多天線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多天線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 MIMO-OFDM Systems. 研究生：徐進發指導教授：沈文和. 中. 華. 民. 國. 九. 十. 四. 博士. 年. 七. 月.

(2) 多天線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 MIMO-OFDM Systems 研究生：徐進發. Student：Jin-Fa Shyu. 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 2004 Hsinchu, Taiwan, Republic of China.. 中. 華. 民. 國. 九. 十. 四. 年. 七. 月.

(3) 多天線正交分頻多工系統之通道估計與 I/Q 失衡補償研究生：徐進發. 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，而多天線正交分頻多工(MIMO-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(Radio Frequency)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，而 I/Q 失衡問題將是我們討論的重點。此論文中將同時考慮通道估測以及 I/Q 失衡補償兩項議題，並提出一共同最大可能性估計(Joint Maximum Likelihood Estimation)之估計和補償方法。再者，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，我們會針對兩者各自提出估計及補償的方法。另外，論文中的通道估計和 I/Q 失衡補償演算法不僅可應用於單天線(SISO)正交分頻多工系統，更可應用在多天線正交分頻多工系統，例如 IEEE802.11n。論文內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。. i.

(4) Joint Maximum Likelihood Estimation of Channel and I/Q Imbalance for MIMO-OFDM Systems Student: Jin-Fa Shyu. Advisor: Dr. Wern-Ho Sheen. Department of Communication Engineering National Chiao Tung University. Abstract Wireless communication systems with high data rate are strongly desired by a variety of applications, and MIMO-OFDM is one of the promising technologies satisfying the need for high throughput. A direct-conversion receiver (DCR) architecture becomes a trend in industry and academic world nowadays because it is small, cheap and less 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, and etc. In this thesis, the RF impairment of I/Q mismatch is addressed; moreover, we propose a joint maximum likelihood (ML) estimation of I/Q imbalance and channel for MIMO-OFDM systems, such as IEEE 802.11n. The problems of I/Q mismatch compensation and channel estimation are considered at the same time, and we offer the compensation algorithms based on not only frequency-independent but also frequency-dependent I/Q imbalances. In addition to SISO-OFDM systems, our algorithms are able to work in the MIMO-OFDM systems with tone-interleaved training sequences for MIMO channel estimation.. ii.

(5) Acknowledgement First of all, I have to offer my heartfelt thanks to my advisor, Dr. Wern-ho Sheen, for helping me develop this paper. During the 2-year master program, the professor provided me with lots of precious instruction and suggestions, and taught me the academic research methodologies in order to help me cultivate the ability to do the research independently and have critical thinking. Furthermore, the professor impressed me with his strict attitude toward doing the research. Based on this kind of attitude, the professor effectively helped me to clarify my mistaken concepts and then to smoothly develop the whole project. Besides, I have to say “Thank you” to all the members in BRASLAB: Dr. Jeng-shin Hsu, Dr. Jan-shin Ho, Dr. Rodger Tseng, Chang-lung Hsiao, Chih-cheng Kuo, I-kang Fu, Eason Lee, Kai-yi Fang, Sin-lang Lai, Min-yu Lin, Liang-wei Huang, Chang-hsin Chen, Yen-wen Yang, Ting-che Tseng and Cathy Lin. Especially thanks my classmate, Ting-che Tseng, who usually discussed with me and cooperated with me to verify the simulation platform we built together. Last but not least, thanks Hui-hsuan Chen for revising my language in this paper. Also, thanks for the invaluably great support from my family. Dad and Mom, because of you, I learned the importance of being responsible and the correct attitude toward my life. Because of you, I could completely devote myself to the study without any worry. All in all, I feel immense gratitude to those who had ever helped me to accomplish this paper.. By Jin-fa Shyu July, 2005. iii.

(6) Contents 摘要.................................................................................................................................i Abstract ..........................................................................................................................ii Acknowledgement ....................................................................................................... iii Contents ........................................................................................................................iv List of Figures ................................................................................................................v Chapter 1 Introduction ...................................................................................................1 Chapter 2 System Model................................................................................................6 Chapter 3 Frequency-Independent I/Q Imbalance .......................................................14 3.1 Modified System Model ....................................................................................14 3.2 Joint ML Estimation...........................................................................................16 3.2.1 ML Estimation of Channel..........................................................................16 3.2.2 ML Estimation of I/Q Imbalance ................................................................18 3.3 Performance of ML Estimation..........................................................................19 3.3.1 Mean and MSE of ML Estimation of I/Q Imbalance Parameter ................19 3.3.2 Mean and MSE of ML Estimation of Channel in Time Domain ................21 3.3.3 Mean and MSE of ML Estimation of Channel in Frequency Domain .......24 Chapter 4 Frequency-Dependent I/Q Imbalance .........................................................29 4.1 Modified System Model ....................................................................................29 4.2 Joint ML Estimation...........................................................................................31 4.2.1 ML Estimation of Channel..........................................................................33 4.2.2 ML Estimation of I/Q Imbalance ................................................................33 Chapter 5 Simulation Results.......................................................................................37 5.1 PER Performance...............................................................................................37 5.2 MSE of ML Estimation......................................................................................43 Chapter 6 Conclusions .................................................................................................46 References....................................................................................................................47 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.........................................6 Fig.2-2 Mathematical model of a direct-conversion receiver with I/Q imbalance. .......6 Fig.5-1 PER performance with perfect I/Q for different Lh ........................................38 Fig.5-2 Amplitude imbalance g = 1.1 and phase imbalance θ = 5° ...........................40 Fig.5-3 Amplitude imbalance g = 1dB and phase imbalance θ = 8° ..........................40 Fig.5-4 PER with frequency-independent I/Q imbalance for different Lh ...................42 Fig.5-5 MSE of ML estimation with different values of Lh ........................................44. v.

(8) Chapter 1 Introduction Wireless communication systems with high data rate are strongly desired by a variety of applications. Then, MIMO-OFDM is one of the promising technologies satisfying the need for high throughput. Orthogonal frequency division multiplexing (OFDM) [1][2] possesses the ability to resist multipath channel by simple frequency-domain equalizer; thus, it is adopted by many wireless standards, e.g. IEEE 802.11a/g/n, IEEE 802.16a, digital video broadcasting (DVB), and etc. Multiple-input multiple-output (MIMO) [3] makes use of multiple transmitter and receiver antennas to transmit independent data streams simultaneously for increasing diversity and spectral efficiency. Consequently, the combination of MIMO and OFDM is widely discussed in recent years and has been used by some wireless broadband systems, such as IEEE 802.11n [4][5] and IEEE.802.16a. To popularize this communication technology, we need a receiver architecture that is small, cheap and less power-consuming. A direct-conversion receiver (DCR)[6] is a good choice that meets these requirements so that it becomes a trend in industry and academic world nowadays. This kind of architecture, however, introduces some radio frequency (RF) imperfections such as the direct current (DC) offset, frequency offset, in-phase/quadrature (I/Q) imbalance, phase noise, and etc [2][7][8][9][10]. Some digital signal processing algorithms are needed to deal with these RF imperfections in baseband to ease the requirements of analog front-end devices. In this thesis, we are going to dig into the problems of compensation for I/Q imbalance and estimation for channel in MIMO-OFDM systems. The I/Q imbalance means the mismatch between I and Q branches in amplitudes and phases [10][11]. It is commonly mentioned in the direct-conversion receiver, which uses analog quadrature down-mixing. Then, there are two major factors in I/Q imbalance. For one thing, the imperfect local oscillator (LO) results in 1.

(9) 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[4][5]. 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. In the next chapter, the system model and effects of I/Q imbalance are about to be explained in detail. S*2. S*-1 S1*. S*-2. f. − fc. 0. fc. (a) Received signal in passband. S*2. S1*. * S*-1 S-2. f. 0 (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 2.

(10) In fact, there are tons of literatures focusing on the problem of I/Q imbalance and the compensation scheme; see [11]-[24] and references therein. However, most of them put emphasis on frequency-independent I/Q imbalance, and only a few of them [11]-[17] provide compensation schemes for frequency-dependent I/Q imbalance, which is much more complicated. Based on the assumption that desired signal and image interference are statistically independent, some methods make use of the blind source separation techniques to extract the desired signal [11][12][18][19]. However, the finite training sequences used to estimate the coefficients for compensation do not always satisfy the assumption. Therefore, decision-directed methods are usually needed for these algorithms to adaptively converge the desired solutions. It, nevertheless, can not be applied directly in the MIMO systems because the MIMO systems involve multi-user detection (MUD) procedures. Some other adaptive compensation methods, such as [20], also encounter the same difficulty. Then, [16] derives an adaptive MMSE solution of joint MUD and I/Q mismatch cancellation in frequency domain for MIMO-OFDM systems. It independently calculates the coefficients of detection and compensation for each subcarrier, but the coefficients of neighboring subcarriers are highly correlated. Accordingly, we should take account of this property to improve speed of convergence. On the other hand, some researchers develop the compensation algorithms by means of the training symbols known by the receiver in advance, like [21]-[24]. Reference [21] analyzes the frequency-dependent I/Q imbalance in the presence of frequency-offset. Still, some restrictions on the training sequence exist and a finite impulse response (FIR) filter is adopted to correct the frequency-dependent I/Q mismatch. Truly, a FIR filter is not suitable for OFDM systems because it increases the effective channel length experienced by transmitted signal. If the effective channel 3.

(11) length is larger than the guard interval (GI), we have to show the great concern about the inter-symbol interference (ISI) which is a troublesome issue in OFDM systems. Then, speaking of Reference [22], it only discusses frequency-independent I/Q imbalance problem on the basis of channel smoothness criterion. Moreover, both [21] and [22] ignore the noise contribution when they analyze these RF imperfections, and it doesn’t conform to the real case. Different from [21] and [22], assuming the noise at receiver is an additive white Gaussian noise (AWGN), references [23][24] propose techniques that jointly estimates the I/Q imbalance and other RF impairments based on the maximum likelihood (ML) criterion. However, the joint estimators of [23][24] are only for the frequency-independent I/Q mismatch and can not be directly applied to MIMO-OFDM systems. In this thesis, we propose a joint ML estimators of I/Q imbalance and channel for MIMO-OFDM systems, such as IEEE 802.11n [4][5], and the problems of I/Q mismatch compensation and channel estimation are considered at the same time. The basic idea of our I/Q compensation scheme is similar to [23][24]. What is more, we acquire three advantages of our algorithms. First of all, we derived the compensation algorithms not only for frequency-independent but also for frequency-dependent I/Q imbalances. Secondly, in addition to SISO-OFDM systems, it is able to work in the MIMO-OFDM systems with tone-interleaved training sequences for MIMO channel estimation [4]. Thirdly, the property of high correlation between neighboring subcarriers is adopted when we both calculate the channel response and I/Q imbalance parameter of each subcarrier. The remainder of this thesis organized as follows. Chapter 2 introduces the system model of I/Q imbalance. Then, we propose ML channel estimators and the compensation schemes not only for frequency-independent I/Q imbalance in Chapter 3, but also for frequency-dependent I/Q imbalance in Chapter 4. Computer simulation 4.

