Chapter 1 Introduction Wireless communications technologies are now prevalent throughout today's society and growing in demand. The whole world are planning and installing radio networks to support communications requirements, the success of these networks may be driven by the availability of the radio frequency spectrum. The radio frequency spectrum, a finite natural resource, has greater demands placed on it every day. In an effort to make the most efficient use of this resource, various technologies have been developed so that multiple, simultaneous users can be supported in a finite amount of spectrum. This concept is called "multiple access." To ensure profit grows parallel with the demand for wireless technologies, manufacturers have had to develop methods of putting more users in the same spectrum space. In this thesis, we focus on the discussion of Orthogonal Frequency Division Multiplex (OFDM), a multiple access technology which has drawn increasing attention recently. 1.1 Advantage of OFDM Systems Frequency division multiplexing (FDM) is a technology that transmits multiple signals simultaneously over a single transmission path, such as a cable or wireless system. Each signal travels within its own unique frequency range (carrier), which is modulated by the data (text, voice, video, etc.). OFDM spread spectrum technique distributes the data over a large number of carriers that are spaced apart at precise frequencies. This spacing provides the "orthogonality" in this 1 technique which prevents the demodulators from seeing frequencies other than their own. In a typical terrestrial broadcasting scenario, there are multipath-channels (i.e. the transmitted signal arrives at the receiver using various paths of different length). Since multiple versions of the signal interfere with each other (inter symbol interference (ISI)), it becomes very hard to extract the original information. The “orthogonality” between sub-carrier makes OFDM outperform other wireless systems in terms of high spectral efficiency, resiliency to RF interference, and lower multi-path distortion. Due to these benefits, OFDM systems have received increasing attention. There is a great interest in using OFDM for high-speed wireless local area network applications. Development is ongoing for wireless point-to-point and point-to-multipoint configurations using OFDM technology. In a supplement to the IEEE 802.11 standard, the working group published IEEE 802.11a, which outlines the use of OFDM in the 5.8 GHz band[42]. OFDM forms the basis for the Digital Audio Broadcasting (DAB) and Digital Video Broadcasting (DVB) standard in the European market. OFDM also forms the basis for the global ADSL (asymmetric digital subscriber line) standard. 1.2 Advantage of MIMO-OFDM Systems The major challenges in future wireless communications system design are increased spectral efficiency and improved link reliability. The wireless channel constitutes a hostile propagation medium, which suffers from fading (caused by destructive addition of multipath components) and interference from other users. Diversity provides the receiver with several (ideally independent) replicas of the transmitted signal and is therefore a 2 powerful means to combat fading and interference and thereby improve link reliability. Common forms of diversity are time diversity (due to Doppler spread) and frequency diversity (due to delay spread). In recent years the use of spatial (or antenna) diversity has become very popular, which is mostly due to the fact that it can be provided without loss in spectral efficiency. Receive diversity, that is, the use of multiple antennas on the receive side of a wireless link, is a well-studied subject [4]. Driven by mobile wireless applications, where it is difficult to deploy multiple antennas in the handset, the use of multiple antennas on the transmit side combined with signal processing and coding has become known under the name of space-time coding and is currently an active area of research. The use of multiple antennas at both ends of a wireless link (multiple-input multiple-output (MIMO) technology) has recently been demonstrated to have the potential of achieving extraordinary data rates. The corresponding technology is known as spatial multiplexing and yields an impressive increase in spectral efficiency. The main motivation for using OFDM in a MIMO channel is the fact that OFDM modulation turns a frequency-selective MIMO channel into a set of parallel frequency MIMO channels. Besides spatial diversity, broadband MIMO channels offer higher capacity and frequency diversity due to delay spread. Orthogonal frequency division multiplexing significantly reduces receiver complexity in wireless broadband systems. The use of MIMO technology in combination with OFDM, i.e., MIMO-OFDM [5,6,7], therefore becomes an attractive solution for future broadband wireless systems. 3 MIMO-OFDM is a technology that uses multiple antennas to transmit and receive radio signals. It allows service providers to deploy a Broadband Wireless Access (BWA) system that has Non-Line-of-Sight (NLOS) functionality. Specifically, MIMO-OFDM takes advantage of the multipath properties of environments using base station antennas that do not have LOS. The MIMO systems use multiple antennas to simultaneously transmit data, in small pieces to the receiver, which can process the data flows and put them back together. This process, called spatial multiplexing, proportionally boosts the data-transmission speed by a factor equal to the number of transmitting antennas. In addition, since all data is transmitted both in the same frequency band and with separate spatial signatures, this technique utilizes spectrum very efficiently. 1.3 Problem in OFDM Systems One of the arguments against OFDM is that, it is sensitive to synchronization errors. There are two main kinds of synchronization errors: time symbol error and Carrier Frequency Offset (CFO). In this thesis, only the effect of CFO is studied. CFO is the difference between the carrier frequency of the received signal and the frequency of the receiver oscillator. It is caused by the Doppler shift and oscillator instabilities. There are two types of carrier frequency offset: Integer CFO and fractional CFO. Carrier frequency errors result in a shift of the received signal’s spectrum in the frequency domain. With the frequency errors as an integer multiple of the subcarrier 4 spacing, the subcarriers are still mutually orthogonal. But the received data symbols, which are mapped to the OFDM spectrum, are in the wrong position in the demodulated spectrum. Fractional CFO spills the energy over the subcarriers, resulting in loss of their mutual orthogonality and hence causes inter-carrier interference(ICI). Both SISO-OFDM and MIMO-OFDM systems suffer from the loss of orthogonality between the sub-carriers due to CFO. CFO attenuates the desired signal, adds phase, and reduces the signal to noise ratio (SNR)[8]. As a result, performance of the systems is severely downgraded. Accurate carrier offset estimation and compensation is more critical in OFDM communication systems than other modulation schemes. In this dissertation the author develop a decoupled maximum likelihood blind carrier offset estimator. The performance of the estimator will be analyzed and compared with other estimators (ESPRIT, CP and hopping pilot approach) in the literature for both SISO-OFDM and MIMO-OFDM systems. Compared to the existing methods, the advantage of the proposed CFO estimator is that 1) It has better spectrum efficiency as it does not require any additional training sequence or pilot symbol. 