊗ h[n] + n[n]. ωn *. (3.11). Take FFT of (3.11). RComp − Φ. RIm age−Comp = RComp − DRIm age−Comp Φ = SH + N. where RComp = FFT ( e − j. ωn. r[n]) , RIm age−Comp = FFT ( e− j. (3.12). r [ n]). ωn *. Equation (3.12) similar to equation (2.14) illustrates how to apply proposed algorithm in frequency domain with CFO compensation. Note that we assume CFO has been estimated prior to our joint estimation. Therefore, we can get RComp and RIm age−Comp to pre-compensate CFO effect in (3.12) and then do joint estimation based on the signal model the same with (2.14) derived previously.. 18.

(26) Chapter 4 Performance Analysis In this chapter, the mean and mean square error (MSE) of the ML estimators are examined under the assumption of a high signal-to-noise ratio (SNR). Section 4.1 provides the analysis of the mean and MSE of our proposed estimator of I/Q imbalance Φ , and the mean and MSE of our proposed estimator of channel H are analyzed in section 4.2.. 4.1 Mean and MSE of I/Q Imbalance Φ ˆ In this section, the mean and MSE of Φ ML defined in (3.10) are analyzed.. Following from (2.14). R - DR* Φ = DS H + N ⇒ R = DR* Φ + DS H + N = DR* Fϕϕ + DS Fh h + N. (4.1). and (3.10). ϕˆML = ⎡( D−H1 DR Fϕ ) ˆ Φ ML. I ⎢⎣ = FϕϕˆML. (. (. H. *. = Fϕ ⎡ D−H1I D R* Fϕ ⎣⎢. ). (. H. (. = Fϕ ⎡ D−H1I D R* Fϕ ⎢⎣. ). ). I N − PD−1 D F D−H1I DR* Fϕ ⎤ HI S h ⎥⎦. ). I N − PD−1 D F D−H1I D R* Fϕ ⎤ HI S h ⎦⎥. H. (. ). I N − PD−1 D F D−H1I D R* Fϕ ⎤ HI S h ⎥⎦. −1. −1. (D. −1. (D. (D. −1 HI. −1 HI. −1 HI. DR* Fϕ. ). D R* Fϕ. D R* Fϕ. ). D R* Fϕ. ). H. (I. ). H. H. (I. (I. ). − PD−1 D F D−H1I R. N. HI. S h. ). − PD−1 D F D−H1I ( D R* Fϕϕ + DS Fh h + N ). N. HI. S h. ). − PD−1 D F D−H1I D R* Fϕ ϕ. N. HI. S h. I Lϕ ×Lϕ. (. + Fϕ ⎡ D−H1I D R* Fϕ ⎣⎢. ). H. (I. N. ). − PD−1 D F D−H1I D R* Fϕ ⎤ HI S h ⎦⎥. −1. (D. −1 HI. (∵ I (. + Fϕ ⎡ D−H1I D R* Fϕ ⎣⎢. ). (. H. (I. N. = Fϕϕ + Fϕ ⎡ D−H1I D R* Fϕ ⎣⎢. (. ). − PD−1 D F D−H1I D R* Fϕ ⎤ HI S h ⎦⎥. ). = Φ N ×1 + Fϕ ⎡ D−H1I D R* Fϕ ⎢⎣. H. ). (. H. −1. (D. −1 HI. H. (I. N. − PD−1 D F HI. S h. 19. ). HI. S h. − PD−1 D F is a projection matrix of D−H1I DS Fh. D R* Fϕ. ). N. ). − PD−1 D F D S D−H1I Fh h 0 N ×Lh. HI. S h. ). (I. H. I N − PD−1 D F D−H1I D R* Fϕ ⎤ HI S h ⎦⎥. (I. N. N. −1. D D R* Fϕ ⎤ ⎥⎦ −1 HI. ). ). − PD−1 D F D−H1I N HI. (D. −1. −1 HI. (D. S h. (I − P ) D N F ) (I − P )D N. D R* Fϕ. −1 HI. D R*. ). H. N. D−H1I D S Fh. H. ϕ. N. D−H1I DS Fh. −1 HI. −1 HI.

(27) ˆ The relation between the true I/Q imbalance Φ N×1 and its estimates Φ ML can be. expressed as (4.2) derived from above equation.. (. ˆ = Φ + F ⎡ D−1 D * F ∴ Φ ML N ×1 ϕ ⎢ ⎣ HI R ϕ. ). H. (. ). I N − PD−1 D F D−H1I D R* Fϕ ⎤ HI S h ⎥⎦. −1. (D. −1 HI. D R* Fϕ. ). H. (I. N. ). − PD−1 D F D−H1I N HI. S h. (4.2). To proceed, we define Ra which is the deterministic part of R in (2.14). {( s[n] ⊗ h [n] + n [n]) ⊗ h [n] + ( s[n] ⊗ h [n] + n [n]) ⊗ h [n]} = FFT {s[n] ⊗ h [n] ⊗ h [n] + ( s[n] ⊗ h [n]) ⊗ h [n]}. R = FFT. *. +. 0. CH. −. 0. CH. *. +. CH. −. CH. Ra. {. }. + FFT n0 [n] ⊗ h+ [n] + ( n0 [ n]) ⊗ h− [n] *. Ra + DΓ0 D H I N 0 + DΨ 0 D H I N 0*. {. }. Ra = FFT s[n] ⊗ hCH [n] ⊗ h+ [n] + ( s[n] ⊗ hCH [n]) ⊗ h− [n] *. The unbiasedness of estimators in (4.1) can be seen if R is replaced with Ra . Therefore, the estimators are approximately unbiased when SNR. 1 . Specifically,. ˆ the bias of Φ ML may be approximated by ˆ − Φ ⎤ = E ⎧⎨F ⎡ D−1 D * F E ⎡⎣ Φ ϕ ML HI R ϕ ⎦ ⎩ ⎢⎣. (. (. Fϕ ⎡⎢ D−H1I D R* Fϕ a ⎣. From (3.3). N. Because E [ N ]. (I. N. ). H. ) (I H. N. (. ). I N − PD−1 D F D−H1I D R* Fϕ ⎤ HI S h ⎥⎦. ). − PD−1 D F HI. S h. D D R* Fϕ ⎤⎥ a ⎦ −1 HI. − DΦ D*Φ ) DΓ0 D H I N 0. E ⎡⎣( I N − DΦ D*Φ ) DΓ0 D H I N 0 ⎤⎦ = 0. ˆ − Φ ⎤ = 0 in high SNR ∴ E ⎣⎡Φ ML ⎦. 20. −1. −1. (D. (D. −1 HI. −1 HI. D R* Fϕ. D R* Fϕ a. ). H. ) (I H. N. (I. N. ). ⎫ − PD−1 D F D−H1I N ⎬ HI S h ⎭. ). − PD−1 D F D−H1I E [ N ] HI. S h.

