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## Material Information- Title:
- Co-channel interference suppression for OFDM systems
- Creator:
- Won, Jonghyun (
*Author, Primary*) - Publication Date:
- 2003
- Copyright Date:
- 2003
## Subjects- Subjects / Keywords:
- Antennas ( jstor )
Channel noise ( jstor ) Covariance ( jstor ) Matrices ( jstor ) Parametric models ( jstor ) Signals ( jstor ) Spacetime ( jstor ) Supernova remnants ( jstor ) Thermal noise ( jstor ) White noise ( jstor )
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- 9/9/1999
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- 53322787 ( OCLC )
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CO-CHANNEL INTERFERENCE SUPPRESSION FOR OFDM SYSTE7\iS By JONGHYUN WON A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003 Copyright 2003 by Jonghyun Won To my family. ACKNOWLEDGMENTS First and foremost, I thank my adviser, Dr. Erik G. Larsson, for having con- fidence in me and for allowing me to pursue this research. The work presented in this thesis would not be possible without his continuous guidance. I would like to thank my committee members, Dr. John M. Shea and Dr. Tan F. Wong, for taking the time to read and critique my thesis. I also thank the professors in Korea from Electrical Engineering Department at Dongguk University for helping me to study abroad at University of Florida. I thank my beloved parents for being there and believing in me for all the years. They have provided me the strength and inspiration to continue my studies at the University of Florida. I want to thank all of my lovely friends for ahv--,- motivating me to be a better person and for providing me with guidance and advice when I needed it. TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................ iv LIST OF FIGURES ................... ........ vii ABSTRACT ...................... ........... ix CHAPTER 1 INTRODUCTION ............................. 1 2 WIRELESS COMMUNICATIONS OVER FADING CHANNELS ... 3 2.1 Error Probability for Communication System ......... .. 3 2.2 Antenna Diversity ........... .............. 5 2.3 Interference . . . . . . . 7 3 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING (OFDM) 8 3.1 Data Transmission and Detection in OFDM ...... ... .. 9 3.2 ML Detection for OFDM with White Noise ............ 11 3.3 ML Detection for OFDM with Colored Noise ........... 12 4 NEW ALGORITHM FOR INTERFERENCE SUPPRESSION IN O FD M . ... . . . ..... . . 16 4.1 C('!t i ,, I Estimation ................... .... 16 4.2 Estimation of R(n) with Piecewise Constant Model ...... 18 4.3 Automatic Selection of Model Order ....... ... .... 20 4.4 Numerical Examples for the Piecewise Constant Model ..... 22 4.5 Estimation of R(n) with Moving Average Process ....... 26 4.6 Auto Selection of Model Order ..... . . ..... 28 4.7 Numerical Examples for Moving Average Process . ... 28 5 INTRODUCTION TO SPACE TIME-OFDM (ST-OFDM) . 33 5.1 MIMO Systems and Space-Time Block Coding . ... 33 5.2 Space-Time OFDM .................. .... .. 35 6 APPLICATION OF OUR ALGORITHM TO ST-OFDM ...... ..38 6.1 C'!i i !,, I Estim ation .................. ..... .. 38 6.2 Estimation of R(n) with Moving Average Process ...... .. 40 6.3 Numerical Examples for the Moving Average Process ..... 41 7 CONCLUSION AND FUTURE WORK . . .. 45 REFERENCES ................... ..... ........ 47 BIOGRAPHICAL SKETCH .................. ........ .. 49 LIST OF FIGURES Figure page 3-1 Data transmission and reception scheme for an OFDM system 9 4-1 Example of the piecewise constant model for a SISO system . 19 4-2 Symbol-error-rate (SER) for the piecewise constant model when there is only thermal noise present. ............... ..23 4-3 Symbol-error-rate (SER) for the piecewise constant model when there are one strong interference signal (C/I 10dB) and thermal noise present. .................. .. ...... 24 4-4 Autonomous selection of model order, SNR = 30dB, for piecewise- constant model. .................. ..... 25 4-5 Comparison between detection (using piecewise constant model and coherent detection using perfect knowledge of the channel and the noise covariance). ............... .. ..26 4-6 Symbol-error-rate (SER) for the moving average process model when there is only thermal noise present. ............. ..29 4-7 Symbol-error-rate (SER) for the moving average process model when there are one strong interferer (C/I=10dB) and thermal noise present. .................. ...... 30 4-8 Autonomous selection of model order, SNR = 30dB, for moving- average process. .................. ..... 31 4-9 The comparison between detection using moving average process and coherent detection using perfect knowledge of the channel and the noise covariance. ................ ..... 32 6-1 Symbol-error-rate (SER) for the moving average process model when there is only thermal noise is present. . ..... 42 6-2 Symbol-error-rate (SER) for the moving average process model when strong interferer comes from a single antenna user. . 43 6-3 Symbol-error-rate (SER) for the moving average process model when strong interferers comes from co-channel user's multiple antennas. .................. ............ .. 44 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science CO-CHANNEL INTERFERENCE SUPPRESSION FOR OFDM SYSTE7iS By Jonghyun Won August 2003 C('! In: Erik G. Larsson Major Department: Electrical and Computer Engineering We propose new methods for co-channel interference suppression in orthog- onal frequency division multiplexing (OFDM) systems with multiple receive antennas. Our methods use a low-order time-domain model to parameterize the matrix-valued spectral density of the noise and interference. A penalized maximum likelihood-function approach is used to choose the proper model order automatically in each received data frame. We show that these methods signif- icantly improve the symbol-error-rate when interference exists in the received data. Numerical examples are provided to illustrate the performance of our new methods and compare with conventional methods. We also discuss the difficulties associated with the application of our method to space-time OFDM (ST-OFDM). CHAPTER 1 INTRODUCTION Orthogonal frequency division multiplexing (OFDM) has become one of the most promising transmission techniques for wireless local area networks because of its low equalization complexity in frequency selective fading channels, and its robustness to inter-symbol interference (ISI). OFDM transforms a frequ'-i-Tv- selective fading channel into multiple independent flat fading subchannels. By transmitting one symbol in each subchannel, the symbol duration is increased; hence the effects of ISI caused by a frequency selective fading environment are reduced. Therefore, an OFDM system can achieve a high data rate and a reliable transmission in a fading channel. However, due to limited and congested frequency bands in cellular radio environments, the system performance is often limited by interference. Interference is one of the major limiting factors on the performance of wireless communication systems, as it typically increases the bit-error-rate (BER). There are two 1i i, ri types of interference: so-called .I.1i i:ent channel interference (ACI) and co-channel interference (CCI). ACI comes from signals that are transmitted by a communication system using a frequency band .,I.i ient to the desired user's frequency band in the same area. This interference can be minimized by using a strictly restricted signal bandwidth or a proper channel assignment. Guey et al. [1] proposes ACI rejection methods for land-mobile radio systems. CCI is caused by the frequency reuse scheme in wireless networks, for example, when signals come from a co-channel cell which uses the same frequency band as the desired user. As CCI consists of the same frequencies, it cannot be rejected by simply increasing the transmit power or using narrow channel filtering. There are many statistical CCI interference suppression methods for radio communication networks. For example, Jongren et al. [2] proposed an interference rejection method using low-rank features of the spatial covariance matrix; Karlsson and Heinegard [3] -,-.- -1. .1 an "interference rejection" method for the GSM system. These techniques have been quite successful (see Dam et al. [4]). In this thesis, we propose new CCI suppression methods for OFDM and space-time OFDM (ST-OFDM) with multiple receive antennas. The basic idea of our algorithms is that the (matrix-valued) power spectrum of the interference is smooth, and hence we can use a low-order time-domain model to parameterize the spectral density of the noise and interference. One of our methods proposes a simple model for the covariance of the interference and noise. This model assumes that pairs of M .,I.i i'ent subchannels have the same spectral density. Thus, the performance of our model depends on how we select the number of subchannels, M. A penalized maximum-likelihood function approach is used to choose M automatically in each received data frame. Another method uses a multidimensional moving average (I! A) process to model the power spectrum density of the interference and noise with a variable number of parameters, M. Also here, a penalized maximum-Likelihood function approach is used to select the proper number of parameters. We also apply these algorithms in the ST-OFDM case and show numerical examples. CHAPTER 2 WIRELESS COMMUNICATIONS OVER FADING CHANNELS In a wireless environment, transmitted signals usually decrease in power or become distorted in shape because of the time-variation of the channel and the propagation environment. Communication in a moving car can be one example of a fading channel, since the surrounding environment of the car varies so that the propagation channel changes according to the velocity of the car. 2.1 Error Probability for Communication System Let us consider a simple example of a wireless communication system to see the effect of fading. We assume, for simplicity, that the propagation channel is linear, time-invariant, and frequency flat; also assume that the system consists of a single transmit and receive antenna. Such a system is called a single-input single-output (SISO) system. Consider a transmitted symbol s from a finite constellation S. Then the received signal can be expressed as y=h s+e (2.1) where h denotes the channel gain and e denotes zero-mean complex Gaussian noise with variance a2. The probability of density function (p.d.f) for this linear model is p(y)= 2 exp hs 2 (2.2) We can detect s using the maximum likelihood rule, which maximizes (2.2) and is equivalent to s = argmin ly hs 2 (2.3) sES From Equation (2.3), we can derive the detection error probability for the system. Suppose we detect a wrong symbol So from a finite constellation S instead of the true symbol s. Then the pairwise probability for an incorrect decision is derived as P(s slh) =P(|y hs,2 < y hs 2) P(ly|2 2Re(y*hs,) + |hs,12 < y12 2Re(y*hs) + Ihs 2) (2.4) P(\h 2(Is2 2 _- 2) 2 Re{y*h(s, s)} < 0) where (.)