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(1)國 立 交 通 大 學 電信工程學系碩士班 碩 士 論 文. 多傳送多接收位元交錯調變碼系統之 低複雜度迭代訊號偵測設計. On the Design of Low Complexity Iterative Signal Detection for MIMO BICM Systems. 研 究 生:曾鼎哲 指導教授:沈文和 博士. 中. 華. 民. 國. 九 十 四 年 七 月.

(2) 多傳送多接收位元交錯調變碼系統之 低複雜度迭代訊號偵測設計 On the Design of Low Complexity Iterative Signal Detection for MIMO BICM Systems 研 究 生:曾鼎哲. Student:Ting-Che Tseng. 指導教授:沈文和 博士. Advisor:Dr. Wern-Ho Sheen. 國 立 交 通 大 學 電 信 工 程 學 系 碩 士 班 碩 士 論 文 A Thesis Submitted to Institute of Communication Engineering College of Electrical Engineering and Computer Science National Chiao Tung University in Partial Fulfillment of the Requirements for the Degree of Master of Science in Communication Engineering July 2005 Hsinchu, Taiwan, Republic of China 中華民國九十四年七月.

(3) 多傳送多接收位元交錯調變碼系統之 低複雜度迭代訊號偵測設計. 研究生: 曾鼎哲. 指導教授: 沈文和 博士. 國立交通大學 電信工程學系碩士班. 摘要 隨著數位多媒體時代的來臨,用戶對於資料的需求急速增加。下一代的無線 通訊系統,如無線區域網路(802.11n)、第四代行動通訊系統,將可能採用多根 天線傳送及接收(MIMO)技術以提高資料傳輸率。然而,要如何在現有的通訊系統 中實現此新的技術是近年來熱門的研究。現行的通訊系統使用正交分頻多工 (OFDM)技術和位元交錯調變碼(BICM)來克服多重路徑、瑞雷(Rayleigh)衰落通 道,以提升系統效能。因此,此篇論文主要是在下一代通訊系統中,設計低複雜 度迭代訊號偵測。在低複雜度零強制(ZF)和最小均值平方差(MMSE)訊號偵測器 中,利用近似方法推導位元度規(bit metrics)的計算。另外,藉由渦輪(Turbo) 原理,設計低複雜度迭代 MMSE 偵測器,並提出幾個近似的方法減少偵測器的運 算。最後,採用下一代無線區域網路 802.11n 提案的系統架構,作為系統模擬環 境。利用電腦模擬方式,印證使用近似的位元規度計算,能有效地提高系統效能。 此外,在迭代 MMSE 偵測器中,從模擬結果顯示,利用這些近似的方法能降低計 算的複雜度,但不會減弱系統效能。. i.

(4) On the Design of Low Complexity Iterative Signal Detection for MIMO BICM Systems Student: Ting-Che Tseng. Advisor: Dr. Wern-Ho Sheen. Department of Communication Engineering National Chiao Tung University. Abstract With the advent of digital multimedia communications era, the amount of the demand for data stream of subscribes is increasing rapidly. The next generation of wireless communications, such as 802.11n wireless local area networks (WLAN), 4G mobile communications, may utilize multiple input multiple output (MIMO) approach to enhance data rate. Existing communications use orthogonal frequency division multiplexing (OFDM) and bit-interleaved coded modulation (BICM) techniques to overcome multipath Rayleigh fading channels. Hence, the theme of my thesis is to design low complexity iterative signal detection for next generation of wireless communications. We derive the bit metrics based on zero-forcing (ZF) and minimum mean squared error (MMSE) detector by approximation. Besides, we design low complexity iterative MMSE detector with turbo principle and propose some methods of approximation to reduce computation complexity. Finally, we apply them to the system model of 802.11n Proposal. From simulation results, it proves that using approximated bit metrics can improve the performance, and employing the approximation of iterative MMSE detector can reduce the computation complexity without performance deterioration.. ii.

(5) 誌. 謝. 本篇論文得以順利完成,首先要特別感謝我的指導教授 沈文和 博士,在兩年研究過程中給予非常多的指導與建議,並教導我進行研 究的方法與態度,尤其沈文和教授做研究謹慎小心的態度,讓我印象 深刻,對我這兩年在研究所以及未來做研究有影響很大。在本篇論文 中,有許多的觀念及推導方式都學習沈教授做研究的方法和教導,才 使得研究工作能夠順利進行。另外,要感謝口試委員 祁忠勇教授和 李大嵩教授的指正與建議,使我的碩士論文更加完善。 除此之外,我要感謝無線寬頻接取系統實驗室的學長許正欣博 士、郭志成、傅宜康、蕭昌龍,以及同學蔡政龍與林愷昕,另外,還 有其他已畢業的學長姐們,學長曾俊傑博士、何建興博士、李育瑋、 黃亮維、陳長新、方凱易和學姐楊雁雯等,以及許多曾經幫助過我的 人。此外,要特別感謝徐進發同學,和我頻繁的討論,共同架設以及 驗證模擬平台,並給我許多建議。在這兩年研究生活中,感謝大家所 給予的鼓勵與幫忙,讓我克服種種困難,論文得以順利完成。 最後,感謝我的家人,尤其是我的父母和阿姨,在求學的過程中, 給我無憂無慮的生活,讓我能專心致力讀書與研究。還有感謝姐姐為 我禱告,使我能順利完成學業。另外,要感謝我最親愛的雙胞胎弟弟, 聖哲,是和我相處最久的人,一起讀書討論,在生活上也互相照應。 無論遭遇任何挫折你們總是給我最大的支持並陪伴我渡過難關,謝謝 你們。. 民國九十四年七月 研究生曾鼎哲謹識於交通大學. iii.

