Reduced Complexity Signal Detection and Channel Estimation for Iterative MIMO-OFDM Systems

Transcription

1 Reduced Complexity Signal Detection and Channel Estimation for Iterative MIMO-OFDM Systems Licai Fang This thesis is presented for the degree of Doctor of Philosophy School of Electrical, Electronic and Computer Engineering May 2016

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3 Abstract Multi-Input Multi-Output (MIMO) is a key technology in broadband wireless communications, and it has been used in WiMax, LTE and WiFi (802.11n/ac). As Orthogonal Frequency Division Multiplexing (OFDM) can transform a frequency selective fading channel into a set of parallel frequency flat fading channels and thus greatly reduce the complexity of equalization, MIMO is typically combined with OFDM in practical applications. For a MIMO-OFDM system, the channel estimation and signal detection algorithms based on linear-minimum-mean-square-error (LMMSE) are often employed because of their good performance. But conventional algorithms typically require a matrix inversion with cubic level complexity, which is a major obstacle for practical implementation. To reduce the complexity, in this thesis, we focused on algorithms design by reducing the number of costly operations and the cost of each operation. Due to the law of large numbers, the matrix to be inverted, in both the LMMSE channel estimation of an OFDM system and the uplink signal detection of a massive MIMO system (i.e., both the number of transmit and receive antennas are large), approaches a diagonally dominant matrix. By exploiting this special structure, the Neumann series expansion was employed to reduce the complexity of matrix inversion from cubic to quadratic level. At the same time, we found that in a massive MIMO-OFDM system there are strong correlations between the matrix inversions in uplink LMMSE detection of adjacent subcarriers. Similar correlations were also found between different iterations of an LMMSE detector in a turbo MIMO-OFDM system. By exploiting the correlations between adjacent subcarriers or different iterations, interpolation based methods can effectively reduce the number of costly operations. Specifically, in this thesis, an LMMSE detection algorithm for turbo-mimo systems, which exploits the correlation of matrix inversion between different iterations, was proposed to reduce the complexity of non-first iterations from O(Nt 3 ) to O(Nt 2 ) where N t is the number of transmit antenna. Then a Partial Gaussian method was proposed to be employed for spatially correlated channels, and a branch-and-bound algorithm was proposed to reduce the complexity of the Partial Gaussian algorithm. For LMMSE chan-

4 iv nel estimation of OFDM systems, a low complexity algorithm based on Neumann series expansion was investigated. This proposed algorithm can achieve mean-square error (MSE) performance close to the optimal LMMSE estimator but with only O(N logl) complexity where N is the number of subcarriers and L is the number of time domain channel coefficients taps. With the aid of turbo processing, we also proposed a data-aided channel estimator which can track time-varying channels caused by terminals movement (up to 100 km/hour) with very low pilot overhead. We also investigated medium-sized massive MIMO systems. A low cost LMMSE detection algorithm based on Neumann series expansion for uplink applications was proposed. Compared to alternative algorithms, the algorithm can significantly reduce the total detection complexity to O(KN t N r ) where N r is the number of receive antenna and K (typically K < 3) is the number of Neumann series expansion. The computation saving comes from the fact that proposed algorithm can not only avoid computing matrix inversion but also replace matrix-matrix multiplications with matrix-vector multiplications.

5 List of Publications [1] L. Fang, and D. Huang. Neumann Series Expansion Based LMMSE Channel Estimation for OFDM Systems. IEEE Communications Letters, vol. 20, no. 4, pp , April (Chapter 4) [2] L. Fang, L. Xu, and D. Huang. Low complexity iterative MMSE-PIC detection for medium-size massive MIMO. IEEE Wireless Communications Letters, 5(1): , Feb (Chapter 5) [3] Licai Fang, Lu Xu, Qinghua Guo, Defeng Huang, and S. Nordholm. A low complexity iterative soft-decision feedback MMSE-PIC detection algorithm for massive MIMO. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages , (Chapter 2) [4] Licai Fang, Lu Xu, Qinghua Guo, D.D. Huang, and S. Nordholm. A hybrid iterative MIMO detection algorithm: Partial Gaussian approach with integer programming. In 2014 IEEE/CIC International Conference on Communications in China (ICCC), pages , (Chapter 3)

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7 Acknowledgements First, I would like to thank my supervisors Prof. David (Defeng) Huang and Dr. Qinghua Guo for their support, for giving me the opportunity to pursue my Ph.D. Without their directions, enlightenments and encouragements, this thesis would have been impossible. Then I would like to thank the colleagues in the Signal Processing Wireless Communication Laboratory (SPWCL) research group at the University of Western Australia, namely, Dr. Lu Xu, Dr. Jindan Yang, Dr. Hang Li and Dr. T.-U. I. Khandoker. Their insightful academic discussion is invaluable to my research. Most importantly, my sincere thanks go to my wife Dr. Wei Hou and our families. Their consistent supports are the main driving force for me to finish this thesis during my 40s.