* denotes the complex conjugate. We divide both sides by ih 2 and insert y = hs + e into the equation (2.4). Then we get P(s sh) =P( ,2 + 2 2 -2Re{s*s + 2Re {(*(s )} < ) =P(|s S2 + 2Re {()*( s) < 0) (2.5) _Q 20_ 2 The Q-function in the last step is a consequence of the fact that 2Re{(e/h)*(s - So)} is a Gaussian random variable with zero mean and variance 2s So 2 2/lh12 The above result is the error probability for a constant channel gain. If the channel suffers from Rayleigh fading, h can be modeled as a complex Gaussian random variable with zero-mean and variance p2. We can compute an average error probability for the fading channel by using the C'I, i ir.!.1 bound. Due to the Rayleigh distribution assumption, the probability density function of the channel h can be expressed as p(| 2) exp ( 12 (2.6) p Ipli Averaging (2.5) over h using (2.6), we get Eh[P(s -+ slh)] = P(s sh)P(ht2dh 2 < exp(- S s- 2 2h1 exp(- I)dlh12 (2.7) (s- s2 52 1 02)12 2-1 4 2 42 2 The signal-to-noise (SNR) for the received signal is p2/a2, and the last step of (2.7) shows that the average error probability for the fading channel behaves as P(error) o SNR-1 (2.8) We can compare this result to that of a constant channel. If we use the fact that the SNR equals p2/o2 and the ('I, i i, ll' bound in (2.5), then the error probability for a deterministic channel behaves as P(error) o exp(-SNR) (2.9) 2.2 Antenna Diversity Since the fading is Rayleigh distributed and time-variant, sometimes the channel can be useless. One of the most popular methods for combating the fading effect is antenna diversity. By employing multiple antennas on the transmitter or the receiver side, the communication system can transmit the same information over multiple channels, which are independent of each other. Such a system is called a multiple-input multiple-output (\MI \ O) system. In a MIMO system, the fading effect can be significantly mitigated and also the channel capacity can be increased as another advantage. Equipping multiple antennas at the base station, which gives receive diversity for the uplink, is done in many communication systems. Considering n, receive antennas instead of one antenna in a SISO system, we can see how receive antenna diversity counteracts the fading effect. In this case, the channel h becomes an 1 x n, vector and thus, we let h = [hi .. h,]T be the propagation channel vector in which each value corresponds to the channel gain between the transmit antenna and each receive antenna. Here (.)T denotes the transpose. Then, with the same assumptions as for the SISO system example, the received data can be written as y h s + e (2.10) where y = [ylr y,J]T is an received data vector and e = [e ... en]T is a complex Gaussian noise vector with zero mean and covariance matrix a2j ,. The pairwise error probability, obtained by a similar calculation as Section 2.1, is P(s-I sh) 2 Q o2 ) (2.11) If we assume that the n, channels are independent and complex Gaussian random variables with zero mean and variance p2, respectively, the probability density function for h can be expressed as P(h) f J 2exp(- 2 (2.12) n-1 Pn n92 Again, using the ('!. i 11l' bound, (2.11) and (2.12), we can derive a bound on the average error probability for MIMO fading channels as follows Eh[P(s solh)] f= J J 1 P(s soh)P(h)dlhl12 ... dlh 2 | 2 0_ 2 01 | | 2 |s s, 2|? 2\ \h, 2 2 (2.13) < exp ( IS So121h, 1 1 t ( hi12 \j\ 2.. 422 2 1( 1 2 2 1) r 1 it - S4 +t 4 ni1 ni1 The last part of Equation (2.13) shows that the more antennas there are on the receiver side, the more mitigation of fading effect is achieved. Also, it shows that the average probability of a detection error over the fading channel behaves as -(diversity order) P(error) oc SNR =- (2.14) Also, a diversity effect as well as other benefits can be obtained by equipping multiple antennas at transmitter side (this is called transmit diversity). Accord- ing to recent research, it is possible to reproduce the diversity from multiple transmit antennas while preserving the transmission data rate and a simple decoding complexity. This method, which is called Space-Time Coding, will be introduced in ('! Ilpter 5. 2.3 Interference Recently, the population of mobile users has dramatically increased due to the fast development of wireless communication technologies; hence, using carefully chosen channel reuse schemes has become more important in cellular communication systems. Because of channel reuse, interference between signals from the same frequency band is often generated. This interference, which is called co-channel interference (CCI), is the one of the n ii' ,r limiting factors for the performance of wireless communication systems. Unlike thermal noise which can be overcome by increasing the transmit power, CCI cannot be overpowered by increasing the carrier power. Moreover, an increase in carrier transmit power can augment the interference. However, the effect of interference can often be reduced by suitable signal processing method at receiver side. CHAPTER 3 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING (OFDM) As wireless communications have experienced great success and are widely used everywhere, the demands for good quality and variety of service have increased. To fulfill these desires, development for high speed wireless commu- nication systems has been investigated. One successful approach to achieving reliable and efficient high speed data transmission scheme is OFDM. The main idea of OFDM is that by dividing the entire channel bandwidth into independent multiple narrow bands, each sub-band becomes relatively flat. We can compare OFDM with a conventional single carrier system. A con- ventional system transmits each data symbol one by one in a whole occupied channel. One of the disadvantages using a single carrier is that if the channel suf- fers from frequency selective f ,lii- the channel equalization usually is complex. On the other hand, in an OFDM system, each subchannel occupies only a small fraction of the total bandwidth, and OFDM transmits one data symbol in each subchannel in parallel. Since a small bandwidth of each subchannel makes the symbol durations much longer than the memory of the channel, OFDM can use simple channel equalization and also obtain the same data rate as a conventional system. This property allows OFDM to support high speed rates. A detailed explanation of OFDM can be found in Larsson and Stoica [5], Heiskala and Terry [6] and Li et al. [7]. Here, we introduce the fundamental principles of an OFDM system following Larsson and Stoica [5]. Transmit Pre (equal) I 0 Data NO-Npre-1 N0-1 I Receive Figure 3 1: Data transmission and reception scheme for an OFDMI) ---- (I: Larsson and Stoica [5], '. ,ter5) 3.1 Data Transmission and Detection in OFDM Consider a general OFDM system with a single transmit antenna and n, receive antennas. Suppose we have s(n) symbols for n = 0,..., N 1 to be transmitted. After taking inverse Fourier transform for the symbols, the encoded data can be expressed as 1N- x(n) =k)e, = 0,..., N 1 (3.1) k0 Prior to transmission, a cyclic prefix (CP) is added at the beginning of the sequence x(n). The CP is a copy of the last Npre symbols, where Npre > L and L is the channel delay spread. The CP makes the transmitted signal periodic with period N so that the linear convolution induced by the propagation channel is transformed into a circular convolution, which corresponds to multiplication of channel gains and symbols in the frequency domain. We can express the encoded data with a cyclic prefix as follows x(n) =x(n+N), n Np e... -1 (3.2) 10 Figure 3-1 illustrates the transmission scheme. From the encoded data x(n), we can observe that each symbol s(k) is carried on each subcarrier c which are orthogonal each other. The encoded data are transmitted through a channel, which has L d 1 iv taps, and after discarding CP, the received data vector from the multiple antennas can be expressed as linear convolution of the channel and the encoded data plus thermal noise and interference L y(n) = hkx(n k)+ e(n) k0 L N- (3.3) s(m)hk exp 7(n k)m + e(n) k= 0 m= 0 where hk denotes kth vector-valued channel tap. After taking the Fourier transform of the received data, we get the multiplication of each frequency flat fading transformed channel and data (see, e.g., Larsson and Stoica [5], ('! ipter 5) N-1 z () -= Y Y() e - N-1N-1 L N-1 ^jEEE8snrnk "C6 15e(l)eC 1=0 m=0 k=0 =0 1 +1 m in Sn)hke- (n m)l k e(l)e- (3.4) 1=0 m= k= 0 0 0 L N-1 L e(l)e=0 kc-O 0 1- 1=0 N-l (n)( -h n) +)e e(n) where we have defined the channel transfer function h(2n h Cke- (3.5) k 0 From Equation (3.4), we can detect the symbols s(n) using simple equalization. 3.2 ML Detection for OFDM with White Noise First, we need to consider the noise term before deriving the ML detector for OFDM. We assume for simplicity that the noise vectors e(n) are spatially white circular Gaussian random variables with zero mean and variance a2. This assumption is proper when only thermal noise exist. Let consider the noise covariance in Equation (3.4) as follows N-1 N-1 E[e(ienH(k)] = E -- e() e- i e(m)e- k HI 0o m o (3.6) N-1N-1 SlSE[e(l)eH(m)]e- e- 1=0 m=0 From the assumption on e(n), N-1 E[e(n)eH(k)] = Y E[e(l)eH(1)]- - l=0 a2I, n k (3.7) 0, n/k Equation (3.7) shows that the noise term in (3.4) is white Gaussian noise with covariance a21, and thus, z(n) is a sufficient statistic for the detection of s(n). Hence, ML detection for OFDM is equivalent to minimizing the following metric Iz(n) s(n) h(n) 12 (3.8) where L h(n)= hie--' (3.9) l=0 3.3 ML Detection for OFDM with Colored Noise A more general assumption on the noise e(n) is to ,-v it is circularly symmetric Gaussian with the following covariance function R, = E[e(n)eH(n 1)1 (3.10) where (.)H denotes the conjugate transpose. Then the power spectrum of e(n) is R(n) = Rie~ (3.11) 1=-oo Such colored noise can be used to model co-channel interference. 1 If N is reasonably large in (3.4), we show below that e(n) and e(m) are independent for n / m and hence the ML detection for an OFDM system is equivalent to minimizing the following metric mmin R-12(n)(z(n)- h()s(n)) (3.12) s(n) ES We now show that R(n) is the covariance of the noise term in Equation (3.4) and that e(n) is independent of e(m), n / m. First we consider the auto-covariance function of e(n) as follows N-1 N-1 H E[e(n)eH(n)] = E 1 e(k)e 1 e(k)e- kL O (3.~3) N-1 N-1(k k) Y Y E[e(k)eH(k)1e- k-) k=0 k=o Note that, due to the insertion of the prefix, a signal transmitted in an OFDM ---.---- is not statisti, : iy white even if the source signal is white. .* 'ct this effect in this thesis. Let k- k then N-1 E[e(n)eH (n1=i R Ie- k=0 l=k-N+1 / 0 l+N-1 N-lN- ( le-i2n i2 n) l=-N+1 k 0 l=1 k= ( )Ni (3.14) S(N + 1)Rle- + (N +)Re- l -N+I l1- N-1 N-1 l -(N-1) l=-(N-1) Under the mild assumption that the covariance sequence decays sufficiently rapidly, the last term in (3.14) goes to zero when N -0 oc. Hence N-1 E[e(n)eH() R Ifn (3.15) l=-(N-l) Hence, in an OFDM system, the auto-covariance of the noise term after taking the Fourier transform of the received data is the same as the power spectral density of the noise in the received data. Next, we calculate the cross-covariance of e(n) 1 N-1 N-I kH- E[e(n)eH(m)] = E 1 e(k)e-i k k e()k-ei (3.16) N-1N-1 Z E[e(k)eH()]ei V-(nk- k=0 k=o If we let k k = then Equation (3.16) can be expressed as follows N-1 k E[e(n)eH(m = RIC Rm-l) k 0 l=k-N+l 0 l+N-1 Ne l=-N+1 k=0 A(n-m) + N iN N-1 (n- m)kf-m B(n-m) Note that (n-m)k 0 N-1 k C k=0 and l-1 k-0 If we take the norm of (3.17), can be expressed as follows N-1 (n-m)k (n-m)k (3.19) k 1 the norm of first summation term, ||A(n m)}\, u I|A(n -m)|| 1 1=-N+1 1-1 |R111 le -- )k k-0 < 111^^ ^^^ ^^ ^ ^ (3.20) N 1NR1 -N + l--N+1 (3.17) (3.18) Also, IB(n m)| can be expressed as follows with (3.19) N-1 N-1 IIB(n n)ll NR e (n m) l=1 k=l N-1 l-1 l11 kO0 1 R N-I 1-1I (3.21 l=1 k=-O < Y 111111 -Nv l -N+I Under the mild assumption that the covariance sequence decays sufficiently rapidly, when N -- oc, (3.20) and (3.21) go to zero. As a consequence, the Fourier transformed noise is temporally white and has covariance R(n). CHAPTER 4 NEW ALGORITHM FOR INTERFERENCE SUPPRESSION IN OFDM When there is strong interference in the received signals, it can be beneficial to suppress the interference, so that correct signal decisions can be made. One approach to estimate the covariance of the interference is to instantly estimate the covariance of each subchannel, but normally such instantaneous covariance estimates do not work well if there is only a short training sequence. One approach to cancel the co-channel interference is -, i-.- -1. 1 in Li and Sollenberger [8], which uses a minimum mean-square error (\1\ SE) diversity combiner (DC). They estimate the channels and the noise covariance in the frequency domain, and use these estimated values to suppress the interference. In this chapter, we introduce novel methods for interference suppression in an OFDM system. These algorithms are based on a low-order time domain model to parameterize the matrix-valued spectral density of the interference. 4.1 ('i! ,i ,, I Estimation Let us assume that we received more than one training sequence (T > 1). After taking the Fourier transform of the received training data, they can be expressed as L N-1 t(n) = h exp i ) s(n) + e()e (4.1) 1=0 k O for n = 0,... 