(12) results are demonstrated in Chapter 5. Finally, Chapter 6 concludes the thesis.. 5.

(13) Chapter 2 System Model A general architecture of a direct-conversion receiver is presented in Fig.2-1. Based on it, 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. The mismatch among all the analog devices of I/Q branches results in the effective amplitude and phase imbalances. To be more specifically, the I/Q imbalance due to imperfect LO is frequency-independent whereas the imbalance produced by the following amplifiers, LPF and A/D converters is frequency-dependent. By means of the mathematical model of a direct-conversion receiver in Fig.2-2, we can analyze the effects of the frequency-independent and frequency-dependent I/Q imbalance.. Equation Section 2. Fig.2-1 General architecture of a direct-conversion receiver.. v I (t ) r (t ). 2 Re {r (t )e j 2π f ct }. y I (t ). y I [n ]. yQ (t ). yQ [n ]. cos(2π f c t ) − g sin(2π f ct + θ ) vQ (t ). Fig.2-2 Mathematical model of a direct-conversion receiver with I/Q imbalance. 6.

(14) The. output. of. imperfect. LO. is. expressed. as. CLO (t ) = cos(2π f c t ) − 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 equal to one. CLO (t ) = cos(2π f c t ) − jg sin(2π f ct + θ ). (2.1). = γ 0 e − j 2 π f c t + ϕ 0 e j 2π f c t. where γ 0. 1 1 + ge − jθ ) and ϕ 0 ( 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. r (t ). 2 Re {r (t )e j 2π fct } = r (t )e j 2π fct + r * (t )e − j 2π fct. (2.3). where r (t ) = a (t ) ⊗ hCH (t ) + n (t ), a (t ) is the transmitted signal, hCH (t ) is the impulse response of channel, n(t ) is an AWGN.. The received signal r (t ) is multiplied by the output of LO CLO (t ) , which is not. orthogonal perfectly.. v (t ) = r (t ) × CLO (t ) = ⎡⎣ r (t )e j 2π fct + r * (t )e − j 2π fct ⎤⎦ × ⎡⎣γ 0e − j 2π fct + ϕ 0e j 2π fct ⎤⎦ = γ 0 r ( t ) + ϕ 0 r * ( t ) + γ 0 r * ( t ) e − j 4 π f c t + ϕ 0 r ( t ) e j 4π f c t will be removed by the following LPFs. 7. (2.4).

(15) = γ 0 r (t ) + ϕ 0 r * (t ). (2.5). If the last two terms in (2.4), which will be removed by the following LPF, are ignored, we would find out that the desired signal is multiplied by a factor γ 0 and interfered by its image signal. This kind of I/Q imbalance effect due to the imperfect LO is frequency-independent.. Because the following analog components of I/Q branches are mismatch, I part and Q part signals are processed differently. We have to rewrite (2.5) into (2.6) to express the relationship of I/Q branches between r (t ) and v (t ) more clearly. v I (t ) + j ⋅ vQ (t ). v (t ). = γ 0 r (t ) + ϕ 0 r * (t ) 1 1 1 + ge − jθ ) r (t ) + (1 − ge jθ ) r * (t ) ( 2 2 1 1 = ⎡⎣ r (t )+r * (t ) ⎤⎦ + g ⎡⎣ e − jθ r (t ) − e jθ r * (t ) ⎤⎦ 2 2 = Re {r (t )} + j ⋅ g Im {e − jθ r (t )} =. = rI (t ) + j ⋅ g ⎡⎣cos θ rQ (t ) − sin θ rI (t ) ⎤⎦ = vI ( t ). (2.6). = vQ ( t ). where v I (t ) vQ (t ). Re {v (t )} = rI (t ), Im {v (t )} = g ⎡⎣ cos θ rQ (t ) − sin θ rI (t ) ⎤⎦ ,. rI (t ). Re {r (t )} ,. rQ (t ). Im {r (t )}.. We could express (2.6) in the frequency domain, and that results in. VI ( f ) + j ⋅ VQ ( f ). V( f ) =. RI ( f ) VI ( f ). + j ⋅ g ⎡⎣ cos θ RQ ( f ) − sin θ RI ( f ) ⎤⎦ VQ ( f ). 8. (2.7).

(16) FT {v (t )} ,. where V ( f ). FT {v I (t )},. VI ( f ) VQ ( f ). FT {vQ (t )},. R( f ). FT {r (t )} , FT {rI (t )} ,. RI ( f ) RQ ( f ). FT {rQ (t )}. Then, in terms of Fourier transform, the signal y (t ). y I (t ) + j ⋅ yQ (t ) after I/Q. branch imbalance is given by Y ( f ) YI ( f ) + j ⋅ YQ ( f ) = VI ( f ) H I ( f ) + j ⋅ VQ ( f ) H Q ( f ) = RI ( f ) H I ( f ) + j ⋅ g ⎡⎣cos θ RQ ( f ) − sin θ RI ( f ) ⎤⎦ H Q ( f ) = RI ( f ) ⋅ ⎣⎡ H I ( f ) − j ⋅ H Q ( f ) g sin θ ⎤⎦ + j ⋅ RQ ( f ) ⎡⎣ H Q ( f ) g cos θ ⎤⎦ 1 1 ⎡⎣ R( f ) + R* ( − f ) ⎤⎦ ⎡⎣ H I ( f ) − j ⋅ H Q ( f ) g sin θ ⎤⎦ + j ( − j ) ⎡⎣ R( f ) − R* ( − f ) ⎤⎦ ⎡⎣ H Q ( f ) g cos θ ⎤⎦ 2 2. =. RI ( f ). =. RQ ( f ). 1 R ( f ) ⎣⎡ H I ( f ) − j ⋅ H Q ( f ) g sin θ + H Q ( f ) g cos θ ⎦⎤ 2 1 + R* ( − f ) ⎡⎣ H I ( f ) − j ⋅ H Q ( f ) g sin θ − H Q ( f ) g cos θ ⎤⎦ 2. =. 1 1 R( f ) ⎡⎣ H I ( f )+H Q ( f ) ge− jθ ⎤⎦ + R* ( − f ) ⎡⎣ H I ( f ) − H Q ( f ) ge jθ ⎤⎦ 2 2. =. 1 ⎡ H Q ( f ) − jθ ⎤ 1 ⎡ H ( f ) jθ ⎤ * ge ⎥ R( f ) H I ( f ) + ⎢1 − Q ge ⎥ R ( − f ) H I ( f ) 1+ ⎢ 2 ⎣ HI ( f ) 2 ⎣ HI ( f ) ⎦ ⎦ γ0( f ). (2.8). ϕ0 ( f ). = γ 0 ( f ) R( f ) H I ( f ) + ϕ 0 ( f ) R ( − f ) H I ( f ) *. = H I* ( − f ). = γ 0 ( f ) R( f ) H I ( f ) + ϕ0 ( f ) R* (− f ) H I* (− f ) where γ 0 ( f ). ϕ0 ( f ). (2.9). 1 ⎡ H Q ( f ) − jθ ⎤ 1+ ge ⎥ , 2 ⎢⎣ H I ( f ) ⎦. ( P.S.. γ 0( f ) ≈ 1. 1 ⎡ H Q ( f ) jθ ⎤ 1− ge ⎥ 2 ⎢⎣ H I ( f ) ⎦. ( P.S.. ϕ0 ( f ) ≈ 0. 2. 2. ). (2.10). ). The equation (2.9) shows that the desired signal R( f ) H I ( f ) is multiplied by a frequency-dependent. factor. γ 0( f ). and 9. interfered. by. its. image. signal.

(17) frequency-dependently due to the frequency-dependent factor ϕ 0 ( f ) .. Let us see (2.8) in time domain. 1 1 y (t ) = r (t ) ⊗ ⎡⎣ hI (t )+hQ (t ) ge − jθ ⎤⎦ + r * (t ) ⊗ ⎡⎣ hI (t ) − hQ (t ) ge jθ ⎤⎦ 2 2 h+ ( t ). h− ( t ). = r (t ) ⊗ h+ (t ) + r * (t ) ⊗ h− (t ) = [ a (t ) ⊗ hCH (t ) + n(t )] ⊗ h+ (t ) + [ a (t ) ⊗ hCH (t ) + n(t ) ] ⊗ h− (t ) *. * = a (t ) ⊗ hCH (t ) ⊗ h+ (t ) + n(t ) ⊗ h+ (t ) + a * (t ) ⊗ hCH (t ) ⊗ h− (t ) + n* (t ) ⊗ h− (t ) (2.11). where hI (t ) hQ (t ) h+ (t ) h− (t ). FT −1 { H I ( f ) } ,. FT −1 { H Q ( f ) },. 1 ⎡ hI (t )+hQ (t ) ge − jθ ⎤⎦ , 2⎣ 1 ⎡ hI (t ) − hQ (t ) ge jθ ⎤⎦ 2⎣. Assume y (t ) is sampled with a symbol rate 1 Ts , that results in * y[n ] = a[n ] ⊗ hCH [n ] ⊗ h+ [n ] + n[n ] ⊗ h+ [n ] + a *[n ] ⊗ hCH [n ] ⊗ h− [n ] + n*[n ] ⊗ h− [n ]. (2.12). Then we concentrate on one OFDM symbol after removing guard interval (GI). Assuming that GI is larger than the effective length of hCH [n ] ⊗ h+ [n ] as well as * hCH [n ] ⊗ h− [n ] , the convolutions in (2.12) are equivalent to circular convolutions. The. signal model after FFT could be easily calculated as (2.13) and (2.14) present.. * y[n ] = a[n ] ⊗ hCH [n ] ⊗ h+ [n ] + n[n ] ⊗ h+ [n ] + a *[n ] ⊗ hCH [n ] ⊗ h− [n ] + n* [n ] ⊗ h− [n ]. = ( a[n ] ⊗ hCH [n ] + n[n ]) ⊗ h+ [n ] + ( a[n ] ⊗ hCH [n ] + n[n ]) ⊗ h− [n ] *. ⇒ r[i ]. FFT { y[n ] }. = ( s[i ] kCH [i ] + w[i ]. ) × k+ [i ] + ( s[−i ] kCH [−i ] + w[−i ] ) 10. *. × k− [i ]. (2.13).