2) The proposed scheme has better BER performance especially when SNR is low. 5 1.4 Thesis Outline The intention of this chapter is to outline the simulated system environment and highlight the subject matters that are pertinent to the dissertation. Chapter 1 of this dissertation has provided a concise coverage of the relevant materials that are required for the understanding of the subject matter of this dissertation. A more in-depth study of SISOOFDM and MIMO-OFDM communication systems are covered in Chapter 2. In Chapter 3, the effect of CFO to OFDM systems is analyzed. The two main types of CFO estimation schemes in the literature, the data-aided and non-data aided schemes[9], are introduced and compared. Performance of the proposed DEML (Decoupled Maximum Likelihood) blind carrier offset estimator for SISO-OFDM and MIMO-OFDM systems is analyzed in Chapter 4 and Chapter 5, respectively. Finally, Chapter 6 concludes the report with a summary of the results that are obtained and recapitulates the objective of this dissertation. 6 Chapter 2 OFDM Systems In this chapter, OFDM is first compared with other multiple access techniques in order to analyze the benefits of the system. The advantage of OFDM systems over multipath frequency selective fading channel is then addressed. The last part of this chapter gives an overview of OFDM and MIMO-OFDM systems. 2.1 Comparison of OFDM With Other Multiple Access Techniques Multiple access schemes are used to allow many simultaneous users to use the same fixed bandwidth radio spectrum. The bandwidth allocated to any communication system is always limited. For mobile phone systems, the total bandwidth is typically 50 MHz, which is split in half to provide the forward and reverse links of the system. Sharing of the spectrum is required in order to increase the user capacity. Frequency Division Multiple Access (FDMA), Time Division Multiple Access (TDMA) and Code Division Multiple Access (CDMA) are the three major methods of sharing the available bandwidth to multiple users in wireless communication system. 2.1.1 FDMA The first generation of multiple access technique is the analog FDMA systems such as AMPS (Advanced Mobile Phone Services). For a system of FDMA, the available bandwidth is subdivided into a number of sub-channels with narrower bandwidth. Each user is allocated a unique frequency band in which to transmit and receive on. During a call, no other user can share the same frequency band. 7 The main shortcoming of FDMA systems is the bandwidth inefficiency. In FDMA systems, the bandwidth of each channel allocated to each user is typically 10 kHz-30 kHz for voice communications. However, the minimum required bandwidth for speech is only 3 kHz. The extra bandwidth between adjacent signal spectra is called guard band, which is maintained in order to prevent sub-channels from interfering with each other. In a typical FDMA system, up to 50% of the total spectrum is wasted due to the extra spacing between sub-channels. Moreover, precise narrowband filters are necessary for FDMA systems to filter out interference signals from neighboring sub-channels. 2.1.2 TDMA The second generation consists of the first mobile digital communication systems such as the TDMA based GSM (Global System for Mobile Communication). Unlike FDMA system, one user in TDMA system takes all the frequency bandwidth but during a precise interval of time. TDMA divides the available spectrum into multiple time slots, by giving each user a time slot in which they can transmit or receive. In reality, only one person is actually using the channel at any given moment, but he or she only uses it for short bursts. He then gives up the channel momentarily to allow the other users to have their turn. TDMA partly overcomes the problem of low bandwidth inefficiency in FDMA system by using wider bandwidth channels, which are shared by several users. Multiple users access the same channel by transmitting their data in different time slots. TDMA systems are more bandwidth efficient as compared to FDMA systems, since no extra guard band is needed. 8 There are however, two main problems with TDMA. There is an overhead associated with the change-over between users due to time slotting on the channel. A change-over time must be allocated to allow for any tolerance in the start time of each user, due to propagation delay variations and synchronization errors. This limits the number of users in each channel, results in lower system capacity. Another problem in TDMA systems is the multipath delay spread, which is an important parameter to access the performance capabilities of wireless systems. Because there are obstacles and reflectors in the wireless propagation channel, the transmitted signal arrivals at the receiver from various directions over a multiplicity of paths. Such a phenomenon is called multipath. Multiple reflections of the transmitted signal may arrive at the receiver at different time, the time dispersion of the channel is called multipath delay spread. For a reliable communication without using adaptive equalization or other anti-multipath techniuques, the transmitted data rate should be much smaller than the inverse of multipath delay spread[10]. Otherwise, the multipath delay spread will result in Inter Symbol Interference (ISI) (or bits "crashing" into one another) which the receiver cannot sort out. The symbol rate of each channel is high in TDMA systems (as the channel handles the information from multiple users) resulting in problems with multipath delay spead. 9 2.1.3 CDMA Code Division Multiple Access (CDMA) is a spread spectrum technique that uses neither frequency channels nor time slots. With CDMA, the narrow band message (typically digitized voice data) is multiplied by a large bandwidth signal that is a pseudo random noise code (PN code). All users in a CDMA system use the same frequency band and transmit simultaneously. This is possible because the signal of each user is modulated by a unique PN code. It is like everybody is talking at the same time but using different languages, there is no interference between each other because none of the listeners understand any language other than that of the individual to whom they are listening. One of the main advantages of CDMA systems is the capability of using multipath signals that arrive in the receivers with different time delays. FDMA and TDMA, which are narrow band systems, cannot discriminate between the multipath arrivals, and resort to equalization to mitigate the negative effects of multipath. While CDMA systems can make use of the multipath signals and combine them to make an even stronger signal at the receivers by using different technologies, e.g. RAKE receiver [11]. 