1, t = 1,... T, where zt(n) stands for Fourier-transformed known data. We can estimate the channel from (4.1), assuming known training data, using various methods. The ]' i-ii, ilr: principle [9], explains that among models describing data well, the model with the smallest number of parameters is a better choice than others. For this reason, it is more desirable to estimate time domain channels instead of frequency domain channels, because the former consist of n,(L + 1) parameters; however in the frequency domain, we would need to estimate nN parameters. ('!i ,ii!, I estimation can be done by the following procedure. Denote the transmitted and received training data before Fourier trans- formation by xt(n) and yt(n). Using Toeplitz matrices, the training data Xt, channel H, noise Et and received data sequence Yt can be expressed by Yt = XtH + Et (4.2) where xt(0) xt(-1) ... .. xt(-L) Xt() Xt(0) xt( L) (4.3) xt(N 1) xt(N 2) ... ... xt(N 1 L) SeL(0) y (0) H = Et= Yt (4.4) h e(N -1) y(N 1) The negative log likelihood function for (4.2) becomes, assuming the noise is white T min IYt- XtH| 2 (4.5) t=1 and the ML estimate of channel H is determined by T T H (= XHX,) XHYY (4.6) t=1 t=1 After estimating the channel H, the frequency domain h(n) channels can be easily calculated by using Fourier transformation L h(n) hle-i2WNnl (4.7) l=0 4.2 Estimation of R(n) with Piecewise Constant Model The interference signal in e(n) consists of the same frequencies as desired signal, and it has propagated over a channel with similar properties as the channel h(n). Hence, we can expect that the power spectrum of e(n) is smooth and this characteristics can be used to estimate R(n). Following this reasoning, we can expect that the covariance in a portion of the frequency band consist of approximately the same values. The piecewise constant model assumes that the instant covariance R(n) of M .il1i i:ent subchannels consists of the same value or approximately the same value. With this model, we partition the frequency band in N/M segments, calculate R(n) for each subchannel, take an average of R(n) values in each segment which consists of M .,Ii] ient subchannels and apply each averaged value R(n) to M subchannels in each segment instead of R(n). More detailed explanations are as follows Using an estimated channel H from (4.6), we calculate residuals from received training data Yt = (Yt XtH)T (4.8) where Yt consists of Y = t(0) .. yt(N l) (4.9) 2.5 2- 1.5- I I\ I / I 0.5- / " 10 20 30 40 50 60 Figure 4 1: of the constant model for a SISO .-em. We take the Fourier transform of yt(n) as follows N-1 rt(n) =- yt(k)e- (4.10) k-0 We next estimate R(n) for each subchannel by ST R(n) j_ rt(n)rt"(f) (4.11) t= 1 and then, take the average in each segment i = 0,... N/M 1, according to the parameter M M-1 RM(t) =M R(Mi + m) (4.12) m=o0 We use RM(i) for segment i to suppress the interference instead of the instan- taneous moment based estimate R(n). Figure 4-1 shows an example of the piecewise constant model for a SISO system. The dash-dotted line is the true covariance of the interference and the thermal noise, and the solid line is the es- timated covariance using our method. Hence, according to (3.12), the maximum likelihood detection for information data is expressed as -1/2 2 min M (i)(z(Mi + m) h(Mi + m)s(Mi + m)) (4.13) s(Mi+m)ES for i 0,...,N/M- 1 and m 0,...,M- 1. 4.3 Automatic Selection of Model Order To get the best performance, we need to choose the model order, the parameter M, autonomously in each sequence of data. As a sub-optimal way, we use a maximum-likelihood (\! I) approach for selecting M. After estimating the covariance for different M, we insert those values into the negative log- likelihood function with training symbols, compare and select the minimum one as the proper model order for the information data sequence. It is desirable to choose the model, that explains the data (interference and noise) well and has a parametrization as simple as possible. But ML .i.- li- chooses the highest model order, when there is interference in the training sequence. To counterbalance this tendency, we add a penalty term inspired by the minimum description length (\!I)L) method (more detailed discussion can be found in Kay [10], Section 6.8) to the ML rule MDL(M) InP(Yt;RM H) + N (4.4) 2 where nM is the number of estimated parameters in RM. Thus, this method finds the model order M which minimizes T N/M-1 M-1 mn {m Y R 2(i)(zt(Mi + m) (Mi+ m)st(M )) t=1 i=0 m=0 (4.15) N/M-1T M n + TM- log RM(i + 2 In i=0 where st(n) is the known received data. Note that RM(i) is the average value of the moment based covariance matrices of M .il.i i:ent subchannels and zt(Mi + m) h(Mi + m)st(Mi + m) = rt(Mi + m) (4.16) Note that the norm square term of equation (4.15) can be written T N/M-1 M-1 S R-1 2(i)(z,(Mi + m) h(Mi + m)st (Mi + mn)) t=l i=0 m=0 T N/M-1 M-1 -1/2 -1H/2 E E M Tr{RM/(i)t(Mi + m) (Mi + m)R )} t=l i=0 m=0 T N/M-1M-1 E E Tr{RM 2()R /(i)rt(Mli+m)rHMi+m)} t=l i=0 m=0 N/M-1M-1 T = E Tr{RMI(i)YRt(Mi+ m)}H(M + )} i=O m=O t=l (4.17) N/M-1 M- T-1 T Y yTr{RM ()R(Mi +nnm)} i=O m=0 NIM- 1 T Tr{RM (n)(R(Mk)+R(Mk+ 1)+..-+R(Mk + M- ))} i=0 N/M-1 NIM-1 T Y M- T{RM(i)RM() i=0 = TN Tr{I} where Tr{-} stands for the trace of a matrix. Following this result, Equation (4.15) can be rewritten in the more simplified form N/M-1 minM log RM(i) + n nN (4.18) Mr 2 i=0 We use this simplified method with training sequences to select the proper value of M and apply the chosen model to suppress the interference in the data sequences. 4.4 Numerical Examples for the Piecewise Constant Model In this section, we show some simulation examples for our new algorithm for an OFDM system. In our experiment, N = 64 subcarriers, 1 training sequence and n, = 2 receive antennas are used. The channel consists of L + 1 = 5 independent Rayleigh fading channel taps, and has an exponentially decaying power delay profile: E[l 112] O2 1.5-, 0, ... L. The training sequence is an impulse to estimate channel, and the constellation size for our simulation is QPSK. Figure 4-2 shows the estimated Symbol-error-rate (SER) versus the Carrier- to-Noise ratio (C/N) when the received signals are not affected by interference, and when only thermal noise is present. We present the result for the cases when M = 2, 4, 8, and 64, and with the autonomous selection. we also compare the result to coherent detection and conventional detection ( In this case, we use noise covariance R(n) = I for all n. ) The non-coherent detection and our method with M = 64 show almost same result. The reason for this is that setting M = 64 implies that the noise spectrum is independent of n. With the other values of M, we see a performance loss relative to conventional detection. Our auto-selection method chooses M = 64 model order for each sequence as we expected. U) I I a I . Est. channel I \. 10 -15 F) s E Coherent \v 0 -- Est. channel (Conventional) + ,.o,. Est. channel and noise cov. M=64 e da a p* a. Est. channel and noise cov. M=8 ". . f0-3. s oo Est. channel and noise cov. M=4 ad .. +. Est. channel and noise cov. M=2 \ 0. .. x Est. channel and noise cov. auto ' 0 5 10 15 20 Signal-to-noise ratio (SNR) [dB] Figure 4-2: 'mbol-error-rat.e (SER) for the piece@wise constant model when there is ,'.--! thermal noise present. In Figure 4-3, we present the case when there is one strong interferer present. The co-channel user has the same channel delay profile and power delay profile as the desired user has, and the signal to interference ratio is 10dB. This figure shows poor performances of coherent detection and conventional detection without estimation of interference, but our method successfully mitigates the co-channel interference. For high SNR, the M = 4 model order outperforms the other models, but in the region of less than 20dB, the M = 8 model order shows somewhat better performance than M = 4. For each SNR, our auto-selection 24 10-1 lo~r----------------------- 10 I "o I S\.^ a) E. ....as. .. . 0 S Coherent .o.. Est. channel and noise cov. M=64 .3 Est. channel and noise cov. M=8. .. Est. channel and noise cov. M=4 i "'.... 10 15 20 25 30 35 Signal -to-nnoise ratio (SNR) dB] Figure 4-3: bol-error-rat e (SER) anr the piecewise constant model when there are one strong interference signal (Cl/I 0dB) and thleral noise present. method does not al-- t. choose the best model, which has the lowest symbol- error-rate. However, the auto-selection method significantly outperfrms the coherent method without estimating the covariance of interference. Figure 4-4 shows how our autonomous selection method performs when the signal-to-noise ratio is 30dB. Our model a-- chooses M 64 when only thermal noise is present as we expect. When one strong interferer and thermal noise are present, our method usually selects M = 8 and M = 4. Noise only when SNR = 30dB 100 80- 60 40 20- 0 II I 24 8 64 interference with no!e when SNR = 30dB 100 80- 60 40 20- 24 8 64 2 4 8 64 Figure 4-4: Autonomous selection of model order, SNR constant model. 'dB, .: piecewise- We summarize the comparison between detection using our method and coherent detection ( using perfect knowledge of the channel and the noise covariance ) in Figure 4-5. In the case when the received signals are only affected by thermal noise, Figure 4-5 indicates that conventional detection and our new method have the same performance. In the case when received signals are affected by strong interference, our method shows a significant improvement compared to conventional detection. I I I a. , ',;.. -e- ,. .. , 0. * * Coherent (C/I=-) Conventional (C/I=o) New method (C/I=o) Coherent (C/l=10 dB) Conventional (C/1=10 dB) New method (C/I=10 dB) 0... 01 1.. 0 ...... 0 .0...0. 0 x ********o ..., ..,,*.,*o w 0_2 L0 I 0 I- E >S U) 10-3 15 20 25 30 35 Carrier-to-noise ratio (C/N) [dB] Figure 4-5: Comparison betw een detection (using pieceNwise constant model and coherent detection using ...r.. knowledge of the channel and the noise covariance). 4.5 Estimation of R(n) with Moving Average Process In this section, we introduce an alternative new method for interference suppression. This method uses a low-order time-domain model to parameterize the matrix-valued spectral density of the noise and interference. We still use the channel estimation technique described in Section 4.1. Then, we calculate residuals from the training sequence using estimates of the channel, H, in (4.6) 10 'X, as follows Yt (Yt- XtH)T (4.19) From these residuals, we estimate RI using the biased auto-covariance estimates in ( see, e.g., Soderstrom and Stoica [9] and Stoica and Moses [11]) T S NT y+ (YL )H (4.20) t=1 for = 0,... where Y+I is the matrix Y with the first I columns removed, and Y-f is the equal to Y with the last I columns removed. We next estimate R(n) via a truncated weighted Fourier sum as follows I M R(n) 1-m )Re- (.21) m= -M H Because the covariance function is conjugate symmetric we can set R-m Rm .Truncating the sum amounts to assuming that the power spectrum of e(n) is smooth, which is probably a reasonable assumption in practice. To use R for detection in (3.12), it is necessary that R is positive definite for n = 0,..., N 1. Fortunately, this is the case. To see why this is so, note first that T 1 H R(n) fYtEX()Yf (4.22) t=1 where E(n) is an N x N matrix whose (k,/)th element is equal to Ek,l(n) (1 Ik 1//(M + 1))e-i2(k-1)n/N when Ik I < M and zero otherwise. But (1 Ik l/(M + 1)) e-"" is the covariance function of a moving average process with coefficients {1, e- ,..., e-i(M-1)} and hence E(n) is positive semi-definite. Because of the noise in Yt, R(n) is positive definite with probability one. 4.6 Auto Selection of Model Order Also for the method using a moving average process, it is necessary to find the proper model order M. As for the piecewise constant model, we can apply penalized ML to obtain an autonomous selection method. But, unfortunately, this does not appear to work that well for the moving average process model. We do not know the exact reason for why this does not work for the moving average process model (in our simulation, the value of the ML metric is much 'i. -.