(28) ˆ The MSE of Φ ML can be approximated as following. ˆ −Φ ⎤ E⎡Φ ⎢⎣ ML ⎥⎦ ⎪⎧ = E ⎨ Fϕ ⎡ D−H1I D R* Fϕ ⎢ ⎪⎩ ⎣ 2. (. ⎧⎪ E ⎨ Fϕ ⎢⎡ D−H1I D R* Fϕ a ⎪⎩ ⎣. (. {. = E QN 0. 2. ). H. (. )( H. I N − PD−1 D F. ). D D R* Fϕ ⎤ ⎦⎥. I N − PD−1 D F. ). D D R* Fϕ ⎥⎤ a ⎦. HI. HI. S h. S h. −1 HI. −1. −1. −1 HI. (D. −1 HI. (. −1 HI. D R* Fϕ. D D R* Fϕ a. (. ⎫⎪ I N − PD−1 D F D N ⎬ HI S h ⎪⎭. )(. ⎫⎪ I N − PD−1 D F D N ⎬ HI S h ⎪⎭. ). H. H. ). ). 2. −1 HI. 2. −1 HI. }. {. = tr QE ⎡⎣ N 0 N 0H ⎤⎦ Q H. }. Note that N 0 is AWGN noise in frequency domain. = σ 02tr {QQ H } = σ 02 Q. (. F. where Q =Fϕ ⎡⎢ D−H1I D R* Fϕ a ⎣. denotes Fronbeinus norm. F. ) (I H. N. ). − PD−1 D F D−H1I D R* Fϕ ⎤⎥ HI S h a ⎦. −1. (D. −1 HI. D R* Fϕ a. ) (I H. N. ). − PD−1 D F D−H1I ( I N − DΦ D*Φ ) DΓ0 D H I HI. S h. 4.2 Mean and MSE of Channel H The mean and MSE of H defined in (3.8) can be derived in the way similar to the above procedure. Start from (3.8) :. Hˆ ML = Fh hˆML = Fh ⎡⎣(DS Fh ) H W H (ϕ ) W (ϕ ) (DS Fh ) ⎤⎦ (DS Fh ) H W H (ϕ ) W (ϕ ) ( R − D R* FϕϕˆML ) −1. (. −1 ˆ = Fh ⎡⎣(DS Fh ) H W H (ϕ ) W (ϕ ) (DS Fh ) ⎤⎦ (DS Fh ) H W H (ϕ ) W (ϕ ) R − D R* Φ ML. (4.3). Substitute (4.1) into (4.3). 21. ).

(29) −1 ˆ ⎤ Hˆ ML = Fh ⎡⎣(DS Fh ) H W H (ϕ ) W (ϕ ) (DS Fh ) ⎤⎦ (DS Fh ) H W H (ϕ ) W (ϕ ) ⎡⎣( D R* Φ + DS H + N ) − D R* Φ ML ⎦. (. ). −1 ˆ ⎤ = Fh ⎡⎣(DS Fh ) H W H (ϕ ) W (ϕ ) (DS Fh ) ⎤⎦ (DS Fh ) H W H (ϕ ) W (ϕ ) ⎡ DS H + D R* Φ − Φ ML + N ⎦ ⎣ −1. = Fh ⎡⎣(DS Fh ) H W H (ϕ ) W (ϕ ) (DS Fh ) ⎤⎦ (DS Fh ) H W H (ϕ ) W (ϕ ) DS Fh h = I Lh×Lh. (. ). −1 ˆ ⎤ + Fh ⎡⎣(DS Fh ) H W H (ϕ ) W (ϕ ) (DS Fh ) ⎤⎦ (DS Fh ) H W H (ϕ ) W (ϕ ) ⎡ D R* Φ − Φ ML + N ⎦ ⎣ † ˆ ⎤ = Fh h + Fh ⎡⎣ W (ϕ ) (DS Fh ) ⎤⎦ W (ϕ ) ⎡ D R* Φ − Φ ML + N ⎦ ⎣. (. ). The relation between the true channel H and its estimates Hˆ ML can be expressed as (4.3) derived from above equation.. (. ). † ˆ ⎤ Hˆ ML = H + Fh ⎡⎣ W (ϕ ) (DS Fh ) ⎤⎦ W (ϕ ) ⎡ D R* Φ − Φ ML + N ⎦ ⎣. (4.3). Again, R is replaced with Ra in high SNR in (4.2) :. (. (I F ) (I. ˆ = Φ + F ⎡ D−1 D * F Φ ML N ×1 ϕ ⎢ ⎣ HI R ϕ. (. Φ N ×1 + Fϕ ⎡⎢ D−H1I D R* a ⎣ = Φ N ×1 + QN 0. ). H. N. H. ϕ. N. ) )D. − PD−1 D F D−H1I DR* Fϕ ⎤ HI S h ⎥⎦ − PD−1 D F HI. S h. −1 HI. DR* Fϕ ⎤⎥ a ⎦. −1. −1. (D. (D. (I F ) (I. −1 HI. DR* Fϕ. −1 HI. D R* a. ). H. − PD−1 D F D−H1I N. N. − PD−1 D F. H. ϕ. ) )D. N. HI. HI. S h. S h. ˆ − Φ = QN ⇒ Φ ML N ×1 0. N. (4.4). Substitute (4.4) and (3.3) into (4.3) Therefore, E ⎣⎡ Hˆ ML − H ⎦⎤. −1 HI. {. † E Fh ⎡⎣ W (ϕ ) (DS Fh ) ⎤⎦ W (ϕ ) ⎡ D R* Q + ( I N − DΦ D*Φ ) DΓ0 D H I ⎤ N 0 ⎣ a ⎦. }. † = Fh ⎡⎣ W (ϕ ) (DS Fh ) ⎤⎦ W (ϕ ) ⎡ D R* Q + ( I N − DΦ D*Φ ) DΓ0 D H I ⎤ E [ N 0 ] ⎣ a ⎦ =0. 22.

(30) E ⎡ Hˆ ML − H ⎤ ⎢⎣ ⎥⎦ 2. 2 ⎧ ⎫ † ⎪ ⎪ * E ⎨ Fh ⎡⎣ W (ϕ ) (DS Fh ) ⎤⎦ W (ϕ ) ⎡ D R* Q + ( I N − DΦ DΦ ) DΓ0 D H I ⎤ N 0 ⎬ ⎣ a ⎦ ⎪ ⎪ Define as P ⎩ ⎭. {. = E PN 0 = σ 02 P. 2. }. F. 23.