2.1.4 OFDM Similar to FDMA, OFDM systems achieve multiple user access by subdividing the available bandwidth into multiple channels. However, OFDM uses the spectrum much more efficiently by spacing the channels much closer together. This is achieved by 10 making all the carriers orthogonal to one another, preventing interference between the closely spaced carriers. OFDM overcomes most of the problems with both FDMA and TDMA. It splits the available bandwidth into many narrow band channels (typically 64-4096). The carriers for each channel are made orthogonal to one another, allowing them to be spaced very close together, without guard band as required in the FDMA systems. There is no overhead associated with switching between users, since users in OFDM systems do not need to be time multiplexed as in TDMA systems. Fig 2.1. Spectrum of a single OFDM sub-carrier and OFDM symbol Figure 2.1 shows the spectrum of a single OFDM sub-channel and the spectrum of an OFDM symbol, which are characterized by the fact that spectrum of different sub-carriers overlaps. As shown in the figure, at the centre frequency of each carrier, the amplitude of all other carriers’ signals are zero. This is called the “orthogonality” between sub-carriers. Although the spectrum of different sub-carriers is overlapping with each other, there is no interference caused by other sub-carriers as long as the OFDM signal is transmitted and received at the precise center frequency of each sub-carrier. The orthogonality between 11 sub-carriers allows them to be spaced as close as theoretically possible. This overcomes the problem of large carrier spacing required in FDMA. Each carrier in an OFDM signal has a very narrow bandwidth (i.e. 1 kHz), thus the resulting symbol rate is low. As mentioned in the previous section, signal with low symbol rate has a high tolerance to multipath delay spread, as the delay spread must be very long to cause significant inter-symbol interference. Compared to CDMA, OFDM is more resistant to frequency selective fading since its parallel nature allows errors in sub-carriers to be corrected. OFDM performs much better than CDMA in a multipath environment since it is better at overcoming Inter Symbol Interference (ISI), which happens when reflected signals overlap with the transmitted signal. However, OFDM is more sensitive to frequency offset, which results in Inter Carrier Interference (ICI). 2.2 Multipath Frequency Selective Fading Channel Before getting into the structure of OFDM systems, we would like to discuss the performance of OFDM systems over a multipath frequency selective fading channel. In an ideal radio channel, the received signal would consist of only a single direct path signal, which would be a perfect reconstruction of the transmitted signal. However in a real channel, the signal is modified during transmission in the channel. The received signal consists of a combination of attenuated, reflected, refracted, and diffracted replicas of the transmitted signal. On top of all this, the channel adds noise to the signal and can cause a shift in the carrier frequency if the transmitter, or receiver is moving (Doppler 12 effect). In a radio link, the RF signal from the transmitter may be reflected from objects such as hills, buildings, or vehicles. This gives rise to multiple transmission paths at the receiver. 2.2.1 Advantage of OFDM Systems in Frequency Selective Fading Channel In any radio transmission, the channel spectral response is not flat. It has dips or fades in the response due to reflections causing cancellation of certain frequencies at the receiver. Reflections of near-by objects (e.g. ground, buildings, trees, etc) can lead to multipath signals which have comparable signal power as the direct signal. This can result in deep nulls in the received signal power due to the strong interference signal. For narrow bandwidth transmissions if the null in the frequency response occurs at the transmission frequency then the entire signal can be lost. This can be partly overcome in two ways. By transmitting a wide bandwidth signal or spread spectrum as CDMA system do, any dips in the spectrum only result in a small loss of signal power, rather than a complete loss. Another method is to split the total transmission bandwidth into many narrow-bandwidth carriers. This is exactly what is done in OFDM systems. The original signal is spread over a wide bandwidth, so most likely nulls in the spectrum only affect a small number of carriers rather than the entire signal. The information carried by those lost carriers can be recovered by using some error correction techniques such as Forward Error Correction (FEC) [12]. 13 2.2.2 Guard Interval in OFDM Systems In order to overcome the effect of mulitpath fading channel, guard interval is necessary in OFDM sytems. One of the most important properties of OFDM transmissions is its high level of robustness against multipath delay spread [13][14]. This is a result of the long symbol period used, which minimizes the inter-symbol interference. The level of multipath robustness can be further increased by the addition of a guard period between transmitted symbols. The guard period allows time for multipath signals from the pervious symbol to die away before the information from the current symbol is gathered. The most effective guard period to use is a cyclic extension of the symbol. Part of the end of the symbol waveform is put at the start of the symbol as the guard period, this effectively extends the length of the symbol, while maintaining the orthogonality of the waveform. The cyclic extension of the symbol is called Cyclic Prefix (CP). Figure 2.2 shows the example of CP in OFDM systems. cyclic prefix Time Fig 2.2. Cyclic prefix – a copy of the last part of OFDM symbol CP enables cyclic convolution for each symbol, thus orthogonality is preserved even with imperfect timing and channel impairment. In wireless environment, sub-carriers are still 14 orthogonal as long as the length of CP exceeds the time dispersion of wireless channels (no ISI). This provides multipath immunity as well as symbol time synchronization tolerance. As long as the multipath delay echoes stay within the guard-period duration, there is strictly no limitation regarding the signal level of the echoes: they may even exceed the signal level of the shorter path! The signal from all paths is combined at the input of the receiver and result in a stronger combined signal. Since the FFT is energy conservative, the energy of the combined signal is the summation of all the signal energy from different paths. On the other hand, the delay spread begins to cause inter-symbol interference when it is longer than the guard interval. However, they do not cause significant problems as long as the echoed signal is sufficiently weak. This is true most of the time as multipath echoes delayed longer than the guard period will have been reflected of very distant objects. Other variations of guard periods are possible. One possible variation is to insert zeroamplitude signal into adjacednt OFDM systems. The OFDM symbols can be easily identified by using this method. It also allows for symbol timing to be recovered from the signal, simply by applying envelop detection. The disadvantage of using this guard period method is that the zero period does not give any multipath tolerance. Throughout this work, the CP based guard interval is adopted. 15 2.3 Generation of OFDM Systems In this section, it is introduced that how data is modulated and demodulated in OFDM systems. Details are given on how data is modulated and transmitted and then recovered at the receiver in a digital approach of the OFDM scheme. 