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9 Table of contents List of Publications List of figures List of tables v xiii xv 1 Introduction Background MIMO Turbo Principle OFDM Turbo MIMO-OFDM System Channel LDPC Encoder and Decoder Soft Mapper and Soft Demapper Signal Detection Channel Estimation Motivations and Contributions Signal Detection Channel Estimation Notations A Low Complexity Soft-Decision Feedback MMSE-PIC Detection Algorithm System Model Gaussian Model Based MMSE Detection Algorithm Complexity Reduction Low Complexity Matrix Inversion A Heuristic Approach to Solve the Stability Problem

10 x Table of contents Computational Complexity Comparison Iterative Method to Improve First-pass Performance Simulation Results Simulation Setup BER Performance Conclusion MIMO Detection Algorithm: Partial Gaussian Approach with Integer Programming Introduction System Model Partial Gaussian Approach with Integer Programming PGA Detection Algorithm Simplified Marginalization Calculation Resolving QIP with the Branch-and-Bound algorithm Simulation Results Simulation Setup BER Performance Complexity Conclusion A Low Cost LMMSE Channel Estimator for OFDM Systems Introduction System Model LMMSE Channel Estimation Newmann Series Expansion Based Channel Estimation Neumann Series Expansion Computational Complexity Comparison Simulation Results Mean-Square Error (MSE) Performance for Time-Invariant Channels Bit Error Rate (BER) Performance for Iterative Systems Discussion The Power Delay Profile (PDP) The Assumption of Quasi-static Channel Conclusion

11 Table of contents xi 5 Low Complexity Iterative MMSE-PIC Detection for Medium-Size Massive MIMO Introduction System Model MMSE Detection Based on Neumann Series Expansion Neumann Series Expansion Computational Complexity Comparison Discussion Simulation Results Conclusion A Novel Interpolation Algorithm for Massive MIMO OFDM System Detection Introduction System Model and Soft-output MMSE Detector MMSE Detection Based on Interpolation Correlation of Matrix Inversion for Massive MIMO-OFDM Systems Interpolation Based Matrix Inversion Computational Complexity Comparison BER Performance Conclusion Summary and Future Work Summary Future Work Channel estimation for MIMO-OFDM systems Channel estimation for Massive MIMO Uplink Signal Detection for Massive MIMO-OFDM Appendix A Proof of the Equality of Algorithm 1 and Algorithm 2 89 Bibliography 93

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13 List of figures 1.1 An Iterative MIMO-OFDM Communication System QC-LDPC Base Parity Check Matrix PAM Constellation Diagram Iterative Detection and Decoding of a MIMO Communication System Iterative Soft-in Soft-Out MMSE Detector BER Performance Comparison Between Exact Implementation and Proposed Approximation for a MIMO System BER Performance Comparison Between Different Number of Selfiterations for MIMO BER Performance Comparison Between Different Number of Selfiterations for MIMO BER Performance Comparison Between Different Number of Selfiterations for 4 4 MIMO Iterative Detection and Decoding of a MIMO Communication System An example of the proposed branch and bound algorithm where d is the tree level, lb means low bound, ub means upper bound and m is the vector that minimizes f (m). Because the first heuristic solution happens to be the final solution, there are only 6 nodes visited BER performances of 16-QAM MIMO with correlation factor ρ = 0.5 and ρ = BER performance comparison between PGA-Exact and PGA-IP under 16-QAM MIMO correlated channel (ρ = 0.4) MSE performance with different L at SNR of 14dB MSE performance for the 10-tap COST259_RAx channel BER performance for 10-tap COST259_RAx Channel at speed of 100 km/hour

14 xiv List of figures 4.4 MSE Under Channel No BER performance comparison for exact MMSE, proposed and SOR based [1] with MIMO size of K M = Correlations of C h (d) and C g (d) of adjacent subcarriers with N = 64, N t = 20, different ρ and different subcarrier distance d Correlations of C h (d) and C g (d) of adjacent subcarriers (with different d) under different channel models with N = 64, N t = 20 and ρ = Complexity comparison with ρ = 8, N = 128 and I = BER performance comparison for exact MMSE, Matched filter, Proposed V p n with exact H n and Proposed V p n with interpolated H n for N t N r = MIMO

15 List of tables 2.1 Computational Complexity Comparison Average CPU run time (s) comparison between MMSE_PIC, PGA_IP and PGA_Exact for detecting 2000bits with 3 iterations under MIMO with 16-QAM on a X86 Linux PC Simulated Channel Models [2] Computational Complexity Comparison Simulated Channel Models [2] Computational Complexity Comparison

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17 Chapter 1 Introduction 1.1 Background MIMO In late 1980 s, the multiple-input multiple-output antenna (MIMO) systems was proposed for wireless communications. By using multiple antennas at both transmitter and receiver side, MIMO can create multiple parallel channels using the same radio spectrum [3] [4]. MIMO techniques can improve communications performance by either increasing reliability or maximizing throughput. In order to increase reliability, some form of space-time coding (STC) is typically employed to combat multipath scatting by creating spatial diversity [5]. While for improving throughput, spatial multiplexing techniques [6] [7] are employed to exploit multipath scatting. It was shown that the achievable transmitting rate of MIMO systems scales as min(n t,n r )log(1 + SNR) and the link outage scales as SNR N tn r [8] where N t and N r are the numbers of transmitter and receiver antennas, respectively. MU-MIMO For the cellular systems, the conventional MIMO technology has some limitations because the terminals can not employ many antennas due to the cost, power and size