(31) Chapter 5 Derivation of Cramer-Rao Lower Bounds The CRLBs of H and Φ can be derived following the procedure in [19]. Define vectors θ = (ϕ T hT ). and ω = ( ΦT HT ) = Fϕ , hθ. T. T. ⎡ f1 ⎤ ⎛ Fϕ 0( l −1)×( l −1) ⎞ where Fϕ ,h = ⎜ ⎟ , Fϕ = ⎢⎢ ⎥⎥ ⎜ 0( l −1)×( l −1) Fh ⎟ ⎝ ⎠ ⎢⎣ f l −1 ⎥⎦ fi is the i ' th row vector of Fϕ , and Φ[i] = f iϕ. The approximated conditional probability density function is shown below:. {. }. p R − D R* Fρϕ ϕ , h. {. }. H 1 exp − ( R − D R* Fϕϕ − SFh h ) Σ −1 ( R − D R* Fϕϕ − SFh h ) π det ( Σ ) n. where Σ = E { NN H } = σ 02 ( I N − DΦ D*Φ ) DΓ0 D H I D HH I DΓH0 ( I N − DΦ D*Φ ). ( First order approximation ). σ 02 DΓ D H D HH DΓH 0. From (3.13) Γ 0 [k ] =. I. I. H. 0. 1 1+ Φ [N − k] *. Hence, the k'th diagonal of Σ refers to Σ kk. Define log-likelihood function. σ 02 H I [k ]. {. 2. 1 1 + Φ[ N − k ]. 2. }. f (θ ) = ln p R − D R* Fϕϕ ϕ , h . Because ω is a. function of θ , the CRLBs of ω in frequency domain can be derived from f (θ ) by (5.1). CRLB(ωk ) = ( Fϕ ,h I −1FϕH,h ). (5.1). kk. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ I= E⎢ * ⎜ * ⎟ ⎥ ⎢⎣ ∂θ ⎝ ∂θ ⎠ ⎥⎦ ⎡ ∂f (ω ) ⎤ ∂f (ω ) ⎢ ∂ϕ * ⎥ ⎥ =⎢ ∂θ * ⎢ ∂f (ω ) ⎥ ⎢⎣ ∂h* ⎥⎦. 24.

(32) A.. We express f (θ ) in scalar form :. {. }. f (θ ) = ln p R − D R* Fϕϕ ϕ , h = ln. {. }. H 1 exp − ( R − D R* Fϕϕ − SFh h ) Σ −1 ( R − D R* Fϕϕ − SFh h ) π det ( Σ ) n. = ln (π − n det −1 ( Σ ) ) − ( R − D R* Fϕϕ − DS Fh h ) Σ −1 ( R − D R* Fϕϕ − DS Fh h ) H. (. l −1. = ∑ ln π − nσ 0−2 H I [k ] k =1. −. B.. 1. σ. ∑ ( H [k ] l −1. −2. I. 2 0 k =1. −2. 1 + Φ[l − k ]. 2. ). 1 + Φ[l − k ] R[k ] − R*[l − k ]Φ[k ] − S[k ]H [k ] 2. 2. ). Take derivative of f (θ ) with respect to ϕ * using chain rules ( ϕ is a function of Φ ). Then, we obtain (5.2) from. (. ∂f (ω ) l −1 ∂ ln 1 + Φ[l − k ] =∑ ∂ϕ * ∂Φ*[l − k ] k =1. 2. ) ∂Φ [l − k ] *. ∂ϕ *. (. ). 2 ⎛ ∂ 1 + Φ[l − k ] ∂Φ*[l − k ] ⎞ ⎜ H [k ] −2 R[k ] − R*[l − k ]Φ[ k ] − S [ k ]H [k ] 2 ⎟ I * * l −1 ⎜ ⎟ ∂Φ [l − k ] ∂ϕ 1 ⎟ − 2 ∑⎜ 2 σ 0 k =1 ⎜ ∂ R[k ] − R*[l − k ]Φ[ k ] − S[ k ]H [ k ] ∂Φ*[k ] ⎟ ⎜ + H [k ] −2 1 + Φ[l − k ] 2 ⎟ I * * ⎜ ⎟ ϕ ∂Φ ∂ [ k ] ⎝ ⎠ l −1 fl −Hk =∑ * k =1 1 + Φ [l − k ]. ). (. ⎛ H [k ] −2 R[ k ] − R*[l − k ]Φ[ k ] − S[ k ]H [k ] 2 (1 + Φ[l − k ]) f H ⎞ I l −k ⎜ ⎟ − 2∑ σ 0 k =1 ⎜ − H [k ] −2 1 + Φ[l − k ] 2 ( R[k ] − R*[l − k ]Φ[k ] − S[k ]H [k ]) R[l − k ] f H ⎟ I k ⎠ ⎝ 1. = FϕH Φ. l −1. −. 1 1+Φ. *. 1. σ. 2 0. FϕH D−H1. I. H I*. N* +. DΦ1+Φ N. 1. σ. 2 0. FϕH D−H1. I. H I*. DΦ. 1+Φ. Φ1*+Φ. R. N. (5.2) where. denotes elementwise product Φ. 1 1+Φ*. ⎡ 1 =⎢ * ⎣1 + Φ [1]. ⎤ 1 ⎥ * 1 + Φ [l − 1] ⎦. 25. T. , Φ1+Φ = [1 + Φ[1] 1 + Φ[l − 1]]. T.