--r than that of the penalty term): but it is probably related to the fact that the estimates R(n) and h(n) we use are not maximum likelihood. Instead, we have used the detected data sequence to determine the model order. For example, if we transmit one training sequence and five data sequences, we estimate the channel and covariance using the training sequence and then detect first data sequence. Then, we use the detected data sequence to select model order, and the chosen model order is used to detect the other data sequences. Thus, our autonomous selection can be expressed as follows N-1 min y log |(n)|+ (n) (n) h(n)s(n) + log N (4.23) n=0 where s(n) is the detected data. 4.7 Numerical Examples for Moving Average Process In this section, we show some simulation examples using the new algorithm with the moving average process. This experiment was done under the same conditions as described in Section 4.4. Figure 4-6 shows the estimated Symbol-error-rate (SER) versus the Carrier- to-Noise ratio (C/N) when only thermal noise is present. We demonstrate the results for the model orders M = 0, 2, 4, 6, 8 and for autonomous selection among these models (in contrast to the piecewise constant method, increasing the model 29 1 0 -.. 10 -. L o___________ o .- Coherent >, ** Est. channel and noise cov. M=2 *-- Est. channel and noise cov. M=4 10-3 o n+ Est. channel and noise cov. M=6 S-o- Est. channel and noise cov. M=8 ... Auto 0 5 10 15 20 Carrier-to-noise ratio (C/N) [dB] Figure 4-6: 'mbol-error-rate (SER) for the moving average process model when there is (h ce.- thermal noise present. order here means increasing the number of parameters we estimate). Also, to compare our method with a conventional detection scheme, we show results for coherent detection and conventional detection. This figure shows that there is not much difference between conventional detection and the method using moving average process. Figure 4-7 shows the case when there is one strong interferer present ( C/I = 10dB ). From this figure, we see that our method successfully mitigates co-channel interference. Like the piecewise constant model, this moving average LU L. 103 -0 I- E >S 0) 10-3 S. .... . . .... [3 ........ E3 .... .. . + . ., "4 ........ ; ': , I ........ 0. .. i ' 10 15 20 25 30 35 Carrier-to-noise ratio (C/N) [dB] Figure 4-7: 1m-bol-error-rate (SER) for the moving average process model when there are one strong interfere (C/Il10dB) and thermal noise present. process model suppresses the interference more effectively by increasing the number of parameters. Autonomous selection does not ah v--i- choose the model order which shows the best SER. Figure 4-8 di pl- how our autonomous selection model performs when the signal-to-noise ratio is 30dB. Our method chooses M = 0 most of the time when only thermal noise is present. When one strong interferer and thermal noise are present, our model selects mostly M = 6 but also M = 8. 10-1 Coherent Est. channel (Conventional) Est. channel and noise cov. M=0 Est. channel and noise cov. M=2 Est. channel and noise cov. M=4 Est. channel and noise cov. M=6 Est. channel and noise cov. M=8 Auto .. ] . ..0-- 0 --. "O ' I I I I I _ \0 Noise only when SNR = 30dB 1u I I I I 80 60 40 20 0 I I I 0 2 4 6 8 interference with no!e when SNR = 30dB 601 1 I I I I I 0) M40 2 20 CF Figure 4-8: Autonomous selection of model order, SNR average process. l:JB, 0: : moving- Figure 4-9 shows a comparison between detection using our method and co- herent detection using perfect knowledge of the channel and the noise covariance. Note the similarity between Figure 4-9 and Figure 4-5. rI I -J-- b __ LI ___ I r' I ,, I I .x -. 'o. I I -, x | | Coherent (C/I=-) -e- Conventional (C/I=o) New method (C/I=-) ..... Coherent (C/I=10 dB) .o' Conventional (C/I=10dB) New method (C/I=10 dB) S.. .... o.... 0 ........ .. .... 0 ... x. 15 20 25 Carrier-to-noise ratio (C/N) [dB] T' F comparison between detection using moving average process and coherent detection using : knowledge t he channel and the noise covariance. 10- LU U) aO 10 >, CO 10-3 Figure 4 9: *" 'X....... x, CHAPTER 5 INTRODUCTION TO SPACE TIME-OFDM (ST-OFDM) In C'!i pter 2, we introduced receive diversity, which is a popular method to combat fading in wireless communications. By installing multiple antennas at receiver side (usually at the base station), we can mitigate channel fading as well as increase the channel capacity. In recent years, researchers have found that it is possible to obtain the same advantage as receive diversity by using multiple antennas at the transmitter side (this is called transmit diversity). Using multiple antennas at both sides, we can get much more benefits than with receive diversity or transmit diversity alone. However, to exploit these benefits, we need a coding scheme, which is called Space-Time Coding. In this chapter, we briefly introduce a MIMO system, space-time block coding (STBC) and space-time OFDM (ST-OFDM), which combines space-time coding and OFDM. 5.1 MIMO Systems and Space-Time Block Coding Consider a MIMO system which consists of nt transmit antennas and n, receive antennas. We assume that the propagation channels are linear, time- invariant, and frequency flat. Let X and Y be nt x N transmitted data and n, x N received data matrices, respectively. Then, the received data can be expressed in a linear model as follows Y HX+E (5.1) where H is an n, x nt channel matrix in which the (m, n)th element corresponds to the channel gain between the receive antenna m and the transmit antenna n; E is an n, x N noise matrix in which each element is independent and complex Gaussian with zero mean and variance a2. Then, given Y, ML detection of the code matrix X from a finite constellation X is equivalent to X =argmin mY HX 12 (5.2) XEX Like for the SISO system in Section 2.1, we can derive the detection error probability from Equation (5.2). By a similar calculation as in Section 2.1 P(X, |H(X xo) P(XX-+XH) =2a2 ) (5.3) Next, we assume, for simplicity, that all channels in H are independent and complex Gaussian with zero mean and variance p2; hence, the channel is Rayleigh fading. The probability density function for the Rayleigh fading channels can then be expressed as p(H) f= l2t exp ( H 2 (5.4) pnnt W 2 Using the C'!l ~i i l' bound, we get the average error probability as follows (a more detailed derivation can be found in [5, Section 4.3]) EH[P(X XoH)] j P(X -X XoH)p(H)dH p2 42(X Xo)(X Xo)" + I- (5.5) 4j2 < (X- Xo)(X- Xo)lH I (492) The above equation shows that the average error probability for a MIMO system behaves as P(error) oc SNR-"'" (5.6) which means that receive diversity of order n, and transmit diversity of order nt are obtained, as long as X Xo has full rank. Hence, the length of the transmission N has to be longer than the number of transmit antennas, nt. It is not easy to achieve transmit diversity without losing transmission data rate and increasing the decoding complexity. Space-time coding is a way to achieve high data rate and low decoding complexity. One of well known space- time coding schemes is the Alamouti code [12], which is a transmission scheme for two transmit antennas. Another case is the linear orthogonal space-time block code (OSTBC), which has a simple structure. Consider the transmission of n8 symbols {si, s s,}. Then, encoded linear STBC has following structure ns X = (snA + isnB,) (5.7) n=l where An and B, are fixed code matrices, and s, and s9 are the real and imaginary parts of the symbol s,, respectively. OSTBC has the special merit that the MIMO channel decouples into n8 independent scalar channels, because of the following orthogonality property of OSTBC XXH Sn 2 I (5.8) n=l A more detailed explanation for OSTBC can be found in Larsson and Stoica [5], C!i lpter 7, Tarokh et al. [13] and Larsson et al. [14]. 5.2 Space-Time OFDM ST-OFDM is a combination of space-time block coding and OFDM to obtain transmit diversity for an OFDM system. The main idea of ST-OFDM is identical to that of OFDM. Because a ST-OFDM system uses multiple transmit antennas, it sends No encoded nt x N matrices in which n, symbols are encoded. Hence, nNo symbols are transmitted during NNo symbol transmit time intervals. The transmission and detection procedures for ST-OFDM are as follows. First, from nsNo symbols to be transmitted, encode each group of n, symbols to No code matrices G(n) for n = 0, No 1. Second, take the inverse Fourier transform of the code matrices as follows X(n)= (G(k) exp ( nk n = 0,...,N 1 (5.9) Then, add the CP which is taken from Np,, last code matrices of the transmitted data matrices, and select the first columns of all code matrices to transmit them sequentially through channels, assuming that each channel has L dl 1 i taps. Next, select the next columns for the transmission until all the columns of the code matrices have been sent out. At the receiver, discard the CP. Then the received code matrices can be expressed as follows L Y(n) = HkX(n k) + E(n) k=o (5.10) 1 E N HkG(k'. -'" +E(n) k=-0 m=0 After taking Fourier transform of (5.10) and following similar derivation as in (3.4), the received data become Z(n) = 1 Y(l)e l =0 H( G() + E(511) 1 No- 1 H(n) G(n) + 1- E(1)e-- i 1=0 where H(n) >= Hke- (5.12) k-0 Hence, ML detection for ST-OFDM is equivalent to minimizing the following metric IIZ(n)- H(n)G(n) 12 (5.13) More about ST-OFDM can be found in Larsson and Stoica [5], ('!i ilter 8 and Li et al. [15]. CHAPTER 6 APPLICATION OF OUR ALGORITHM TO ST-OFDM In this chapter, we extend our interference suppression algorithm to the ST-OFDM case. 6.1 ('CI ,i,, I Estimation Because a ST-OFDM system uses multiple transmit antennas and transmits an nt x N inverse Fourier transformed code matrix X, the number of training sequences, which consist of No symbols each, is at least N. Let us assume that we have received NT training sequences (T > 1) and that each channel has L d.1 iv taps. Denote transmitted and received code matrices by Xt(n) and Yt(n) for n = 0, No 1, t = 1, T. Then, the received training data can be expressed as follows L Yt(n) = HiXt(n-1) + E(n), n 0,...,No 1 (6.1) l=0 where Hi stands for Ith matrix valued delay channel tap. By using a block- Toeplitz matrix, Equation (6.1) can be rewritten as [Y,() Yt(No-) (6.2) =Ho ... HL Xt + Et(0) ... Et(No- 1) .. ... Xt(N L- ) Xt(No L) Xt(-L) Xt(-L- ) ... X,(O) ... Xt(No- 1) vec(ABC) (CT A) vec(B) (6.4) where 0 denotes the Kronecker product and vec(-) is the vectorization of a matrix. By vectorizing both sides of (6.3), we get (6.5) where vec(Yt(0)) vec(Yt(No 1)) Maximum likelihood estimate of h following metric vec(Ho) vec(Et(0)) S et (6.6) vec(HL)_ vec(Et(N 1)) in (6.5) is equivalent to minimizing the minE Yt -(XT Il,)h 2 t=1 and the ML estimate of the channel h is determined by h =vec(Ho)T .. vec(H)T = T [((X,)(XtX,) )T t=1 where Xt(O) Xt(-1) Xt(1) Xt(O) Note that (6.3) (6.7) In,] Yt (6.8) Yt = (X & I,,)h + et Also, note that after estimating the channel h and reconstructing H, H(n) for the flat fading subchannels can be calculated by taking the Fourier transformation L H(n) = Hle-i2-F Nl (6.9) 0=o 6.2 Estimation of R(n) with Moving Average Process First, note that the received code matrix, Yt(n), consists of N number of n, x 1 column vectors. Vectorization of Yt(n) gives Yt (n) vec(Yt(n)) = yt(n) = (6.10) Ly(n) Next, we calculate the residuals from the vectorization of the received data using the estimated channel h and the block-Toeplitz matrix (6.3) as follows 9 t (XT & I,n)h (6.11) Following (6.10), we can express it as it = yT(0) ... y[(N 1) = y(0)) T -. ( ( )) T -. (N.- 1)) .. ( N(N 1)) (6.12) Then, we reconstruct the matrix Yt(n) as follows Yt(n) = ~r(0) ... YT( 1) = 0,..., N- 1 (6.13) Following the same procedures as (4.20), we estimate R, using the biased auto- covariance estimates T N -NNo E (k) (6.14) t= 1 k= for 1 0,... where Y+(k) is the matrix Y(k) with the first I columns removed, and Y -(k) is the equal to Y(k) with the last I columns removed. We next estimate R(n) via a truncated weighted Fourier sum as follows R(n)= (i M- l)me -, mn (6.15) m=-M = -- M Using R(n), the maximum likelihood detection rule for the information symbols is expressed as 2 argmin R(n)(Z(n) H(n)G(n)) (6.16) G To choose a proper model order, the autonomous selection method can be expressed as follows N-1 min {log |R()|+ R-1/2(Z() ()G()} + logN (6.17) n=0 where G(n) is the detected code matrix. 