(6) Contents 摘要.................................................................................................................................i Abstract ..........................................................................................................................ii 誌 謝.......................................................................................................................... iii Contents ........................................................................................................................iv List of Tables.................................................................................................................vi List of Figures ..............................................................................................................vii Chapter 1: Introduction ............................................................................................. - 1 Chapter 2: System Model.......................................................................................... - 5 2.1 Introduction to TGn Sync Proposal .............................................................. - 5 2.1.1 Preamble Format.............................................................................. - 6 2.1.2 Encoder and Puncturing................................................................... - 7 2.1.3 Bit Interleaving .............................................................................. - 10 2.1.4 Signal Mapping.............................................................................. - 11 2.2 MIMO Channel Model ............................................................................... - 13 2.3 Signal-to-Noise Ratio (SNR) Definition..................................................... - 17 2.4 Notation of MIMO-OFDM Systems .......................................................... - 18 Chapter 3: Linear Multi-Stage Detection................................................................ - 21 3.1 Bit Metrics for BICM ................................................................................. - 21 3.2 ZF Criterion ................................................................................................ - 24 3.2.1 Approximation of Bit Metrics........................................................ - 25 3.2.2 Simulation Results ......................................................................... - 27 3.3 MMSE Criterion ......................................................................................... - 30 3.3.1 Approximation of Bit Metrics........................................................ - 31 3.3.2 Simulation Results ......................................................................... - 33 iv.

(7) 3.4 Conclusions................................................................................................. - 44 Chapter 4: Low-Complexity Iterative Detection .................................................... - 45 4.1 Optimal Receiver Based on MAP Algorithm ............................................. - 48 4.1.1 MAP Detector ................................................................................ - 49 4.1.2 MAP (BCJR) Decoder ................................................................... - 50 4.2 Iterative MMSE Detector............................................................................ - 54 4.2.1 Approximation I of the proposed iterative MMSE detector .......... - 59 4.2.2 Approximation II of the proposed iterative MMSE detector ......... - 60 4.2.3 Approximation III of the proposed iterative MMSE detector........ - 62 4.2.4 Approximation IV of the proposed iterative MMSE detector........ - 64 4.3 Simulation Results ...................................................................................... - 64 4.4 Conclusions................................................................................................. - 71 Chapter 5: Conclusions and Future Works.............................................................. - 72 5.1 Conclusions................................................................................................. - 72 5.2 Future Works............................................................................................... - 73 Appendix A: Multistage Detection for A Linear MMSE Receiver......................... - 74 Appendix B: Multistage Detection for Iterative MMSE Receiver ......................... - 76 Appendix C: Modulation-Coding Scheme (MCS) ................................................. - 78 Appendix D: IEEE 802.11n Channel Model B....................................................... - 80 References............................................................................................................... - 81 -. v.

(8) List of Tables Table 2-1: Timing related parameters ....................................................................... - 7 Table 2-2: Frequency rotation ................................................................................. - 11 Table 2-3: Summary of model parameters for LOS/NLOS conditions. ................. - 14 Table 2-4: Path loss model parameters ................................................................... - 14 Table C-1: Modulation-coding scheme................................................................... - 79 -. vi.

(9) List of Figures Fig. 2-1: Transmitter diagram of TGn Sync proposal for MIMO-OFDM systems in 20MHz ...................................................................................................... - 6 Fig. 2-2: PPDU format for 2x20 mandatory basic MIMO transmission .................. - 6 Fig. 2-3: The convolutional encoder (K=7, R=1/2) .................................................. - 8 Fig. 2-4: The bit-stealing and bit-insertion procedure for code rate Rc =2/3 ........... - 9 Fig. 2-5: The bit-stealing and bit-insertion procedure for code rate Rc =3/4 .......... - 9 Fig. 2-6: Bit interleaver for MIMO systems in TGn Sync proposal ....................... - 10 Fig. 2-7: BPSK, QPSK, and 16-QAM constellation bit encoding.......................... - 12 Fig. 2-8: 64-QAM constellation bit encoding......................................................... - 12 Fig. 2-9: The block diagram of the MIMO channel model..................................... - 13 Fig. 2-10: Multipath MIMO channels with two clusters ........................................ - 15 Fig. 2-11: Power delay profile (PDP) in channel model B ..................................... - 15 Fig. 2-12: CDF of channel model B........................................................................ - 16 Fig. 2-13: “Bell” shape Doppler power spectrum................................................... - 17 Fig. 2-14: Notations of a MIMO-OFDM transmitter.............................................. - 19 Fig. 2-15: Notations of a MIMO-OFDM receiver .................................................. - 20 Fig. 3-1: To group M interleaved-coded bits to map a modulated symbol for MIMO-OFDM systems ......................................................................................... - 22 Fig. 3-2: PER of bit metrics calculation with equal and weighted coefficients by ZF detector for BPSK and QPSK in channel B, 2x2 .................................................. - 28 Fig. 3-3: PER of bit metrics calculation with equal and weighted coefficients by ZF detector for 16-QAM and 64-QAM in channel B, 2x2 ......................................... - 29 Fig. 3-4: BER of bit metrics calculation with equal and weighted coefficients by ZF detector for BPSK and QPSK in channel B, 2x2 .................................................. - 29 vii.

(10) Fig. 3-5: BER of bit metrics calculation with equal and weighted coefficients by ZF detector for 16-QAM and 64-QAM in channel B, 2x2 ......................................... - 30 Fig. 3-6: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector in AWGN channel, 2x2 ............................................................... - 34 Fig. 3-7: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector in AWGN channel, 2x2 ............................................................... - 35 Fig. 3-8: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x2...................................... - 36 Fig. 3-9: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x2............................. - 36 Fig. 3-10: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x2...................................... - 37 Fig. 3-11: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x2............................. - 37 Fig. 3-12: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x3...................................... - 38 Fig. 3-13: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x3............................. - 39 Fig. 3-14: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x3...................................... - 39 Fig. 3-15: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x3............................. - 40 Fig. 3-16: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 3x3...................................... - 41 Fig. 3-17: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 3x3............................. - 41 viii.