(18) where r[i]. FFT {a[n ]} ,. w[i ]. FFT {n[n ]} ,. kCH [i ]. FFT {h[n ]} ,. k I [i ]. FFT {hI [n ]} ,. kQ [i ]. FFT {hQ [n ]},. k + [i ]. FFT {h+ [n ]} = FFT {hI [n] + hQ [n]ge − jθ } = k I [i ] + jge − jθ kQ [i ],. k − [i ]. FFT {h− [n ]} = FFT {hI [n] − hQ [n]ge jθ. } = k [i ] − jge. ⇒ r[i ] = k + [i ] ⋅ ( s[i ] kCH [i ] + w[i ]. ) + k− [i ] ⋅ ( s[−i ] kCH [−i ] + w[−i ] ) = ( k I [i ] + ge − jθ kQ [i ]) ( s[i ] kCH [i ] + w[i ] ) + ( k I [i ] − ge jθ kQ [i ]) ( s[ −i ] kCH [ −i ] + w[−i ]. ). k [i ] ⎞ ⎛ ⎛ + ⎜ 1 − ge jθ Q ⎟ ⎜ s[−i ] kCH [−i ]k I [−i ] + w[−i ] k I [−i ] k I [i ] ⎠ ⎜ ⎝ z0 [ − i ] k0 [ − i ] ⎝ hI [n ] is real ⇒ k I [i ] = k I* [ −i ]. = γ 0 [i ] × ( s[i ] k0 [i ] + z0 [i ] where γ 0 [i ]. 1 + ge − jθ. ϕ 0 [i ]. 1 − ge + jθ. kdiff [i ]. kQ [i ] k I [i ]. kQ [i ] k I [i ] kQ [i ] k I [i ]. *. ⎞ ⎟ ⎟ ⎠. γ 0 [i ]. ( P.S.. kQ [i ]. *. k [i ] ⎞ ⎛ ⎛ = ⎜ 1 + ge − jθ Q ⎟ ⎜ s[i ] kCH [i ]k I [i ] + w[i ]k I [i ] k I [i ] ⎠ ⎜ ⎝ z0 [ i ] k0 [ i ] ⎝. ϕ0 [i ]. jθ. I. ⎞ ⎟ ⎟ ⎠. *. ). ) + ϕ0 [i ] × ( s[−i ] k0 [−i ] + z0 [−i ] ). *. (2.14). = 1 + ge − jθ kdiff [i ] ,. (. P.S . γ 0 [i ] ≈ 1. ). (2.15). = 1 + ge + jθ kdiff [i ],. (. P.S . ϕ 0 [i ] ≈ 0. ). (2.16). ,. k0 [i ]. kCH [i ] ⋅ k I [i ],. z0 [i ]. w[i ] ⋅ k I [i ]. The equation (2.14) indicates that the desired signal on subcarrier i is multiplied by factor γ 0 [i ] and that the interference term comes from the image signal carried by 11.

(19) subcarrier –i. Undoubtedly, the power of the interference term is related to the factor. ϕ 0 [i ] . Also these two parameters, γ 0 [i ] and ϕ 0 [i ] , depend on the imperfect LO and the difference of frequency response on I/Q branches. Since the above-mentioned difference is frequency-dependent, γ 0 [i ] and ϕ 0 [i ] are varied on different subcarriers.. To simplify the analysis, we rewrite (2.14) in matrix form as r = γ0. (s. = γ0. k 0 + z0 ) + ϕ0. (Sk 0 + z 0 ) + ϕ 0. E(s. k 0 + z 0 )*. E(Sk 0 + z 0 )*. = Γ0 ( Sk 0 + z 0 ) +Φ0E(Sk 0 + z 0 )* where s k0 z0. ( P.S.. [ s[0] , s[1], [ k0 [0] , k0 [1], [ z0 [0] , z0 [1],. (2.17). , s[ N − 1] ] , T. , k0 [ N − 1] ] , T. , z0 [ N − 1] ] , T. Assuem z 0 is a zero-mean Gaussian vector with convariance matrix σ 02I N ). [ γ 0 [0] , γ 0 [1], , γ 0 [ N − 1] ]T , T ϕ 0 [ ϕ 0 [0] , ϕ 0 [1], , ϕ 0 [ N − 1] ] , ( S ) N ×N diag { s }, ( Γ0 ) N ×N diag {γ 0 }, ( Φ0 ) N ×N diag {ϕ 0 }, γ0. N is defined as the size of FFT.. e0. (. ⎡1 ⎤ ⎢0⎥ ⎢ ⎥ ⎢0⎥ , ⎢ ⎥ ⎢ ⎥ ⎣⎢0⎦⎥ N ×1. e1. ⎡0⎤ ⎢1 ⎥ ⎢ ⎥ ⎢0⎥ , ⎢ ⎥ ⎢ ⎥ ⎣⎢0⎦⎥ N ×1. ⎡ 0⎤ ⎢ 0⎥ ⎢ ⎥ ⎢ ⎥ , and E ⎢ ⎥ ⎢0⎥ ⎣⎢1⎦⎥ N ×1. , e N −1. P.S . E is a permutation matrix and E ⋅ E = I N ×N. ). ⎡ eT0 ⎤ ⎢ T ⎥ ⎢ e N −1 ⎥ ⎢eTN −2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ e1T ⎥⎦ N ×N. And the matrices Γ0 and Φ0 have the following important properties if the differences of I/Q branches is not large.. 12.

(20) Γ0Γ0H ≈ I N. (2.18). Φ0Φ0H ≈ 0. (2.19). 13.

(21) Chapter 3 Frequency-Independent I/Q Imbalance A. complete. system. model. of. direct-conversion. receiver. with. frequency-dependent I/Q imbalance has been presented in Chapter 2. However, if the individual analog amplifiers, LPFs , and A/D converters on I/Q branches are well-designed or the bandwidth of the system is not large, the difference of H I ( f ) and. H Q ( f ) then are almost neglectable. It means that we could assume. H I ( f ) = H Q ( f ) so that the problem of frequency-dependent I/Q imbalance would become a frequency-independent problem. In other words, frequency-independent I/Q imbalance is a special case of frequency-dependent I/Q imbalance. The frequency-independent. problem. is. discussed. in. this. chapter,. and. the. frequency-dependent one will be solved in chapter 4.. Equation Section (Next) 3.1 Modified System Model In the case of H I ( f ) = H Q ( f ) , (2.17) are modified as. r = γ 0 ( Sk 0 + z 0 ) + ϕ 0E ( Sk 0 + z 0 ) where. ( P.S.. γ0. *. 1 1 + ge − jθ ) and ϕ 0 ( 2. ϕ 0 ≈ 0 and γ 0 ≈ 1. 1 1 − ge jθ ) ( 2. (3.1). ). The amplitude of γ 0 approximates to one. while the amplitude of ϕ 0. approximates to zero. Worthy of notice is that both γ 0 and ϕ 0 are constant for all subcarriers. That is, the effects of I/Q imbalance due to imperfect LO are frequency-independent.. In order to compensate I/Q imbalance and estimate channel response easily, we. 14.

(22) define a new I/Q imbalance parameter ϕ . After some calculations, the equation (3.4) could give us a method to perform the compensation and estimation jointly.. Define ϕ. ϕ0 γ 0*. ( P.S . ϕ ≈ 0 ). (3.2). ϕ0 ⎡ * * ⋅ E γ 0 ( Sk 0 + z 0 ) + ϕ 0*E ( Sk 0 + z 0 ) ⎤ * ⎦ ⎣ γ0 ϕ ϕ * = 0* ⋅ γ 0*E ( Sk 0 + z 0 ) + 0* ⋅ ϕ 0* EE ( Sk 0 + z 0 ) γ0 γ0 I. ϕ E ⋅ ( r* ) =. (3.3). N. = ϕ 0E ( Sk 0 + z 0 ) + *. ϕ0 (Sk 0 + z 0 ) γ 0* 2. 2 ⎡ ⎤ ϕ0 * * ⎡ ⎤ ⇒ r − ϕ E ⋅ ( r ) = γ 0 ( Sk 0 + z 0 ) + ϕ 0E ( Sk 0 + z 0 ) − ⎢ϕ 0E ( Sk 0 + z 0 ) + * ( Sk 0 + z 0 ) ⎥ ⎣ ⎦ γ0 ⎢⎣ ⎥⎦ *. ϕ = γ 0 ( Sk 0 + z 0 ) − 0* ( Sk 0 + z 0 ) γ0 2. (. ) (Sk + z ) (1 − ϕ ) k ⎤⎦ + γ (1 − ϕ ) ⋅ z. = γ0 1− ϕ = S ⋅ ⎡γ 0 ⎣. 2. 0. 0. 2. 2. 0. 0. z. k. ⇒ r − ϕ E ⋅ ( r* ) = Sk + z ,. where k. (. γ0 1− ϕ. 2. )k. 0. (3.4). 0. (. and z γ 0 1 − ϕ. 2. )z. 0. The right-hand-side of (3.4) represents a signal without I/Q imbalance. On the premise that I/Q imbalance parameter ϕ is estimated, the left-hand-side of (3.4) provides an easy solution to compensating frequency-independent I/Q imbalance. There are two unknown parameters, ϕ and k , which have to be estimated, in (3.4). The unknown k is the effective channel response in frequency domain after compensation of I/Q imbalance using ϕ . When the training sequence is transmitted, r and S are deterministic and known by the receiver. The only random term is z ,. which is a zero-mean Gaussian vector with covariance matrix σ 2I N . The equation 15.

(23) (3.5) shows the definition of σ 2 .. Suppose E {z 0 z 0H } = σ 02 I N. { (. ⇒ E {zz H } = E γ 0 1 − ϕ. 2. ) z z (1 − ϕ ) γ } = σ H 0 0. 2. * 0. 2 0. γ0. 2. (1 − ϕ ) I 2 2. N. = σ 2I N. (3.5). σ2. where σ 2. σ 02 γ 0. 2. (1 − ϕ ). 2 2. The following sections are going to provide the ML estimators of I/Q imbalance parameter ϕ and effective channel response k .. 3.2 Joint ML Estimation The r − ϕ E ⋅ ( r* ) in (3.4) is a Gaussian vector with mean Sk and covariance matrix σ 2I N , for the given ϕ and k . The conditional probability density function (pdf) of r − ϕ E ⋅ ( r * ) is written as p(r − ϕ Er* ϕ , k ) =. 1. (πσ ). 2 N. 2⎞ ⎛ 1 exp ⎜ − 2 r − ϕ Er* − Sk ⎟ ⎝ σ ⎠. (3.6). where ϕ and k are trial value of ϕ and k , respectively. The ML estimates of ϕ and k are chosen by maximizing the following likelihood function over ϕ and k :. Λ (ϕ , k ) = − r − ϕ Er* − Sk. 2. (3.7). 3.2.1 ML Estimation of Channel First, ML estimator kˆ ML is derived from maximizing (3.7). It is a typical problem about linear estimator. Given some trial ϕ , ML estimator of k is −1 kˆ ML = ( S H S ) S H ( r − ϕ Er * ) = S −1 ( r − ϕ Er* ). 16. (3.8).