(33) C.. Define Ra as the deterministic part of R : Take FFT of (2.7) :. {( s[n] ⊗ h [n] + n [n]) ⊗ h [n] + ( s[n] ⊗ h [n] + n [n]) ⊗ h [n]} = FFT {s[n] ⊗ h [n] ⊗ h [n] + ( s[n] ⊗ h [n]) ⊗ h [n]}. R = FFT. *. CH. +. 0. CH. −. 0. *. +. CH. Ra. {. −. CH. }. + FFT n0 [n] ⊗ h+ [n] + ( n0 [n]) ⊗ h− [n] *. Ra + DΓ0 D H I N 0 + DΨ 0 D H I N 0*. R. N. (I. N. From (3.3). (R + D a. Γ0. N. − DΦ D*Φ ) DΓ0 D H I N 0. D H I N 0 + DΨ 0 D H I N 0*. ) (I. N. − DΦ D*Φ ) DΓ0 D H I N 0. = ( I N − DΦ D*Φ ) DΓ0 D H I D Ra N 0. + ( I N − DΦ D*Φ ) DΓ0 DΓ0 D H I D H I N 0. + ( I N − DΦ D*Φ ) DΓ0 DΨ 0 D H I D H * N 0 I. Substitute (3.3) N and (5.3) R ⎡ 1 ∂f (ω ) H F V1 N 0 = ρ ⎢Φ 1 − * 2 ∂ϕ ⎣ 1+Φ* σ 0. (5.3) N0 N 0*. ,. (H. I. = H I*. ). N into (5.2), we get : N 0* +. 1. σ 02. V2 D Ra N 0 +. 1. σ 02. V3 N 0. N0 +. 1. σ 02. V4 N 0. (5.4) where V1 = DΦ1+Φ ( I N − DΦ D*Φ ) DΓ0 ⎡⎣( I N − DΦ D*Φ ) DΓ0 ⎤⎦ V2 = D−HHI ( I N − DΦ D*Φ ) DΓ0 DΦ V3 = D H I D−HHI DΦ. 1+Φ. V4 = DΦ. 1+Φ. D.. Φ1*+Φ. (I. N. Φ1*+Φ. (I. 1+Φ. N. *. Φ1*+Φ. − DΦ D*Φ ) DΓ0 DΓ0 ,. − DΦ D*Φ ) DΓ0 DΨ0. Take derivative of f (θ ) with respect to h* :. ∂f (ω ) = FhH DS Σ −1 ( I N − DΦ D*Φ ) DΓ0 DH I N 0 * ∂h. E.. From the derivation of C. and D., we have expressed. (5.5). ∂f (ω ) ∂f (ω ) and as * ∂h* ∂ϕ. function of AWGN N 0 in frequency domain. Then, we can use (5.4) and (5.5) 26. ⎤ N 0* ⎥ ⎦.

(34) to calculate. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ E⎢ ⎥ , * ⎜ * ⎟ ⎢⎣ ∂ϕ ⎝ ∂ϕ ⎠ ⎥⎦. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ E⎢ * ⎜ * ⎟ ⎥ ⎢⎣ ∂ϕ ⎝ ∂h ⎠ ⎥⎦. and. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ E⎢ . In order to compute the above three terms, we should * ⎜ * ⎟ ⎥ ⎣⎢ ∂h ⎝ ∂h ⎠ ⎦⎥ first compute the expectation of the following functions of AWGN N 0 terms : Note that for complex value N 0 [ k ] ∀k E { N 0 [k ]} = 0 ,. { } =σ , E { N [k ] } = 2σ E N 0 [k ]. 2. 2 0. 4. 4 0. 0. ∀k , j and k ≠ j E { N 0 [k ]N 0 [ j ]} = 0 and all the third moment of N 0 [k ] is 0 , i.e. E { N 03 [k ]} = E. {( N [k ]) } = E { N [k ] * 0. 3. 2. 0. } {. }. N 0*[k ] = E N 0 [k ] N 0 [k ] = 0 2. To be more detail, we express those functions of N 0 as scalar form : N 0 = [ N 0 [1], N 0 [2],. , N 0 [l − 1]]. T. T. ). N 0* = ⎡ N 0 [l − 1] , N 0 [l − 2] , ⎣. N0. N 0 = [ N 0 [1]N 0 [l − 1], N 0 [2]N 0 [l − 2],. , N 0 [l − 1]N 0 [1]] ,. N 0* = ⎡ N 0 [1] , N 0 [2] , ⎣. T. N0. 2. 2. 2. 2. 2 , N 0 [1] ⎤ , ⎦. (N. N0. 2 , N 0 [l − 1] ⎤ ⎦. 0. N 0* T. (N. ,. 0. k. = N 0 [l − k ]. (N. 0. N 0* ) = N 0 [k ]. N0. ). 2. k. = N 0 [k ]N 0 [l − k ]. 2. k. Then, use the above scalar form to calculate their self and cross expectation as following : ⎛ ⎜ ⎜ Define 1 = [1,1, ⎜ ⎜⎜ ⎝. ,1]l×1 T. ⎡ ⎢ , P=⎢ ⎢ 1 ⎢ ⎣1. Self expectation :. a.. E { N 0 N 0H } = σ 02I l −1 27. 1⎤ ⎥ ⎥ ⎥ ⎥ ⎦. ⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠.

(35) b.. {. N 0* ⎡⎣ N 0. N 0* ⎤⎦. {. N 0* ⎡⎣ N 0. N 0* ⎤⎦. ∵ E N0 ⇒ E N0. H. }. kj. { {. }=σ. H. }. ⎧ E N [l − k ] 2 N [l − j ] 2 = σ 4 for k ≠ j 0 0 0 ⎪ =⎨ 4 ⎪ E N 0 [l − k ] = 2σ 04 for k =j ⎩ 4 0. }. (11. T. + I l −1 ). where E { }kj is its k'th row and j'th column value. c.. {. N 0 ⎡⎣ N 0. ∵ E N0. N 0 ⎤⎦. }. H. {. kj. = E { N 0 [k ]N 0 [l − k ] N 0*[ j ]N 0*[l − j ]}. }. ⎧⎪ E N [k ] 2 N [l − k ] 2 = σ 4 0 0 0 =⎨ ⎪⎩ 0. {. ⇒ E N0. d.. N 0 ⎡⎣ N 0. N 0 ⎤⎦. {. N 0* ⎡⎣ N 0. N 0* ⎤⎦. {. N 0* ⎡⎣ N 0. N 0* ⎤⎦. ∵ E N0 ⇒ E N0. H. H. }=σ. }. H. kj. 4 0. for k =j and j =l -k otherwise. ( I l −1 + P ). { {. }. ⎧ E N [k ] 2 N [ j ] 2 = σ 4 for k ≠ j 0 0 0 ⎪ =⎨ 4 ⎪ E N 0 [k ] = 2σ 04 for k =j ⎩. }=σ. 4 0. }. (11. T. + I l −1 ). Cross expectation :. e.. {. ∵ E N 0 ⎡⎣ N 0. N 0* ⎤⎦. H. }. kj. {. = E N 0 ⎡⎣ N 0. ( All Third moment of N 0 ⇒E. f.. { N ⎣⎡ N. { ⇒ E{ N. 0. ∵ E N0. 0. 0. N 0* ⎤⎦. N 0* ⎣⎡ N 0. N 0* ⎡⎣ N 0. H. N 0 ⎤⎦. H. is 0 ). } = E { N ⎡⎣ N 0. N 0 ⎤⎦. 0. H. } = E { N [l − k ] N ⎤⎦ } = 0. N 0 ⎦⎤ * 0. H. 0. kj. H. 28. }. kj. {. = E N 0 ⎡⎣ N 0. } = E { N ⎡⎣ N 0. 2. N 0* ⎤⎦. N 0* ⎤⎦. 0. }. N 0*[ j ]N 0*[l − j ] = 0. H. H. }. =0 kj. }=0.