2.3.1 FFT and IFFT in OFDM Systems In order to achieve a high spectral efficiency, the frequency response of the sub-channels are overlapping and orthogonal to each other, which gives the name of OFDM. To generate OFDM symbols successfully, the relationship between all the carriers must be carefully controlled to maintain the orthogonality of the carriers. OFDM is generated by firstly choosing the spectrum required, based on the input data, and modulation scheme used. Each carrier to be produced is assigned some data to transmit. The required amplitude and phase of the carrier is then calculated based on the modulation scheme (typically differential BPSK, QPSK, or QAM). The required spectrum is then converted back to its time domain signal using an Inverse Fourier Transform. In most applications, an Inverse Fast Fourier Transform (IFFT) is used. The IFFT performs the transformation very efficiently, and provides a simple way of ensuring the carrier signals produced are orthogonal. The FFT transforms a cyclic time domain signal into its equivalent frequency spectrum. This is done by finding the equivalent waveform, generated by a sum of orthogonal sinusoidal components. The amplitude and phase of the sinusoidal components represent the frequency spectrum of the time domain signal. The IFFT performs the reverse process, transforming a spectrum (amplitude and phase of each component) into a time domain 16 signal. An IFFT converts a number of complex data points, of length that is a power of 2, into the time domain signal of the same number of points. Each data point in frequency spectrum used for an FFT or IFFT is called a bin. The orthogonal carriers required for the OFDM signal can be easily generated by setting the amplitude and phase of each frequency bin, then performing the IFFT. Since each bin of an IFFT corresponds to the amplitude and phase of a set of orthogonal sinusoids, the reverse process guarantees that the carriers generated are orthogonal 2.3.2 Digital Approach of OFDM A possible realization of an OFDM scheme can be dramatically simplified if a digital approach is used. The approach is based on the use of FFT to generate and to demodulate the transmitted signal. The flowing figure shows the digital implementation of OFDM systems. cos( 2 π f c t ) sk S Bit Stream / P QAM Encoder xk Nc-IFFT Real D/A x Re (U (t ) ) P / S Im (U (t ) ) Imag D/A Transmitter + BPF x s(t) − sin( 2π f c t ) n(t) r(t) cos( 2 π f c t ) S I (t ) x LPF Bit Stream P / S QAM Decoder Nc-FFT S / P BPF A/D SQ (t ) x LPF Receiver − sin( 2π f c t ) Figure 2.3 A digital approach of OFDM systems 17 Consider a sequence of N symbols, each symbol being represented by a point in a 2-D constellation. These symbols can be written as: sk = rk e jϕk = ak + jbk (2.1) where ak , bk are the coordinates of the point that represents the symbol k. Then an inverse fast Fourier transform is computed on this set of symbols. N −1 ∑ sk exp( j 2π nk / N ); xk = n = 0,1….N-1 (2.2) k =0 The signal xk feeds a digital to analog converter to give the complex baseband signal U (t ) . U (t ) = N −1 ∑ xk exp( j 2π f k t ) k =0 (2.3) where f k = k / T , 0 ≤ t ≤ T and T is the symbol duration Then the signal is converted to radio frequency and transmitted through channel. S (t ) = Re (U (t ) ) cos(2π f c t ) − Im (U (t ) ) sin(2π f c t ) N −1 = ∑ k =0 (2.4) ak cos ( 2π ( f c + f k )t ) − bk sin ( 2π ( f c + f k )t ) where f c is the center frequency. S (t ) is the transmitted signal during one modulation period, one complete OFDM symbol S u (t ) in the time domain is described as S u (t ) = ∞ ∑S j = −∞ j (t − jT ) × Π (t − jT ) (2.5) 18 where Π (t ) = 1 . 0≤t > 1 (4.8) where K is the total number of blocks collected for the estimation of carrier frequency offset and I Nc is an identity matrix of size N c × N c . Similarly, R nn ( K ) = 1 K K −1 ∑ ηk ηk H = σ η 2 I N k =0 c for K >> 1 (4.9) It is reasonable that we assume the signal and noise are uncorrelated, i.e., lim K →∞ 1 K K −1 ∑ x k ηk H = 0 N k =0 c (4.10) The covariance matrix of received signal at the nth receiver is 38 R yy ( K ) = = 1 K 1 K K −1 ∑ yk yk H for K >> 1 k =0 K −1 ∑ ( EHxk + ηk )( EHxk + ηk ) H k =0 = EHR xx ( K )H H E H + ση 2 I Nc = EHH H E H + ση 2I N c (4.11) From equation (4.11), we can observe that the transmitted sequence is eliminated by the correlation of received data sequence y k . We can rewrite R yy ( K ) as the Hadamard Product of two matrices ( ) R yy ( K ) = Pφ • HH H + ση 2 I N c (4.12) where ⎡ 1 ⎢ jφ ⎢ e ⎢ ⎢ Pφ = ⎢ ⎢ ⎢e j ( L −1)φ ⎢ ⎢ ⎢ ⎢⎣ 0 e− jφ e − j ( L −1)φ 1 e − jφ e jφ 1 e j ( L −1)φ ⎤ ⎥ ⎥ ⎥ ⎥ e − j ( L −1)φ ⎥ ⎥ ⎥ ⎥ e − jφ ⎥ ⎥ 1 ⎥⎦ 0 e jφ Invoking an approach similar to that used in Jiun H. Yu and Yu T. Su’s paper [34], we set z = e jφ (4.13) and the parameter matrix 39 ⎡ 1 ⎢ ⎢ z ⎢ ⎢ P( z ) = ⎢ ⎢ ( L −1) ⎢z ⎢ ⎢ ⎢⎢ 0 ⎣ z −1 z − ( L −1) z −1 1 z 1 z ( L −1) ⎤ ⎥ ⎥ ⎥ ⎥ − ( L −1) ⎥ z ⎥ ⎥ −1 ⎥ z ⎥ 1 ⎥⎥⎦ 0 z Equation (4.12) can be rewritten as ( ) R yy ( K ) = P ( z ) • HH H + ση 2I N c (4.14) Define ( ) F ( z ) = R yy ( K ) − P( z ) • HH H − ση 2 I N c (4.15) The desired value of z should minimize the Frobenius norm of F ( z ) , that is z = arg min(|| F ( z ) ||F ) = arg min Q( z ) . (4.16) Q( z ) = Tr (F ( z )F H ( z )) (4.17) z z and is a polynomial of z . The minimization can be achieved by adaptive method or an exhaustive searching over the region φ ∈ [0,2π ) . However, the former suffers from a slow convergence rate as well as a local minimum problem while the latter is quite computational expensive since it involves blind search in the whole region [0, 2π ) . Consequently, we propose a polynomial rooting method that is well known to be highly efficient yet does not have a convergence problem. Replacing z by e jφ , 40 Q( z ) = Q(e jφ ) = Tr (F(e jφ )F H (e jφ )) . Define Q(e jφ ) as the differentiation of Q(e jφ ) with respect to φ , then φ can be obtained from one root of Q(e jφ ) Q(e jφ ) = ∂Q(e jφ ) φ =0 (4.18) Again, after representing e jφ by z , Q(e jφ ) can be expressed as Q( z ) . Then, z = e jφ should be one root of Q( z ) and many polynomial rooting algorithm can be applied to calculate the roots of Q( z ) . Note that there may exist multiple roots for equation (4.18). However, the desired root is selected following two criterions: 1. The root z should stay around the unit circle. 2. The root should be chosen as the one that minimizes Q( z ) 4.3 Theoretical Analysis In order to theoretically analyze the performance of the estimator, we derive Cramér-Rao Bound (CRB) for the proposed estimator. CRB describes the ability to estimate a specific set of parameters, without regard to an unknown set of other parameters that influence the measured data. 4.3.1 Cramér-Rao Bound and Fisher Information Matrix The concept of Fisher Information Matrix (FIM) is introduced to derive the Cramér-Rao Bound. Fisher Information is a measure of the information content of the measured signal relative to a particular parameter. The Cramér-Rao Bound is a lower bound on the error variance of the best estimator for estimating this parameter with the given system. 