(24) This estimate of channel, however, is not good enough, because it estimates channel responses of all subcarriers independently. Actually, the channel responses of neighbor subcarriers are highly correlated. We could exploit the property of finite length of effective channel impulse response in time domain. It is a reasonable assumption because the length of GI is supposed to be larger than the effective channel impulse response, which makes OFDM system work. This property in time domain results in smoothing change of channel response in frequency domain. Now, we switch our target to estimate channel impulse response h in time domain instead of channel frequency response k in frequency domain. The relationship of h and k are shown below.. k N ×1. ( Fh ) N ×L. h. where {Fh }i ,n. ( P.S.. h Lh ×1. (3.9). ⎛ 2π ⋅ i ⋅ n ⎞ exp ⎜ − ⎟ , 0 ≤ i ≤ N − 1, 0 ≤ n ≤ Lh − 1 N ⎠ ⎝ Fh is part of Fourier matrix.). N is the size of FFT, Lh is estimated length of channel impulse response.. Then the likelihood function (3.7) can be rewritten as. Λ (ϕ , k ) = Λ (ϕ , h) = − r − ϕ Er* − SFh h. 2. (3.10). By (3.10), new ML estimates of channel h and k are obtained as −1 # hˆ ML = ( FhH S H SFh ) FhH S H ( r − ϕ Er* ) = ( SFh ) ( r − ϕ Er* ). (3.11). # kˆ ML = Fh ⋅ hˆ ML = Fh ⋅ ( SFh ) ( r − ϕ Er* ). (3.12). In the case of WLAN system, the training sequences possess the property. S H S = I N . For example, short training sequences and long training sequences of IEEE 17.

(25) 802.11a and IEEE 802.11g are satisfied this property. The ML estimates (3.11) and (3.12) could be simplified as −1. ⎛ ⎞ # hˆ ML = ⎜ FhH S H S Fh ⎟ FhH S H ( r − ϕ Er * ) = ( Fh ) ( r − ϕ Er* ) IN ⎝ ⎠. (3.13). # kˆ ML = Fh ⋅ hˆ ML = Fh ⋅ ( Fh ) ( r − ϕ Er* ). (3.14). 3.2.2 ML Estimation of I/Q Imbalance We have derived the ML estimate for channel from (3.12). Substitute (3.12) into (3.10), which yields Λ (ϕ ). Λ (ϕ , h) h=hˆ. ML. 2. = − r − ϕ Er * − SFh hˆ ML. 2. = − r − ϕ Er * − SFh ( SFh ) ( r − ϕ Er * ) #. PSF. = − ( I N − PSF ) ( r − ϕ Er* ) , where PSF. 2. (3.15). SFh ( SFh ) is a projection matrix. We could get the ML estimate for #. I/Q imbalance parameter ϕ by maximizing (3.15). −1. ϕˆ ML. ⎧ ⎫ H ⎪ * H * ⎪ = ⎨ ⎣⎡( I N − PSF ) Er ⎦⎤ ⎣⎡( I N − PSF ) Er ⎦⎤ ⎬ ⎣⎡( I N − PSF ) Er * ⎦⎤ ( I N − PSF ) r ⎪⎩ ⎪⎭ N ×1 1× N =. (. (. ( I N − PSF ) Er*. ). 2 −1. (r ). * H. EH ( I N − PSF ). H. ( I N − PSF ) r. ( I N − PSF ). P.S . ( I N − PSF ) is a projection matrix. ). ⎡( I N − PSF ) Er ⎤⎦ =⎣ ⋅r 2 ( I N − PSF ) Er* * H. ⇒ ϕˆ ML = p H ⋅ r. , where p. ( I N − PSF ) Er* 2 ( I N − PSF ) Er* 18. (3.16).

(26) 3.3 Performance of ML Estimation To the simplicity of the following calculation, we define some parameters before analyzing the mean and mean square error (MSE) of ϕˆ ML and kˆ ML . First, rd is defined as. rd. γ 0Sk 0 +ϕ 0E ( Sk 0 ). *. (3.17). The equation (3.1) can be represented in termed of rd as the following shows.. r = γ 0 ( Sk 0 + z 0 ) + ϕ 0E ( Sk 0 + z 0 ). *. = γ 0Sk 0 +ϕ 0E ( Sk 0 ) + γ 0 z 0 + ϕ 0Ez 0* *. Noise part. Deterministic Part. =rd + γ 0 z 0 + ϕ 0Ez 0* , = rd +. 1 1− ϕ. 2. z+. where rd. γ 0Sk 0 +ϕ 0E ( Sk 0 ). *. ϕ Ez* 2 1− ϕ. (3.18). ⎛ ⎞ 1 ϕ 0 / γ 0* * ϕ 2 * * = = ϕ z and z z z ⎜ Proof : z γ 0 1 − ϕ z 0 ⇒ γ 0 z 0 = ⎟ 0 0 2 2 2 ⎜ ⎟ ϕ ϕ ϕ − 1 1 1 − − ⎝ ⎠. (. ). By inspecting (3.18), we observed that rd is the deterministic part of r . Also, it is worthwhile to define. pd. ( I N − PSF ) Erd* 2 ( I N − PSF ) Erd*. (3.19). In high SNR case, these two new variables r and p respectively approximate rd and pd .. 3.3.1 Mean and MSE of ML Estimation of I/Q Imbalance Parameter In this section, the mean and MSE of ϕˆ ML , defined in (3.16), are analyzed. The relationship between ϕˆ ML and ϕ is expressed in (3.20). The AWGN contributes to a random term p H z in ML estimate ϕˆ ML . The equation (3.21) proves that ϕˆ ML is an unbiased estimate when the SNR is high. The MSE of ϕˆ ML in high SNR case is 19.

(27) predicted by (3.22). The correctness of MSE analysis is proved by the computer simulations. The analyzing procedures are as follows:. r − ϕ Er* = SFh h + z ⇒ r = Er * + SFh h + z. ( I N − PSF ) Er* 2 ( I N − PSF ) Er*. ϕˆ ML = p H r , where p. = p H (ϕ Er * + SFh h + z ) = ϕ p H Er * + p H SFh h + p H z (a). (b). ⎡⎣( I N − PSF ) Er* ⎤⎦ ( a ) p Er = Er* 2 ( I N − PSF ) Er* H. H. *. ( Er ) ( I = * H. − PSF ) Er* H. N. ( I N − PSF ) Er*. ( Er ) ( I = * H. N. − PSF ). 2. H. ( I N − PSF ) Er*. ( I N − PSF ) Er*. =. ( I N − PSF ) Er* ( I N − PSF ) Er*. 2. ( ∵ (I. N. − PSF ) is a projection matrix. ). 2 2. =1 ⎡⎣( I N − PSF ) Er* ⎤⎦ (b) p SFh h = SFh h 2 ( I N − PSF ) Er* H. H. ( Er ) ( I = * H. N. − PSF ). ( I N − PSF ) Er*. 2. H. SFh h. ( Er ). * H. I − PSF ) SFh h 2 ( N ( I N − PSF ) Er* 0 =0 ( ∵ ( I N − PSF ) is a orthogonal projection matrix ) =. N ×1. By ( a ) and (b), ϕˆ ML = ϕ p H Er * + p H SFh h + p H z = ϕ + p H z 1. 0. 20. (3.20).

(28) E {ϕˆ ML − ϕˆ } = E {p H z} ≈ E {p H z}. ( P.S.. = p dH E {z}. In high SNR case, p ≈ p d ). 0. ⇒ E {ϕˆ ML − ϕˆ} ≈ 0. {. E ϕˆ ML − ϕˆ. 2. (. In high SNR case. ). (3.21). } = E{p z } 2. H. = E {p H zz H p} ≈ E {p dH zz H p d }. ( P.S.. In high SNR case, p ≈ p d ). = p dH E {zz H }p d = σ 2p dH p d = σ 2 pd. 2. =σ2. ( I N − PSF ) Er ( I N − PSF ) Erd*. }. σ2. {. ⇒ E ϕˆ ML − ϕˆ. 2. ≈. 2. * d. ( I N − PSF ) Erd*. 2. 2. (. In high SNR case. ). (3.22). 3.3.2 Mean and MSE of ML Estimation of Channel in Time Domain In this section, the mean and MSE of hˆ ML , which is defined in (3.11), are analyzed. The relationship between hˆ ML and h is expressed as (3.23). The AWGN contributes a random term. (ϕˆ − ϕ )( SFh ). #. ( SFh ). #. z to ML estimate hˆ ML . Another random term. Er* in hˆ ML comes from the error in ϕˆ ML . Our compensation scheme. is to compensate I/Q imbalance first by using estimated I/Q imbalance parameter. ϕˆ ML , rather than the real ϕ ; then estimate the effective channel response. Thus, the above-mentioned is the reason that mean and MSE of hˆ ML depends on the error term 21.

(29) (ϕˆML − ϕ ) .. The equation (3.24) points out that if ϕˆ ML is unbiased, hˆ ML is an. unbiased estimate in high SNR case, too. The MSE of hˆ ML in high SNR case is also predicted by (3.26). The error of estimated ϕˆ ML increased the MSE of effective channel estimate hˆ ML due to the second term in (3.26). The correctness of MSE analysis is proved by the simulation. The following are the analyzing procedures.. r − ϕˆ Er* = (ϕ Er* + SFh h + z ) − ϕˆ Er* = SFh h − (ϕˆ − ϕ ) Er* + z # hˆ ML = ( SFh ) ( r − ϕˆ Er * ). = ( SFh ) ⎡⎣SFh h − (ϕˆ − ϕ ) Er * + z ⎤⎦ #. = ( SFh ) SFh h − (ϕˆ − ϕ )( SFh ) Er * + ( SFh ) z #. #. #. I Lh. = h − (ϕˆ − ϕ )( SFh ) Er* + ( SFh ) z #. {. {. }. #. (3.23). }. # # E hˆ ML − h = E − (ϕˆ − ϕ )( SFh ) Er* + ( SFh ) z. {. }. = − E (ϕˆ − ϕ )( SFh ) Er * + ( SFh ) E {z}. {. #. = − E (ϕˆ − ϕ )( SFh ) Er *. {. }. #. #. 0 N ×1. }. # ⇒ E hˆ ML − h ≈ − E {ϕˆ − ϕ }( SFh ) Erd*. ( P.S.. In high SNR case, r * ≈ rd* ) (3.24). If ϕˆ is an unbiased estimator, that means that E {ϕˆ − ϕ } = 0 .. {. }. ⇒ E hˆ ML − h ≈ 0 Lh ×1. (3.25). In high SNR case, hˆ ML is also an unbiased estimator, too.. 22.