(36) g.. {. ∵ E N0. N 0* ⎡⎣ N 0. { {. N 0* ⎤⎦. H. }. }. {. = E N 0 [l − k ] N 0 [ j ] kj. 2. 2. }. ⎧ E N [l − k ] 4 = 2σ 4 for j = l − k 0 0 ⎪ =⎨ 2 2 otherwise ⎪ E N 0 [l − k ] N 0 [ j ] = σ 04 ⎩. {. ⇒ E N0. h.. F.. { ⇒ E{ N. ∵ E N0. 0. }. N 0* ⎡⎣ N 0. N 0* ⎤⎦. N 0 ⎡⎣ N 0. N 0* ⎤⎦. N 0 ⎡⎣ N 0. H. }=σ. 4 0. (11. T. + P). } = E {N [k ]N [l − k ] N [ j] } = 0 N ⎤⎦ } = 0 * 0. H. 2. 0. 0. 0. kj. H. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ E and Now we can calculate E ⎢ , ⎥ ⎢ * ⎜ * ⎟ * ⎜ * ⎟ ⎥ h ϕ ∂ ∂ ⎝ ⎠ ⎢⎣ ∂ϕ ⎝ ∂ϕ ⎠ ⎥⎦ ⎣⎢ ⎦⎥. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ E⎢ using above a. ~ h. expectation * ⎜ * ⎟ ⎥ ⎢⎣ ∂h ⎝ ∂h ⎠ ⎥⎦. 1.. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ E⎢ ⎥ * ⎜ * ⎟ ⎢⎣ ∂ϕ ⎝ ∂ϕ ⎠ ⎥⎦. ⎧ H⎡ ⎤ ⎫ 1 1 1 1 * * ⎪Fϕ ⎢Φ 1 − 2 V1 N 0 N 0 + 2 V2 D Ra N 0 + 2 V3 N 0 N 0 + 2 V4 N 0 N 0 ⎥ ⎪ σ0 σ0 σ0 ⎪ ⎣ 1+Φ* σ 0 ⎦ ⎪ = E⎨ ⎬ H ⎤ ⎪⎡ ⎪ 1 1 1 1 * * ⎪⋅ ⎢ Φ 1 − 2 V1 N 0 N 0 + 2 V2 D Ra N 0 + 2 V3 N 0 N 0 + 2 V4 N 0 N 0 ⎥ Fϕ ⎪ σ0 σ0 σ0 ⎦ ⎩ ⎣ 1+Φ* σ 0 ⎭. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪⎪ = E ⎨FϕH ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪. H 1 1 ⎫ ⎡ H H * * H H H ⎤ ⎢Φ 1 * Φ 1 * + σ 4 V1 N 0 N 0 ⎡⎣ N 0 N 0 ⎤⎦ V1 + σ 4 V2 D Ra N 0 N 0 V2 D Ra ⎥ ⎪ 0 0 1+Φ ⎢ 1+Φ ⎥ ⎪ H H ⎢ 1 ⎥ ⎪ 1 H H * * ⎢ + 4 V3 N 0 N 0 ⎡⎣ N 0 N 0 ⎤⎦ V3 + 4 V4 N 0 N 0 ⎡⎣ N 0 N 0 ⎤⎦ V4 ⎥ ⎪ σ σ 0 0 ⎢ ⎥ ⎪⎪ H ⎢ ⎧ ⎥ Fϕ ⎬ ⎫ ⎧ ⎫ H H H⎪ ⎪ 1 H⎪ * * ⎢ − ⎨⎪ 1 Φ ⎥ ⎪ ⎡ N 0 ⎦⎤ V1 ⎬ − ⎨ 2 Φ 1 ⎣⎡ N 0 N 0 ⎦⎤ V1 ⎬ 1 ⎣ N0 ⎢ ⎪⎩σ 02 1+Φ ⎥ ⎪ * ⎪⎭ ⎪⎩σ 0 1+Φ* ⎪⎭ ⎢ ⎥ ⎪ H ⎢ ⎪⎧ 1 ⎥ ⎪ ⎫ ⎧1 ⎫ ⎢ + ⎨ 2 Φ 1 N 0H V2H D HR ⎬⎪ + ⎨⎪ 2 Φ 1 N 0H V2H D HR ⎬⎪ ⎥ ⎪ a a ⎭⎪ ⎩⎪σ 0 1+Φ* ⎭⎪ ⎣⎢ ⎩⎪σ 0 1+Φ* ⎦⎥ ⎭⎪. 29.