(11) Fig. 3-18: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 3x3...................................... - 42 Fig. 3-19: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 3x3............................. - 42 Fig. 3-20: PER of bit metrics calculation with weighted coefficients by MMSE detector and ZF detector for BPSK and QPSK in channel B, 2x2 ........................ - 43 Fig. 3-21: PER of bit metrics calculation with weighted coefficients by MMSE detector and ZF detector for 16-QAM and 64-QAM in channel B, 2x2 ............... - 43 Fig. 4-1: A MIMO transmitter................................................................................. - 45 Fig. 4-2: The MIMO channel.................................................................................. - 46 Fig. 4-3: A MIMO iterative receiver ....................................................................... - 47 Fig. 4-4: A inner detector and a outer decoder........................................................ - 49 Fig. 4-5: To group log 2 M interleaved-coded bits to map a modulated symbol for MIMO systems ...................................................................................................... - 56 Fig. 4-6 : The block diagram of the proposed iterative MMSE receiver ................ - 58 Fig. 4-7: The block diagram of the approximation I of the proposed iterative MMSE receiver .................................................................................................................. - 59 Fig. 4-8: The block diagram of the approximation II of the proposed iterative MMSE receiver .................................................................................................................. - 62 Fig. 4-9: The block diagram of the approximation III of the proposed iterative MMSE receiver .................................................................................................................. - 63 Fig. 4-10: Performance of the proposed iterative MMSE detector (64QAM, Rc=3/4, 2x2)........................................................................................................................ - 65 Fig. 4-11: Performance of proposed iterative MMSE detector with approximation I (BPSK, Rc=1/2, 2x2) ............................................................................................. - 66 Fig. 4-12: Performance of proposed iterative MMSE detector with approximation I ix.

(12) (64-QAM, Rc=3/4, 2x2)......................................................................................... - 67 Fig. 4-13: Performance of proposed iterative MMSE detector with approximation I (64-QAM, Rc=3/4, 3x3)......................................................................................... - 67 Fig. 4-14: Compare the performance of the proposed iterative MMSE detector with approximation II to the proposed iterative MMSE detector (64-QAM, Rc=3/4, 2x2) . 68 Fig. 4-15: Compare the performance of the proposed iterative MMSE detector with approximation III to the proposed iterative MMSE detector (64-QAM, Rc=3/4, 2x2) 69 Fig. 4-16: Compare the performance of the proposed iterative MMSE detector with approximation IV to the proposed iterative MMSE detector (64-QAM, Rc=3/4, 2x2) 70 -. x.

(13) Chapter 1: Introduction With the advent of digital multimedia communication era, such as wireless local area networks (WLAN), digital audio broadcasting (DAB), digital video broadcasting television (DVB-T), mobile communications, and video conference, the amount of the demand for data stream of subscribers is increasing rapidly. The existing wireless communication systems may not satisfy the users. Increasing the transmission bandwidth is a method to enhance data rate. However, the available spectrum is limited and precious so the mean of increasing the transmission bandwidth to raise data rate is inefficiency. Recently, advances in coding, for example turbo code [7] and low density parity check (LDPC) code [8], are used to approach the Shannon bound [9] and then to enhance the capacity of channel. Nevertheless, those advances need a high-complexity receiver. Multiple-input multiple-output (MIMO) technique can enhance the data rate without increasing transmission bandwidth.. The MIMO techniques use multiple antennas to transmit and receive signals. The utility of multiple antennas offers extended range, improved reliability, or higher throughputs. Two main functions of multiple antennas are diversity and multiplexing. If all transmitter antennas send identical data simultaneously with the same bandwidth, such as smart antenna based systems or space-time code (STC) based systems, the systems can provide antenna gain, interference suppression and diversity gain in a fading channel. Smart antenna based systems may have array of multiple antennas only at one end of communication link, such as multiple-input -1-.

(14) single-output (MISO) and single-input multiple-output (SIMO). STC based systems, such as Alamouti code based systems, can provide diversity for MIMO channels. In my thesis, we focus on the other function of MIMO techniques-multiplexing. In spatial multiplexing-based MIMO systems, each transmit antenna can broadcast an independent signal sub-stream at the same time and in the same bandwidth. Using MIMO techniques with n transmitter antennas and n receiver antennas can increase n times data rate than those in systems with single-antenna. This technique is going to be implemented in the growing demand for future high data rate WLAN, WAN, PAN and 4G systems. In order to overcome fading channel, our system design is based on the orthogonal frequency division multiplexing (OFDM) and bit-interleaved coded modulation (BICM) [13] techniques. The two techniques are widely used in existing wireless communications, such as DAB, DVB, WLAN and wireless metropolitan area networks (Wireless MAN). OFDM technique was proposed in 1967 [12]. Due to the difficult and expansive hardware implementation of orthogonal multiple carriers and the lack of digital signal processing (DSP), this technique was not popular at that time. Until the discrete Fourier transform (DFT) was proposed by Weistein and Ebert in 1971, people paid more attention to OFDM technique again. Zehavi used bit-interleaver between encoder and modulator in 1992 [23]. Then the diversity order of coding could be increased by the minimum number of distinct coded bits. This technique was called as BICM in 1998 [13]. It has better performance than symbol interleaver over fading channels with the same coding and decoding architecture.. Since 1998, there have been more and more papers and documents to discuss -2-.

(15) and analyze MIMO techniques. Telatar and Foschini discuss the fundamental capacity limits for MIMO channels in [10] and [11], respectively. For MIMO multiplexing systems, all spatial streams would interfere with one another and be mixed at the receiver. All signals are not separated easily, especially in correlated channels. How to separate and detect data from blended received signals is a critical issue. There are many kinds of detector, such as a maximum likelihood (ML) detector, a minimum mean square errors (MMSE) detector, and a zero-forcing (ZF) detector. The main goal in my thesis is to design a low complexity detector. A MMSE detector is used popularly in MIMO systems. It has higher performance than the other linear detectors and lower complexity compared to the ML detector. Hence, we design a detector based on the MMSE criterion for MIMO-BICM systems.. In the paper [14], author expanded the BICM technique to multiple antenna transmission to obtain its merits in fading channels and derived the optimal bit metrics computations for MIMO-OFDM BICM systems. It is based on a ML detector and has more complex computation. In the paper [15], Butler presented the weight of bit metrics calculation based on a ZF detector. However, the performance of the ZF detector is poor. Hence, we derive the approximation of bit metrics for MIMO-OFDM-BICM systems based on popular MMSE detector by Gaussian approximation. We are going to discuss and analyze the improvement of performance of MMSE detector with the approximated bit metrics. The second topic in my thesis is to design a low complexity iterative MMSE detector based on the turbo principle to improve performance. We derive the bit metrics and coefficients of a iterative MMSE detector. We propose some methods of approximation to reduce computation of a iterative MMSE detector. Those methods of approximation can decrease much computation without deteriorating performance for many iterations -3-.