(30) {. E hˆ ML − h. 2. }. {(. = E hˆ ML − h. { {(. ) ( hˆ H. ML. −h. )}. }. H. # # # # = E ⎡ ( SFh ) z − (ϕˆ − ϕ )( SFh ) Er* ⎤ ⎡( SFh ) z − (ϕˆ − ϕ )( SFh ) Er * ⎤ ⎣ ⎦ ⎣ ⎦. =E. 2. SFh ) z #. (a). } { +E. (ϕˆ − ϕ )( SFh ). #. Er*. 2. }. (b). {. }. H. # # − 2 Re E ⎡( SFh ) z ⎤ ⎡(ϕˆ − ϕ )( SFh ) Er * ⎤ ⎣ ⎦ ⎣ ⎦ (c). (a ). ( SFh ). #. 2. H. # # #H # z = ⎡( SFh ) z ⎤ ⎡ ( SFh ) z ⎤ = z H ( SFh ) ( SFh ) z ⎣ ⎦ ⎣ ⎦. { = tr {( SF ) zz = tr z H ( SFh ) #. ⇒E. {. #H. H. h. ( SFh ). #. z. 2. } E {tr (. ( SFh ) ( SFh ). #. } }. z. #H. (. P.S . scalar = tr {scalar}. ). (. P.S . tr { AB} = tr {BA}. ). }. ⎡ SFh ) # zz H ( SFh ) # H ⎤ ⎣ ⎦. =. {. = σ 2 tr ( SFh ) ( SFh ) #. }. #H. { tr { (. H = σ 2 ⋅ tr ⎡( SFh ) ( SFh ) ⎤ ⎣ ⎦. −1. ⎡ SFh ) H ( SFh ) ⎤ ⎣ ⎦. −1. =σ2 ⋅. ( SFh ) ⋅ ( SFh ) ⎡⎣( SFh ) ( SFh ) ⎤⎦ H. H. }. −1. }. ( b). (ϕˆ − ϕ )( SFh ). ⇒E. {. #. Er *. (ϕˆ − ϕ )( SFh ). #. 2. = ϕˆ − ϕ. 2. ( SFh ). ≈ ϕˆ − ϕ. 2. ( SFh ). Er *. 2. }. {. ≈ E ϕˆ − ϕ =σ2. #. 2. Er *. #. Erd*. 2. ( P.S.. 2. } ⋅ (SF ) Er #. h. ( SFh ). #. Erd*. * d. In high SNR case, r * ≈ rd* ). 2. 2. ( I N − PSF ) Erd*. 2. (c) H. ⎡( SFh ) # z ⎤ ⎡ (ϕˆ − ϕ )( SFh ) # Er * ⎤ ⎣ ⎦ ⎣ ⎦ = (ϕˆ − ϕ ) z H ( SFh ). ( SFh ) Er* #H # = p H z ⋅ z H ( SFh ) ( SFh ) Er * ≈ p dH zz H ( SFh ). #H. #H. ( SFh ). #. #. Erd*. ( P.S. ( P.S.. If ϕˆ − ϕ = p H z. ). In high SNR case, r ≈ rd and p ≈ p d ) 23.

(31) {. }. H. # # ⇒ E ⎡( SFh ) z ⎤ ⎡(ϕˆ − ϕ )( SFh ) Er* ⎤ ⎣ ⎦ ⎣ ⎦. {. ≈ E p dH zz H ( SFh ). #H. ( SFh ). = p dH ⋅ E {zz H } ⋅ ( SFh ) = σ 2 p dH ( SFh ) =σ =. 2. ( Er ) ( I * H d. ⋅ ( SFh ). ( I N − PSF ) Er. * 2 d. ( I N − PSF ) Er. * 2 d. ⎛ ⎜ P.S . ⎜⎜ ⎝. #. }. Erd*. Erd*. − PSF ). N. σ2. =0. #. Erd*. ( SFh ). #H. ( SFh ). #H. #. ( Er ) ⎡⎣( I * H d. #H. N. ( SFh ). #. Erd*. − PSF )( SFh ). #H. ⎤ ( SFh ) # Erd* ⎦. 0 N × Lh. ( I N − PSF )( SFh ). #H. = ( I N − PSF )( SFh ) ⎡( SFh ) ⎣. ⎞ ( SFh ) ⎤⎦ = 0 N ×Lh ⎟⎟ ⎟ ⎠ −1. H. 0 N × Lh. By (a ), (b), and ( c ). {. ⇒ E hˆ ML − h. 2. } = E { (SF ) z } + E { (ϕˆ − ϕ )(SF ) Er } #. 2. #. h. (a). 2. *. h. (b). {. }. H. # # − 2 Re E ⎡( SFh ) z ⎤ ⎡(ϕˆ − ϕ )( SFh ) Er * ⎤ ⎣ ⎦ ⎣ ⎦ (c). {. H = σ 2 ⋅ tr ⎡ ( SFh ) ( SFh ) ⎤ ⎣ ⎦. {. + E p H zz H ( SFh ). {. #H. {. {. ⇒ E hˆ ML − h. }. −1. }. #. Er *. {. ( SFh ). 2. #. Er *. 2. }. +E ϕˆ − ϕ. 2. } ⋅ (SF ) Er #. h. }. * d. 2. In high SNR case, r ≈ rd and p ≈ p d ). H =σ 2 ⋅ tr ⎡ ( SFh ) ( SFh ) ⎤ ⎣ ⎦. 2. } {. + E ϕˆ − ϕ. ( SFh ). H ≈ σ 2 ⋅ tr ⎡( SFh ) ( SFh ) ⎤ ⎣ ⎦. ( P.S.. −1. −1. }+σ. 2. ( SFh ). #. Erd*. ( I N − PSF ) Erd*. 2 # ⎧ * SF Er ( ) 1 − h d H ⎪ ≈ σ 2 ⎨tr ⎡( SFh ) ( SFh ) ⎤ + ⎣ ⎦ ( I N − PSF ) Erd* ⎪ ⎩. {. }. 2. 2. ⎫ ⎪ 2⎬ ⎪ ⎭. 3.3.3 Mean and MSE of ML Estimation of Channel in Frequency Domain 24. (3.26).

(32) In this section, the mean and MSE of kˆ ML , which is defined in (3.12), are analyzed. The relationship between kˆ ML and k is expressed as (3.27). The AWGN # contributes to a random term Fh ( SFh ) z in ML estimate kˆ ML like hˆ ML . Another. random term. (ϕˆ − ϕ ) Fh ( SFh ). #. Er* in kˆ ML comes from the error in ϕˆ ML . Our. compensation scheme is to first compensate I/Q imbalance by using estimated I/Q imbalance parameter ϕˆ ML rather than the real ϕ ; then estimate effective channel response. Therefore, the above-mentioned is the reason that mean and MSE of kˆ ML contains the error term (ϕˆ ML − ϕ ) . Furthermore, the equation (3.28) shows that if. ϕˆ ML is unbiased, kˆ ML is an unbiased estimate in high SNR case, too. The MSE of kˆ ML in high SNR case is also predicted by (3.30). The error of estimated ϕˆ ML. increases the MSE of effective channel estimate kˆ ML due to the second term in (3.30). The correctness of MSE analysis is proved by the computer simulations. The following are the analysis procedures.. # # hˆ ML = h − (ϕˆ − ϕ )( SFh ) Er * + ( SFh ) z. ⇒ kˆ ML. Fh ⋅ hˆ ML # # = Fh ⎡ h − (ϕˆ − ϕ )( SFh ) Er * + ( SFh ) z ⎤ ⎣ ⎦. = Fh h − (ϕˆ − ϕ ) Fh ( SFh ) Er * + Fh ( SFh ) z #. #. k. = k − (ϕˆ − ϕ ) Fh ( SFh ) Er* + Fh ( SFh ) z #. #. 25. (3.27).

(33) {. {. }. }. # # E kˆ ML − k = E − (ϕˆ − ϕ ) Fh ( SFh ) Er* + Fh ( SFh ) z. {. #. {. #. }. = − E (ϕˆ − ϕ ) Fh ( SFh ) Er * + Fh ( SFh ) E {z} = − E (ϕˆ − ϕ ) Fh ( SFh ) Er *. {. #. 0 N ×1. }. {. }. # ⇒ E kˆ ML − k ≈ − E (ϕˆ − ϕ ) Fh ( SFh ) Erd*. } ( P.S. In high SNR case, r. *. ≈ rd* ). = − E {ϕˆ − ϕ } ⋅ Fh ( SFh ) Er #. (3.28). * d. If ϕˆ is an unbiased estimator, that means that E {ϕˆ − ϕ } = 0 .. {. }. ⇒ E kˆ ML − k = 0 Lh ×1. (3.29). In high SNR case, hˆ ML is also an unbiased estimator.. # # kˆ ML − k = − (ϕˆ − ϕ ) Fh ( SFh ) Er * + Fh ( SFh ) z. {. ⇒ E kˆ ML − k. 2. {(. }. = E kˆ ML − k. { {. ) ( kˆ H. ML. −k. )}. }. H. # # # # = E ⎡ Fh ( SFh ) z − (ϕˆ − ϕ ) Fh ( SFh ) Er* ⎤ ⎡ Fh ( SFh ) z − (ϕˆ − ϕ ) Fh ( SFh ) Er* ⎤ ⎣ ⎦ ⎣ ⎦. = E Fh ( SFh ) z #. 2. } { (ϕ. ˆ − ϕ ) Fh ( SFh ) Er*. +E. #. (a ). (b). {. 2. }. }. H. # # − 2 Re E ⎡ Fh ( SFh ) z ⎤ ⎡(ϕˆ − ϕ ) Fh ( SFh ) Er * ⎤ ⎣ ⎦ ⎣ ⎦ (c). (a ) 2. H. # # # Fh ( SFh ) z = ⎡ Fh ( SFh ) z ⎤ ⎡ Fh ( SFh ) z ⎤ ⎣ ⎦ ⎣ ⎦. = z H ( SFh ). #H. FhH Fh ( SFh ) z #. F F ( SF ) z} { = tr {F ( SF ) zz ( SF ) F }. = tr z H ( SFh ). #H. #. h. h. H h. #. h. h. #H. H. h. 26. H h. (. P.S . scalar = tr {scalar}. ). (. P.S . tr { AB} = tr {BA}. ).

(34) ⇒E. { (SF ) z } = E {tr ⎡⎣F (SF ) zz 2. #. #. h. h. H. h. {. ( SFh ). = σ 2 tr Fh ( SFh ) ( SFh ) #. #H. }. FhH ⎤ ⎦. #H. FhH. }. ( SF ) { = σ ⋅ tr {F ⎡( SF ) ( SF ) ⎤ F } ⎣ ⎦ H = σ 2 ⋅ tr Fh ⎡( SFh ) ( SFh ) ⎤ ⎣ ⎦. h. h. H. h. −1. H. 2. −1. h. ⋅ ( SFh ) ⎡ ( SFh ) ⎣. H. ( SFh )⎤⎦. −1. FhH. }. H h. ( b). (ϕˆ − ϕ ) Fh ( SFh ). #. Er *. 2. = ϕˆ − ϕ ≈ ϕˆ − ϕ. ⇒E. {. (ϕˆ − ϕ )( SFh ). #. Er*. 2. }. 2. Fh ( SFh ) Er *. 2. #. #. {. ≈ E ϕˆ − ϕ. 2. }. ⋅ Fh ( SFh ) Erd* #. Fh ( SFh ) Erd*. 2. ( I N − PSF ) Erd*. 2. #. =σ2. ( P.S.. 2. Fh ( SFh ) Erd*. 2. In high SNR case, r * ≈ rd* ) 2. (c) H. ⎡ Fh ( SFh ) # z ⎤ ⎡ (ϕˆ − ϕ ) Fh ( SFh ) # Er* ⎤ ⎣ ⎦ ⎣ ⎦ = (ϕˆ − ϕ ) z H ( SFh ). FhH Fh ( SFh ) Er*. #H. #. = p H zz H ( SFh ). #H. FhH Fh ( SFh ) Er*. ≈ p dH zz H ( SFh ). #H. FhH Fh ( SFh ) Erd*. {. ( P.S. If ϕˆ − ϕ = p z ) ( P.S. In high SNR case,. #. H. #. r ≈ rd and p ≈ p d ). }. H. # # ⇒ E ⎡ ( SFh ) z ⎤ ⎡(ϕˆ − ϕ )( SFh ) Er * ⎤ ⎣ ⎦ ⎣ ⎦. {. ≈ E p dH zz H ( SFh ) = σ 2p dH ( SFh ) =σ =. 2. * H d. FhH Fh ( SFh ) Erd* #. }. FhH Fh ( SFh ) Erd*. #H. ( Er ) ( I. #H. #. − PSF ). N. ( I N − PSF ) Er. * 2 d. σ2. ( I N − PSF ) Er. * 2 d. ⋅ ( SFh ). ( Er ) ⎡⎣( I * H d. #H. N. FhH Fh ( SFh ) Erd* #. − PSF )( SFh ). #H. ⎤ FhH Fh ( SFh ) # Erd* ⎦. 0 N × Lh. =0 ⎛ ⎜ P.S . ⎜⎜ ⎝. ( I N − PSF ) ( SFh ). #H. ⎞ −1 H ⎡ ⎤ = ( I N − PSF )( SFh ) ( SFh ) ( SFh ) = 0 N ×Lh ⎟ ⎣ ⎦ ⎟⎟ 0 N × Lh ⎠. 27.