41 Let the unknown system parameters of a given system be denoted by the length d vector α α = [α1 , α 2 ,..., α d ]T (4.19) Define the noiseless measurement x(α ) = f(α1 , α 2 ,..., α d ) , where f(•) is some unkown vector function. The actual measurement in any real system will always be corrupted by noise. The limit of this noise will be signal dependent shot noise or detector quantization noise. Let the noisy measurement given by y (α ) , where y (α ) = x(α ) + η . Without loss of generality, assume η is a zero mean white Gaussian noise with variance ση2 . With the given system, the ability to estimate α is bounded by the Cramér-Rao Bound [35][36]. The variance of any unbiased estimator of one component of α , say αi , is bounded as below var(α i ) ≥ J ii−1 (α ) (4.20) where J (α ) is the FIM of the parameter vector α , and J ii−1 is the i th diagonal element of J −1 . Let p(y (α )) be the probability density function for the observed noisy data y . The FIM is then given by ⎡∂ ⎤⎡ ∂ ⎤ J(α ) = E ⎢ ln p (y (α )) ⎥ ⎢ ln p(y (α )) ⎥ ⎣ ∂α ⎦ ⎣ ∂α ⎦ T (4.21) 42 It is computationally expensive to directly apply the equation above. Petre Stoic, Eric G. Larasson and Alex B. Gershman proposed a simplifed FIM under the assumption that the signals are temporally white and the noise is both spatially and temporally white [37]. Equation (4.21) is simplified to the following ⎛ dR −1 dR −1 ⎞ J p ,k = KTr ⎜ R R ⎟ ⎜ da p ⎟ da k ⎝ ⎠ p, k = 1,..., d (4.22) where R = E[y (α )y (α ) H ] and K is the number of data snapshots. 4.3.2 CRB of the DEML Estimator Since we assume perfect channel knowledge and there is only one unknown in matrix E . Equation (4.19) is simplified to a = [α1 , α 2 ,..., α d ]T = [φ ση2 ]T (4.23) The Fisher Information Matrix J can be derived from the following equation ⎛ dR yy ⎞ dR yy J p ,k = KTr ⎜ R yy −1 R yy −1 ⎟ , p, k = 1, 2 ⎜ da p ⎟ dak ⎝ ⎠ dR yy da1 = dR yy dφ = jDEHH H E H − jEHH H E H D (4.24) (4.25) Where D is a diagonal matrix 43 ⎡0 ⎢ D=⎢ ⎢ ⎢ ⎣⎢0 dR yy da2 = 0 ⎤ ⎥ ⎥ ⎥ ⎥ N c − 1⎦⎥ 1 dR yy d ση2 (4.26) = I Nc (4.27) = jDEHH H E H − jEHH H E H D (4.28) Define R yy _ φ = dR yy dφ the FIM for the particular OFDM model defined in section 4.1 can be represent by J12 ⎤ ⎡J J = ⎢ 11 ⎥ ⎣ J 21 J 22 ⎦ ⎡Tr (R yy _ φ R yy −1R yy _ φ R yy −1 ) Tr (R yy _ φ R yy −1R yy −1 ) ⎤ =K⎢ ⎥ −1 −1 Tr (R yy −1R yy −1 ) ⎥⎦ ⎢⎣ Tr (R yy R yy _ φ R yy ) (4.29) And the Cramér-Rao Bound matrix is CRB = J −1 ( ⎡ J − J J −1J = ⎢ 11 12 22 21 ⎢ ∆ ⎣ ) −1 ∆⎤ ⎥ ∆ ⎥⎦ (4.30) According to equation (4.20), the variance of any unbiased estimator of φ , is bounded below as −1 var(φ ) ≥ J11 = CRB11 ( = J11 − J12 J 22 −1J 21 ) −1 (4.31) 44 4.4 Numerical Results Simulations are performed to illustrate the efficiency of the proposed algorithm. Simulation results for the normalized mean square error (MSE) against the SNR and the value of L are presented. Because of the property of IFFT, the number of subcarriers, N c has to be exponential of 2. N c is taken as 64 in this work. All 64 subcarriers are used. Perfect channel knowledge is assumed in the simulation. The normalized mean square error defined below is employed as the performance measure of the estimator 1 MSE = Nt ⎛ φˆj − φ j ∑ ⎜⎜ ω j =1 ⎝ Nt ⎞ ⎟ ⎟ ⎠ 2 (4.32) where ω = 2π N c is the channel spacing and N t is the number of Monte-Carlo (MC) trials. The signal-to-noise ratio (SNR) is defined as SNR = 20 log || y k || || ηk || (4.33) Figure4.2 shows the normalized MSE versus SNR values for different K value of 1, 50 and 100. The actual carrier offset is fixed at the value of 0.1 ω . The dotted line shows the simulation result of Ufuk Tureli and Hui Liu’s method (ESPRIT) [1]. The result of CPbased maximum likelihood method by Van de Beek [2] is also plotted in Figure4.2. The number of Monte Carlo trials N t is 200. In order to compare the performance fairly, we adopt the same multipath channel impulse response as the one in [25], which is h = [0.77 + 0.38 j ,0,0,0,0,0,0,0,0.58 j ,−0.58 − 0.67 j ] . Obviously, the proposed decoupled method has better performance than ESPRIT and CP based methods. 45 It is noted that the decoupled algorithm works well at very low SNR value, this can be explained using the equation (4.8) in section 4.3. If K is sufficiently large, the correlation of the additive white Gaussian noise vector is equal to the identity matrix multiplied by a constant. The effect of the AWGN noise is eliminated for the ideal case K → ∞ , so our method works even when the SNR is very low. Figure 4.2. MSE versus SNR with fixed carrier offset 0.1 ω 46 Figure 4.3. MSE versus K with fixed carrier offset 0.1 ω Figure4.3 shows the MSE versus the value of K with SNR of 5dB, 10dB and 20dB. As shown in Figure 4.2, the normalized MSE decrease when SNR increases. However, when the SNR reaches a quite high value, error floor occurs. This is also shown in Fig 4.3, the normalized MSE difference between the two curves with SNR=5dB and SNR=10dB is obviously larger than the difference between SNR=10dB and SNR=20dB. The reason is that decoupled method is not optimum. The precondition of the decoupled method is given in equation (4.7). Practically, the correlation of transmitted sequence is only approximately, but not exactly equal to the identity matrix. For the larger values of K , ( ) ( ( R xx ( K ) is close to an identity matrix. As a result, R yy ( K ) − Pφ • HH H ) ) becomes smaller with the increase of the value of K . Hence, a more accurate carrier offset 47 estimation is obtained. This conclusion is proved by the simulation results shown in Figure 4.3 . The error floor can also be explained by the theory of Cram é r-Rao Boundderived in the next section. In summary, there are two factors causing the error in the decoupled method, one is the additive noise, the other is the non-orhogonality of the transmitted sequence. At low SNR value, the additive noise is the dominating factor causing the error, so the MSE decreases with the increases of SNR. But at high SNR value, the noise is no longer the dominating factor to cause the error. Instead, the difference between the correlation matrix of transmitted sequence R xx ( K ) and the identity matrix leads to the error of carrier offset estimation. In order to get a more precise estimation at high SNR value, the only way is to increase the value of K . Figure 4.4. MSE versus SNR with random carrier offset φ 48 Figure 4.5 MSE versus actual carrier offset φ (SNR=10dB,K=100) Figure 4.4 shows the simulation results of normalized MSE with a randomly generated actual carrier offset φ , which is uniformly distributed over the interval [−0.5ω ,0.5ω ] . The performance is similar to the case with a fixed carrier offset. The normalized MSE versus the actual carrier offset is plotted in Figure 4.5. The results show that the proposed algorithm works for arbitrary values of carrier offset. Figure 4.6 and Figure 4.7 compare the simulation results with the derived Cramer-Rao Bound. 