6.3 Numerical Examples for the Moving Average Process We show simulation results for our algorithm for an ST-OFDM system with N = 64 subcarriers, nt = 2 transmit and n, = 2 receive antennas. The Alamouti code matrix is used in our experiment, and the channel has the same d,1 iv taps and the same average power delay profile as in Sections 4.4 and 4.7. Two training sequences, which each sequence being an impulse, are used, and the constellation is QPSK. S; ,.+ Est. channel and noise cov. M=6 W -2 o0 Est. channel and noise cov. M=8 10 Auto 2 -3 10 - .0 E 0 5 10 15 20 Carrier-to-noise ratio (CN) [dB] Figure 6 -1: m" bol-error-rate (SER) for the moving average process model when there is -. thermal noise is present. Figure 6-1 shows that the Symbol-error-rate performance versus the Carrier- to-Noise ratio (C/N) for an ST-OFDM system when only thermal noise is present. This figure shows that transmit diversity effectively combats the channel fading compared to Figure 4-6. Figure 6-2 shows results for the case when one strong interferer comes from a single antenna user. CCI has the same channel taps and average power d. 1liv profile as the desired user has, and the signal to interference ratio is 10dB. The interference suppressing algorithm outperforms the one that does not 10 Est. channel and noise cov. M=U *o: Est. channel and noise cov. M=2 S-0.- Est. channel and noise cov. M=4 .. -+ Est. channel and noise cov. M=6 S-2 \ o-- Est. channel and noise cov. M=8 S10 ' U -2.. Est..hanne an noiex .coy. M, = ...... Auto 2 -3 10 SE I ",' K..... -4 CO 10 - 10-5 - 0 5 10 15 20 25 30 35 Carrier-to-noise ratio (C/N) [dB] Figure 6-2: "'-mbol-error-rate (SER) for the moving average process model when strong i -..:.... comes from a single antenna user. take interference into account. Also, the algorithm suppresses the interference more effectively by increasing the number of parameters we estimate. However, Figure 6-3 shows that when strong interferers comes from multiple transmit antennas, our method can not mitigate interference well. Also, in this case, increasing the model order does not have a significant effect on the interference suppression. 0 5 10 15 20 25 Carrier-to-noise ratio (C/N) [dB] 30 35 Symbol-error-rate (SEl) for the moving average process model when strong interferers comes from co-channel user's multiple antennas. Figure 6 3: CHAPTER 7 CONCLUSION AND FUTURE WORK Co-channel interference is a major limiting factor for cellular communication systems, and it is expected to become a major bottleneck also for wireless local area networks. In this thesis, we have presented a conceptually very simple and computationally cheap method for suppression of co-channel interference in systems that use OFDM modulation and which have multiple receive antennas. Our method comes in two versions: one that parameterizes the interference and noise spectrum via a piecewise-constant model, and one that uses a low-order moving-average model. In either case, a technique inspired by MDL is used to automatically select the model order, hence avoiding any user parameters to be selected prior to the application of the algorithm. Simulation results show that our method can substantially improve the performance compared to conventional training-based detection. Several problems remain which could be the topic of future studies. First, the model order selection strategy used is by no means optimal (optimal such selection methods in general do not exist, see Lanterman [16]) and it is possible that an alternative methods for choosing the model order may outperform ours. Second, the estimates of the channel and noise covariance that we use are not maximum-likelihood, but nevertheless they are computationally simple to obtain. It is likely that using maximum-likelihood estimates in lieu of our proposed estimates may lead to a method that works better than ours. The maximum- likelihood estimates generally do not exist in closed form, and obtaining them numerically is computationally burdensome. Yet, some low-complexity approx- imations to maximum-likelihood for the problem under study can be found in Jeremic et al. [17]; using the estimates proposed therein instead of our subopti- mal estimates might improve performance. Finally, as the interference signals are Gaussian only to within a certain degree of approximation, using a model that allows for non-Gaussian interference may also improve the error rate. We also studied the application of our algorithm in a system that uses space- time OFDM. This scenario is more difficult, because interference signals that originate from a user with multiple transmit antennas have a richer correlation structure than signal originating from a single-antenna transmitter. For space- time OFDM, our algorithm was able to suppress co-channel interference only to a small degree. It is possible that the use of a more accurate model, perhaps along with a larger number of receive antennas, may improve the situation. This topic is also left for future work. REFERENCES I1 J. C. Guey, A. K! ,lirallah, and G. E. Bottomley, "A, li ,. in channel interference rejection for land mobile radio systems," in Proc. of IEEE Vehicular T I,,.. i,/;/ Conference, Ottawa, Canada, vol. 3, pp. 1715-1719, 1998. 12 G. Jongren, D. Astily, and B. Ottersten, "Structured spatial interference rejection combining," in Proc. of the European S.:,.i1rl Processing Conference, Tampere, Finland, Sept. 2000. [31 J. Karlsson and J. Heinegard, I1i I i, i ,,. rejection combining for GSM," in Proc. of IEEE Conference on Universal Personal Communications, Cambridge, MA, vol. 1, pp. 433-437, 1996. 14] H. Dam, M. Berg, S. Andersson, R. Bormann, M. Fredrich, F. Ahrens, and T. Henss, "Performance evaluation of adaptive antenna base stations in a commercial GSM network," in Proc. of IEEE Vehicular T 1 ,. .' .i; Conference, Amsterdam, Netherlands, vol. 1, pp. 47-51, 1999. 15 E. G. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications, Cambridge University Press, Cambridge, UK, 2003. J. Heiskala and J. Terry, OFDM Wireless LANs: A Theoretical and Practical Guide, SAMS, Indianapolis, IN, 2001. 17 Y. Li, Jr. L. J. Cimini, and N. R. Sollenberger, "Robust channel estima- tion for OFDM systems with rapid dispersive fading channels," IEEE Transactions on Communications, vol. 46, pp. 902-915, July 1998. Y. Li and N. R. Sollenberger, "Adaptive antenna arrays for OFDM systems with cochannel interference," IEEE Transactions on Communications, vol. 47, pp. 217-229, 1999. T. Soderstrom and P. Stoica, System II. ,./.:'. /,,/.n, Prentice Hall Interna- tional, Hemel Hempstead, UK, 1989. [10] S. M. Kay, Fundamentals of Statistical S.:j,,,i Processing: Detection Th.., Prentice Hall, Englewood Cliffs, 1998. [11] P. Stoica and R. Moses, Introduction to Spectral A,., .l;;,:- Prentice Hall, Upper Saddle River, NJ, 1997. [12 S. M. Alamouti, "A simple transmit diversity technique for wireless communications," IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451-1458, Oct. 1998. 13] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Space-time block codes from orthogonal designs," IEEE Transactions on Information Th(. , vol. 45, no. 5, pp. 1456-1467, July 1999. [14] E. G. Larsson, P. Stoica, and J. Li, "Orthogonal space-time block codes: Maximum likelihood detection for unknown channels and unstructured interferences" IEEE Transactions on S.:,,,Il Processing, vol. 51, pp. 362-372, Feb. 2003. [15] Y. Li, N. Seshadri, and S. Ariyavisitakul, "C('I I, I estimation in OFDM systems with transmitter diversity in mobile wireless channels," IEEE Journal on Selected Areas in Communications, vol. 17, no. 3, pp. 461-471, Mar. 1999. [16] A. D. Lanterman, "Schwarz and Wallace and Rissanen: Intertwining themes in theories of model order estimation," International Statistical Review, vol. 69, no. 2, pp. 185-212, Aug. 2001. [17] A. Jeremic, T. Thomas and A. Nehorai, "OFDM channel estimation in the presence of interference," Proc. of IEEE Sensor A ,,.,; and Multichannel S.:,.Irl Processing Workshop, Rosslyn, VA, pp. 154 158, Aug. 2002. BIOGRAPHICAL SKETCH Jonghyun Won was born in Buchon, Korea. He received his B.Sc. degree in 2000 from the Department of Electrical Engineering, Dongguk University, Seoul, Korea. Since 2001, he has been pursuing a Master of Science degree in the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL. |

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PAGE 3 Tomyfamily. PAGE 4 iv PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS............................ iv LISTOFFIGURES............................... vii ABSTRACT................................... ix CHAPTER 1INTRODUCTION............................. 1 2WIRELESSCOMMUNICATIONSOVERFADINGCHANNELS... 3 2.1ErrorProbabilityforCommunicationSystem........... 3 2.2AntennaDiversity.......................... 5 2.3Interference............................. 7 3ORTHOGONALFREQUENCYDIVISIONMULTIPLEXING(OFDM) 8 3.1DataTransmissionandDetectioninOFDM........... 9 3.2MLDetectionforOFDMwithWhiteNoise............ 11 3.3MLDetectionforOFDMwithColoredNoise........... 12 4NEWALGORITHMFORINTERFERENCESUPPRESSIONIN OFDM.................................. 16 4.1ChannelEstimation......................... 16 4.2Estimationof R ( n )withPiecewiseConstantModel....... 18 4.3AutomaticSelectionofModelOrder................ 20 4.4NumericalExamplesforthePiecewiseConstantModel..... 22 4.5Estimationof R ( n )withMovingAverageProcess........ 26 4.6AutoSelectionofModelOrder................... 28 4.7NumericalExamplesforMovingAverageProcess........ 28 5INTRODUCTIONTOSPACETIME-OFDM(ST-OFDM)...... 33 5.1MIMOSystemsandSpace-TimeBlockCoding......... 33 5.2Space-TimeOFDM........................ 35 6APPLICATIONOFOURALGORITHMTOST-OFDM....... 38 v PAGE 6 6.1ChannelEstimation......................... 38 6.2Estimationof R ( n )withMovingAverageProcess....... 40 6.3NumericalExamplesfortheMovingAverageProcess..... 41 7CONCLUSIONANDFUTUREWORK................ 45 REFERENCES.................................. 47 BIOGRAPHICALSKETCH........................... 49 vi PAGE 7 LISTOFFIGURES Figure page 3{1DatatransmissionandreceptionschemeforanOFDMsystem... 9 4{1ExampleofthepiecewiseconstantmodelforaSISOsystem..... 19 4{2Symbol-error-rate(SER)forthepiecewiseconstantmodelwhen thereisonlythermalnoisepresent.................. 23 4{3Symbol-error-rate(SER)forthepiecewiseconstantmodelwhen thereareonestronginterferencesignal(C/I=10dB)andthermal noisepresent.............................. 24 4{4Autonomousselectionofmodelorder,SNR=30dB,forpiecewiseconstantmodel............................. 25 4{5Comparisonbetweendetection(usingpiecewiseconstantmodel andcoherentdetectionusingperfectknowledgeofthechannel andthenoisecovariance)...................... 26 4{6Symbol-error-rate(SER)forthemovingaverageprocessmodel whenthereisonlythermalnoisepresent............... 29 4{7Symbol-error-rate(SER)forthemovingaverageprocessmodel whenthereareonestronginterferer(C/I=10dB)andthermal noisepresent.............................. 30 4{8Autonomousselectionofmodelorder,SNR=30dB,formovingaverageprocess............................. 31 4{9Thecomparisonbetweendetectionusingmovingaverageprocess andcoherentdetectionusingperfectknowledgeofthechannel andthenoisecovariance....................... 32 6{1Symbol-error-rate(SER)forthemovingaverageprocessmodel whenthereisonlythermalnoiseispresent............. 42 6{2Symbol-error-rate(SER)forthemovingaverageprocessmodel whenstronginterferercomesfromasingleantennauser..... 43 vii PAGE 8 6{3Symbol-error-rate(SER)forthemovingaverageprocessmodel whenstronginterfererscomesfromco-channeluser'smultiple antennas................................ 