(35) By ( a ), (b), and ( c),. {. E kˆ ML − k. 2. } {. = E Fh ( SFh ) z #. 2. } { +E. (ϕˆ − ϕ ) Fh ( SFh ). (a). #. Er *. 2. }. (b). {. }. H. # # − 2 Re E ⎡ Fh ( SFh ) z ⎤ ⎡ (ϕˆ − ϕ ) Fh ( SFh ) Er * ⎤ ⎣ ⎦ ⎣ ⎦ (c). {. −1. } {. 2 # H = σ 2 ⋅ tr Fh ⎡( SFh ) ( SFh ) ⎤ FhH + E ϕˆ − ϕ Fh ( SFh ) Er * ⎣ ⎦. {. + E p H zz H ( SFh ). {. #H. FhH Fh ( SFh ) Er * #. −1. }. }. {. }. 2. }. 2 # H ≈ σ 2 ⋅ tr Fh ⎡( SFh ) ( SFh ) ⎤ FhH +E ϕˆ − ϕ ⋅ Fh ( SFh ) Erd* ⎣ ⎦. ( P.S.. In high SNR case, r ≈ rd and p ≈ p d ). {. = σ 2 ⋅ tr Fh ⎡( SFh ) ⎣. {. ⇒ E kˆ ML − k. 2. }. 2. H. ( SFh )⎤⎦. −1. }. Fh ( SFh ) Erd*. 2. ( I N − PSF ) Erd*. 2. #. FhH + σ 2. 2 # ⎧ ⎫ * F SF Er ( ) −1 h h d H ⎪ H 2 ⎪ ≈ σ ⎨tr Fh ⎡ ( SFh ) ( SFh ) ⎤ Fh + ⎬ ⎣ ⎦ * 2 ( I N − PSF ) Erd ⎪ ⎪ ⎩ ⎭. {. }. 28. (3.30).

(36) Chapter 4 Frequency-Dependent I/Q Imbalance In this chapter, we focus on the case of frequency-dependent I/Q imbalance and propose the corresponding compensation algorithm. The main procedures of channel estimation and I/Q mismatch correction are similar to those introduced in the chapter 3. However, the degrees of the desired signal that is disturbed by the image signal are different from each subcarrier as (2.17) presents, so the I/Q imbalance parameter ϕ for compensation in (3.4) should not remain the same for all subcarriers. Due to this reason, it is supposed to have individual I/Q imbalance parameter ϕ [i ] belonging to the subcarrier i. We are about to go through the analyzing procedures again to obtain the new ML estimates of frequency channel response and frequency-dependent I/Q imbalance.. Equation Section (Next) 4.1 Modified System Model The system model of frequency-dependent I/Q imbalance expressed in (2.17) is r = Γ0 ( Sk 0 + z 0 ) +Φ0E(Sk 0 + z 0 )* , where ( Γ0 ) N × N. diag {γ 0 },. ( S ) N ×N. diag {s} , and ( Φ0 ) N ×N. diag {ϕ 0 }. After some calculations, the equation (4.5) provides us a method to perform the compensation and estimation jointly. * Φ0E ( Γ*0 ) r* = Φ0E ( Γ*0 ) ⎡ Γ0Sk 0 +Φ0E ( Sk 0 + z 0 ) ⎤ ⎣ ⎦ −1. −1. *. * = Φ0E ( Γ*0 ) ⎡ Γ*0 ( Sk 0 + z 0 ) +Φ*0E ( Sk 0 + z 0 ) ⎤ ⎣ ⎦ −1. = Φ0E ( Γ*0 ) Γ*0 ( Sk 0 + z 0 ) + Φ0E ( Γ*0 ) Φ*0E ( Sk 0 + z 0 ) −1. −1. *. IN. = Φ0E ( Sk 0 + z 0 ) + Φ0E ( Γ*0 ) Φ*0E ( Sk 0 + z 0 ) *. −1. 29. (4.1).

(37) r − Φ0E ( Ω*0 ) r* −1. = Γ0 ( Sk 0 + z 0 ) − Φ0E ( Ω*0 ) Φ*0E ( Sk 0 + z 0 ) −1. −1 = ⎡⎢ Γ0 − Φ0E ( Γ*0 ) Φ*0E⎤⎥ ( Sk 0 + z 0 ) ⎣ ⎦. ⎡ ⎤ −1 * * −1 ⎥ ⎢ = I N − Φ0E ( Γ0 ) Φ0 EΓ0 Γ0 ( Sk 0 + z 0 ) ⎢ ⎥ Λ ⎣ ⎦ = ( I N − Λ ) Γ0 ( Sk 0 + z 0 ). where Λ. Φ0E ( Γ0−1 ) Φ*0EΓ0−1 (diagonal matrix) *. = ( I N − Λ ) Γ0Sk 0 + ( I N − Λ ) Γ0 z 0 = S ( I N − Λ ) Γ0k 0 + ( I N − Λ ) Γ0 z 0 k. ( P.S. ∵ Both S and ( I = Sk + z. z. N. − Λ ) Ω0 are diagonal matrices. ). ( I N − Λ ) Γ0k 0. where k. and z. ( I N − Λ ) Γ0 z 0. (4.2). Then, rewrite Φ0E ( Ω*0 ) r* to another form (4.4) that is more easily understood. −1. Define Φ Φ0E ( Γ*0 ) E , which is a diagonal matrix.. (4.3). Φ0E ( Γ*0 ) r* = Φ0E ( Γ*0 ). (4.4). −1. −1. −1. −1 EE r* = ⎡⎢ Φ0E ( Γ*0 ) E⎤⎥ ( Er* ) = Φ ( Er* ) ⎣ ⎦ I N. Φ. Finally, the equation (4.2) can be expressed as. r − Φ ( Er* ) = Sk + z where Φ k z. (4.5). Φ0E ( Γ*0 ) E, −1. ( I N − Λ ) Γ0k 0 , ( I N − Λ ) Γ0 z 0. (diagonal matrix) (column vector) (column vector). The form of equation (4.5) resembles the one of equation (3.4), except that the scalar ϕ is replaced with a diagonal I/Q imbalance matrix Φ . The right-hand-side of (4.5) represents a signal without I/Q imbalance. On the premise that the I/Q imbalance matrix Φ is estimated, the left-hand-side of (4.5) develops an easy solution to compensating frequency-dependent I/Q imbalance. However, there are two unknown parameters, Φ and k , which have to be estimated, in (4.5). The first 30.

(38) unknown k is an effective channel response in frequency domain followed by compensation of I/Q mismatch which utilizes Φ . Secondly, another unknown I/Q imbalance matrix Φ is a diagonal matrix; that means only the diagonal elements of Φ. need. estimating.. Different. from. the. compensation. scheme. for. frequency-independent case mentioned in chapter 3, we should utilize different coefficients to correct frequency-dependent I/Q imbalance effects upon different subcarriers. When the training sequence is transmitted, r and S are deterministic and identified by the receiver. However, the only random term is z , which is a zero-mean Gaussian vector with diagonal covariance matrix Σ as (4.6) expresses.. Suppose E {z 0 z 0H }= σ 02I ⇒Σ. E {zz H }. {. H. }. -Λ ). H. = E ( I N -Λ ) Γ0 z 0 z 0H Γ0H ( I N -Λ ) = ( I N -Λ ) Γ0 E {z z. H 0 0. }Γ ( I H 0. = σ 02 ( I N -Λ ) Γ0Γ0H ( I N -Λ ). N. (4.6). ( a diagonal matrix ). H. ≈σ I. 2 0 N. The following sections are going to provide the ML estimators of I/Q imbalance matrix Φ and effective channel response k .. 4.2 Joint ML Estimation The r − ΦE ⋅ ( r* ) in (4.5) is a Gaussian vector with mean Sk and covariance matrix Σ , for the given Φ and k . The conditional probability density function of. r − ΦE ⋅ ( r* ) is written as. {. }. p r − ΦEr * Φ, k =. {. }. H 1 exp − ( r − ΦEr * − Sk ) Σ −1 ( r − ΦEr* − Sk ) π det ( Σ ) n. 31. (4.7).

(39) where Σ. E {z H z} = σ 02 ( I N − Λ ) Γ0Γ0H ( I N − Λ ). H. ≈ σ 02 I N. where Φ and k are trial value of Φ and k respectively. The ML estimates of Φ and k are chosen on the account of maximizing the following likelihood. function (4.9) over Φ and k . H Λ (Φ, k ) = − ( r − ΦEr * − Sk ) ⎡ ( I N − Λ ) Γ0Γ0H ( I N − Λ ) ⎤ ⎣ ⎦ H. −1. (r − ΦEr. *. − Sk ). −1 −1 = − ( r − ΦEr* − Sk ) ⎡ Γ0−1 ( I N − Λ ) ⎤ ⎡ Γ0−1 ( I N − Λ ) ⎤ ( r − ΦEr* − Sk ) ⎣ ⎦ ⎣ ⎦ H. H. = − Γ0−1 ( I N -Λ ). −1. (r − ΦEr. Λ ⎛ ⎞ −1 = − Γ0 ⎜ I N - ΦEΦ*E ⎟ ⎜ ⎟ ⎝ ⎠. ⎛ ⎜ P.S . Λ ⎜ ⎝. *. − Sk ). 2. 2. −1. (r − ΦEr. *. − Sk ). (4.8). ⎞ * * Φ0E ( Γ0−1 ) Φ*0EΓ0−1 = Φ0E ( Γ0−1 ) ⋅ E E ⋅ Φ*0EΓ0−1 ⋅ E E = ΦEΦ*E ⎟ (4.9) ⎟ Φ* Φ ⎠. By the property Γ0Γ0H ≈ I N , the equation (4.8) can be approximated to Λ (Φ, k ) ≈ − ( I N − Λ ). −1. (r − ΦEr. *. − Sk ). 2. (4.10). Then, in order to improve the performance of channel estimator, we turn to estimate channel impulse response h in time domain instead of channel frequency response k in frequency domain. The reason for this transformation has been explained in section 3.2.1 in detail. Λ (Φ, k ) = Λ (Φ, h). where k = Fh h. = − Γ0−1 ( I N − Λ ) ≈ − (IN − Λ). −1. −1. (r − ΦEr. (r − ΦEr. *. *. − SFh h ). − SFh h ). 2. 2. 32. (4.11).