49 Figure 4.6 Comparison of simulation result with CRB (fix SNR) Figure 4.7 Comparison of simulation result with CRB (fix K) 50 As shown in Fig 4.6 and 4.7, there are gaps between the simulation result and the theoretical Cramér-Rao Bound. This is because the DEML estimator is sub-optimum. It is assumed that R xx ( K ) = 1 K K −1 ∑ xk xk H = I N k =0 (4.34) c Actually, the equation only holds when infinite blocks are collected for estimation. It is also observed that there exist error floor in Fig 4.7. This can be explained from the theoretical derivation of CRB. R yy _ φ = jDEHH H E H − jEHH H E H D is independent of the variance of noise, i.e, is independent of SNR. ( R yy −1 = EHH H E H + σ η 2 I N c ) −1 (4.35) When SNR increases to a very large value, the second part of R yy −1 , ση 2 I N c become neglectable. ( ≈ ( EHH R yy −1 = EHH H E H + ση 2 I N c H EH ) ) −1 (4.36) −1 R yy −1 is also independent of SNR value when SNR increase to a very high value. As mentioned previously, −1 var(φ ) ≥ J11 = CRB11 ( = J11 − J12 J 22 −1J 21 ) −1 (4.37) Where 51 ⎡Tr (R yy _ φ R yy −1R yy _ φ R yy −1 ) Tr (R yy _ φ R yy −1R yy −1 ) ⎤ J=K⎢ ⎥ −1 −1 Tr (R yy −1R yy −1 ) ⎥⎦ ⎢⎣ Tr (R yy R yy _ φ R yy ) (4.38) So, the Cramér-Rao Bound becomes independent of SNR value when SNR is very high. This is consistent with the simulation results shown in Fig 4.7. In this chapter, a blind carrier offset estimation method based on decoupled maximum likelihood is proposed for Single Input Single Output (SISO) OFDM communication systems over multipath fading channels. In this method, no additional training sequence or pilot symbols are required. Moreover, the method doesn’t involve any matrix inversion as many other algorithms do. We have compared the performance with CP based scheme and the ESPRIT-like blind estimation algorithm raised by Ufuk Tureli and Hui Liu. It has been shown that the decoupled method performs well especially at very low SNR value. As a result, the carrier offset is estimated accurately even if the signal is transmitted with very low signal energy. Hence, the energy efficiency of the OFDM systems is highly improved. Moreover, the proposed algorithm is more bandwidth efficient as compared to the CPbased scheme when multipath fading channel is involved, since it doesn’t require any excess CP. In the next Chapter, we will analyze the performance of the DEML for Multiple Input Multiple Output (MIMO) OFDM communication systems. 52 Chapter 5 DEML Blind CFO Estimator For MIMO-OFDM systems The previous chapter introduces and analyzes the DEML carrier frequency offset estimator for a single-input and single-output (SISO) OFDM system. As introduced in the first chapter, Both of SISO-OFDM and MIMO-OFDM systems suffer from the loss of orthogonality between the sub-carriers due to carrier frequency offset. This chapter analyzed the DEML estimator for a MIMO-OFDM system. 5.1 MIMO-OFDM Systems Model Baseband equivalent representation of a MIMO-OFDM system is shown in Figure1. Let us define s m,k mth antenna. T ⎡⎣ sm,k (0),.....sm,k ( N c − 1) ⎤⎦ as k th block of data to be transmitted at the There are M and N antenna at the transmitter and receiver respectively. The input data is first fed into a space division multiplexer to yield blocks {s m ,k } M m =1 . In a practical OFDM systems, the number of sub-carriers that are used to transmit data is generally smaller than the size of the discrete Fourier transform (DFT) block, to prevent aliasing by oversampling and/or for transmit filtering. In this paper, we assume that all Nc sub-carriers carry information. OFDM modulation is implemented by applying an inverse discrete Fourier Transform (IDFT) operator to the data streams s m,k . 53 Space Input Division Data Multiplexing S/P IDFT IDFT P/S P/S Add CP Add CP Remove CP Remove CP S/P S/P DFT DFT P/S P/S CFO Estimator S/P Output Data Space Division Multiplexing Figure 5.1 MIMO-OFDM discrete equivalent baseband model 54 The transmitted N c -point time domain signal is given by x m,k = Ws m,k = ⎡⎣ x m,k (0),.....x m,k ( N c − 1) ⎤⎦ T (5.1) where W is the Nc × N c inverse DFT (IDFT) matrix: 1 ⎡1 ⎢ e jω 1 ⎢ W= ⎢ ⎢ ⎢⎣1 e j ( Nc −1) ω ⎤ ⎥ e ⎥ ⎥ ⎥ e j ( Nc −1)×( Nc −1) ω ⎥⎦ 1 j ( N c −1) ω (5.2) where ω = 2π N c is the channel spacing. For DFT-based OFDM, a cyclic prefix is added to the output of the IDFT before it is transmitted through a fading channel. The length of the cyclic prefix, N g samples, is chosen such that it is longer than the impulse response of the channel to avoid inter symbol interference (ISI) and to preserve the orthogonality between subchannels. The baseband signal is upconverted to radio frequency (RF) before being transmitted over a frequency selective fading channel. In this chapter, we employ the same channel model used in SISO-OFDM system. The finite impulse response of the multipath slowly time varying fading channel is defined as hm,n (l ), l ∈ [0, L − 1], m ∈ [1, M ], n ∈ [1, N ] . At the receiver, the received RF signal is downconverted using a receiver local oscillator. In the absence of the carrier frequency offset (CFO), which due to the mismatch between transmit-receive oscillators, the k th received block at the nth receiver after the removal of the cyclic prefix is given by M L −1 yn ,k (i ) = ∑∑ hm ,n (l ) xm,k ( (i − l ) mod N c ) +ηn ,k (i ) m =1 l = 0 i = 0,......, N c − 1 n = 1,......, N (5.3) 55 where hm,n (l ) (0 ≤ l ≤ L − 1) are independent complex-valued Rayleigh fading random variables between mth transmitter and nth receiver, and ηn,k (i ) (i = 0,.., N c − 1) are independent complex-valued Gaussian random variables with zero mean and variance ση2 . L is the length of the channel impulse response. The circular convolution of the channel with the data given by equation (5.3) can also be represented using matrix representation as follows M y n ,k = ∑ H m,n x m,k + ηn ,k , n = 1,......, N (5.4) m=1 where 0 0 hm,n (L −1) hm,n (2) hm,n (1) ⎤ ⎡ hm,n (0) ⎢ h (1) h (0) ⎥ 0 0 m,n ⎢ m,n ⎥ ⎢ ⎥ hm,n (1) 0 0 hm,n (L −1) ⎢ ⎥ 0 hm,n (L −1)⎥ Hm,n = ⎢hm,n (L −1) ⎢ 0 ⎥ hm,n (0) hm,n (0) 0 ⎢ ⎥ hm,n (0) 0 ⎥ ⎢ 0 ⎢⎢ 0 0 hm,n (L −1) hm,n (L − 2) hm,n (1) hm,n (0) ⎥⎥⎦ ⎣ is a lower triangular matrix with dimension N c × N c . In the presence of a carrier offset, e jφ , the receiver input is modulated by E = diag ⎡⎣1, e jφ ,.....e j ( Nc −1) φ ⎤⎦ and the nth receiver input becomes y n ,k = M ∑ EH m ,n x m ,k e m =1 j φ ( k −1)( N c + N g ) + η n ,k (5.5) Since W H EW ≠ I , the E matrix destroys the orthogonality among sub-carriers and thus introduces inter-channel interference (ICI). In order to recover the transmitted signal, the carrier offset, φ must be estimated and compensated before performing the DFT. 56 5.2 Blind Carrier Offset Estimator In this section, we derive the cost function for estimating the carrier frequency offset φ . The channel here is assumed to be slowly time varying which is valid for most broadcasting OFDM systems. So the channel response matrix H m,n remains unchanged for certain block intervals. The noise is assumed to be additive, white and Gaussian (AWGN). As proved in chapter 4, xm,k (i ) ’s orthogonal to each other and it is reasonable to assume the data sequences in different transmitter are uncorrelated. This leads to the key idea of the decoupled method, i.e., R xm x ( K ) = m 1 K ⎧⎪0N H = x x ∑ m , k m , k ⎨I c k =0 ⎪⎩ Nc K −1 m≠m m=m (5.6) for K >> 1 , K is the number of blocks collected to estimate the carrier frequency offset. I N is an identity matrix of size Nc × N c . Since xm,k (i ) are i.i.d Gaussian distributed c random variables, it is not difficult to show the validity of equation (5.6). The result of equation (5.6) shows that it is possible to estimate the CFO without the knowledge of transmitted sequence. Similarly, 1 R ηn η ( L) = n K K −1 ∑ ηn,k ηn,k k =0 H 2 ⎪⎧ση I Nc =⎨ ⎪⎩0Nc n=n n≠n for K >> 1 . (5.7) It is also reasonable that we assume the signal and noise are uncorrelated, i.e. lim K →∞ 1 K K −1 ∑ x m , k ηn , k H = 0 N k =0 c (5.8) The covariance matrix of received signal at the nth receiver is 57 R ynyn ( K ) = = 1 K 1 K K −1 ∑ y n ,k y n ,k H for K >> 1 k =0 ⎧M ⎫ jφ ( k −1)( N c + N g ) + ηn , k ⎬ ∑ ⎨∑ EH m,n xm,k e k =0 ⎩ m =1 ⎭ K −1 ⎧M ⎫ jφ ( k −1)( Nc + N g ) + ηn , k ⎬ ⎨ ∑ EH m,n x m,k e ⎩ m=1 ⎭ M ⎧ ⎫ = E ⎨ ∑ H m,n R xm xm ( K )H m,n H ⎬ E H + ση2I Nc ⎩ m=1 ⎭ H = EG n E H + ση2I Nc (5.9) where Gn M ∑ H m,n H m,n H (5.10) m =1 From equation (5.9), we can observe that the transmitted sequence is eliminated by the ensemble correlation of received data sequences y n,k . Again we can rewrite R y n y n ( K ) as the Hadamard Product of two matrices Pφ and G n R y n y n ( K ) = Pφ • G n + ση 2I N c (5.11) where ⎡ 1 ⎢ jφ ⎢ e ⎢ ⎢ Pφ = ⎢ ⎢ ⎢e j ( L −1)φ ⎢ ⎢ ⎢ ⎣⎢ 0 e− jφ e − j ( L −1)φ 1 e − jφ e jφ 1 e j ( L −1)φ ⎤ ⎥ ⎥ ⎥ ⎥ − j ( L −1)φ ⎥ e ⎥ ⎥ ⎥ e − jφ ⎥ ⎥ 1 ⎦⎥ 0 e jφ Let R yy ( K ) be the mean of covariance matrices got from each antenna 58 R yy ( K ) = 1 N N −1 ∑ R y y (K ) n n n=0 ⎛ 1 N −1 ⎞ = E ⎜ ∑ G n ⎟ E H + ση2 I N c ⎝ N n=0 ⎠ N −1 ⎛1 ⎞ = Pφ • ⎜ ∑ G n ⎟ + ση2I N c ⎝ N n =0 ⎠ (5.12) Using the same approach adopted in last chapter, set z = e jφ , and the parameter matrix ⎡ 1 ⎢ ⎢ z ⎢ ⎢ P( z ) = ⎢ ⎢ ( L −1) ⎢z ⎢ ⎢ ⎢⎢ 0 ⎣ z −1 z − ( L −1) 1 z −1 z 1 z ( L −1) ⎤ ⎥ ⎥ ⎥ ⎥ z − ( L −1) ⎥ ⎥ ⎥ −1 ⎥ z ⎥ 1 ⎥⎥⎦ 0 z Equation (5.12) can be written as R yy ( K ) = P ( z ) • U + ση 2I N c (5.13) where U= 1 N N −1 ∑ Gn (5.14) n =0 the desired value of z should minimize the Frobenius norm of F( z ) , that is z = arg min(|| F ( z ) ||F ) = arg min Q( z ) . (5.15) Q( z ) = Tr (F( z )F H ( z ))) (5.16) z z Where 59 is a polynomial of z . Similar to the method adopted in the previous chapter for SISOOFDM systems, φ can be obtained from the root of Q(e jφ ) , which is the differentiation of Q(e jφ ) with respect to φ Q(e jφ ) = ∂Q(e jφ ) φ =0 (5.17) Again, there may exist multiple roots for equation (5.17) However, the desired root is selected following two criterions: 1. The root z should stay around the unit circle. 2. The root should be chosen as the one that minimizes Q( z ) . 5.3 Theoretical Analysis Similar to the case of SISO-OFDM, we assume perfect channel knowledge and there is only one unknown in matrix E . Equation (4.20) is simplified to a = [α1 , α 2 , α d ]T = [φ ση2 ]T (5.18) And the Fisher information matrix is ⎛ dR yy ⎞ dR yy J p ,k = KTr ⎜ R yy −1 R yy −1 ⎟ , p, k = 1, 2 ⎜ da p ⎟ dak ⎝ ⎠ dR yy da1 = dR yy dφ = jDEUE H − jEUE H D (5.19) (5.20) Where 60 ⎡0 ⎢ D=⎢ ⎢ ⎢ ⎣⎢0 dR yy da2 = 0 ⎤ ⎥ ⎥ ⎥ ⎥ N c − 1⎦⎥ 1 dR yy d ση2 = I Nc (5.21) = jDEUE H − jEUE H D (5.22) Define R yy _ φ = dR yy dφ The Fisher Information Matrix is ⎡Tr (R yy _ φ R yy −1R yy _ φ R yy −1 ) Tr (R yy _ φ R yy −1R yy −1 ) ⎤ J=K⎢ ⎥ −1 −1 Tr (R yy −1R yy −1 ) ⎥⎦ ⎢⎣ Tr (R yy R yy _ φ R yy ) (5.23) And the Cramér-Rao Bound matrix is given by CRB = J −1 ( ⎡ J − J J −1J = ⎢ 11 12 22 21 ⎢ ∆ ⎣ ) −1 ∆⎤ ⎥ ∆ ⎥⎦ (5.24) The variance φ , is bounded below as −1 var(φ ) ≥ J11 = CRB11 ( = J11 − J12 J 22 −1J 21 ) −1 (5.25) 61 5.4 Numerical Results Simulations are performed to illustrate the performance of proposed CFO estimator. Simulation results for the Normalized mean square error (MSE) against the SNR and the value of K are presented. The number of sub-carriers, N c is taken as 64. All 64 subcarriers are used. Perfect channel knowledge is assumed in the simulation. The normalized mean square error defined below is employed as the performance measure of In this chapter, we consider exponential power profile, Rayleigh fading independent taps, that have, σl2 = σ12e − l −1 L' (5.26) L ∑ σl2 = 1 (5.27) l =1 The parameter L ' in (5.26) decides the rate of power decay on successive paths, with L' = L , we get σ12 = 1 − e −1/ L 1 − e −1 (5.28) 62 Figure 5.2. MSE versus SNR with M=N=2, L=4 Figure5.2 shows the normalized MSE versus SNR values for different K value of L + 1 , Nc , 2 Nc and 3N c . The carrier frequency offset is randomly selected. The dotted line shows the simulation results of Mi-Kyung Oh et al., hopping pilots method [3]. Similar to the proposed estimator, this hopping pilot approach also estimate the CFO by collecting several blocks instead of estimating per block basis. In order to compare the simulation results fairly, we adopt the same parameter as in [3], where L = 4 , M = N = 2 , N c = 64 . The number of Monte Carlo trials Nt is 200. The solid and dotted lines represent the result of the proposed method and the hopping pilots approach respectively. As shown in Figure 5.2, the proposed method works much better than the hopping pilots algorithm when the number of collected blocks is small. When the number of collected blocks is quite large like 2 N c or 3N c , the simulation result of the hopping pilot approach is slightly better than the proposed one. This may be due to more pilot symbols obtained when the 63 number of collected blocks is increased. However, the bandwidth efficiency of Mi-Hyung et al., approach suffers because of the inserted pilot symbols. This is the trade-off between bandwidth efficiency and performance of the estimator. Diversity gain is a measure of improved estimator robustness to channel fading. In order to show the diversity gain of the proposed estimator for MIMO-OFDM systems, Figure 5.3 is the plot to compare the NMSE of CFO for MIMO-OFDM ( M = N = 2 ) and for single-antenna OFDM ( M = N = 1 ). Figure5.3. MSE versus SNR for MIMO and SISO OFDM 64 As expected, the proposed CFO estimator works better for the case of MIMO systems. When one of the channels suffer from deep fading, the CFO still can be recovered from other channels which do not suffer from fading that much. This is why the estimator outperforms in MIMO systemss than SISO systems. This result also proves that MIMOOFDM is more robust to multipath fading than SISO-OFDM. It is noted that there is an error floor in Figure 5.3. The reason is that decoupled method is not optimum. The precondition of the decoupled method is given in equation (5.5). The transmitted sequence will be orthogonal when K approaches infinity. But practically, the correlation of transmitted sequence is only approximately, but not exactly equal to the identity matrix. There are two factors causing the error in the decoupled method, one is the additive noise, the other is the non-orhogonality of the transmitted sequence. At low SNR value, the additive noise is the dominating factor causing the error, so the MSE decreases with the increases of SNR. But at high SNR value, the noise is no longer the dominating factor to cause the error. Instead, the difference between the correlation matrix of transmitted sequence Rx m xm (K ) and the identity matrix leads to the error of carrier offset estimation. In order to get a more precise estimation at high SNR value, the only way is to increase the value of K . This is also shown in Figure 5.4. 65 Figure 5.4. MSE versus K with SNR=5,10,20dB Figure 5.4 shows the MSE versus the value of K with SNR of 5dB, 10dB and 20dB. As shown in Figure5.4, the normalized MSE difference between the two curves with SNR=5dB and SNR=10dB is obviously larger than the difference between SNR=10dB and SNR=20dB. The reason is same as the one stated in the last paragraph. The Cramér-Rao Bound is plotted to compare the performance with the simulation results in Fig 5.5 and 5.6. There are gaps between the simulation result and the theoretical CRB because of the same reason stated in the previous chapter. 