(37) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + E ⎨FϕH ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩. H ⎡ ⎪⎧ 1 ⎤ ⎫ ⎫ ⎪⎧ 1 ⎫ H H H⎪ H⎪ ⎢ + ⎨ 2 Φ 1 ⎡ N 0 N 0 ⎤ V3 ⎬ + ⎨ 2 Φ 1 ⎡ N 0 N 0 ⎤ V3 ⎬ ⎥ ⎪ ⎦ ⎣ ⎦ ⎢ ⎩⎪σ 0 1+Φ* ⎣ ⎥ ⎪ ⎭⎪ ⎩⎪σ 0 1+Φ* ⎭⎪ ⎢ ⎥ ⎪ H ⎢ ⎧⎪ 1 ⎥ ⎪ ⎫ ⎧ ⎫ H H 1 ⎪ * * H⎪ H⎪ ⎢ + ⎨ 2 Φ 1 ⎡⎣ N 0 N 0 ⎤⎦ V4 ⎬ + ⎨ 2 Φ 1 ⎡⎣ N 0 N 0 ⎤⎦ V4 ⎬ ⎥ ⎪ ⎢ ⎩⎪σ 0 1+Φ* ⎥ ⎪ ⎭⎪ ⎩⎪σ 0 1+Φ* ⎭⎪ ⎢ ⎥ ⎪ H ⎢ ⎧1 ⎥ ⎪ * * H ⎫ H H H ⎫ ⎧ 1 H H ⎢ − ⎨σ 4 V1 N 0 N 0 N 0 V2 D Ra ⎬ − ⎨σ 4 V1 N 0 N 0 N 0 V2 D Ra ⎬ ⎥ ⎪ ⎭ ⎭ ⎩ 0 ⎢ ⎩ 0 ⎥ ⎪ H⎥ ⎪ ⎢ ⎧1 H H ⎫ ⎫ ⎧ ⎢ + ⎨ 4 V1 N 0 N 0* ⎡ N 0 N 0 ⎤ V3H ⎬ + ⎨ 14 V1 N 0 N 0* ⎡ N 0 N 0 ⎤ V3H ⎬ ⎥ ⎪ ⎣ ⎦ ⎣ ⎦ ⎢ ⎩σ 0 ⎭ ⎥F ⎪ ⎭ ⎩σ 0 ⎢ ⎥ ϕ⎬ H H H ⎢ ⎧1 ⎥ ⎪ ⎫ ⎫ ⎧ 1 * * H H ⎢ + ⎨ 4 V1 N 0 N 0 ⎡⎣ N 0 N 0 ⎤⎦ V3 ⎬ + ⎨ 4 V1 N 0 N 0 ⎡⎣ N 0 N 0 ⎤⎦ V3 ⎬ ⎥ ⎪ ⎭ ⎩σ 0 ⎭ ⎥ ⎪ ⎢ ⎩σ 0 H ⎢ ⎥ ⎪ H H ⎧1 H⎫ H⎫ ⎢+ ⎧ 1 V D N ⎡ N ⎥ ⎪ N 0 ⎤⎦ V3 ⎬ + ⎨ 4 V2 D Ra N 0 ⎡⎣ N 0 N 0 ⎤⎦ V3 ⎬ ⎢ ⎨⎩σ 04 2 Ra 0 ⎣ 0 ⎥ ⎪ σ ⎭ ⎩ 0 ⎭ ⎥ ⎪ ⎢ H ⎥ ⎪ ⎢ ⎧1 H H ⎫ ⎧1 ⎫ ⎥ ⎪ ⎢ + ⎨ 4 V2 D Ra N 0 ⎡⎣ N 0 N 0* ⎤⎦ V4H ⎬ + ⎨ 4 V2 D Ra N 0 ⎡⎣ N 0 N 0* ⎤⎦ V4H ⎬ ⎥ ⎪ ⎢ ⎩σ 0 ⎭ ⎩σ 0 ⎭ ⎪ ⎢ H⎥ ⎧1 ⎪ ⎢ ⎧1 * H * H H⎫ H⎫ ⎥ ⎢ + ⎨σ 4 V3 N 0 N 0 ⎡⎣ N 0 N 0 ⎤⎦ V4 ⎬ + ⎨σ 4 V3 N 0 N 0 ⎡⎣ N 0 N 0 ⎤⎦ V4 ⎬ ⎥ ⎪ ⎭ ⎩ 0 ⎭ ⎦ ⎭ ⎣ ⎩ 0. T H H T H⎫ ⎧Φ Φ H 1 1 + V1 ( 11 + I l −1 ) V1 + V3 ( I l −1 + P ) V3 + V4 ( 11 + I l −1 ) V4 ⎪ 1+Φ ⎪ * 1+Φ* ⎪ ⎪ H H ⎪ ⎪⎪ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎪ ⎡ = FϕH ⎨− ⎢Φ 1 1T V1H ⎥ − ⎢Φ 1 1T V1H ⎥ + ⎢Φ 1 1T V4H ⎥ + ⎢Φ 1 1T V4H ⎥ ⎬ Fϕ ⎪ ⎣ 1+Φ* ⎦ ⎣ 1+Φ* ⎦ ⎣ 1+Φ* ⎦ ⎣ 1+Φ* ⎦ ⎪ ⎪ ⎪ H 1 ⎪− ⎡ V1 (11T + P ) V4H ⎤ − ⎡ V1 (11T + P ) V4H ⎤ + 2 V2 D Ra V2H D HRa ⎪ ⎦ ⎣ ⎦ σ0 ⎪⎩ ⎣ ⎪⎭. 2.. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ E⎢ * ⎜ * ⎟ ⎥ ⎣⎢ ∂ϕ ⎝ ∂h ⎠ ⎦⎥. ⎧ H⎡ 1 1 1 * ⎪⎪Fϕ ⎢ Φ 1 − 2 V1 N 0 N 0 + 2 V2 D Ra N 0 + 2 V3 N 0 σ0 σ0 = E ⎨ ⎣ 1+Φ* σ 0 H ⎪ H −1 * ⎪⎩⋅ ⎣⎡Fh DS Σ ( I N − DΦ DΦ ) DΓ0 D H I N 0 ⎦⎤. 30. N0 +. 1. σ 02. V4 N 0. ⎤⎫ N 0* ⎥ ⎪ ⎪ ⎦⎬ ⎪ ⎪⎭.

(38) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ = E ⎨FϕH ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪. = =. 1. σ. 2 0. 1. σ. 2 0. ( P.S =. 3.. H ⎡ ⎤ ⎫ H −1 * ⎢ Φ 1 N 0 ⎡⎣ DS Σ ( I N − DΦ DΦ ) DΓ0 D H I ⎤⎦ ⎥ ⎪ ⎢ 1+Φ* ⎥ ⎪ ⎪ H ⎥ ⎢ 1 H * −1 * ⎢ − 2 V1 N 0 N 0 N 0 ⎡⎣ DS Σ ( I N − DΦ DΦ ) DΓ0 D H I ⎤⎦ ⎥ ⎪ ⎢ σ0 ⎥ ⎪ ⎢ 1 ⎥ ⎪⎪ H H −1 * ⎢ + 2 V2 D Ra N 0 N 0 ⎡⎣ DS Σ ( I N − DΦ DΦ ) DΓ0 D H I ⎤⎦ ⎥ Fh ⎬ ⎢ σ0 ⎥ ⎪ ⎢ 1 ⎪ H ⎥ ⎢ + 2 V3 N 0 N 0 N 0H ⎡⎣ DS Σ −1 ( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦ ⎥ ⎪ ⎢ σ0 ⎥ ⎪ ⎢ 1 ⎪ H ⎥ ⎢ + 2 V4 N 0 N 0* N 0H ⎡⎣ DS Σ −1 ( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦ ⎥ ⎪ ⎢⎣ σ 0 ⎥⎦ ⎭⎪. FϕH V2 D Ra E ⎡⎣ N 0 N 0H ⎤⎦ ⎡⎣ DS Σ −1 ( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦ Fh H. FϕH V2 D Ra DSH σ 02 ⎡⎣ Σ −1 ( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦ Fh H. Σ = σ 02 ( I N − DΦ D*Φ ) DΓ0 D H I D HH I DΓH0 ( I N − DΦ D*Φ ). H. ). F H V V D H ⎡ D ( I − DΦ D*Φ ) DΓ0 ⎤⎦ Fh σ ϕ 2 Ra S ⎣ H I N −1. 1. 2 0. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ E⎢ * ⎜ * ⎟ ⎥ ⎢⎣ ∂h ⎝ ∂h ⎠ ⎥⎦. { = E {F D Σ. = E ⎡⎣FhH DS Σ −1 ( I N − DΦ D*Φ ) DΓ0 D H I N 0 ⎤⎦ ⎡⎣FhH D S Σ −1 ( I N − DΦ D*Φ ) DΓ0 D H I N 0 ⎤⎦ H h. S. −1. (I. − DΦ D*Φ ) DΓ0 D H I N 0 N 0H ⎡⎣ DS Σ −1 ( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦ Fh H. N. = FhH DS DSH Σ −1Σ −1 σ 02 ( I N − DΦ D*Φ ) DΓ0 D H I ⎡⎣( I N − DΦ D*Φ ) DΓ0 D H I ⎤⎦ Fh H. =∑. = FhH DS DSH Σ −1Fh =. {. F H DS DSH D H I ( I N − DΦ D*Φ ) DΓ0 ⎡⎣ D H I ( I N − DΦ D*Φ ) DΓ0 ⎤⎦ 2 h σ. 1. 0. 31. }. * −1. Fh. H. }. }.