18 2 Introduction constraint. Another issue of conventional MIMO is the propagation limitations; in case of LOS (line-of-sight) propagation, channel rank loss or antenna correlation, the spatial multiplexing gain in conventional MIMO will be severely degraded [9]. To achieve the gain of multiple access capacity and overcome above two issues, the multi-user MIMO (MU-MIMO) scheme had been proposed and researched in recent years. By treating every user s terminal as a virtual MIMO antenna, the MIMO spatial multiplexing gain can be preserved. Although the individual users will not experience increased throughput by MU-MIMO, but the overall system performance will improve dramatically. So many state-of-the-art wireless communication standards have adopted MU-MIMO, like 3GPP long-term evolution advanced (3GPP LTE-A)(Release 10) [10], IEEE m (WiMAX Profile 2.0) [11] and Wifi (802.11ac) [12]. Massive MIMO With the maturing of MU-MIMO, by making the number of antennas much larger at Base Station side, comes the concept of massive MIMO, which is characterized with hundreds of antennas at Base Station and can serve tens of terminals simultaneously. Massive MIMO can reap all the benefits of conventional MIMO and MU-MIMO in a much greater scale [13] [14]. Firstly, high energy efficiency can be obtained by focusing the energy with extreme sharpness into small regions in space. Specifically, by appropriately shaping the signals sent out by the antennas, all radio wave fronts collectively emitted by all antennas interfere constructively at the intended terminals, but destructively almost everywhere else. [15] illustrated that the energy focus effect by comparing M = 10 transmit antennas (M-element Uniform Linear Array (ULA)) and M = 100 antennas. It shows that when the number of transmit antenna at the transmitter is 100, by applying spatial precoding, the field strength can be focused to a point rather than in a certain direction as done in conventional MIMO or MU-MIMO. This energy focus property can greatly reduce the interference between spatially separated users and reduce the total radiated signal power, thereby the Base Station can benefit from this property to greatly reduce the total output RF power. At the same time, based on information theory [15],

19 1.1 Background 3 massive MIMO can increase the spectral efficiency 10+ times from the aggressive spatial multiplexing. Besides the above base station scenario where the communication is multipointto-point for uplink or point-to-multipoint for downlink, there is also point-to-point applications like the back-haul connections between base stations. For this kind of configuration, a large number of antennas can be used both at transmit and receive base stations. It is also worth noting that when the number of receive antennas at the base station is large and much larger than the total number of transmit antennas in user terminals, a simple detection algorithm such as a matched filter can achieve very good performance, as with the assumption of i.i.d. entries for channel matrix H, the channel vectors become orthogonal to each other and H H H converges to a scaled identity matrix. But from practical implementation point of view, medium size antenna arrays are also of interest Turbo Principle Nearly at the same time as the emerging of MIMO technology, the invention of turbo codes and iterative decoding [16] paved the way for achieving system performance close to the Shannon limit. By exchanging information between several decoding units iteratively, the system performance was shown to be close to optimal decoding, but with feasible complexity. Then the turbo principle [16] was used to improve performance of other tasks in the wireless receiver, e.g., equalization [17] [18] [19], channel estimation [20], multi-user detection [21] and MIMO detection [22] [23] [24]. For a coded communication system, as the complexity of the optimal receiver is exponential in the length of the data transmitted, most practical receivers include two separate blocks: signal detection and channel decoding. The signal detectors have been designed to process the received observations to account for the effects of the channel and to estimate the transmitted channel symbols that best fit the observed data. Then the soft information (in the form of Log-Likelihood Ratio (LLR)) is passed to the channel decoder for decoding.

20 4 Introduction Applying the turbo principle to this kind of receiver, comes the iterative detection and decoding (IDD) system. In IDD, a soft-input and soft-out detector is required which can accept soft information from the decoder and output soft information to the decoder. In general, only extrinsic information can be exchanged between the detector and the decoder [25]. Turbo principle can also be applied to the task of channel estimation. In order to track channel variation caused by movement of terminals, data-aided scheme is often employed. For slow fading channel with preamble-type pilots, the channel coefficients copied from last symbol can be improved by exploiting the soft or hard information feedback from the decoder as the virtual pilot [26]. Similarly, for superimposed-type pilot, it is common to perform iterative channel estimation and decoding by exploiting data fed back from the channel decoder [20] OFDM Most modern wireless communication systems are broadband systems which have high data rates. As a result, the symbol rate is much higher than the channel coherence bandwidth and thus the channel is frequency selective. The major issue about frequency selective fading is the inter-symbol interference (ISI), which is caused by the fact that the symbol period is shorter than the delay spread. To combat with ISI, one way is to employ equalization with single carrier. As the computational complexity of equalization is quite high, another popular technique for coping with frequency-selective fading effects is using orthogonal frequency division multiplexing (OFDM). The idea behind OFDM is to split a broadband signal that experiences frequencyselective fading into multiple narrow sub-bands (subcarrier) so that each subcarrier experiences flat fading. Because the bandwidths of the sub-bands is less than the coherence bandwidth of the channel, each sub-stream is far less vulnerable to the ISI than the original input stream. At the same time, although each OFDM subcarrier is narrowband, the bandwidth of the OFDM symbol is greater than the coherence bandwidth of a frequency selective channel. To mitigate the effects of the ISI between

21 1.2 Turbo MIMO-OFDM System 5 OFDM symbols, guard intervals are inserted between OFDM symbols so that time dispersion of current OFDM symbol will not interfere with subsequent OFDM symbols. In practice, an OFDM symbol is obtained by taking the inverse discrete Fourier transform (IDFT) of a block of modulation symbols at the transmitter. Then at the receiver the forward discrete Fourier transform (DFT) is performed to restore the modulated symbols. As both the IDFT and DFT can be implemented using fast Fourier transform (FFT) algorithms, OFDM is considered as a low cost technique. 1.2 Turbo MIMO-OFDM System Fig. 1.1 An Iterative MIMO-OFDM Communication System The research interest of this thesis is to reduce the complexity of iterative MIMO- OFDM systems which combine all major benefits of above three key technologies. Fig. 1.1 is a block diagram of the iterative MIMO-OFDM system. At the transmit side, a convolution code encoder or LDPC code encoder is employed for channel encoder. Then the serial encoded bits sequence is split to N s parallel sub-streams. Each sub-stream will be scrambled by an interleaver and followed by constellation mapper to map a chunk of bits to a constellation symbol. Then all the sub-streams data pass the pre-coding block to map N s sub-streams to N t transmit chains. After spatial mapping, each transmit chain has OFDM modulation applied to it by processing it through an IFFT block that converts