66 Figure 5.5 Comparison of CRB and simulation results (fix K) Figure 5.6 Comparison of CRB and simulation results (fix SNR) 67 In this chapter, the blind carrier offset estimation method is adopted for MIMO-OFDM communication systems over multipath fading channels. In this method, no additional training sequence or pilot symbols are required. Moreover, the method doesn’t involve any matrix inversion as many other algorithms do. It has been proved that the performance of the proposed estimator is comparable with the hopping pilot method in [30]. Moreover, the proposed algorithm is more bandwidth efficient as compared to most of existing method based on the pilot or CP scheme when multipath fading channel is involved since it doesn’t require any excess CP. 68 Chapter 6 Conclusions and Future Work In this dissertation, we proposed a decoupled maximum likelihood blind estimation (DEML). method which does not involve any pilot or training symbols. Unlike most of traditional maximum likelihood methods, the proposed method doesn’t rely on the structure of cyclic prefix either. The proposed method is compared with two well-known existing methods in the literature: Ufuk Tureli and Hui Liu’s method (ESPRIT) and Cyclic Prefix based maximum likelihood method by Van de Beek. It is proved that the decoupled algorithm works well especially at very low SNR value. This is because the effect of AWGN noise is eliminated when there are sufficient number of blocks were collected. In chapter five, the performance of the proposed estimator for MIMO-OFDM systems is compared with the simulation results of Mi-Kyung Oh et al., hopping pilots method. The proposed method works much better than the hopping pilots algorithm when the number of collected blocks is small. When the number of collected blocks is quite large and more pilots symbols were obtained for the hopping pilots method, the simulation result of the hopping pilot approach is slightly better than the proposed one. However, the bandwidth efficiency of Mi-Hyung et al., approach suffers because of the inserted pilot symbols. This is the trade-off between bandwidth efficiency and performance of the estimator. The performance of the estimator for SISO-OFDM and MIMO-OFDM systems are also compared. The proposed CFO estimator works better for the case of MIMO-OFDM systems. When one of the channels suffer from deep fading, the CFO still can be 69 recovered from other channels which do not suffer from fading that much. This is why the estimator outperforms in MIMO systemss than SISO systems. This result also proves that MIMO-OFDM is more robust to multipath fading than SISO-OFDM. The simulation results for both SISO-OFDM and MIMO-OFDM systems show that there exists error floor when the SNR value is large. The reason is that decoupled method is not optimum. The transmitted sequence will be orthogonal when the number of collected blocks approaches infinity. But practically, the correlation of transmitted sequence is only approximately, but not exactly equal to the identity matrix. The Cramér-Rao Bound is derived and compared with the simulation results for both SISO-OFDM and MIMO-OFDM systems. The reason stated in the last paragraph also explains the difference between the simulation result and the Cramer-Rao Bound. In conclusion, there are two factors causing the error in the decoupled method, one is the additive noise, the other is the non-orhogonality of the transmitted sequence. At low SNR value, the additive noise is the dominating factor causing the error, so the MSE decreases with the increases of SNR. But at high SNR value, the noise is no longer the dominating factor to cause the error. Instead, the difference between the correlation matrix of transmitted sequence R xx ( K ) and the identity matrix leads to the error of carrier offset estimation. In order to get a more precise estimation at high SNR value, the only way is to collect more data blocks. 70 An extension of this thesis can be the research on the joint estimation of CFO and channel information. In this thesis, we have assumed full knowledge of channel information. 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Tech., Vol. 19, No. 5, pp.628-634, Oct. 1971 75 [...]... that the actual signal to noise ratio is reduced when carrier frequency offset exsists Hence the performance of the OFDM systems is degraded In order to improve the performance of OFDM systems, the carrier frequency offset must be estimated and compensated before performing the FFT demodulation There are mainly two types of carrier frequency offset estimator available in the Literature: Data-aided scheme... at transmitter and multiple antennas at receiver (MIMO) technology in combination with OFDM, i.e., MIMO- OFDM, therefore becomes an attractive solution for future broadband wireless systems Similar to SISO -OFDM systems, MIMO- OFDM systems exhibits great sensitivity to CFO Several data-aided schemes were proposed in the literature Optimal training for MIMO channel estimation was considered in [26] It... of MIMO- OFDM Systems Various schemes that employ multiple antennas at the transmitter and receiver are being considered to improve the range and performance of communication systems By far the most promising multiple antenna technology today is called multiple-input multipleoutput (MIMO) system MIMO systems employ multiple antennas at both the transmitter and receiver MIMO- OFDM combines OFDM and MIMO. .. (BER) of OFDM systems is analyzed in detail Then, methods to estimate and compensate the CFO available in the literature are introduced and compared 3.1 Effect of CFO in OFDM Systems It has been demonstrated that OFDM systems is much more sensitive to Carrier Frequency Offset than other single carrier systems [18][19] As introduced in the first chapter, CFO can be normalized with respect to the subcarrier... exploited to estimate the carrier frequency offset in [32] by using a single input multi-output (SIMO) system This method is extended to a MIMO- OFDM systems in [33] By exploiting MIMO diversity, significant gains over single transmit diversity at low SNR can be achieved 32 Chapter 4 DEML Blind CFO Estimator for OFDM systems As discussed in the last chapter, a few carrier frequency estimation algorithm... matrix destroys the orthogonality among sub-carriers and thus introduces inter-channel interference (ICI) In order to recover the transmitted signal, s k , the carrier offset, φ must be estimated and compensated before performing the DFT 4.2 DEML Blind Carrier Offset Estimator In this section, we derive the cost function for estimating the carrier frequency offset φ As mentioned above, the channel is... users, since users in OFDM systems do not need to be time multiplexed as in TDMA systems Fig 2.1 Spectrum of a single OFDM sub -carrier and OFDM symbol Figure 2.1 shows the spectrum of a single OFDM sub-channel and the spectrum of an OFDM symbol, which are characterized by the fact that spectrum of different sub-carriers overlaps As shown in the figure, at the centre frequency of each carrier, the amplitude... SNR + 1 2 o 2 < C (ε) SNRo (3.9) ... is reduced when carrier frequency offset exsists Hence the performance of the OFDM systems is degraded In order to improve the performance of OFDM systems, the carrier frequency offset must be... receiver (MIMO) technology in combination with OFDM, i.e., MIMO- OFDM, therefore becomes an attractive solution for future broadband wireless systems Similar to SISO -OFDM systems, MIMO- OFDM systems. .. introduced and compared Performance of the proposed DEML (Decoupled Maximum Likelihood) blind carrier offset estimator for SISO -OFDM and MIMO- OFDM systems is analyzed in Chapter and Chapter 5, respectively