(39) G.. Then, define. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ A= E⎢ ⎥ * ⎜ * ⎟ ⎢⎣ ∂ϕ ⎝ ∂ϕ ⎠ ⎥⎦. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ B=E⎢ * ⎜ * ⎟ ⎥ ⎣⎢ ∂h ⎝ ∂h ⎠ ⎦⎥ ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ C=E⎢ * ⎜ * ⎟ ⎥ ⎢⎣ ∂ϕ ⎝ ∂h ⎠ ⎥⎦. ⎡ ∂f (ω ) ⎛ ∂f (ω ) ⎞ H ⎤ Fisher Information Matrix of θ : I = E ⎢ * ⎜ * ⎟ ⎥ ⎣⎢ ∂θ ⎝ ∂θ ⎠ ⎦⎥. ⎡ ⎡ ∂f (ω ) ⎤ ⎤ ⎢ ⎢ ∂ϕ * ⎥ ⎡⎛ ∂f (ω ) ⎞ H ∂f (ω ) H ⎤ ⎥ ⎛ ⎞ ⎥ ⎥ ⎢⎜ = E ⎢⎢ ⎥ * ⎟ ⎜ * ⎟ ⎢ ⎢ ∂f (ω ) ⎥ ⎢⎣⎝ ∂ϕ ⎠ ⎝ ∂h ⎠ ⎥⎦ ⎥ ⎢ ⎢⎣ ∂h* ⎥⎦ ⎥ ⎣ ⎦ ⎡A C⎤ =⎢ H ⎥ ⎣C B ⎦. ∴ CRLB(ωk ) = ( Fϕ ,h I −1FϕH,h ) where. ( )kk. kk. which can be nurmerically computed through computer simulation. is its k'th diagnoal value. 32.

(40) Chapter 6 Simulation Results Computer simulations were conducted to evaluate the performance of proposed scheme. In the first part of simulation, the analytical results and CRLBs were confirmed and compared to the simulation results. Then, we examined performances and sensitivities of the proposed compensation scheme in the second and third part of simulation.. An OFDM system designed for IEEE 802.11a WLAN standards is considered in our simulation. The OFDM symbol is based on total 64 carriers (48 for data, 4 for pilot and others left open) uniformly distributed in 20 MHz channel bandwidth in RF band. The modulations on each carrier range from BPSK, QPSK, 16-QAM to 64-QAM.. Cyclic prefix is copy of the last quarter of each OFDM symbol. The 802.11a standards also specify one short preamble for synchronization and one long preamble for channel estimation. Our estimation is using the long preamble as prior information to do joint estimation. Following model described in Equation 1, a multi-path channel effect is constructed as a three-taps complex-valued FIR whose phases are uniformly random distributed and magnitudes are Rayleigh distributed with averaged power decaying exponentially.. For frequency-independent IQ imbalance, amplitude g and phase imbalance θ are set to be 1.08 and 5 . As to frequency-dependent IQ imbalance, it is modeled in terms of impulse responses of baseband IQ branches hI [n] and hQ [n] . Due to variation of analog components, these two filters are modeled based on order-3 of low-pass Butterworth filters with different cutoff frequencies at 8.5 MHz and 8.2 MHz. The sampling rate is set to be 20 MHz. 33.

(41) 6.1 MSE of ML Estimates The MSE analytical expressions derived in chapter 4 were checked using fixed channel given by h=[0.7047 + 0.7047i , 0.0578 + 0.0578i , 0.0047 + 0.0047i]. (L=3) Also, the CRLBs derived in chapter 5 were compared to our estimators. The MSE of I/Q imbalance Φ is calculated as its average MSE per subcarrier, and channel H is also calculated as its average MSE per subcarrier.. Figure 6-1 and Figure 6-2 show means-squared-error (MSE) of IQ imbalance and delay-spreading channel estimation. The estimation error is computed averagely in frequency domain per subcarrier. It was observed that the computer simulation results of proposed estimators coincide with the MSE analyses in high SNR, and there still exists slight mismatch in the low SNR level due to some approximations that can not be made in low SNR.. Furthermore, the MSE were almost identical to the. CRLBs.. Figure 6-2 also compare the performance of channel estimator with and without smoothing. The “Smoothing” refers to our proposed estimator which estimates channel impulse response in time domain, while the “No Smoothing” refers to estimating the channel independently between subcarriers in frequency domain. As expected, the smoothing property indeed obtains a large performance gain over channel estimation.. 34.

(42) MSE of φ (per subcarrier), Simulation V.S Analysis CRLB. 0. 10. Simulation MSE of φ Analysis MSE of φ CRLB of φ. -1. 10. -2. MSE. 10. -3. 10. -4. 10. -5. 10. 0. 5. 10. 15. 20 SNR(dB). 25. 30. 35. 40. Fig.6-1 MSE of IQ imbalance estimation MSE of H (per subcarrier), Simulation V.S Analysis CRLB. 0. 10. Simulation MSE of H (No Smoothing) Simulation MSE of H (Smoothing) Analysis MSE of H CRLB of H. -1. 10. -2. MSE. 10. -3. 10. -4. 10. -5. 10. 0. 5. 10. 15. 20 SNR(dB). 25. 30. Fig.6-2 MSE of delay-spreading channel estimation 35. 35. 40.