(16) and long packets.. The reset of this thesis is organized as follows: in chapter 2, we depict our simulation scenario, channel models, and system architectures. Moreover, we give the notations for MIMO-OFDM systems and definition of the signal-to-noise ratio (SNR). In chapter 3, we present the approximation of bit metric calculation based on a ZF detector and a MMSE detector for MIMO-OFDM BICM systems. In chapter 4, we design a low-complexity iterative MMSE detector and use some methods of approximation to reduce the computation complexity of the detector. Finally, we give some conclusions and future works in chapter 5.. -4-.

(17) Chapter 2: System Model The next generation of wireless local area networks (WLAN), IEEE802.11n, is based on multiple input multiple output (MIMO) and orthogonal frequency division multiplexing (OFDM) techniques to provide a point-to-point high throughput transmission. The working group of IEEE802.11n holds a conference on the odd months. There are four complete proposals proposed last year by four different groups, TGn Sync, WWiSE, MitMot, and Qualcomm. In the beginning of this year, the Qualcomm gave up their proposal and joined the TGn Sync group which is composed of Agere Systems Inc., Intel Corporation, Marvell Semiconductor Inc., and etc. Mitsubish and Motorola gave up their proposal (MitMot) and joined the TGn Sync and the WWiSE groups, respectively. Hence, there are tow major groups, the TGn Sync and the WWiSE to compete in order to make their own proposal to become the standard of IEEE802.11n.. The physical layer of two proposals of TGn Sync and WWiSE are based on the same MIMO-OFDM systems, but whole system design are different, especially the preamble format and transmission mode. Here, our simulation platform is based on version 3 of the TGn Sync proposal to IEEE 802.11n. Equation Section 2. 2.1 Introduction to TGn Sync Proposal. The block diagram of transmitter in TGn Sync Proposal for throughput enhancement is shown in Fig. 2-1. -5-.

(18) Constellation Mapper. iFFT 64 tones in 20MHz 52 populated tones 48 data tones 4 pilot tones. Insert GI window. RF BW ~ 17MHz. Frequency Interleaver across 48 data tones. Constellation Mapper. iFFT 64 tones in 20MHz 52 populated tones 48 data tones 4 pilot tones. Insert GI window. RF BW ~ 17MHz. spatial parser. Puncturer. Channel Encoder. Frequency Interleaver across 48 data tones. Fig. 2-1: Transmitter diagram of TGn Sync proposal for MIMO-OFDM systems in 20MHz. The basic configuration of this proposal delivers a maximum mandatory rate of 243 Mbps with only two antennas. This rate is 5 times the rate of 802.11a/g (54Mbps). The proposal also includes options for higher rates beyond 600 Mbps. In order to achieve the higher data rates, the PHY techniques use MIMO techniques with spatial division multiplexing of spatial streams and evolution of 802.11 OFDM PHY. The proposal uses wider bandwidth options, 40MHz channelization, to increase data rate. Timing related parameters is shown in Table 2-1.. 2.1.1 Preamble Format. The PPDU format for transmission with 2 antennas in a 20MHz channelization is shown in Fig. 2-2.. ANT_2. 20MHz. ANT_1. 20MHz. 8us L-STF. L-STF. 8us. 4us. L-LTF. L-SIG. L-SIG. L-LTF. 8us. 2.4us. 7.2us. 7.2us. HT-SIG. HT-STF. HT-LTF. HT-LTF. HT-DATA. HT-SIG. HT-STF. HT-LTF. HT-LTF. HT-DATA. Fig. 2-2: PPDU format for 2x20 mandatory basic MIMO transmission. -6-.

(19) The high through (HT) preamble of TGn Sync proposal is a concatenation of the legacy preamble (802.11.a) and a HT-specific preamble. The functions performed by the preamble include start of packet detection, auto-gain-control (AGC), coarse frequency offset estimation, coarse timing offset estimation, fine frequency offset estimation, fine frequency offset estimation, and channel estimation.. Parameter. Value for 20 MHz. Value for 40 MHz. Channel. Channel. N SD : Number of data subcarriers. 48. 108. N SP : Number of pilot subcarriers. 4. 6. N SN : Number of center null subcarriers. 1 (tone = 0). 3 (tones = -1,0,+1). N SR : Subcarrier range. 26. 58. (index range). (-26 … +26). (-58 … +58). Δ F : Subcarrier frequency spacing. 0.3125 MHz. 0.3125 MHz. (= 20 MHz / 64). (= 40 MHz / 128). 3.2 μsec. 3.2 μsec. 0.8 µsec. 0.8 µsec. : Short GI duration. 0.4 µsec. 0.4 µsec. TGI 2 : Legacy LongTraining symbol GI. 1.6 µsec. 1.6 µsec. TSYM : Symbol interval. 4 µsec. 4 µsec. TLONG : Long training field duration. 8 µsec. 8 µsec. THT − LONG :HT Long training field duration. 7.2 µsec. 7.2 µsec. TSHORT : Short training field duration. 8 µsec. 8 µsec. THT − SHORT : HT Short training field duration. 2.4 µsec. 2.4 µsec. TS : Nyquist sampling interval. 50 nsec. 25 nsec. TFFT : IFFT/FFT period TGI. : GI duration. TShortGI. duration. Table 2-1: Timing related parameters. 2.1.2 Encoder and Puncturing. A mandatory encoder is a convolutional encoder and a optional encoder is a -7-.

(20) low-density-parity-check (LDPC) encoder. In our simulation platform, the transmitter is implemented by the mandatory encoder. The convolutional encoder should work by the industry-standard generator polynomials, g0 =1338 and g1 =1718 , with the constraint length 7 of the code rate Rc = 1 2 , as shown in Fig. 2-3.. Fig. 2-3: The convolutional encoder (K=7, R=1/2). In order to achieve high data rate and different coding rate Rc with the same the industry-standard convolutional encoder, the transmitter would employ a puncturing method. Puncturing the coded bits is shown in Fig. 2-4 and Fig. 2-5 to reach coding rate Rc = 2/3 and Rc = 3/ 4 , respectively. In our receiver design, we choose the soft Viterbi decoding to decode information bits. However, we use a MAP decoder to design an iterative receiver.. -8-.

(21) Fig. 2-4: The bit-stealing and bit-insertion procedure for code rate Rc =2/3. Fig. 2-5: The bit-stealing and bit-insertion procedure for code rate Rc =3/4 -9-.