22 6 Introduction a block of modulated constellation points to a time domain block of symbols followed by adding the cyclic prefix (CP). The resulting baseband sequence of symbols in each chain are then passed to the analog and RF blocks before being applied to a transmit antenna. At the receive side, after the CP of the data received on every receive antenna is removed, FFT will be performed to generate the frequency domain symbols. Then the channel estimator estimates the frequency domain channel coefficients based on the received pilot data. With the frequency domain channel coefficients, the MIMO detection is performed on every subcarrier. The detected data is then de-mapped to soft information (typically in LLR format) and sent to the channel decoder. In an IDD system, the decoded bits (or soft information) will be sent back (after re-mapping them to symbols) to the channel estimator and/or the symbol detector to purify the results of last iteration Channel The nature of the wireless environment results in the transmitted signal experiencing various forms of corruption including noise and fading. The background noise and thermal noise of the channel are the major contributors of noise which is commonly modelled as additive white Gaussian noise (AWGN). Fading, which is the variation of the signal amplitude over time and frequency, may either be due to multipath propagation, referred to as multi-path fading, or to shadowing from obstacles that affect the propagation of a radio wave, referred to as shadow fading. The fading phenomenon can be broadly classified into two different types: largescale fading and small-scale fading. The large-scale fading is characterized by average path loss and shadowing. On the other hand, small-scale fading refers to the result of multipath propagation. In a wireless environment, the transmitted signal may be scattered into multiple paths as a result of reflection and refraction off environmental obstacles and atmospheric effects. An attenuated version of the transmitted signal propagates through each path and arrives at the receiver at different times. Consequently, the received signal

23 1.2 Turbo MIMO-OFDM System 7 is distorted by one symbol interfering with subsequent symbols, which is commonly referred as inter symbol interference (ISI). Characteristics of a multipath fading channel are often specified by a power delay profile (PDP). Using a PDP, different signal paths are characterized by their relative delay (τ i ) and average power (P(τ i )). Then the RMS delay spread σ τ can be calculated by the square root of the second central moment of PDP as σ τ = τ 2 τ 2 where the mean excess delay τ is given by the first moment of PDP as τ = k τ k P(τ k ) k P(τ k ) and τ 2 = k τ 2 k P(τ k) k P(τ k ). In general, the coherence bandwidth, denoted as B c, is inversely-proportional to the RMS delay spread, that is, B c 1 σ τ. Fading Due to Time Dispersion Due to time dispersion, a transmit signal may undergo fading over a frequency domain either in a selective or non-selective manner, which is referred to as frequency-selective fading and frequency-flat fading. For the given channel frequency response, frequency selectivity is generally governed by signal bandwidth. When the signal bandwidth (B s 1/T s, T s is the symbol period) is narrow compared with the coherence bandwidth (B c ) of the channel, the signal experiences flat fading; otherwise, it experiences frequencyselective fading. Fading Due to Frequency Dispersion Variation in the time domain is closely related to movement of the transmitter or receiver, which incurs a spread in the frequency domain, known as a Doppler shift. The maximum Doppler shift can be calculated by f m = v max f C /c 0 where v max is the maximum velocity between the receiver antenna and the transmitter antenna, f C is the frequency of carrier and c 0 is the speed of electromagnetic wave. Depending on the extent of the Doppler spread, the received signal undergoes fast or slow fading. When the coherence time T c f 1 m is smaller than the symbol period T s (T s > T c ), a channel impulse response quickly varies within the symbol period. Under this condition, the transmit signal is subject to fast fading.

24 8 Introduction LDPC Encoder and Decoder Low-density parity-check (LDPC) codes are linear block codes which can provide nearcapacity performance. They were proposed by Gallager in his dissertation [27] in Then in 1981 Tanner generalized LDPC codes and introduced a graphical representation of LDPC codes in [28]. In mid-1990 s Mackay, Luby and others [29] [30] [31] also independently discovered the advantages of spare parity-check matrices. The most obvious character of LDPC codes is that the parity-check matrix has a low density of 1 s for binary LDPC codes. For a LDPC code with (n k) n parity-check matrix H, if the number of 1 s in each column w c equals to the number of 1 s in each row w r, this code is called regular LDPC code and otherwise called irregular LDPC code with the code rate of k/n. A special subclass of LDPC codes, called Quasi-Cyclic LDPC (QC-LDPC) codes has received much attention because of their superb error correction performance [27]. QC-LDPC codes is characterized that a cyclic shift of one codeword results in another codeword and due to this regular structure their encoding is proved to be linear with code length. As QC-LDPC has near capacity performance and can be decoded by low-complexity iterative decoding algorithm, it has been adopted by many industrial standards like IEEE n, IEEE ac and IEEE e, as an error correction code [32] [12] [33]. LDPC Encoder Algorithm Fig. 1.2 QC-LDPC Base Parity Check Matrix

25 1.2 Turbo MIMO-OFDM System 9 The base parity check matrix of rate 5/6 length-1944 QC-LDPC codes (employed in IEEE n/ac standards) is defined in Fig The digits indicate the cyclic shift values of identity sub-matrices. The - indicates a zero matrix and the sub-matrix size Z is defined as 81. The base parity check matrix can be partitioned into the two sub-matrices as shown in Fig Let H = [H 1 H 2 ] be the partitioned base parity check matrix, where H 1 is an (n k) k matrix, and H 2 is an (n k) (n k) matrix. Let c = [m p] be a codeword block, where m and p denote information and parity bit sequences, respectively. From the property that the correct codeword satisfies the parity check equation, the parity bit sequence p can be derived as follows, Hc T = H 1 m T + H 2 p T = 0, (1.1) p T = H 1 2 H 1m T. (1.2) From (1.2), it is clear that this encoding requires to compute an inverse of matrix with size of (n k) (n k) and the direct computation has big computational complexity of O((n k) 3 ). But when we check the structure of H 2 carefully, we can see that this matrix has a very regular structure which can be exploited for low complexity implementation. It can be seen from Fig. 1.2 that H 2 contains either identity submatrix (with some shift factor) or zero submatrix. More importantly, two of the three sub-matrices of the first columns have the same value and every other column contains two same value. Therefore, if we let H 1 m T = [λ 0,λ 1,...,λ n k 1 ] T and p = [p(0),p(1),...,p(n k 1)], the first subvector of p 0 can be easily obtained with p(0) T = n k 1 λ i (1.3) i=0 Then the remainder of the parity bits can be obtained by forward substitution. This algorithm leads to linear complexity solution for QC-LDPC encoding. Actually, many