(43) 6.2 Sensitivities of Proposed Estimators In practical situation, the effective length of channel and I/Q imbalance ϕ need to be estimated, thus could not be optimal and suffers from some modeling error in a real front-end filter equivalent to an infinite impulse response (IIR) filter.. It can be observed in Fig.6-3 and 6-4 that there indeed exists an optimal length Lh = 11 for our given I/Q imbalance and channel case. In addition, the performance will be saturate when Lh is chosen shorter than its effective optimal length, however, it suffers from just a little acceptable performance loss when Lh is chosen larger than its equivalent optimal length. Hence, this result suggests us to choose larger channel length to maintain a good performance level.. 0. 10. MSE of φ (Lh=9) MSE of φ (Lh=11) MSE of φ (Lh=13). -1. MSE. 10. MSE of φ (Lh=15). -2. 10. -3. 10. -4. 10. 0. 5. 10. 15. 20 SNR(dB). 25. 30. Fig.6-3 Channel estimation with various Lh. 36. 35. 40.

(44) 0. 10. MSE of H (Lh=9) MSE of H (Lh=11) MSE of H (Lh=13). -1. MSE. 10. MSE of H (Lh=15). -2. 10. -3. 10. -4. 10. 0. 5. 10. 15. 20 SNR(dB). 25. 30. 35. 40. Fig.6-4 I/Q imbalance Φ estimation with various Lh. 0. 10. MSE of φ (L =7) φ. MSE of φ (L =9) φ. MSE of φ (L =11) φ. -1. 10. MSE of φ (L =13). MSE. φ. -2. 10. -3. 10. -4. 10. 0. 5. 10. 15. 20 SNR(dB). 25. 30. 35. Fig.6-5 I/Q imbalance Φ estimation with various Lϕ 37. 40.

(45) 0. 10. MSE of φ (No CFO) MSE of φ (CFO = 0.005 ) MSE of φ (CFO = 0.01 ) MSE of φ (CFO = 0.02 ) MSE of φ (CFO = 0.04 ) MSE of φ (CFO = 0.04 with Compensation). -1. MSE. 10. -2. 10. -3. 10. -4. 10. 0. 5. 10. 15. 20 SNR(dB). 25. 30. 35. 40. Fig.6-6 I/Q imbalance Φ estimation with various CFO effect. In Fig.6-5, different length Lϕ is checked. When SNR is smaller, AWGN noise dominates interferences, and it suggests us to chose smaller length Lϕ . On the contrast, when SNR is larger, I/Q imbalance may dominate interferences, then the performance will be better in choosing longer length Lϕ . Hence, the length Lϕ depends on operating SNR.. Finally, the severity of CFO effect on MSE and effectiveness of our proposed CFO compensation scheme in section 3.4 are examined in fig.6-6. The CFO is normalized to subcarrier spacing. It is shown that our estimation scheme can suffer CFO effect lower than 0.01. Thus, the residue of estimated CFO should be smaller 38.

(46) than 0.01 to make our algorithm work. “ CFO =0.04 with Compensation” legend means that applying proposed CFO pre-compensation at receiver with CFO being 0.04. It can be seen that our proposed CFO compensation is effective to combat CFO effect.. 39.

(47) 6.3 Uncoded BERs Performance The performances of the estimators were also examined in terms of the uncoded bit error rate (BER). “IQ imbalance/No Comp” refers to a receiver with I/Q imbalance but no compensation. “IQ imbalance/Comp with No smoothing” refers to ordinary OFDM channel estimation in frequency domain; whereas, “IQ imbalance/Comp with smoothing” refers to our proposed estimators in time domain. “Ideal Receiver” refers to a receiver with no I/Q imbalance. As expected, the BER curve becomes saturated in the presence of I/Q imbalance effect; on the other hand, the performances of our proposed algorithms is close to ideal receivers and outperforms the traditional OFDM channel estimation done in frequency domain.. 0. 10. IQ Imbalance/No Comp IQ Imbalance/Comp with No Smoothing IQ Imbalance/Comp with Smoothing Ideal Receiver(No IQ imbalance). -1. Uncoded BER. 10. -2. 10. -3. 10. -4. 10. 10. 15. 20. 25 SNR(dB). 30. Fig.6-7 Uncoded BERs with 64 QAM. 40. 35. 40.

(48) 64 QAM Uncoded BER with CFO effect. 0. 10. -1. Uncoded BER. 10. -2. 10. Uncoded BER (Ideal,No IQ Imbalance/CFO) Uncoded BER (CFO = 0.005 ) Uncoded BER (CFO = 0.01 ) Uncoded BER (CFO = 0.02 ) Uncoded BER (CFO = 0.04 ) Uncoded BER (CFO = 0.04) with Comp. -3. 10. -4. 10. 0. 5. 10. 15. 20 SNR(dB). 25. 30. 35. 40. Fig.6-8 CFO sensitivities on 64 QAM Uncoded BERs. The last Fig.6-2 shows the sensitivity of CFO effect on uncoded BER performance and effectiveness of our proposed CFO compensation scheme in section 3.4. The same results with MSE performance can be seen in terms of uncoded BERs. The BERs get saturate again when estimated CFO residue error is larger than 0.01. In addition, our proposed CFO compensation scheme combined with our proposed algorithm also makes it work to achieve performance target assuming CFO has been estimated previously within certain level.. 41.

(49) Chapter 7 Conclusions An algorithm for joint estimation of channel and IQ imbalance effects (both of frequency-independent and frequency-dependent) for OFDM system was developed. To make use of characteristics of OFDM symbol and channel smoothing property, we construct our likelihood function using frequency data to estimate time-domain channel and IQ imbalance effects.. Also, we develop carrier frequency offset. compensation method combined with our algorithm assuming that carrier frequency offset has been estimated prior to our estimation. The estimation requires only one OFDM symbol as prior information to reaches performance target. structure for such OFDM symbol is assumed.. No special. The performance was investigated. analytically and by computer simulation, which shows that the proposed algorithm reaches CRLBs as the received SNR above certain level. We also observed that our performance depends on prior information about channel length of delay-spreading channel effect and degree of difference of IQ imbalance. Some future topics about this work can be extended and researched. (1) Apply to MIMO-OFDM systems that may take advantage of spatial diversity gains to improve performances. (2) Take into account transmitter side I/Q imbalance and DC offset effect in MIMO technique. (3) Develop CFO estimation method in the presence of I/Q imbalance effect.. 42.

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