44 viii PAGE 9 Weproposenewmethodsforco-channelinterferencesuppressioninorthog-onalfrequencydivisionmultiplexing(OFDM)systemswithmultiplereceiveantennas.Ourmethodsusealow-ordertime-domainmodeltoparameterizethematrix-valuedspectraldensityofthenoiseandinterference.Apenalizedmaximumlikelihood-functionapproachisusedtochoosethepropermodelorderautomaticallyineachreceiveddataframe.Weshowthatthesemethodssignif-icantlyimprovethesymbol-error-ratewheninterferenceexistsinthereceiveddata.Numericalexamplesareprovidedtoillustratetheperformanceofournewmethodsandcomparewithconventionalmethods.Wealsodiscussthedicultiesassociatedwiththeapplicationofourmethodtospace-timeOFDM(ST-OFDM). ix PAGE 10 Orthogonalfrequencydivisionmultiplexing(OFDM)hasbecomeoneofthemostpromisingtransmissiontechniquesforwirelesslocalareanetworksbecauseofitslowequalizationcomplexityinfrequencyselectivefadingchannels,anditsrobustnesstointer-symbolinterference(ISI).OFDMtransformsafrequency-selectivefadingchannelintomultipleindependentatfadingsubchannels.Bytransmittingonesymbolineachsubchannel,thesymboldurationisincreased;hencetheeectsofISIcausedbyafrequencyselectivefadingenvironmentarereduced.Therefore,anOFDMsystemcanachieveahighdatarateandareliabletransmissioninafadingchannel.However,duetolimitedandcongestedfrequencybandsincellularradioenvironments,thesystemperformanceisoftenlimitedbyinterference. Interferenceisoneofthemajorlimitingfactorsontheperformanceofwirelesscommunicationsystems,asittypicallyincreasesthebit-error-rate(BER).Therearetwomajortypesofinterference:so-calledadjacentchannelinterference(ACI)andco-channelinterference(CCI). ACIcomesfromsignalsthataretransmittedbyacommunicationsystemusingafrequencybandadjacenttothedesireduser'sfrequencybandinthesamearea.Thisinterferencecanbeminimizedbyusingastrictlyrestrictedsignalbandwidthoraproperchannelassignment.Gueyetal.[1]proposesACIrejectionmethodsforland-mobileradiosystems. CCIiscausedbythefrequencyreuseschemeinwirelessnetworks,forexample,whensignalscomefromaco-channelcellwhichusesthesamefrequency 1 PAGE 11 bandasthedesireduser.AsCCIconsistsofthesamefrequencies,itcannotberejectedbysimplyincreasingthetransmitpowerorusingnarrowchannelltering.TherearemanystatisticalCCIinterferencesuppressionmethodsforradiocommunicationnetworks.Forexample,Jongrenetal.[2]proposedaninterferencerejectionmethodusinglow-rankfeaturesofthespatialcovariancematrix;KarlssonandHeinegard[3]suggestedan"interferencerejection"methodfortheGSMsystem.Thesetechniqueshavebeenquitesuccessful(seeDametal.[4]). Inthisthesis,weproposenewCCIsuppressionmethodsforOFDMandspace-timeOFDM(ST-OFDM)withmultiplereceiveantennas.Thebasicideaofouralgorithmsisthatthe(matrix-valued)powerspectrumoftheinterferenceissmooth,andhencewecanusealow-ordertime-domainmodeltoparameterizethespectraldensityofthenoiseandinterference.Oneofourmethodsproposesasimplemodelforthecovarianceoftheinterferenceandnoise.ThismodelassumesthatpairsofMadjacentsubchannelshavethesamespectraldensity.Thus,theperformanceofourmodeldependsonhowweselectthenumberofsubchannels,M.Apenalizedmaximum-likelihoodfunctionapproachisusedtochooseMautomaticallyineachreceiveddataframe.Anothermethodusesamultidimensionalmovingaverage(MA)processtomodelthepowerspectrumdensityoftheinterferenceandnoisewithavariablenumberofparameters,M.Alsohere,apenalizedmaximum-Likelihoodfunctionapproachisusedtoselectthepropernumberofparameters.WealsoapplythesealgorithmsintheST-OFDMcaseandshownumericalexamples. PAGE 12 Inawirelessenvironment,transmittedsignalsusuallydecreaseinpowerorbecomedistortedinshapebecauseofthetime-variationofthechannelandthepropagationenvironment.Communicationinamovingcarcanbeoneexampleofafadingchannel,sincethesurroundingenvironmentofthecarvariessothatthepropagationchannelchangesaccordingtothevelocityofthecar. 2.1 Error Probability for Communication System Letusconsiderasimpleexampleofawirelesscommunicationsystemtoseetheeectoffading.Weassume,forsimplicity,thatthepropagationchannelislinear,time-invariant,andfrequencyat;alsoassumethatthesystemconsistsofasingletransmitandreceiveantenna.Suchasystemiscalledasingle-inputsingle-output(SISO)system.ConsideratransmittedsymbolsfromaniteconstellationS.Thenthereceivedsignalcanbeexpressedasy=hs+e(2.1) wherehdenotesthechannelgainandedenoteszero-meancomplexGaussiannoisewithvariance2.Theprobabilityofdensityfunction(p.d.f)forthislinearmodelisp(y)=1 Wecandetectsusingthemaximumlikelihoodrule,whichmaximizes(2.2)andisequivalentto^s=argmins2Sjyhsj2(2.3) 3 PAGE 13 FromEquation(2.3),wecanderivethedetectionerrorprobabilityforthesystem.SupposewedetectawrongsymbolsofromaniteconstellationSinsteadofthetruesymbols.ThenthepairwiseprobabilityforanincorrectdecisionisderivedasP(s!sojh)=P(jyhsoj2 PAGE 14 Averaging(2.5)overhusing(2.6),wegetEh[P(s!sojh)]=Zjhj2P(s!sojh)P(jhj2)djhj2Z10expjssoj2jhj2 Thesignal-to-noise(SNR)forthereceivedsignalis2=2,andthelaststepof(2.7)showsthattheaverageerrorprobabilityforthefadingchannelbehavesasP(error)/SNR1(2.8) Wecancomparethisresulttothatofaconstantchannel.IfweusethefactthattheSNRequals2=2andtheChernoboundin(2.5),thentheerrorprobabilityforadeterministicchannelbehavesasP(error)/exp(SNR)(2.9) 2.2 Antenna Diversity SincethefadingisRayleighdistributedandtime-variant,sometimesthechannelcanbeuseless.Oneofthemostpopularmethodsforcombatingthefadingeectisantennadiversity.Byemployingmultipleantennasonthetransmitterorthereceiverside,thecommunicationsystemcantransmitthesameinformationovermultiplechannels,whichareindependentofeachother.Suchasystemiscalledamultiple-inputmultiple-output(MIMO)system.InaMIMOsystem,thefadingeectcanbesignicantlymitigatedandalsothechannelcapacitycanbeincreasedasanotheradvantage.Equippingmultipleantennasatthebasestation,whichgivesreceivediversityfortheuplink,isdoneinmanycommunicationsystems.ConsideringnrreceiveantennasinsteadofoneantennainaSISOsystem,wecanseehowreceiveantennadiversitycounteracts PAGE 15 thefadingeect.Inthiscase,thechannelhbecomesan1nrvectorandthus,weleth=[h1hnr]Tbethepropagationchannelvectorinwhicheachvaluecorrespondstothechannelgainbetweenthetransmitantennaandeachreceiveantenna.Here()Tdenotesthetranspose.Then,withthesameassumptionsasfortheSISOsystemexample,thereceiveddatacanbewrittenasy=hs+e(2.10) wherey=[y1ynr]Tisanreceiveddatavectorande=[e1enr]TisacomplexGaussiannoisevectorwithzeromeanandcovariancematrix2Inr.Thepairwiseerrorprobability,obtainedbyasimilarcalculationasSection2.1,isP(s!sojh)=Qs IfweassumethatthenrchannelsareindependentandcomplexGaussianrandomvariableswithzeromeanandvariance2nr,respectively,theprobabilitydensityfunctionforhcanbeexpressedasP(h)=nrYn=11 Again,usingtheChernobound,(2.11)and(2.12),wecanderiveaboundontheaverageerrorprobabilityforMIMOfadingchannelsasfollowsEh[P(s!sojh)]=Zjh1j2Zjhnrj2P(s!sojh)P(h)djh1j2djhnrj2Z10expjssoj2jh1j2 PAGE 16 ThelastpartofEquation(2.13)showsthatthemoreantennasthereareonthereceiverside,themoremitigationoffadingeectisachieved.Also,itshowsthattheaverageprobabilityofadetectionerroroverthefadingchannelbehavesasP(error)/SNR(diversityorder| {z }=nr)(2.14) Also,adiversityeectaswellasotherbenetscanbeobtainedbyequippingmultipleantennasattransmitterside(thisiscalledtransmitdiversity).Accord-ingtorecentresearch,itispossibletoreproducethediversityfrommultipletransmitantennaswhilepreservingthetransmissiondatarateandasimpledecodingcomplexity.Thismethod,whichiscalledSpace-TimeCoding,willbeintroducedinChapter5. 2.3 Interference Recently,thepopulationofmobileusershasdramaticallyincreasedduetothefastdevelopmentofwirelesscommunicationtechnologies;hence,usingcarefullychosenchannelreuseschemeshasbecomemoreimportantincellularcommunicationsystems.Becauseofchannelreuse,interferencebetweensignalsfromthesamefrequencybandisoftengenerated.Thisinterference,whichiscalledco-channelinterference(CCI),istheoneofthemajorlimitingfactorsfortheperformanceofwirelesscommunicationsystems.Unlikethermalnoisewhichcanbeovercomebyincreasingthetransmitpower,CCIcannotbeoverpoweredbyincreasingthecarrierpower.Moreover,anincreaseincarriertransmitpowercanaugmenttheinterference.However,theeectofinterferencecanoftenbereducedbysuitablesignalprocessingmethodatreceiverside. PAGE 17 Aswirelesscommunicationshaveexperiencedgreatsuccessandarewidelyusedeverywhere,thedemandsforgoodqualityandvarietyofservicehaveincreased.Tofulllthesedesires,developmentforhighspeedwirelesscommu-nicationsystemshasbeeninvestigated.OnesuccessfulapproachtoachievingreliableandecienthighspeeddatatransmissionschemeisOFDM. ThemainideaofOFDMisthatbydividingtheentirechannelbandwidthintoindependentmultiplenarrowbands,eachsub-bandbecomesrelativelyat.WecancompareOFDMwithaconventionalsinglecarriersystem.Acon-ventionalsystemtransmitseachdatasymbolonebyoneinawholeoccupiedchannel.Oneofthedisadvantagesusingasinglecarrieristhatifthechannelsuf-fersfromfrequencyselectivefading,thechannelequalizationusuallyiscomplex.Ontheotherhand,inanOFDMsystem,eachsubchanneloccupiesonlyasmallfractionofthetotalbandwidth,andOFDMtransmitsonedatasymbolineachsubchannelinparallel.Sinceasmallbandwidthofeachsubchannelmakesthesymboldurationsmuchlongerthanthememoryofthechannel,OFDMcanusesimplechannelequalizationandalsoobtainthesamedatarateasaconventionalsystem.ThispropertyallowsOFDMtosupporthighspeedrates.AdetailedexplanationofOFDMcanbefoundinLarssonandStoica[5],HeiskalaandTerry[6]andLietal.[7].Here,weintroducethefundamentalprinciplesofanOFDMsystemfollowingLarssonandStoica[5]. 8 PAGE 18 9 Figure3{1:DatatransmissionandreceptionschemeforanOFDMsystem(from LarssonandStoica[5],chapter5) 3.1 Data Transmission and Detection in OFDM ConsiderageneralOFDMsystemwithasingletransmitantennaand n r receiveantennas.Supposewehave s ( n )symbolsfor n =0 ;:::;N 1tobe transmitted.AftertakinginverseFouriertransformforthesymbols,theencoded datacanbeexpressedas x ( n )= 1 p N N 1 X k =0 s ( k )exp i 2 N nk ;n =0 ;:::;N 1(3.1) Priortotransmission,acyclicprex(CP)isaddedatthebeginningofthe sequence x ( n ).TheCPisacopyofthelast N pre symbols,where N pre L and L isthechanneldelayspread.TheCPmakesthetransmittedsignalperiodicwith period N sothatthelinearconvolutioninducedbythepropagationchannelis transformedintoacircularconvolution,whichcorrespondstomultiplicationof channelgainsandsymbolsinthefrequencydomain.Wecanexpresstheencoded datawithacyclicprexasfollows x ( n )= x ( n + N ) ;n = N pre ;:::; 1(3.2) PAGE 19 Figure3{1illustratesthetransmissionscheme.Fromtheencodeddatax(n),wecanobservethateachsymbols(k)iscarriedoneachsubcarrierei2 Nnwhichareorthogonaleachother.Theencodeddataaretransmittedthroughachannel,whichhasLdelaytaps,andafterdiscardingCP,thereceiveddatavectorfromthemultipleantennascanbeexpressedaslinearconvolutionofthechannelandtheencodeddataplusthermalnoiseandinterferencey(n)=LXk=0hkx(nk)+e(n)=1 N(nk)m+e(n)(3.3) wherehkdenoteskthvector-valuedchanneltap.AftertakingtheFouriertransformofthereceiveddata,wegetthemultiplicationofeachfrequencyatfadingtransformedchannelanddata(see,e.g.,LarssonandStoica[5],Chapter5)z(n)=1 Nln=1 N(lk)mei2 Nln+1 Nln=1 N(nm)lei2 Nkm+1 Nln=s(n)LXk=0hkei2 Nkn+1 Nln=s(n)h2 Nn+1 Nln| {z }~e(n)(3.4) wherewehavedenedthechanneltransferfunctionh2 Nn=LXk=0hkei2 Nkn(3.5) PAGE 20 FromEquation(3.4),wecandetectthesymbolss(n)usingsimpleequalization. 3.2 ML Detection for OFDM with White Noise First,weneedtoconsiderthenoisetermbeforederivingtheMLdetectorforOFDM.Weassumeforsimplicitythatthenoisevectorse(n)arespatiallywhitecircularGaussianrandomvariableswithzeromeanandvariance2.Thisassumptionisproperwhenonlythermalnoiseexist.LetconsiderthenoisecovarianceinEquation(3.4)asfollowsE[~e(n)~eH(k)]=E1 Nln1 NmkH=1 Nlnei2 Nmk(3.6) Fromtheassumptionone(n),E[~e(n)~eH(k)]=1 Nl(nk)=8>><>>:2I;n=k0;n6=k(3.7) Equation(3.7)showsthatthenoisetermin(3.4)iswhiteGaussiannoisewithcovariance2I,andthus,z(n)isasucientstatisticforthedetectionofs(n).Hence,MLdetectionforOFDMisequivalenttominimizingthefollowingmetrickz(n)s(n)h(n)k2(3.8) whereh(n)=LXl=0hlei2 Nln(3.9) PAGE 21 3.3 ML Detection for OFDM with Colored Noise Amoregeneralassumptiononthenoisee(n)istosayitiscircularlysymmetricGaussianwiththefollowingcovariancefunctionRl=E[e(n)eH(nl)](3.10) where()Hdenotestheconjugatetranspose.Thenthepowerspectrumofe(n)isR(n)=1Xl=Rlei2 Nnl(3.11) Suchcolorednoisecanbeusedtomodelco-channelinterference.1 WenowshowthatR(n)isthecovarianceofthenoiseterminEquation(3.4)andthat~e(n)isindependentof~e(m),n6=m.Firstweconsidertheauto-covariancefunctionof~e(n)asfollowsE[~e(n)~eH(n)]=E"1 Nkn1 Nnk!H#=1 N(kk)n(3.13) PAGE 22 Letkk=l,thenE[~e(n)~eH(n)]=1 Nln=1 Nln+N1Xl=1N1Xk=lRlei2 Nln!