(22) 2.1.3 Bit Interleaving. In order to overcome the Rayleigh fading channel and avoid any transmitter antenna fade, this proposal utilizes a space-frequency bit interleaving shown in Fig. 2-6. Coded and punctured bits are interleaved across spatial streams and frequency tones by two steps- spatial stream parsing and frequency interleaving.. SISO (11a/g). MIMO 2x. 11a Bit interleaver, Permutation Operation 1. 11a Bit interleaver, Permutation Operation 2. 11a Bit interleaver, Permutation Operation 1. 11a Bit interleaver, Permutation Operation 2. 11a Bit interleaver, Permutation Operation 1. 11a Bit interleaver, Permutation Operation 2. parser Frequency Rotation. Fig. 2-6: Bit interleaver for MIMO systems in TGn Sync proposal. Spatial stream parsing uses a round-robin parser to parse coded and punctured bits to multiple spatial streams, defined by s = max { N BPSC / 2,1}. (2.1). where N BPSC is the number of bits per subcarrier and s is the number of QAM bit order values. The parser sends consecutive blocks of s bits to different spatial streams.. The second step is frequency interleaver based on the 802.11a interleaver with certain modifications. It can be divided to three permutations. The first permutation is defined by the rule - 10 -.

(23) i = N row × ( k mod N column ) + floor ( k / N column ) , k = 0,1,… , N CBPS − 1. (2.2). where N CBPS is the number of coded bits per OFDM symbol. The second permutation is defined by the rule. j = s × floor ( i / s ) + ( i + N CBPS − floor ( N column × i / N CBPS ) ) mod s, i = 0,1,… , N CBPS − 1. (2.3). where s is determined by s = max( N BPSC / 2,1) The third permutation is defined by the rule. (. ). r = j − ( ( 2 × iss ) mod 3 + 3 × floor(iss / 3) ) × N rot × N BPSC mod N CBPS j = 0,1,… , N CBPS − 1. (2.4). where iSS = 0,1, … , N SS − 1 is the index of the spatial stream on which this interleaver is operating.. Channelization. 20MHz. Total # of Streams. 1. 2. 3. 4. 1. 2. 3. 4. 0. 0. 0. 0. 0. 0. 0. 0. 22. 22. 22. 58. 58. 58. 11. 11. 29. 29. st. nd. 2 stream Rotation. Frequency. 1 stream. 3rd stream. 40MHz. th. 33. 4 stream. 87. Table 2-2: Frequency rotation. 2.1.4 Signal Mapping. The signal of OFDM subcarriers should be modulated by BPSK, QPSK, 16-QAM, or 64-QAM with the gray labeling. It is the same as the modulation scheme of the standard of IEEE802.11a. The constellations of BPSK, QPSK, and 16-QAM are shown in Fig. 2-7. The constellations of 64-QAM is shown in Fig. 2-8.. - 11 -.

(24) Fig. 2-7: BPSK, QPSK, and 16-QAM constellation bit encoding. Fig. 2-8: 64-QAM constellation bit encoding. - 12 -.

(25) 2.2 MIMO Channel Model. The block diagram of the MIMO indoor channel model proposed by IEEE802.11 TGn is shown in Fig. 2-9.. Fig. 2-9: The block diagram of the MIMO channel model. There are six channel models defined in IEEE 802.11n document [29] for next generation of WLAN. The properties of these channel models are shown in Table 2-3 and Table 2-4. K-factor for LOS conditions applies only to the first tap, for all other taps K= −∞ dB.. - 13 -.

(26) Model. Conditions. K-factor (dB). RMS delay spread (ns). # of clusters. A (optional). LOS/NLOS. 0/ -∞. 0. 1 tap. B. LOS/NLOS. 0 / -∞. 15. 2. C. LOS/NLOS. 0 / -∞. 30. 2. D. LOS/NLOS. 3 / -∞. 50. 3. E. LOS/NLOS. 6 / -∞. 100. 4. F. LOS/NLOS. 6 / -∞. 150. 6. Table 2-3: Summary of model parameters for LOS/NLOS conditions.. New Model. dBP (m). Slope before dBP. Slope after dBP. Shadow fading std. dev. (dB) before dBP (LOS). Shadow fading std. dev. (dB) after dBP (NLOS). A (optional). 5. 2. 3.5. 3. 4. B. 5. 2. 3.5. 3. 4. C. 5. 2. 3.5. 3. 5. D. 10. 2. 3.5. 3. 5. E. 20. 2. 3.5. 3. 6. F. 30. 2. 3.5. 3. 6. Table 2-4: Path loss model parameters. We choose channel model B for our simulation environment. There are 2 clusters shown in Fig. 2-10 and 9 multipaths in channel model B. The power delay profile of channel model B is shown in Fig. 2-11. The cumulative distribution function (CDF) of channel model is shown in Fig. 2-12. The channel model B is a multipath Rayleigh fading channel with the speed of pedestrian v =1.2 km/hr .. - 14 -.

(27) Cluster 1 Cluster 2. R1. R2. LOS Tx Antennas Rx Antennas Fig. 2-10: Multipath MIMO channels with two clusters. Power delay profile (PDP) in channel model B. 1 0.9 0.8 0.7. Power. 0.6 0.5 0.4 0.3 0.2 0.1 0. 1. 2. 3. 4. 5 Tap index. 6. 7. 8. Fig. 2-11: Power delay profile (PDP) in channel model B. - 15 -. 9.

(28) cdf of Channel Model B. 0 -0.2 -0.4. log10 CDF. -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -30. -25. -20. -15. -10 -5 20 log10(h) [dB]. 0. 5. 10. 15. Fig. 2-12: CDF of channel model B. The fading characteristics of the indoor wireless channels are very different from the mobile case. Transmitter and receiver are stationary and people are moving between them in indoor wireless systems, but the user terminals are often moving through an environment in outdoor mobile systems. Therefore, a new function S ( f ) can be defined as (2.5) for indoor scenario to fit the Doppler power spectrum measurements. (in linear values, not dB values): 2 S ( f ) = ⎡1 + A ( f f d ) ⎤ ⎣ ⎦. -1. (2.5). where A is a constant, used to define the 0.1 S ( f ) , at a given frequency f d , being the Doppler Spread.. ( S ( f )). f = fd. = 0.1. where. i fd =. vo. λ. : the Doppler spread. i vo is the environmental speed (default value is 1.2 km/hr) iλ=. c : the wavelength fc - 16 -. (2.6).