(40) 4.2.1 ML Estimation of Channel First, ML estimator kˆ ML is derived from maximizing approximated likelihood function (4.11). It is a typical problem about linear estimator. Given a specific trial Φ , (4.12) and (4.13) demonstrate the ML estimators of channel. −1. hˆ ML. ⎧ ⎫ ⎪ ⎪ −1 H −1 −1 H −1 = ⎨(SFh ) H ⎡⎣( I N − Λ ) ⎤⎦ ( I N − Λ ) (SFh ) ⎬ (SFh ) H ⎡⎣ ( I N − Λ ) ⎤⎦ ( I N − Λ ) ( r − ΦEr * ⎪⎩ ⎪⎭ W W. = ⎡⎣(SFh ) H W(SFh ) ⎤⎦ (SFh ) H W ( r − ΦEr* −1. ). (4.12). −1 ⇒ kˆ ML = Fh ⎡⎣(SFh ) H W(SFh ) ⎤⎦ (SFh ) H W ( r − ΦEr*. where W. ⎡ ( I N − Λ ) −1 ⎤ ⎣ ⎦. H. (IN − Λ). −1. ). (4.13). H = ⎡( I N − Λ )( I N − Λ ) ⎤ ⎣ ⎦. −1. ⎛ P.S . E {zz H } = σ 02 ( I N − Λ ) Γ0Γ0H ( I N − Λ ) H ⎞ ⎜ ⎟ ≈I N ⎜ ⎟ H −1 ⎟ 2 2 ⎜ σ σ I Λ I Λ W ≈ − − = ( )( ) N N 0 0 ⎝ ⎠. 4.2.2 ML Estimation of I/Q Imbalance We have derived the ML estimate for channel from (4.13). Substitute (4.13) into (4.11), which yields. Λ ( Φ). Λ (Φ, k ) k =kˆ. ML. ≈ − (IN − Λ). −1. (. r − ΦEr* − SFh ⎡⎣(SFh ) H W(SFh ) ⎤⎦ (SFh ) H W ( r − ΦEr * −1. ⎛ ⎞ −1 −1 ⎜ H H ⎡ ⎤ = − ( I N − Λ ) I N − SFh ⎣(SFh ) W (SFh ) ⎦ (SFh ) W ⎟ ( r − ΦEr * ⎜ ⎟ ⎜ ⎟ P SFW ⎝ ⎠ = − (IN − Λ). −1. (IN. − PSFW ) ( r − ΦEr *. ). )). 2. 2. ). 2. (4.14). Both Λ and PSFW are functions of Φ , so it is difficult to maximize (4.14) over Φ . 33. ).

(41) Two approximations, (4.15) and (4.16), are needed to derived the ML estimate of I/Q imbalance matrix Φ in a close form.. Λ = ΦEΦ*E is a diagonal matrix and ΦΦ H ≈ 0 N ×N. ⇒ ( I N − Λ )( I N − Λ ) H ≈ I N −1. ⇒ ⎡⎣( I N − Λ)( I N − Λ) H ⎤⎦ ≈ I N. W. ⎡( I N − Λ ) −1 ⎤ ⎣ ⎦. ⇒ PSFW. H. (IN − Λ). −1. (4.15). ≈ IN −1. SFh ⎡⎣ (SFh ) H W(SFh ) ⎤⎦ (SFh ) H W −1. ≈ SFh ⎡⎣(SFh ) H (SFh ) ⎤⎦ (SFh ) H PSF. ⇒ PSFW ≈ PSF ,. where PSF. −1. SFh ⎡⎣ (SFh ) H (SFh ) ⎤⎦ (SFh ) H. (4.16). Then, the likelihood function (4.14) is approximated as. Λ (Φ) ≈ − ( I N − PSF ) ( r − ΦEr*. ). 2. (4.17). The unknown I/Q imbalance matrix Φ is a diagonal matrix, and only its diagonal elements has to be estimated. Therefore, we rewrite the likelihood function in term of the unknown column vector ϕ , which is the diagonal of the matrix Φ . Λ (ϕ ). − ( I N − PSF ) ( r − ϕ where ϕ N ×1. [ϕ1. Er*. 2. ,. ϕ N ] , ϕ n = {Φ}nn and n = 1 ~ N. ϕ2. = − ( I N − PSF )( r − Vϕ. ). T. ). 2. ,. where VN × N. diag {Er* }. (4.18). We could get the ML estimate for I/Q imbalance vactor ϕ by maximizing Λ (ϕ ) as (4.19) indicates. 34.

(42) ϕˆ ML = ⎡ V H ( I N − PSF ) ⎣ ( P.S . I N − PSF. −1. − PSF ) V ⎤ V H ( I N − PSF ) ⎦ is a projection matrix ). H. (IN. H. (IN. − PSF ) r. −1. = ⎡⎣ V H ( I N − PSF ) V ⎤⎦ V H ( I N − PSF ) r. (4.19). However, the inverse matrix in (4.19) does not always exist, so we have to reduce the dimension of the unknown vector ϕ . The coefficients for I/Q compensation are different from each subcarrier, but the changes with frequency are slowly. If we consider ϕ as the parameter in frequency domain, the corresponding parameter in time domain ρ is with narrow spread. Hence, we represent (4.18) in terms of ρ and suppose that the dimension of ρ is Lρ . Lρ should be smaller than the dimension of ϕ and depend on the degree of difference between I/Q branches. The transformation from ϕ to ρ is by means of partial Fourier matrix Fρ , and the similar skill is used when we estimate channel response. The following are the detailed procedures: Λ (ϕ N ×1 ) ≈ Λ (ρ Lρ ×1 ). where. (F ). ϕ N ×1. = − ( I N − PSF ) ( r − VFρ ρ. ρ N ×L ρ. ). ρ Lρ ×1 (4.20). 2. Lρ − 1 L −1 ⎛ 2π ⋅ i ⋅ n ⎞ exp ⎜ − ≤n≤ ρ , ⎟ , 0 ≤ i ≤ N − 1, − N ⎠ 2 2 ⎝ N is the size of FFT,. where [Fρ ]i ,n. Lρ is estimated length of ρ, Lρ < N. ( P.S.. Fh is the part of Fourier matrix.). H H ⇒ ρˆ ML = ⎢⎡( VFρ ) ( I N − PSF ) ( I N − PSF ) ( VFρ )⎥⎤ ⎣ ⎦ ( P.S. I N − PSF is a projection matrix ). = ⎢⎡ ( VFρ ) ⎣. H. ( I N − PSF ) ( VFρ )⎦⎥⎤. −1. −1. ( VF ) ( I. H ⇒ ϕˆ ML = Fρ ⎡⎢( VFρ ) ( I N − PSF ) ( VFρ )⎤⎥ ⎣ ⎦. −1. H. ρ. H. ρ. ( VF ) ( I. N. − PSF ) r. ( VF ) ( I H. ρ. 35. N. N. − PSF ) r. − PSF ). H. (IN. − PSF ) r. (4.21) (4.22).

(43) Finally, the ML estimate of I/Q imbalance vector ϕ is obtained from (4.22), and we ˆ . can calculate the corresponding estimated I/Q imbalance matrix Φ ML. 36.

(44) Chapter 5 Simulation Results 5.1 PER Performance IEEE802.11n is the next generation specification of wireless local area network (WLAN) which is going to adopt the MIMO-OFDM technology. In this section, our simulation scenario is based on the IEEE802.11n proposal proposed by TGn Sync [4], and Table.5-1 contains some corresponding simulation parameters.. Parameters. Value. FFT Size. 64. GI Length. 16 ( 0.8μ s ). BW. 20MHz. Tx × Rx. 2×2. Channel Model. ChB (80 ns) [25]. DeMUD Method. MMSE [3]. Coder. Convolutional code. Decoder. Weighted soft decoder. Packet Size. 1000 Bytes. Synchronization. Perfect. Table.5-1 Simulation parameters of IEEE 802.11n proposal provided by TGn Sync. 37.

(45) PER. 10. Lh=8,16; No I/Q Imbalance. 0. 10. -1. 10. -2. BPSK, 1/2, Lh= 8 BPSK, 1/2, Lh=16 BPSK, 1/2, Perfect QPSK, 1/2, Lh= 8 QPSK, 1/2, Lh=16 QPSK, 1/2, Perfect QPSK, 3/4, Lh= 8 QPSK, 3/4, Lh=16 QPSK, 3/4, Perfect 16QAM, 1/2, Lh= 8 16QAM, 1/2, Lh=16 16QAM, 1/2, Perfect 16QAM, 3/4, Lh= 8 16QAM, 3/4, Lh=16 16QAM, 3/4, Perfect 64QAM, 2/3, Lh= 8 64QAM, 2/3, Lh=16 64QAM, 2/3, Perfect 64QAM, 3/4, Lh= 8 64QAM, 3/4, Lh=16 64QAM, 3/4, Perfect. 0. 5. 10. 15. 20 SNR (dB). 25. 30. 35. Fig.5-1 PER performance with perfect I/Q for different Lh In Fig.5-1, it is supposed that the receivers are with perfect I/Q, which means there is no I/Q mismatch problem. We observe the packet error rate (PER) performance of our channel ML estimates with different values of Lh for 7 modulation and coding (MCS) schemes. Cases of perfect channel state information (CSI) are represented by dash-dot lines. By inspecting Fig.5-1, there is only about 1dB performance loss in the cases of Lh = 16 (dotted lines) from these of perfect CSI. When we choose smaller value of Lh = 8 (solid lines), the performance is improved in low SNR case, such as BPSK and QPSK MCSs. The reason is that our ML channel estimator with smaller Lh could remove more noise components than the one with larger Lh . However, the performances of Lh = 8 degrade seriously in high SNR case 38. 40.

(46) because more channel components are eliminated by the channel estimator with smaller Lh . It is a tradeoff between the degree of channel distortion and the ability to resist noise. The suitable Lh depends on SNR of environment and length of effective channel delay spread. It is a reasonable to set the value of Lh as the length of GI.. 39.