=1 Nln+N1Xl=1(N+l)Rlei2 Nln)=N1Xl=(N1)Rlei2 NlnN1Xl=(N1)jlj Nln(3.14) Underthemildassumptionthatthecovariancesequencedecayssucientlyrapidly,thelasttermin(3.14)goestozerowhenN!1.HenceE[~e(n)~eH(n)]N1Xl=(N1)Rlei2 Nln(3.15) Hence,inanOFDMsystem,theauto-covarianceofthenoisetermaftertakingtheFouriertransformofthereceiveddataisthesameasthepowerspectraldensityofthenoiseinthereceiveddata.Next,wecalculatethecross-covarianceof~e(n)E[~e(n)~eH(m)]=E"1 Nkn1 Nmk!H#=1 N(nkmk)(3.16) PAGE 23 Ifweletkk=l,thenEquation(3.16)canbeexpressedasfollowsE[~e(n)~eH(m)]=1 N(nkmk+ml)=1 N(nm)kei2 Nml| {z }A(nm)+1 N(nm)kei2 Nml| {z }B(nm)(3.17) NotethatN1Xk=0ei2 N(nm)k=0(3.18) andl1Xk=0ei2 N(nm)k=N1Xk=lei2 N(nm)k(3.19) Ifwetakethenormof(3.17),thenormofrstsummationterm,kA(nm)k,canbeexpressedasfollowskA(nm)k1 N(nm)kj| {z }jlj1 PAGE 24 Also,kB(nm)kcanbeexpressedasfollowswith(3.19)kB(nm)k=1 N(nm)k=1 N(nm)k1 N(nm)kj1 Underthemildassumptionthatthecovariancesequencedecayssucientlyrapidly,whenN!1,(3.20)and(3.21)gotozero.Asaconsequence,theFouriertransformednoiseistemporallywhiteandhascovarianceR(n). PAGE 25 Whenthereisstronginterferenceinthereceivedsignals,itcanbebenecialtosuppresstheinterference,sothatcorrectsignaldecisionscanbemade.Oneapproachtoestimatethecovarianceoftheinterferenceistoinstantlyestimatethecovarianceofeachsubchannel,butnormallysuchinstantaneouscovarianceestimatesdonotworkwellifthereisonlyashorttrainingsequence.Oneapproachtocanceltheco-channelinterferenceissuggestedinLiandSollenberger[8],whichusesaminimummean-squareerror(MMSE)diversitycombiner(DC).Theyestimatethechannelsandthenoisecovarianceinthefrequencydomain,andusetheseestimatedvaluestosuppresstheinterference.Inthischapter,weintroducenovelmethodsforinterferencesuppressioninanOFDMsystem.Thesealgorithmsarebasedonalow-ordertimedomainmodeltoparameterizethematrix-valuedspectraldensityoftheinterference. 4.1 Channel Estimation Letusassumethatwereceivedmorethanonetrainingsequence(T1).AftertakingtheFouriertransformofthereceivedtrainingdata,theycanbeexpressedaszt(n)=LXl=0hlexpi2ln Nst(n)+1 Nkn(4.1) forn=0;:::;N1;t=1;:::;T,wherezt(n)standsforFourier-transformedknowndata.Wecanestimatethechannelfrom(4.1),assumingknowntrainingdata,usingvariousmethods. 16 PAGE 26 Theparsimonyprinciple[9],explainsthatamongmodelsdescribingdatawell,themodelwiththesmallestnumberofparametersisabetterchoicethanothers.Forthisreason,itismoredesirabletoestimatetimedomainchannelsinsteadoffrequencydomainchannels,becausetheformerconsistofnr(L+1)parameters;howeverinthefrequencydomain,wewouldneedtoestimatenrNparameters.Channelestimationcanbedonebythefollowingprocedure. DenotethetransmittedandreceivedtrainingdatabeforeFouriertrans-formationbyxt(n)andyt(n).UsingToeplitzmatrices,thetrainingdataXt,channelH,noiseEtandreceiveddatasequenceYtcanbeexpressedbyYt=XtH+Et(4.2) whereXt=266666664xt(0)xt(1)xt(L)xt(1)xt(0)xt(1L).........xt(N1)xt(N2)xt(N1L)377777775 Thenegativeloglikelihoodfunctionfor(4.2)becomes,assumingthenoiseiswhiteminHTXt=1kYtXtHk2(4.5) PAGE 27 andtheMLestimateofchannel^Hisdeterminedby^H=TXt=1XHtXt1TXt=1XHtYt(4.6) Afterestimatingthechannel^H,thefrequencydomain^h(n)channelscanbeeasilycalculatedbyusingFouriertransformation^h(n)=LXl=0^hlei22 Nnl(4.7) 4.2 Estimation of with Piecewise Constant Model Theinterferencesignaline(n)consistsofthesamefrequenciesasdesiredsignal,andithaspropagatedoverachannelwithsimilarpropertiesasthechannelh(n).Hence,wecanexpectthatthepowerspectrumofe(n)issmoothandthischaracteristicscanbeusedtoestimateR(n).Followingthisreasoning,wecanexpectthatthecovarianceinaportionofthefrequencybandconsistofapproximatelythesamevalues.ThepiecewiseconstantmodelassumesthattheinstantcovarianceR(n)ofMadjacentsubchannelsconsistsofthesamevalueorapproximatelythesamevalue.Withthismodel,wepartitionthefrequencybandinN=Msegments,calculateR(n)foreachsubchannel,takeanaverageofR(n)valuesineachsegmentwhichconsistsofMadjacentsubchannelsandapplyeachaveragedvalue^R(n)toMsubchannelsineachsegmentinsteadofR(n).MoredetailedexplanationsareasfollowsUsinganestimatedchannel^Hfrom(4.6),wecalculateresidualsfromreceivedtrainingdata~Yt=(YtXt^H)T(4.8) where~Ytconsistsof~Yt=~yt(0)~yt(N1)(4.9) PAGE 28 Figure4{1:ExampleofthepiecewiseconstantmodelforaSISOsystem. WetaketheFouriertransformof~yt(n)asfollows~rt(n)=1 Nnk(4.10) WenextestimateR(n)foreachsubchannelby^R(n)=1 andthen,taketheaverageineachsegmenti=0;:::;N=M1,accordingtotheparameterM^RM(i)=1 Weuse^RM(i)forsegmentitosuppresstheinterferenceinsteadoftheinstan-taneousmomentbasedestimate^R(n).Figure4{1showsanexampleofthepiecewiseconstantmodelforaSISOsystem.Thedash-dottedlineisthetrue PAGE 29 covarianceoftheinterferenceandthethermalnoise,andthesolidlineisthees-timatedcovarianceusingourmethod.Hence,accordingto(3.12),themaximumlikelihooddetectionforinformationdataisexpressedasmins(Mi+m)2S^R1=2M(i)(z(Mi+m)^h(Mi+m)s(Mi+m))2(4.13) fori=0;:::;N=M1andm=0;:::;M1. 4.3 Automatic Selection of Model Order Togetthebestperformance,weneedtochoosethemodelorder,theparameterM,autonomouslyineachsequenceofdata.Asasub-optimalway,weuseamaximum-likelihood(ML)approachforselectingM.AfterestimatingthecovariancefordierentM,weinsertthosevaluesintothenegativelog-likelihoodfunctionwithtrainingsymbols,compareandselecttheminimumoneasthepropermodelorderfortheinformationdatasequence.Itisdesirabletochoosethemodel,thatexplainsthedata(interferenceandnoise)wellandhasaparametrizationassimpleaspossible.ButMLalwayschoosesthehighestmodelorder,whenthereisinterferenceinthetrainingsequence.Tocounterbalancethistendency,weaddapenaltyterminspiredbytheminimumdescriptionlength(MDL)method(moredetaileddiscussioncanbefoundinKay[10],Section6.8)totheMLruleMDL(M)=lnP(Yt;^RMj^H)+nM PAGE 30 wherenMisthenumberofestimatedparametersinRM.Thus,thismethodndsthemodelorderMwhichminimizesminMTXt=1N=M1Xi=0M1Xm=0^R1=2M(i)(zt(Mi+m)^h(Mi+m)st(Mi+m))2+TMN=M1Xi=0log^RM(i)+TnM wherest(n)istheknownreceiveddata.NotethatRM(i)istheaveragevalueofthemomentbasedcovariancematricesofMadjacentsubchannelsandzt(Mi+m)^h(Mi+m)st(Mi+m)=~rt(Mi+m)(4.16) Notethatthenormsquaretermofequation(4.15)canbewrittenTXt=1N=M1Xi=0M1Xm=0^R1=2M(i)(zt(Mi+m)^h(Mi+m)st(Mi+m))2=TXt=1N=M1Xi=0M1Xm=0Trf^R1=2M(i)~rt(Mi+m)~rHt(Mi+m)^RH=2M(i)g=TXt=1N=M1Xi=0M1Xm=0Trf^RH=2M(i)^R1=2M(i)~rt(Mi+m)~rHt(Mi+m)g=N=M1Xi=0M1Xm=0Trf^R1M(i)TXt=1~rt(Mi+m)~rHt(Mi+m)g=TN=M1Xi=0M1Xm=0Trf^R1M(i)^R(Mi+m)g=TN=M1Xi=0Trf^R1M(n)(^R(Mk)+^R(Mk+1)++^R(Mk+M1))g=TN=M1Xi=0MTrf^R1M(i)^RM(i)g=TNTrfIg(4.17) PAGE 31 whereTrfgstandsforthetraceofamatrix.Followingthisresult,Equation(4.15)canberewritteninthemoresimpliedformminMMN=M1Xi=0log^RM(i)+nM WeusethissimpliedmethodwithtrainingsequencestoselectthepropervalueofMandapplythechosenmodeltosuppresstheinterferenceinthedatasequences. 4.4 Numerical Examples for the Piecewise Constant Model Inthissection,weshowsomesimulationexamplesforournewalgorithmforanOFDMsystem.Inourexperiment,N=64subcarriers,1trainingsequenceandnr=2receiveantennasareused.ThechannelconsistsofL+1=5independentRayleighfadingchanneltaps,andhasanexponentiallydecayingpowerdelayprole:E[khlk2]/1:5l;l=0;:::;L.Thetrainingsequenceisanimpulsetoestimatechannel,andtheconstellationsizeforoursimulationisQPSK. Figure4{2showstheestimatedSymbol-error-rate(SER)versustheCarrier-to-Noiseratio(C/N)whenthereceivedsignalsarenotaectedbyinterference,andwhenonlythermalnoiseispresent.WepresenttheresultforthecaseswhenM=2;4;8,and64,andwiththeautonomousselection.wealsocomparetheresulttocoherentdetectionandconventionaldetection(Inthiscase,weusenoisecovariance^R(n)=Iforalln.)Thenon-coherentdetectionandourmethodwithM=64showalmostsameresult.ThereasonforthisisthatsettingM=64impliesthatthenoisespectrumisindependentofn.WiththeothervaluesofM,weseeaperformancelossrelativetoconventionaldetection.Ourauto-selectionmethodchoosesM=64modelorderforeachsequenceasweexpected. PAGE 32 23 Figure4{2:Symbol-error-rate(SER)forthepiecewiseconstantmodelwhen thereisonlythermalnoisepresent. InFigure4{3,wepresentthecasewhenthereisonestronginterferer present.Theco-channeluserhasthesamechanneldelayproleandpowerdelay proleasthedesireduserhas,andthesignaltointerferenceratiois10 dB .This gureshowspoorperformancesofcoherentdetectionandconventionaldetection withoutestimationofinterference,butourmethodsuccessfullymitigatesthe co-channelinterference.ForhighSNR,the M =4modelorderoutperformsthe othermodels,butintheregionoflessthan20 dB ,the M =8modelordershows somewhatbetterperformancethan M =4.ForeachSNR,ourauto-selection PAGE 33 Figure4{3:Symbol-error-rate(SER)forthepiecewiseconstantmodelwhenthereareonestronginterferencesignal(C/I=10dB)andthermalnoisepresent. methoddoesnotalwayschoosethebestmodel,whichhasthelowestsymbol-error-rate.However,theauto-selectionmethodsignicantlyoutperformsthecoherentmethodwithoutestimatingthecovarianceofinterference. Figure4{4showshowourautonomousselectionmethodperformswhenthesignal-to-noiseratiois30dB.OurmodelalwayschoosesM=64whenonlythermalnoiseispresentasweexpect.Whenonestronginterfererandthermalnoisearepresent,ourmethodusuallyselectsM=8andM=4. PAGE 34 25 Figure4{4:Autonomousselectionofmodelorder,SNR=30dB,forpiecewiseconstantmodel. Wesummarizethecomparisonbetweendetectionusingourmethodand coherentdetection(usingperfectknowledgeofthechannelandthenoise covariance)inFigure4{5.Inthecasewhenthereceivedsignalsareonlyaected bythermalnoise,Figure4{5indicatesthatconventionaldetectionandour newmethodhavethesameperformance.Inthecasewhenreceivedsignalsare aectedbystronginterference,ourmethodshowsasignicantimprovement comparedtoconventionaldetection. PAGE 35 26 Figure4{5:Comparisonbetweendetection(usingpiecewiseconstantmodeland coherentdetectionusingperfectknowledgeofthechannelandthe noise covariance). 4.5 Estimation of R ( n ) with Moving Average Process Inthissection,weintroduceanalternativenewmethodforinterference suppression.Thismethodusesalow-ordertime-domainmodeltoparameterize thematrix-valuedspectraldensityofthenoiseandinterference.Westilluse thechannelestimationtechniquedescribedinSection4.1.Then,wecalculate residualsfromthetrainingsequenceusingestimatesofthechannel, ^ H ,in(4.6) PAGE 36 asfollows~Yt=(YtXt^H)T(4.19) Fromtheseresiduals,weestimate^Rlusingthebiasedauto-covarianceestimatesin(see,e.g.,SoderstromandStoica[9]andStoicaandMoses[11])^Rl=1 forl=0;:::where~Y+listhematrix~Ywiththerstlcolumnsremoved,and~Ylistheequalto~Ywiththelastlcolumnsremoved.WenextestimateR(n)viaatruncatedweightedFouriersumasfollows^R(n)=MXm=M1jmj Nmn(4.21) Becausethecovariancefunctionisconjugatesymmetricwecanset^Rm=^RHm.Truncatingthesumamountstoassumingthatthepowerspectrumofe(n)issmooth,whichisprobablyareasonableassumptioninpractice.Touse^Rfordetectionin(3.12),itisnecessarythat^Rispositivedeniteforn=0;:::;N1.Fortunately,thisisthecase.Toseewhythisisso,noterstthat^R(n)=1 where(n)isanNNmatrixwhose(k;l)thelementisequaltok;l(n)=(1jklj=(M+1))ei2(kl)n=NwhenjkljMandzerootherwise.But(1jklj=(M+1))eiwnisthecovariancefunctionofamovingaverageprocesswithcoecientsf1;eiw;:::;ei(M1)wgandhence(n)ispositivesemi-denite.Becauseofthenoisein~Yt,^R(n)ispositivedenitewithprobabilityone. PAGE 37 4.6 Auto Selection of Model Order Alsoforthemethodusingamovingaverageprocess,itisnecessarytondthepropermodelorderM.Asforthepiecewiseconstantmodel,wecanapplypenalizedMLtoobtainanautonomousselectionmethod.But,unfortunately,thisdoesnotappeartoworkthatwellforthemovingaverageprocessmodel.Wedonotknowtheexactreasonforwhythisdoesnotworkforthemovingaverageprocessmodel(inoursimulation,thevalueoftheMLmetricismuchbiggerthanthatofthepenaltyterm):butitisprobablyrelatedtothefactthattheestimates^R(n)and^h(n)weusearenotmaximumlikelihood.Instead,wehaveusedthedetecteddatasequencetodeterminethemodelorder.Forexample,ifwetransmitonetrainingsequenceandvedatasequences,weestimatethechannelandcovarianceusingthetrainingsequenceandthendetectrstdatasequence.Then,weusethedetecteddatasequencetoselectmodelorder,andthechosenmodelorderisusedtodetecttheotherdatasequences.Thus,ourautonomousselectioncanbeexpressedasfollowsminMN1Xn=0nlogj^R(n)j+^R1=2(n)(z(n)^h(n)^s(n)2o+nM where^s(n)isthedetecteddata. 