(29) i c: the light speed. i f c : the carrier frequency. S ( f ) is similar to the “Bell” shape spectrum, as shown in. -10 dB. fd. f max. Fig. 2-13: “Bell” shape Doppler power spectrum. f max is the maximum frequency component of the Doppler power spectrum.. 2.3 Signal-to-Noise Ratio (SNR) Definition. The signal to noise ratio is defined as the ratio of the signal power in the aggregate of the -10dB signal bandwidths divided by the noise power in the aggregate of the -10dB signal power bandwidths. In addition, the signal power at the receiver is the sum of signal powers from all the transmitter antennas for MIMO systems. ⎧ ⎪ Channel Gain=E ⎨ ⎪ ⎩. (∑. ). P (j) N R ⎫⎪ ⎬ N ⎪ ∑ j=T1 P(j) t ⎭ NR. j=1 r. where. i Pr (j) is the received signal power at j th receiver antenna i P(j) is the transmitted signal power at j th transmitter antenna t - 17 -. (2.7).

(30) 2.4 Notation of MIMO-OFDM Systems. The notations of MIMO-OFDM transmitter is shown in Fig. 2-14. where i bn : the nthinformation bit i cn : the nth coded bit i cnp : the nth interleaved bit at the p th transmitter antenna i s p.k : the modulated signal at the k th subcarrier and the. th. OFDM symbol. th. at the p transmitter antenna i s′p (t): the transmitted signal at time t at the p th transmitter antenna i p ∈ {1,. , NT }: the index of transmitter antenna. i n : the index of bit sequence i. : the index of modulated symbol sequence. i t : time index. i NT : the number of transmitter antennas i Lb : the number of information bits bn i Lc =. Lb. i Lc = ⎡. Rc. : the number of coded bits cn , where R c = nk00 is code rate. Lc ⎤ : the number of interleaved bits cnp per tx antenna NT. i K =52 : the number of OFDM subcarriers. i Ls = ⎡. Lc ⎤ : the number of OFDM symbols per tx antenna for M -QAM K ⋅ log 2 M. - 18 -.

(31) {c }. { }. { }. { }. L -1 1 c n n=0. {bn }n =0. Lb -1. { } cn. cnp. Lc -1. Lc -1 n=0. s 1,k. s p,k. = Ls -1,k= K. s1′(t ) s′p (t ). =0 ,k=0. n =0. {c } NT n. Fig. 2-14: Notations of a MIMO-OFDM transmitter. - 19 -. Lc -1 n=0. { } s N,kT. s′NT (t ).

(32) The notations of MIMO-OFDM receiver is shown in.Fig. 2-15. r1′(t ). { } {Λ ( c )}. {r }. Bit Metrics Calculation. DeMUD. q ,k. {Λ ( c )}. p n. y p,k. Remove GI FFT. n. Deinterleaver. Viterbi Decoder. {H }. Remove GI FFT. q ,p ,k. rn′R (t ). Channel Estimation. Fig. 2-15: Notations of a MIMO-OFDM receiver. where i rq′(t): the received signal at time t at the q th transmitter antenna. i r q.k : the modulated signal at the k th subcarrier and the. th. OFDM symbol. at the q th transmitter antenna i y p.k : the detected signal at the k th subcarrier and the. th. OFDM symbol. th. from the p transmitter antenna. ( ). i Λ cnp : the a posteriori log likelihood ratio of cnp. ( ). i Λ cn : the a posteriori log likelihood ratio of deinterleaved bit cn i bn : the nth estimated information bit i q ∈ {1,. , N R }: the index of receiver antenna. i N R : the number of transmitter antennas. - 20 -. { } bˆn. Lb -1 n =0.

(33) Chapter 3: Linear Multi-Stage Detection This chapter considers MIMO-OFDM systems with bit-interleaved coded modulation (BICM) [7]. The maximum likelihood (ML) receiver has higher computation complexity. Therefore, we design a low-complexity receiver with a linear detector based on zero-forcing (ZF) and minimum mean squared errors (MMSE) algorithms. Here, we propose how to calculate the bit metrics for BICM on a ZF receiver and an MMSE receiver. Equation Section 3. 3.1 Bit Metrics for BICM. BICM technique is suited to multipath fast-fading channels, then the sub-channels of OFDM systems with bit-interleaver can be approximated as independently fast-fading. For better performance, the decoder is implemented by soft Viterbi decoding. Because bit interleaving is applied to the encoded bit before the M-QAM modulator, maximum likelihood decoding of BICM signals would require joint decoding and demodulation. According to the MAP criterion, estimate the coded bit sequence. {cˆ } p n. {c }. p Lc -1 n n=0. Lc -1 n=0. at the p th sub-stream by. {. { } {y }. =arg max p ⎡ cnp L -1 p c c { n }n=0 ⎢⎣. Lc -1. n=0. p ,k. =Ls -1,k=K -1 =0,k=0. ⎤ ⎥⎦. }. (3.1). Thus, all possible coded and interleaved bit sequences would be calculate in (3.1). Zehavi proposed a decoding scheme in [23] to compute sub-optimal simplified bit metrics to be used inside a Viterbi decoder for path metric computation. - 21 -.

(34) Define the bit metrics of coded and interleaved bit c. p ,k,m. for BICM system by. ignoring the noise color, i.e., assuming the real part and the image part of noise are independent. Λ (c. p ,k,m. ⎛ p ⎡ c p,k,m =1 y p,k ⎤ ⎞ ⎦⎟ ln ⎜ ⎣ p p ⎜ p ⎡ c =0 y ,k ⎤ ⎟ ⎦⎠ ⎝ ⎣ ,k,m. ). We redefine coded and interleaved bit. (3.2). cnp to be c p,k,m ,. c p,k,m =cnp ,. (3.3). Interleaved coded bit sequences. cnp. p cn-1. cnp+m. p p c cn+M # -1 n+M #. Every log2 M coded bits to map a modulated symbol. cp,k-1,M#-1 c p,k,0. c p,k,m. c p,k,M # -1 c p,k+1,0. where M # =log 2 M Fig. 3-1: To group M interleaved-coded bits to map a modulated symbol for MIMO-OFDM systems. where i n = ⋅ K +k ⋅ log 2 M + m, i m ∈ ( 0,. ,log 2 M - 1) , the bit index of constellation. i ∈ ( 0,. ,Ls - 1 ) , the OFDM symbol index. i k ∈ ( 0,. ,K - 1) , the subcarrier index. c p,k,m. is the coded bit in the mth bit mapped onto a M-QAM symbolψ ,at the. k th subcarrier, at the. th. OFDM symbol, and at the p th sub-stream. Because the. - 22 -.