(47) PER. Lh=16, g=1.1, θ=5°. 10. 0. 10. -1. BPSK, 1/2, Imbalanced IQ BPSK, 1/2, Perfect IQ QPSK, 1/2, Imbalanced IQ QPSK, 1/2, Perfect IQ QPSK, 3/4, Imbalanced IQ QPSK, 3/4, Perfect IQ 16QAM, 1/2, Imbalanced IQ 16QAM, 1/2, Perfect IQ 16QAM, 3/4, Imbalanced IQ 16QAM, 3/4, Perfect IQ 64QAM, 2/3, Imbalanced IQ 64QAM, 2/3, Perfect IQ 64QAM, 3/4, Imbalanced IQ 64QAM, 3/4, Perfect IQ. 10. -2. 0. 5. 10. 15. 20 SNR (dB). 25. 30. 35. 40. PER. Fig.5-2 Amplitude imbalance g = 1.1 and phase imbalance θ = 5° 10. 0. 10. -1. Lh=16, g=1dB, θ=8°. BPSK, 1/2, Imbalanced IQ BPSK, 1/2, Perfect IQ QPSK, 1/2, Imbalanced IQ QPSK, 1/2, Perfect IQ QPSK, 3/4, Imbalanced IQ QPSK, 3/4, Perfect IQ 16QAM, 1/2, Imbalanced IQ 16QAM, 1/2, Perfect IQ 16QAM, 3/4, Imbalanced IQ 16QAM, 3/4, Perfect IQ 64QAM, 2/3, Imbalanced IQ 64QAM, 2/3, Perfect IQ 64QAM, 3/4, Imbalanced IQ 64QAM, 3/4, Perfect IQ. 10. -2. 0. 5. 10. 15. 20 SNR (dB). 25. 30. 35. Fig.5-3 Amplitude imbalance g = 1dB and phase imbalance θ = 8° 40. 40.

(48) It is supposed that there exists frequency-independent I/Q imbalance with. {g , θ } = {1.1,. 5°} in Fig.5-2 and. {g , θ } = {1dB, 8°}. in Fig.5-3. After we utilize. our I/Q compensation scheme, the PER performances of imbalanced I/Q cases are almost the same with perfect I/Q cases for different MCSs. Therefore, it strongly proves that our compensation algorithm for I/Q imbalance works well.. 41.

(49) PER. 10. 0. 10. -1. Lh=8 and 16, g=1.1, θ=10°. BPSK, 1/2, Lh= 8 BPSK, 1/2, Lh=16 QPSK, 1/2, Lh= 8 QPSK, 1/2, Lh=16 QPSK, 3/4, Lh= 8 QPSK, 3/4, Lh=16 16QAM, 1/2, Lh= 8 16QAM, 1/2, Lh=16 16QAM, 3/4, Lh= 8 16QAM, 3/4, Lh=16 64QAM, 2/3, Lh= 8 64QAM, 2/3, Lh=16 64QAM, 3/4, Lh= 8 64QAM, 3/4, Lh=16. 10. -2. 0. 5. 10. 15. 20 SNR (dB). 25. 30. 35. 40. Fig.5-4 PER with frequency-independent I/Q imbalance for different Lh. Assuming that the frequency-independent I/Q imbalance with. {g , θ } = {1.1,. 5°}. exists in a DCR, we give the different values Lh in our ML estimates in Fig.5-4. The tradeoff between the channel distortion and the ability to resist noise can be inspected not only in Fig.5-1, but also in Fig.5-4.. 42.

(50) 5.2 MSE of ML Estimation In this section, we derived the mean square error (MSE) of ML estimates from computer simulations and equations (3.22)(3.26)(3.30). The multipath channel is modeled by exponential decay channel model [26]. The long training sequences of IEEE 802.11a system [27] are used to estimate channel and I/Q imbalance. It is supposed that channel delay spread TRMS equals 50ns and the I/Q imbalance parameters. {g , θ }. are {1.1, 5°} . We give three values of Lh ,. 16, 11, and 6, to observe the MSE of all ML estimates.. Lh=16; g=1.1; θ=5°; T RMS=25ns; CH Tap =6. 1. 10. MSE MSE MSE MSE MSE MSE. 0. 10. of of of of of of. IQφ (Simulation) IQφ (Analysis) Channel h (Simulation) Channel h (Analysis) Channel k (Simulation) Channel k (Analysis). -1. MSE. 10. -2. 10. -3. 10. -4. 10. 0. 5. 10. 15 SNR (dB). 20. Fig.5-5 (a) Channel Taps =6 and Lh = 16. 43. 25. 30.

(51) MSE. 10. Lh=11; g=1.1; θ=5°; TRMS=25ns; CH Tap =6. 1. 10. 0. 10. -1. 10. -2. 10. -3. 10. -4. MSE MSE MSE MSE MSE MSE. 0. 5. 10. 15 SNR (dB). of of of of of of. 20. IQ φ (Simulation) IQ φ (Analysis) Channel h (Simulation) Channel h (Analysis) Channel k (Simulation) Channel k (Analysis). 25. 30. Fig.5-5 (c) Channel Taps =6 and Lh = 11. MSE. 10. Lh=6; g=1.1; θ=5°; TRMS=25ns; CH Tap =6. 1. 10. 0. 10. -1. 10. -2. 10. -3. 10. -4. MSE MSE MSE MSE MSE MSE. 0. 5. 10. 15 SNR (dB). 20. of of of of of of. IQ φ (Simulation) IQ φ (Analysis) Channel h (Simulation) Channel h (Analysis) Channel k (Simulation) Channel k (Analysis). 25. Fig.5-5 (c) Channel Taps =6 and Lh = 6 Fig.5-5 MSE of ML estimation with different values of Lh. 44. 30.

(52) Fig.5-5 (a)(b)(c) all show that the computer simulation results coincide with our MSE analyses of ML estimates, (3.22), (3.26) and (3.30). However, there still exists slight mismatch between the simulation and analysis curves of ϕ in the low SNR location. The tiny divergence results from the following two approximations, r ≈ rd and p ≈ p d , which are made in the procedures of MSE analyses. If we concentrate on the magnitude of MSE in Fig.5-5 (a)(b)(c), it is obvious that the ML estimates with the least Lh = 6 possess the least MSE. The smaller Lh is, the more noise ML estimate can remove. The taps of exponential decay channel model with TRMS = 25ns equal to 6, so the ML estimates with Lh = 6 do not suffer from channel distortion.. 45.

(53) Chapter 6 Conclusions In this thesis, we propose a joint ML estimators of I/Q imbalance and channel for MIMO-OFDM systems, such as IEEE 802.11n [4][5], and the problems of I/Q mismatch compensation and channel estimation are considered at the same time. What is more, we acquire three advantages of our algorithms. First of all, we derived the compensation algorithms not only for frequency-independent but also for frequency-dependent I/Q imbalances. Secondly, in addition to SISO-OFDM systems, it is able to work in the MIMO-OFDM systems with tone-interleaved training sequences for MIMO channel estimation [4]. Thirdly, the property of high correlation between neighboring subcarriers is adopted when we both calculate the channel response and I/Q imbalance parameter of each subcarrier. For the future works, in order to develop a more complete and robust digital receiver, we may raise more potential issues on RF impairment, e.g. frequency offset and DC offset, which usually exist in the DCR architecture.. 46.

(54) References [1] R. V. Nee and R. Prasad, OFDM for Wireless Multimedia Communications, Artech House Publisher, 2000. [2] M. Engels, Wireless OFDM Systems: How to make them work, Kluwer Academic Publisher, 2002. [3] B. Vucetic and J. Yuan, Space-Time Coding, Wiley Publisher, 2003. [4] TGn Sync Group, IEEE P802.11 Wireless LAN - TGn Sync Proposal Technical. Specification, Proposal of IEEE802.11n, IEEE Document 802.11-04/889r4, January 2005. [5] WWiSE Group, IEEE P802.11 Wireless LAN - WWiSE Proposal: High. throughput extension to the 802.11 Standard, Proposal of IEEE802.11n, IEEE Document 802.11-04/886r6, January 2005. [6] A.A. Abidi; “Direct-conversion radio transceivers for digital communications,”. IEEE Journal Solid-State Circuits, Volume 30, Issue 12, Page(s):1399–1410, Dec. 1995. [7] B. Razavi, “Design considerations for direct-conversion receivers” IEEE. Transactions Circuits and Systems II: Analog and Digital Signal Processing, Volume 44, Issue 6, Page(s):428–435, June 1997. [8] B. Debaillie, B. Come, W. Eberle, S. Donnay, M. Engels, “Impact of front-end filters on bit error rate performances in WLAN-OFDM tansceivers,” IEEE Radio. and Wireless Conference, Page(s):193–196, Aug. 2001 [9] J. Liu, A. Bourdoux, J. Craninckx, B. Come, P. Wambacq, S. Donnay, A. Barel, “Impact of front-end effects on the performance of downlink OFDM-MIMO transmission,” IEEE Radio and Wireless Conference, Page(s):159–162, Sept. 2004. [10] B. Cutler, “Effects of physical layer impairments on OFDM systems,” RF Design 47.

(55) Magazine, May 2002. [11] M. Valkama, M. Renfors, V. Koivunen, “Compensation of frequency-selective I/Q imbalances in wideband receivers: models and algorithms,” IEEE 3rd Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Page(s):42-45, March 2001. [12] K.P. Pun, J.E. Franca, C. Azeredo-Leme, C.F. Chan,C.S. Choy, “Correction of frequency-dependent I/Q mismatches in quadrature receivers,” IEEE Electronics Letters, Volume 37, Issue 23, Page(s):1415–1417, Nov 2001. [13] X. Guanbin, S. Manyuan, L. Hui, “Frequency offset and I/Q imbalance compensation for direct-conversion receivers,” IEEE Transactions Wireless Communications, Volume 4, Issue 2, Page(s):673–680, March 2005. [14] T.M. Ylamurto, “Frequency domain IQ imbalance correction scheme for orthogonal frequency division multiplexing (OFDM) systems,” IEEE Wireless Communications and Networking, Volume 1, Page(s):20–25, March 2003. [15] A. Schuchert, R. Hasholzner, P. Antoine, “A novel IQ imbalance compensation scheme for the reception of OFDM signals,” IEEE Transactions Consumer Electronics, Volume 47, Issue 3, Page(s):313–318, Aug. 2001. [16] R.M. Rao, B. Daneshrad, “I/Q mismatch cancellation for MIMO-OFDM systems,” 15th IEEE International Symposium Personal, Indoor and Mobile Radio Communications (PIMRC), Volume 4, Page(s):2710-2714, Sept. 2004. [17] A. Tarighat, A.H. Sayed, “On the baseband compensation of IQ imbalances in OFDM systems,” IEEE International Acoustics, Speech, and Signal Processing (ICASSP '04) Conference, Volume 4, Page(s):iv-1021-4, May 2004. [18] P. Rykaczewski, J. Brakensiek, F.K. Jondral, “Decision directed methods of I/Q imbalance compensation in OFDM systems,” IEEE 60th Vehicular Technology Conference (VTC2004-Fall), Volume 1, Page(s):484-487, Sept. 2004. 48.