26 10 Introduction efforts now focus on how to improve encoding throughput and reduce implementation complexity at the same time [34] [35]. LDPC Decoder Algorithm Based on the Tanner graph representation of LDPC codes, the iterative massage passing algorithm (MPA) is typically exploited to do the decoding. Tanner graph is a kind of bipartite graph whose nodes can be separated into two types, and edges may only connect two nodes of different types. These two nodes in Tanner graph are the variable nodes (v-node) and the check nodes (c-node). The Tanner graph is drawn based on the following rule: check node j is connected to variable node i whenever element h ji of parity check matrix H is a 1. So, it is easy to know that there are m = n k check nodes for check equations and n variable nodes for code bits. The task of LDPC decoder is to compute the a posteriori probability (APP) for a bit in the transmit codeword c = [c 0,c 1,...,c n 1 ] equals 1 given the received word y = [y 0,y 1,...,y n ] in LLR: ( ) Pr(ci = 0 y) L(c i ) = log. (1.4) Pr(c i = 1 y) When drawing a Tanner graph, typically we put the c-nodes above the v-nodes. Then the message passing from a v-node i to a c-node j is noted as m i j. This extrinsic information message is the probability of Pr(c i = b input message), b {0,1} which comes from channel input and all its neighbours excluding the c-node itself. In the reverse direction, the message passing from a c-node to a v-node m ji is the probability of Pr(check equation f j is satisfied input message). Now we introduce the following notations [36]: V j =v-nodes connected to c-node f j C i =c-nodes connected to v-node c i M v (i) = messages from all v-nodes except node c i M c ( j) = messages from all c-nodes except node f j

27 1.2 Turbo MIMO-OFDM System 11 P i = Pr(c i = 1 y i ) S i = event that the check equations involving c i are satisfied q i j (b) = Pr ( c i = b S i,y i,m c ( j) ), where b {0,1}. For LLR format, m i j = log[q i j (0)]/q i j (1)] r ji (b) = Pr ( check equation f j is satisfied c i = b,m v (i) ), where b {0,1}. For LLR format, m ji = log [ r ji (0)/r ji (1) ] Then, the MPA can be summarized as follows, Step 1: Initialization: For every v-node, initialize p i = Pr(c i = 1 y i ), then q i j (0) = 1 p i and q i j (1) = p i for each h i j = 1. Under AWGN channel, p i = 1/(1 + exp(2y i /σ 2 )). Step 2: For each c-node, update r ji by r ji (0) = r ji (1) = 1 r ji (0). (1 2q i j(1)) and i V j \i Step 3: Update q i j by q i j (0) = K i j (1 P i ) j C i \ j r j i(0), q i j (1) = K i j P i j C i \ j r j i(1) and K i j is selected to ensure that q i j (1) + q i j (0) = 1. Step 4: Update Q i by Q i (0) = K i (1 P i ) r ji (0) and Q i (1) = K i P i r ji (1) j C i j C i and K i j is selected to ensure that Q i (1) + Q i (0) = 1. Step 5: Hard decision: For i = 0,1,...,n 1, if Q i (1) > Q i (0) then ĉ i = 1; else ĉ i = 0. Step 6: If ĉh T = 0 or reaching the maximum iteration number, stop; else, go to Step Soft Mapper and Soft Demapper Soft Mapper The function of a soft mapper module is to calculate the symbol mean and variance from the extrinsic LLRs of code bits coming from the Soft-Input Soft-Output (SISO)

28 12 Introduction decoder [37]. The soft mapper calculates {m n,v n } based on extrinsic LLR L(c n ) using the following equations: m n = E(x n ) = 2 Q α i p(x n = α i ) (1.5) i=1 v n = Cov(x n,x n ) = 2 Q α i 2 p(x n = α i ) m 2 n (1.6) i=1 where each constellation symbol α i corresponds to a binary vector s i = [s i,1,s i,2,...,s i,q ] T, and the symbol s probability p(x n = α i ) can be calculated as: p(x n = α i ) = Q p(c n, j = s i, j ) (1.7) j=1 while p(c n, j = s i, j ) is the probability of a code bit, which is normally represented by LLR: L j = ln p(c n, j = 0) p(c n, j = 1) = ln p(c n, j = 0) 1 p(c n, j = 0). (1.8) With (1.5) - (1.7), the computational complexity is O(Q2 Q ). When high order modulation is exploited, the computational complexity is high and the low complexity algorithms can be found in [38] and [39]. Soft Demapper In a coded system, the soft output from the equalizer (or detector) typically can greatly improve the system BER performance compared to hard output. The symbol output from equalizer (or detector) should be demapped to bit information in LLR format which is the input requirement from most of the channel decoders like the Turbo code or the LDPC code. When the soft output symbols are assumed as Gaussian distributed, they can be described by their mean vector m and auto-covariance diagonal matrix V. The task of the demapper is to compute the LLR for each code bit c n,q, which can be