(35) computation of bit metrics of coded bit c the k th subcarrier, at the. th. p ,k,m. depends only on detected signal. y p,k at. OFDM symbol, and at the p th sub-stream, we can ignore. the subcarrier index k and the OFDM symbol index . Let. cmp =c p,k,m , y p =y p,k. p and the bit metrics of cm is. Λ (c. p m. ). ⎛ p ⎡ cmp =1 y p ⎤ ⎞ ⎦⎟ ln ⎜ ⎣ p p ⎜ p ⎡ c =0 y ⎤ ⎟ ⎦⎠ ⎝ ⎣ m. (3.4). A posteriori probability log likelihood ratio (LLR) can be shown as ⎛ ∑ p ⎡ s p =ψ y p ⎤ ⎞ ⎛ p ⎡ cmp =1 y p ⎤ ⎞ ⎣ ⎦⎟ ⎦ ⎟ =ln ⎜ ψ =Ψ (1)m ln ⎜ ⎣ p ⎜ ⎟ ⎜ p ⎡cm =0 y p ⎤ ⎟ ⎡ s p =ψ y p ⎤ ⎟ p ⎜ ∑ ⎦⎠ ⎦⎟ ⎜ ψ =Ψ ( 0) ⎣ ⎝ ⎣ m ⎝ ⎠. (3.5). where p i Ψ (1) m : the subset of all symbols with cm =1. p i Ψ (0) m : the subset of all symbols with cm =0. By the Bayes rules,. ⎛ ⎜ ψ∑ =Ψ (1) m ln ⎜ ⎜⎜ ψ ∑ ⎝ =Ψ (m0). ⎛ p ⎣⎡ s p =ψ y p ⎦⎤ ⎞ ⎟ ⎜ ψ∑ =Ψ (1) m = ln ⎟ ⎜ p p p ⎡⎣ s =ψ y ⎤⎦ ⎟ ⎟ ⎜⎜ ψ ∑ ⎠ ⎝ =Ψ (m0 ). p ⎣⎡ y p s p =ψ ⎦⎤ p ⎡⎣ s p =ψ ⎤⎦ ⎞ ⎟ ⎟ p ⎡⎣ y p s p =ψ ⎤⎦ p ⎡⎣ s p =ψ ⎤⎦ ⎟ ⎟ ⎠. (3.6). Because all symbols on the constellation are transmitted with equal probability, then equation (3.6) can be modified to. ⎛ ∑ p ⎡ s p =ψ y p ⎤ ⎞ ⎛ ∑ p ⎡ y p s p =ψ ⎤ ⎞ ⎦⎟ ⎦⎟ ⎜ ψ =Ψ (1)m ⎣ ⎜ ψ =Ψ (1)m ⎣ ln ⎜ = ln ⎟ ⎜ ⎟ p p p p ⎜⎜ ∑ p ⎡⎣ s =ψ y ⎤⎦ ⎟⎟ ⎜⎜ ∑ p ⎡⎣ y s =ψ ⎤⎦ ⎟⎟ ⎝ ψ =Ψ (m0) ⎠ ⎝ ψ =Ψ (m0 ) ⎠ By equations (3.5) and (3.7), the bit metrics is equal to. - 23 -. (3.7).

(36) ⎛ ∑ ⎜ ψ =Ψ (1) m Λ ( cmp ) =ln ⎜ ⎜⎜ ∑ ⎝ ψ =Ψ (m0 ). p ⎡⎣ y p s p =ψ ⎤⎦ ⎞ ⎟ ⎟ p ⎡⎣ y p s p =ψ ⎤⎦ ⎟ ⎟ ⎠. (3.8). Sub-optimal simplified LLR can be reduced by the log-sum approximation. ⎛ ⎞ ln ⎜ ∑ xi ⎟ ≈ max ( ln ( xi ) ) i ⎝ i ⎠. (3.9). The log-sum approximation is a good approximation if the summation in the left-hand side of equation (3.9) is dominated by the largest term. Then, at high signal-to-noise ratio (SNR), the bit metrics can be approximated by the log-sum approximation, see. ⎛ max p ⎡ y p s p =ψ ⎤ ⎞ ⎣ ⎦⎟ ψ =Ψ (1) ⎜ m Λ ( c ) ≈ ln ⎜ p p p ⎡⎣ y s =ψ ⎤⎦ ⎟⎟ ⎜ ψmax ( 0) ⎝ =Ψ m ⎠ p m. (3.10). 3.2 ZF Criterion. In this section, use ZF approach to detect signal. Because the MIMO-OFDM systems is used in the indoor WLAN scenario, we can assume the MIMO channel is multipath quasi-static Rayleigh fading channel. The frequency response H k of th. MIMO channel in the k subchannel of OFDM systems is defined as. ⎡ H k1,1 ⎢ Hk = ⎢ ⎢ H kN R ,1 ⎣. H kq,p. H k1,NT ⎤ ⎥ ⎥ N R ,NT ⎥ Hk ⎦. The transmitted signal vector before IFFT/GI is defined as. s .k = ⎡⎣ s 1.k ,. ,s N.kT ⎤⎦. (3.11). s. .k. T. The received signal vector after FFT/remove-GI is defined as r .k - 24 -. (3.12).