4.7 Numerical Examples for Moving Average Process Inthissection,weshowsomesimulationexamplesusingthenewalgorithmwiththemovingaverageprocess.ThisexperimentwasdoneunderthesameconditionsasdescribedinSection4.4. Figure4{6showstheestimatedSymbol-error-rate(SER)versustheCarrier-to-Noiseratio(C/N)whenonlythermalnoiseispresent.WedemonstratetheresultsforthemodelordersM=0;2;4;6;8andforautonomousselectionamongthesemodels(incontrasttothepiecewiseconstantmethod,increasingthemodel PAGE 38 29 Figure4{6:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen thereisonlythermalnoisepresent. orderheremeansincreasingthenumberofparametersweestimate).Also,to compareourmethodwithaconventionaldetectionscheme,weshowresultsfor coherentdetectionandconventionaldetection.Thisgureshowsthatthere isnotmuchdierencebetweenconventionaldetectionandthemethodusing movingaverageprocess. Figure4{7showsthecasewhenthereisonestronginterfererpresent( C=I =10 dB ).Fromthisgure,weseethatourmethodsuccessfullymitigates co-channelinterference.Likethepiecewiseconstantmodel,thismovingaverage PAGE 39 30 Figure4{7:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen thereareonestronginterferer(C/I=10dB)andthermalnoisepresent. processmodelsuppressestheinterferencemoreeectivelybyincreasingthe numberofparameters.Autonomousselectiondoesnotalwayschoosethemodel orderwhichshowsthebestSER. Figure4{8displayshowourautonomousselectionmodelperformswhenthe signal-to-noiseratiois30 dB .Ourmethodchooses M =0mostofthetimewhen onlythermalnoiseispresent.Whenonestronginterfererandthermalnoiseare present,ourmodelselectsmostly M =6butalso M =8. PAGE 40 31 Figure4{8:Autonomousselectionofmodelorder,SNR=30dB,formovingaverageprocess. Figure4{9showsacomparisonbetweendetectionusingourmethodandcoherentdetectionusingperfectknowledgeofthechannelandthenoisecovariance. NotethesimilaritybetweenFigure4{9andFigure4{5. PAGE 41 32 Figure4{9:Thecomparisonbetweendetectionusingmovingaverageprocess andcoherentdetectionusingperfectknowledgeofthechanneland the noise covariance. PAGE 42 InChapter2,weintroducedreceivediversity,whichisapopularmethodtocombatfadinginwirelesscommunications.Byinstallingmultipleantennasatreceiverside(usuallyatthebasestation),wecanmitigatechannelfadingaswellasincreasethechannelcapacity.Inrecentyears,researchershavefoundthatitispossibletoobtainthesameadvantageasreceivediversitybyusingmultipleantennasatthetransmitterside(thisiscalledtransmitdiversity).Usingmultipleantennasatbothsides,wecangetmuchmorebenetsthanwithreceivediversityortransmitdiversityalone.However,toexploitthesebenets,weneedacodingscheme,whichiscalledSpace-TimeCoding.Inthischapter,webrieyintroduceaMIMOsystem,space-timeblockcoding(STBC)andspace-timeOFDM(ST-OFDM),whichcombinesspace-timecodingandOFDM. 5.1 MIMO Systems and Space-Time Block Coding ConsideraMIMOsystemwhichconsistsofnttransmitantennasandnrreceiveantennas.Weassumethatthepropagationchannelsarelinear,time-invariant,andfrequencyat.LetXandYbentNtransmitteddataandnrNreceiveddatamatrices,respectively.Then,thereceiveddatacanbeexpressedinalinearmodelasfollows whereHisannrntchannelmatrixinwhichthe(m;n)thelementcorrespondstothechannelgainbetweenthereceiveantennamandthetransmitantennan; 33 PAGE 43 LikefortheSISOsysteminSection2.1,wecanderivethedetectionerrorprobabilityfromEquation(5.2).ByasimilarcalculationasinSection2.1 Next,weassume,forsimplicity,thatallchannelsinHareindependentandcomplexGaussianwithzeromeanandvariance2;hence,thechannelisRayleighfading.TheprobabilitydensityfunctionfortheRayleighfadingchannelscanthenbeexpressedas UsingtheChernobound,wegettheaverageerrorprobabilityasfollows(amoredetailedderivationcanbefoundin[5,Section4.3])EH[P(X!XojH)]=ZHP(X!XojH)p(H)dH=2 TheaboveequationshowsthattheaverageerrorprobabilityforaMIMOsystembehavesasP(error)/SNRnrnt(5.6) PAGE 44 whichmeansthatreceivediversityofordernrandtransmitdiversityoforderntareobtained,aslongasXXohasfullrank.Hence,thelengthofthetransmissionNhastobelongerthanthenumberoftransmitantennas,nt. Itisnoteasytoachievetransmitdiversitywithoutlosingtransmissiondatarateandincreasingthedecodingcomplexity.Space-timecodingisawaytoachievehighdatarateandlowdecodingcomplexity.Oneofwellknownspace-timecodingschemesistheAlamouticode[12],whichisatransmissionschemefortwotransmitantennas.Anothercaseisthelinearorthogonalspace-timeblockcode(OSTBC),whichhasasimplestructure.Considerthetransmissionofnssymbolsfs1;;snsg.Then,encodedlinearSTBChasfollowingstructureX=nsXn=1(snAn+i~snBn)(5.7) whereAnandBnarexedcodematrices,andsnand~snaretherealandimaginarypartsofthesymbolsn,respectively.OSTBChasthespecialmeritthattheMIMOchanneldecouplesintonsindependentscalarchannels,becauseofthefollowingorthogonalitypropertyofOSTBCXXH=nsXn=1jsnj2I(5.8) AmoredetailedexplanationforOSTBCcanbefoundinLarssonandStoica[5],Chapter7,Tarokhetal.[13]andLarssonetal.[14]. 5.2 Space-Time OFDM ST-OFDMisacombinationofspace-timeblockcodingandOFDMtoobtaintransmitdiversityforanOFDMsystem.ThemainideaofST-OFDMisidenticaltothatofOFDM.BecauseaST-OFDMsystemusesmultipletransmitantennas,itsendsNoencodedntNmatricesinwhichnssymbolsareencoded.Hence,nsNosymbolsaretransmittedduringNNosymboltransmittimeintervals.The PAGE 45 transmissionanddetectionproceduresforST-OFDMareasfollows.First,fromnsNosymbolstobetransmitted,encodeeachgroupofnssymbolstoNocodematricesG(n)forn=0;;No1.Second,taketheinverseFouriertransformofthecodematricesasfollowsX(n)=1 Nnk;n=0;:::;No1(5.9) Then,addtheCPwhichistakenfromNprelastcodematricesofthetransmitteddatamatrices,andselecttherstcolumnsofallcodematricestotransmitthemsequentiallythroughchannels,assumingthateachchannelhasLdelaytaps.Next,selectthenextcolumnsforthetransmissionuntilallthecolumnsofthecodematriceshavebeensentout.Atthereceiver,discardtheCP.ThenthereceivedcodematricescanbeexpressedasfollowsY(n)=LXk=0HkX(nk)+E(n)=1 N(nk)m+E(n)(5.10) AftertakingFouriertransformof(5.10)andfollowingsimilarderivationasin(3.4),thereceiveddatabecomeZ(n)=1 Nln=H(n)G(n)+1 Nln(5.11) whereH(n)=LXk=0Hkei2 Nkn(5.12) PAGE 46 Hence,MLdetectionforST-OFDMisequivalenttominimizingthefollowingmetrickZ(n)H(n)G(n)k2(5.13) MoreaboutST-OFDMcanbefoundinLarssonandStoica[5],Chapter8andLietal.[15]. PAGE 47 Inthischapter,weextendourinterferencesuppressionalgorithmtotheST-OFDMcase. 6.1 Channel Estimation BecauseaST-OFDMsystemusesmultipletransmitantennasandtransmitsanntNinverseFouriertransformedcodematrixX,thenumberoftrainingsequences,whichconsistofNosymbolseach,isatleastN.LetusassumethatwehavereceivedNTtrainingsequences(T>1)andthateachchannelhasLdelaytaps.DenotetransmittedandreceivedcodematricesbyXt(n)andYt(n)forn=0;;No1;t=1;;T.Then,thereceivedtrainingdatacanbeexpressedasfollowsYt(n)=LXl=0HlXt(nl)+E(n);n=0;:::;No1(6.1) whereHlstandsforlthmatrixvalueddelaychanneltap.Byusingablock-Toeplitzmatrix,Equation(6.1)canberewrittenasYt(0)Yt(No1)=H0HLXt+Et(0)Et(No1)(6.2) 38 PAGE 48 whereXt=266666664Xt(0)Xt(1)Xt(NoL1)Xt(1)Xt(0)Xt(NoL).........Xt(L)Xt(L1)Xt(0)Xt(No1)377777775(6.3) Notethatvec(ABC)=(CTA)vec(B)(6.4) wheredenotestheKroneckerproductandvec()isthevectorizationofamatrix.Byvectorizingbothsidesof(6.3),wegetyt=(XTtInr)h+et(6.5) whereyt=266664vec(Yt(0))...vec(Yt(No1))377775;h=266664vec(H0)...vec(HL)377775;et=266664vec(Et(0))...vec(Et(No1))377775(6.6) Maximumlikelihoodestimateofhin(6.5)isequivalenttominimizingthefollowingmetricminhTXt=1yt(XTtInr)h2(6.7) andtheMLestimateofthechannel^hisdeterminedby^h=vec(^Ho)Tvec(^HL)TT=1 PAGE 49 Also,notethatafterestimatingthechannel^handreconstructing^H,^H(n)fortheatfadingsubchannelscanbecalculatedbytakingtheFouriertransformation ^H(n)=LXl=0^Hlei22 Nnl(6.9) 6.2 Estimation of with Moving Average Process First,notethatthereceivedcodematrix,Yt(n),consistsofNnumberofnr1columnvectors.VectorizationofYt(n)givesvec(Yt(n))=yt(n)=266664y1t(n)...yNt(n)377775(6.10) Next,wecalculatetheresidualsfromthevectorizationofthereceiveddatausingtheestimatedchannel^handtheblock-Toeplitzmatrix(6.3)asfollows~yt=yt(XTtInr)^h(6.11) Following(6.10),wecanexpress~ytas~yt=~yTt(0)~yTt(No1)T=~y1t(0)T~yNt(0)T~y1t(No1)T~yNt(No1)TT(6.12) Then,wereconstructthematrix~Yt(n)asfollows~Yt(n)=~ynt(0)~ynt(No1);n=0;:::;N1(6.13) PAGE 50 Followingthesameproceduresas(4.20),weestimate^Rlusingthebiasedauto-covarianceestimates^Rl=1 forl=0;:::where~Y+l(k)isthematrix~Y(k)withtherstlcolumnsremoved,and~Yl(k)istheequalto~Y(k)withthelastlcolumnsremoved.WenextestimateR(n)viaatruncatedweightedFouriersumasfollows^R(n)=MXm=M1jmj Nmn(6.15) Using^R(n),themaximumlikelihooddetectionrulefortheinformationsymbolsisexpressedasargminG^R(n)(Z(n)^H(n)G(n))2(6.16) Tochooseapropermodelorder,theautonomousselectionmethodcanbeexpressedasfollowsminMN1Xn=0nlogj^R(n)j+^R1=2(n)Z(n)^H(n)^G(n)2o+nM where^G(n)isthedetectedcodematrix. 6.3 Numerical Examples for the Moving Average Process WeshowsimulationresultsforouralgorithmforanST-OFDMsystemwithN=64subcarriers,nt=2transmitandnr=2receiveantennas.TheAlamouticodematrixisusedinourexperiment,andthechannelhasthesamedelaytapsandthesameaveragepowerdelayproleasinSections4.4and4.7.Twotrainingsequences,whicheachsequencebeinganimpulse,areused,andtheconstellationisQPSK. PAGE 51 42 Figure6{1:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen thereisonlythermalnoiseispresent. Figure6{1showsthattheSymbol-error-rateperformanceversustheCarrierto-Noiseratio(C/N)foranST-OFDMsystemwhenonlythermalnoiseis present.Thisgureshowsthattransmitdiversityeectivelycombatsthechannel fadingcomparedtoFigure4{6. Figure6{2showsresultsforthecasewhenonestronginterferercomes fromasingleantennauser.CCIhasthesamechanneltapsandaveragepower delayproleasthedesireduserhas,andthesignaltointerferenceratiois10 dB Theinterferencesuppressingalgorithmoutperformstheonethatdoesnot PAGE 52 43 Figure6{2:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen stronginterferercomesfromasingleantennauser. takeinterferenceintoaccount.Also,thealgorithmsuppressestheinterference moreeectivelybyincreasingthenumberofparametersweestimate.However, Figure6{3showsthatwhenstronginterfererscomesfrommultipletransmit antennas,ourmethodcannotmitigateinterferencewell.Also,inthiscase, increasingthemodelorderdoesnothaveasignicanteectontheinterference suppression. PAGE 53 44 Figure6{3:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen stronginterfererscomesfromco-channeluser'smultipleantennas. PAGE 54 Co-channelinterferenceisamajorlimitingfactorforcellularcommunicationsystems,anditisexpectedtobecomeamajorbottleneckalsoforwirelesslocalareanetworks.Inthisthesis,wehavepresentedaconceptuallyverysimpleandcomputationallycheapmethodforsuppressionofco-channelinterferenceinsystemsthatuseOFDMmodulationandwhichhavemultiplereceiveantennas.Ourmethodcomesintwoversions:onethatparameterizestheinterferenceandnoisespectrumviaapiecewise-constantmodel,andonethatusesalow-ordermoving-averagemodel.Ineithercase,atechniqueinspiredbyMDLisusedtoautomaticallyselectthemodelorder,henceavoidinganyuserparameterstobeselectedpriortotheapplicationofthealgorithm.Simulationresultsshowthatourmethodcansubstantiallyimprovetheperformancecomparedtoconventionaltraining-baseddetection. Severalproblemsremainwhichcouldbethetopicoffuturestudies.First,themodelorderselectionstrategyusedisbynomeansoptimal(optimalsuchselectionmethodsingeneraldonotexist,seeLanterman[16])anditispossiblethatanalternativemethodsforchoosingthemodelordermayoutperformours.Second,theestimatesofthechannelandnoisecovariancethatweusearenotmaximum-likelihood,butneverthelesstheyarecomputationallysimpletoobtain.Itislikelythatusingmaximum-likelihoodestimatesinlieuofourproposedestimatesmayleadtoamethodthatworksbetterthanours.Themaximum-likelihoodestimatesgenerallydonotexistinclosedform,andobtainingthem 45 PAGE 55 numericallyiscomputationallyburdensome.Yet,somelow-complexityapprox-imationstomaximum-likelihoodfortheproblemunderstudycanbefoundinJeremicetal.[17];usingtheestimatesproposedthereininsteadofoursubopti-malestimatesmightimproveperformance.Finally,astheinterferencesignalsareGaussianonlytowithinacertaindegreeofapproximation,usingamodelthatallowsfornon-Gaussianinterferencemayalsoimprovetheerrorrate. Wealsostudiedtheapplicationofouralgorithminasystemthatusesspace-timeOFDM.Thisscenarioismoredicult,becauseinterferencesignalsthatoriginatefromauserwithmultipletransmitantennashavearichercorrelationstructurethansignaloriginatingfromasingle-antennatransmitter.Forspace-timeOFDM,ouralgorithmwasabletosuppressco-channelinterferenceonlytoasmalldegree.Itispossiblethattheuseofamoreaccuratemodel,perhapsalongwithalargernumberofreceiveantennas,mayimprovethesituation.Thistopicisalsoleftforfuturework. PAGE 56 [1] J.C.Guey,A.Khayrallah,andG.E.Bottomley,\Adjacentchannelinterferencerejectionforlandmobileradiosystems,"inProc.ofIEEEVehicularTechnologyConference,Ottawa,Canada,vol.3,pp.1715{1719,1998. [2] G.Jongren,D.Astely,andB.Ottersten,\Structuredspatialinterferencerejectioncombining,"inProc.oftheEuropeanSignalProcessingConference,Tampere,Finland,Sept.2000. 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