29 1.2 Turbo MIMO-OFDM System 13 expressed as [18] L(c n,q ) = ln P(c n,q = 0 y) P(c n,q = 1 y) = ln x n A 0 q x n A 1 q P(x n y) P(x n y) (1.9) where A 0 q (A 1 q ) denotes the subset of all α i A corresponding to a binary subsequence with the qth bit given by 0 (1). When IDD is adopted, only extrinsic information will be passed to the channel decoder. The extrinsic LLR [17] L e (c n,q ) = L(c n,q ) L a (c n,q ) = ln x n A 0 q x n A 1 q P(y x n )P(x n ) P(y x n )P(x n ) La (c n,q ) (1.10) will be the input to the decoder, where L a (c n,q ) is the output extrinsic LLR of the decoder in the last iteration and P(x n ) can be calculated from L a (c n,q ). The probability of the data symbol x n being the constellation point α i is given by P(x n = α i ) exp ( α i m e n 2 ). After some manipulation, we can get v e n L e (c n,q ) = ln α i A 0 q α i A 1 q exp ( α i m e n 2 ) v e n P(c n,q = s i,q ) q q exp ( α i m e n 2 ) v e n P(c n,q = s i,q ) q q (1.11) Directly computing LLR in (1.11) needs exhaustively search every constellation point which results high computational complexity if high order constellation is employed. To reduce this complexity, quite a lot of works can be referred although they are implemented in different background [38] [40]. The basic idea of these methods are using the regularity of constellation points and employing the approximation used by max_log_map algorithm in [41] to change the exhaustively search to a piecewise linear combination. If we ignore the a priori information which has been found with little

30 14 Introduction performance penalty and apply this approximation, (1.11) can be represented as L e (c n,q ) ln α i A 0 q α i A 1 q exp ( α i m e n 2 ) v e n exp ( α i m e n 2 ) v e n 1 v e max ( α i m e n 2 ) 1 n α i Aq 0 v e max ( α i m e n 2 ) n α i Aq 1 = 1 [ min v e ( α i m e n 2 ) min ( α i m e n 2 ) ] n α i A 1 q α i A 0 q (1.12) Fig. 1.3 is a 4-PAM constellation diagram and the m e n is located in the point. It is Fig PAM Constellation Diagram easy to see that the results of the two min operation in (1.12) are the white constellation point and the black one, and thus (1.12) can be easily calculated as follows: L e (c n,0 ) = 1 5v e n (4m e n 8) (m e n 0) 1 5v e n ( 4m e n 8) (m e n < 0) (1.13) and 1 5v e n (8m e n 8) (m e n 2) L e 1 (c n,1 ) = (4m e 5v e n) ( m e n < 2) n 1 5v e n (8m e n + 8) (m e n 2). (1.14)

31 1.2 Turbo MIMO-OFDM System Signal Detection After the data symbols are transmitted over a MIMO channel and corrupted by AWGN, the receiver receives superimposed and noised version of these symbols. The data detection block is responsible for recovering those corrupted data symbols based on certain estimation criterion. At the receiver side, in order to improve performance, iterative detection and decoding can be employed based on the turbo principle. From the iterative receiver diagram Fig. 1.1, it can be seen that the SISO decoder and the SISO detector iteratively exchange soft extrinsic information between them. The following is a brief review of conventional detection methods. If there is no a-priori information available, the ML (Maximum Likelihood) method can be employed while the MAP (Maximum a-posteriori ) method can be employed if the a-priori information is available. But, both ML and MAP based methods suffer from the huge computational complexity which is exponential in the number of transmit antenna N t and modulation constellation size Q. In order to reduce complexity, linear methods such as zero forcing (ZF) or Minimum Mean Square Error (MMSE) can be employed. In the family of non-linear detection algorithms, Sphere Decoding (SD) based search algorithms have been deeply studied [42] [43]. Basically, SD algorithms have exponential average complexity [3], and most importantly the complexity depends on channel status and received SNR. In order to make the complexity deterministic, Fixed-Complexity Sphere Decoder (FCSD) has been proposed with medium complexity and near ML performance [43] [44]. Another non-linear detection is called Partial Gaussian method [45]. This algorithm has low and fixed computational complexity and near MAP performance by using an adjustable parameter M. The basic idea behind this method is taking M important symbols as discrete symbols but others as continuous. The continuous symbols can be assumed to be Gaussian distributed which makes the whole computational complexity very low. The last type of detection algorithm is based on factor graph [46] [47] [17] [48] [49] [50] [51] [52]. In this thesis, we will focus on MIMO spatial multiplexing technique which can transmit data at a higher speed than the system employing spatial diversity. Consider

32 16 Introduction a MIMO-OFDM system with spatial multiplexing technique in Fig. 1.1 which has N t antennas at the transmit side, N r antennas at the receiver side and N subcarriers. The cyclic prefixes (CP) are inserted before the IFFT of x(n) to ensure the orthogonality among the subcarriers and prevent inter-symbol interference (ISI) between consecutive OFDM symbols. Considering a quasi-static channel which is constant during one OFDM symbol, this OFDM system can be described as a set of parallel frequency flat additive white Gaussian noise (AWGN) channels. Then the channel H can be denoted by a matrix sized N r N t with its (i, j)th entry h i j denoting the channel gain between the ith transmit antenna and the jth receive antenna where j [1,2,...,N r ] and i [1,2,...,N t ]. So, for every subcarrier, a length-n r observation vector y at the receive side can be written as y = Hx + w (1.15) where w denotes a length-n r circularly symmetric additive white Gaussian noise (AWGN) vector with zero-mean and covariance of σ 2 I. It is worth noting that there are totally N such equations in a MIMO-OFDM system. Conventional Detection Algorithms Linear signal detection algorithms like ZF and MMSE treat all other transmitted signals as interferences and minimize or nullify these interferences when detecting the desired signals. Specifically, according to the system model of (1.15), the ZF detection algorithm can be described as: ˆx ZF = (H H H) 1 H H y (1.16) while MMSE algorithm can be listed as ˆx MMSE = (H H H + σ 2 I) 1 H H y (1.17) where ˆx is the detected transmit symbols. The noise enhancement effect of the above two algorithms is significant when the condition number of the channel matrix is large

33 1.2 Turbo MIMO-OFDM System 17 (the minimum singular value is very small) [53] while the effect of noise enhancement in MMSE algorithm is less critical than that in ZF algorithm. In order to improve performance, Maximum likelihood (ML) is often employed which calculates the Euclidean distance between the received signal vector and the product of all possible transmitted signal vectors with the given channel H and finds the one with the minimum distance. Mathematically, the ML algorithm can be described as: ˆx ML = arg min ( y Hx 2 ). (1.18) x A N t It is obvious that the complexity of ML algorithm is exponential in N t which is too complex for a practical implementation, but its performance is much better than aforementioned ZF and MMSE algorithms, especially for small-size MIMO. But for large MIMO system, linear detection algorithms such as MMSE-PIC can have near optimal performance [54]. To reduce the computational complexity of ML algorithm, search based algorithms like Sphere Decoding (SD) can be exploited. After applying QL decomposition to H (H = QL, Q T Q = I and L is lower triangular), the problem (1.18) can be visualized as a decision tree with N t layers [55] as follows: min f 1(x 1 ) + f 2 (x 1,x 2 ) f Nt (x 1,x 2,...,x Nt ) (1.19) {x 1,x 2,...,x N t } where f k (x 1,x 2,...,x k ) = ( y k k l=1 L k,lx l ) 2 and ỹ = Qy. The basic idea under SD algorithm is to use efficient tree traversal algorithms to eliminate the number of nodes visited and thus reduce the total complexity. Soft-In Soft-Out Detection Algorithms for Turbo MIMO-OFDM Systems The more reliable feedback from the decoder is a good information source to perform interference cancellation. A lot of multi-user detection algorithms can be applied to MIMO detection like the minimum mean square error parallel interference cancellation (MMSE-PIC) algorithms [56] [57]. These algorithm involves a matrix inversion when detecting every symbol. To reduce the complexity, an iterative method to implement

34 18 Introduction the MMSE filter was proposed in [58]. Then [59] presented a method which needs precomputing one matrix inversion only and then detects every symbol with low complexity incremental calculations. In 2011, [60] proposed a well optimized version of MMSE-PIC with only one matrix inversion for detecting a block of data and implemented it in ASIC which has been widely cited as the state-of-the-art MIMO detection implementation benchmark. This algorithm is listed in Algorithm 1: Algorithm 1 MMSE-PIC MIMO Detection Algorithm Input: ŷ,h, L a Output: L e extrinsic LLR value for every bit 1: Compute the Gram matrix G = H H H and the matched filter output y MF = H H y. 2: Compute the a priori soft-symbols m and variances V with (1.5) and (1.6). 3: Perform PIC based on ŷ MF according to ŷ MF i = H H ŷ i = y MF j, j i g j m j, j = 1,...,N t where g j denotes the jth column of G. 4: Compute the matrix inversion of A 1 = (GV + σ 2 I Nt ) 1. 5: Compute the MMSE filter outputs as µ i = a H i g i and ˆx i = a H i ȳ i, i = 1,...,N t, where a H i is the ith row of A 1. 6: Compute the extrinsic variance and extrinsic mean by 7: v e i = 1/µ i 1 8: m e n = ˆx i /µ i 9: Compute LLRs L e (c i,q ) with (1.12), i = 1,...,N t, q = 1,...,Q. Also in 2011, [17] proposed a generic method to implement a Soft-Input Soft-Output (SISO) detector, where the a posteriori distribution of a multivariate Gaussian vector was calculated first, followed by the calculation of the extrinsic information of each individual variable. The calculation of multiple variables together naturally enables sharing of computational units, thereby reducing system complexity. This algorithm is described in Algorithm 2; After applying this algorithm to MIMO detection, we found that although [17] and [60] have very different formulae, they actually can generate the same extrinsic mean and variance and thus the same soft-output to the channel decoder. The proof is given in Appendix A.

35 1.2 Turbo MIMO-OFDM System 19 Algorithm 2 Gaussian model based MMSE detection Input: ŷ,h, L a Output: L e extrinsic LLR value for every bit 1: Compute the Gram matrix G = H H H. 2: Compute the a priori soft-symbols m and variances V with (1.5) and (1.6). 3: Calculate the a posteriori mean m p and variance V p by 4: V p = (V G) 1 2σ 2 5: m p = m + 1 V p (H H y Gm). 2σ 2 6: Calculate the extrinsic mean m e n and variance v e n by 7: v e n = ( 1 vn p v 1 n ) 1 8: m e n = v e n( mp n vn p m n v n ). 9: Compute the LLRs L e (c i,q ) with (1.12), i = 1,...,N t, q = 1,...,Q Channel Estimation In OFDM systems, a long enough cyclic prefixes (CP) insertion before the IFFT can ensure the orthogonality among the subcarriers and prevent inter-symbol interference (ISI) between consecutive OFDM symbols. Considering a quasi-static channel which is constant during one OFDM symbol, this OFDM channel can be described as a set of parallel additive white Gaussian noise (AWGN) channels. The orthogonality allows each subcarrier component of the received signal to be expressed as the product of the transmitted signal and channel frequency response at the subcarrier. Then the channel can be estimated by using a preamble or pilot symbols known to both transmitter and receiver for pilot subcarriers, then various interpolation techniques can be applied to estimate the channel response of the subcarriers between pilot subcarriers. Depending on the arrangement of pilots, four different types of pilot structures are typically employed. 1: Block Type: OFDM pilot symbols at all subcarriers are transmitted periodically. Typically, a time domain interpolation is performed to get the whole channel information. It is suitable for frequency-selective slow fading channels. 2: Comb Type: Every OFDM symbol has pilot tones at the periodically-located subcarriers. It is suitable for fast-fading channels.

36 20 Introduction 3: Lattice Type: As a combination of block type and comb type, pilot tones are inserted along both the time and frequency axes with given periods. 4: Superimposed Pilot: Low power of training (pilots) signal is added to the data signal at the transmitter. The data-aided scheme, where the signal from the detector or the channel decoder, is typically exploited to do interference cancellation for the channel estimation. Channel Estimation for OFDM System After dropping the CP and performing FFT, the received frequency domain signal for OFDM symbol n is given by y(n) = X(n)η(n) + w(n) (1.20) where y(n) denotes a length-n observation vector, X(n) diag{x(n)} denotes an N N diagonal matrix with x(n) (data transmitted in nth OFDM symbol, x(n) = [x 1,x 2,,x N ] T ) on its diagonal, η(n) is the frequency domain channel coefficients and w(n) denotes a length-n circularly symmetric AWGN vector with PDF C N (w;0,σ 2 I). For notation simplicity, from now on we omit the time index n. Pilot based channel estimation When training symbols are available the leastsquare (LS ) and minimum-mean-square-error (MMSE) techniques are widely used for channel estimation. Ĥ LS = X 1 y, LS channel estimation. Ĥ MMSE = NFPF H X H (NXFPF H X H + σ 2 I) 1 y, MMSE channel estimation, where P is the channel power profile, F is the DFT matrix with the (k,l)th element given by (F) k,l = Ne j 2πkl N with j = 1. Although LS channel estimation has very low complexity, it suffers from noise enhancement issue. In order to improve the performance of OFDM channel estimation,

37 1.2 Turbo MIMO-OFDM System 21 the DFT based channel estimation algorithm can be employed. Specifically, after taking IDFT of the estimated frequency domain channel coefficients, we get the time domain channel coefficients with length N. But the actual time domain coefficients only have the length of L and typically L < N. By assigning the coefficients to zero for those with index larger than L and transforming them back to frequency domain, we get the channel estimation with better performance. The MMSE channel estimation algorithm is much robust from noise enhancement but the matrix inversion requires O(N 3 ) complexity. To reduce the cubic complexity, there are many algorithms have been proposed such as [61] [62] and [63] using windowed discrete Fourier transform (WDFT) methods and [64] using Dual-Diagonal LMMS algorithm. Channel Estimation for MIMO-OFDM System Classical channel estimation techniques for OFDM cannot be used in MIMO-OFDM system directly, since the received signal is a superposition of signals transmitted from different antennas for each OFDM subcarrier. The Expectation-Maximization (EM) algorithm can convert a multiple-input channel estimation problem into a number of single-input channel estimation problems [65]. MIMO-OFDM System Model In Fig. 1.1, the received signal on the m R th receive antenna at time n after performing a DFT can be expressed as: y(n) mr = X(n)Fh mr (n) + w (1.21) where y(n) mr = [y mr,1,y mr,2,...,y mr,n], X = [X 1,X 2,...,X NT ] are the transmitted symbols, X mt includes the symbols transmitted over N subcarriers from the m T th transmit antenna on its diagonal, F = I NT F and F is the truncated DFT matrix, with [F] u,s = 1 N e j2πus/n, and u = 0,...,N 1,s = 0,...,L 1, h mr = [h T 1,m R,...,h T N T,m R ] T is the time domain channel vector, with h mt,m R = [h mt,m R,0,...,h mt,m R,l,...,h mt,m R,L 1].

38 22 Introduction LS for MIMO-OFDM The LS channel estimate for (1.21) is expressed as ĥ mr (n) = (F H X H (n)x(n)f) 1 F H X H (n)y(n) mr (1.22) Obviously, the matrix to be inverted is with the size of N T L N T L and involves the complexity of O(N 3 T L3 ). 1.3 Motivations and Contributions Signal Detection In massive MIMO applications [54], as the number of transmit antennas N t is very large, many of the conventional MIMO detection algorithms like Sphere Decoding (SD) [42] have prohibitive complexity. As a result, new algorithms were proposed to reduce the complexity [66]-[67]. In [66] and [68], two local neighborhood search methods known as likelihood ascent search (LAS) and reactive tabu search (RTS) were presented. Both can achieve near-optimal performance for BPSK or QPSK modulations but perform poorly with high-order quadrature amplitude modulation (QAM). To further improve the performance for high-order QAM, layered tabu search (LTS) was presented in [67] but with much higher complexity. Interestingly, when turbo-processing is employed, recent research shows that for massive MIMO and under well conditioned channels, the linear detection method such as iterative minimum mean-squared error with soft interference cancellation can achieve near optimal performance [54]. Together with the iterative detection and decoding (IDD) technology, linear detection algorithm like the minimum mean square error parallel interference cancellation (MMSE-PIC) algorithm [56] [57] is attractive because of its low complexity and good bit error rate (BER) performance. To reduce the burden of performing matrix inversion for detecting every symbol in MMSE- PIC algorithm, some reduced complexity algorithms have been proposed [58] [59] and implemented in ASIC [60] [69] which require only one matrix inversion to detect one block of receive data.