(37) r .k = ⎡⎣ r 1.k ,. ,r N.kR ⎤⎦. T. (3.13). And the received signal vector after FFT/remove-GI can be represented as. r .k =H k s .k +n where n. .k. =[n1.k ,. (3.14). .k. ,n N.kR ] is the received noise vector. th. Define the coefficient of the linear detector in the k subchannel is ⎡ g k1,1 ⎢ G kZF = ⎢ ⎢ g kNT ,1 ⎣. g kp,q. g k1,N R ⎤ ⎥ p p,1 ⎥ and g k = ⎡⎣ g k , g kNT ,N R ⎥⎦. ,g kp,N R ⎤⎦. H. (3.15). To detect signal at the p th sub-stream based on the zero-forcing criterion. ˆ =1 ZF criterion: G kZF H k. (3.16). So,. ( ). ˆ G kZF = ⎡ H ⎣⎢ k. H. -1. ˆ ⎤ H ˆ H k⎥ k ⎦. (3.17). Then, the output signal of the ZF receiver is. y .k =G kZFr .k =G kZF H k s .k +G kZFn. .k. (3.18). Assume there is perfect channel estimation. Then the detected signal vector is. y .k =s .k +G kZFn .k. (3.19). where. y .k = ⎡⎣ y 1.k ,. ,y N.kR ⎤⎦. T. 3.2.1 Approximation of Bit Metrics. Observe the p th sub-stream detected signal. - 25 -. y p.k ,.

(38) ( ). y p.k =s p.k + g kp. H. n. (3.20). .k. ,n N.kR are statistically independent and identically. 1. The received noises n .k ,. distributed complex Gaussian random variables with zero mean and variance σ n2 . Define over all noise term in the equation (3.20) is. ( ). z p.k = g kp. H. n .k =g kp,1n1.k +. +g kp,N R n N.kR. (3.21). Then z p.k is still a complex Gaussian random variable with zero mean and variance σ z2p . .k. σ. 2 z p.k. (. = g. p,1 k. )σ 2. 2 n1.k. (. +. + g. p,N R k. )σ 2. NR. 2 n N.kR. =σ ⋅ ∑ g kp,q. 2. 2 n. (3.22). q=1. where. σ n2 =. =σ n2NR =σ n2. 1 .k. .k. Then the detect signal is shown as. y p.k =s p.k +z p.k The conditional pdf of. y p.k. (3.23). is a complex Gaussian distribution,. (. ). p y p.k s p.k =ψ =. π σz. ⎧ exp ⎨ σ-2 1 y p.k-ψ ⎩ z p.k. (. 1 p .k. ) ⎫⎬ 2. (3.24). ⎭. By the equation (3.10) and (3.24), the bit metrics c p.k ,m is equal to. (. Λ c. p .k , m. ). 2⎫ ⎞ ⎛ ⎧ exp ⎨ σ-2 1 y p.k-ψ ⎬ ⎟ ⎜ ψmax (1) =Ψ m ⎩ z p.k ⎭⎟ =ln ⎜ ⎜ 2 ⎧ -1 p ⎫⎟ exp y - ψ 2 ⎨ ⎬⎟ ⎜ ψmax . k ( 0) σ ⎩ z p.k ⎭⎠ ⎝ =Ψ m 2 1 = 2 - min(1) y p.k-ψ + min( 0 ) y p.k-ψ. σz. p .k. {. ψ =Ψ m. (. ). (. ). (. - 26 -. ). ψ =Ψ m. (. (3.25). )} 2.

(39) Then, define the coefficients of bit metrics c p.k ,m for BICM. p .k. W =. 1. σ z2. p .k. NR ⎡ 2⎤ = ⎢σ n2 ⋅ ∑ g kp,q ⎥ q=1 ⎣ ⎦. -1. (3.26). By the way, the signal-to-noise ratio of y p.k is. { }=. E s p.k SNR=. {. 2. E z p.k 2. }. σ s2 NR. σ ⋅∑ g 2 n. (3.27) p,q 2 k. q=1. So the coefficient of bit metrics c p.k ,m for BICM is directly proportional to the signal-to-noise ratio of detected signal y p.k .. 3.2.2 Simulation Results. Our simulation platform is based on the proposal of TGn Sync. The signal bandwidth (BW) is 20MHz. The transmitter and receiver use 128-points IFFT and FFT, respectively. The antenna spacing in the transmitter and receiver are equal to 0.5 wavelength. The decoder uses soft Viterbi algorithm to decide information bits with trace back length of 128. Assume there are perfect synchronization in the receiver, i.e. without frequency offset, clock offset, and phase rotation. The channel is well-kwon in the receiver. There are 8000 information bits per packet. There are at least 500 packet errors down to 1% packet error rate (PER) or a total of 10,000 packets in our simulation. The detector design in this section is based on the ZF criterion. Compare the equal weight W .pk =1 and weighted W .pk = ⎡σ z2p ⎤ ⎣ .k ⎦. metrics calculation.. - 27 -. -1. of bit.

(40) Case1: Channel B of IEEE802.11n, 2x2. From the simulation results Fig. 3-2 and Fig. 3-3, we can discover that the performance of weighted coefficients for bit metrics computation is much better than. PER. those of equal gain. There are about 4~5 dB improvement under the PER=0.1.. 10. 0. 10. -1. P E R v s . S N R _ d B C h -B 2 x 2 (p e rfe c t C S I) Z F D e te cto r. W=SINR W=1 B P S K ,R c =1 /2 QP S K ,R c =1 /2 QP S K ,R c =3 /4 10. -2. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. S NR _dB. Fig. 3-2: PER of bit metrics calculation with equal and weighted coefficients by ZF detector for BPSK and QPSK in channel B, 2x2. - 28 -.

(41) PER. P E R v s . S N R _ d B C h -B 2x 2 (p e rfe ct C S I) Z F D e tec to r. 10. 0. 10. -1. W=SINR W=1 1 6-Q A M ,R c =1/2 1 6-Q A M ,R c =3/4 6 4-Q A M ,R c =2/3 6 4-Q A M ,R c =3/4 10. -2. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. S N R _dB. Fig. 3-3: PER of bit metrics calculation with equal and weighted coefficients by ZF detector for 16-QAM and 64-QAM in channel B, 2x2. BER vs. SNR_dB Ch-B 2x2 (Perfect CSI) ZF detector. 0. 10. BPSK,R c=1/2 QPSK,R c=1/2 QPSK,R c=3/4. W=SINR W=1 -1. BER. 10. -2. 10. -3. 10. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. SNR_dB. Fig. 3-4: BER of bit metrics calculation with equal and weighted coefficients by ZF detector for BPSK and QPSK in channel B, 2x2. - 29 -.

(42) BER vs. SNR_dB Ch-B 2x2 (Perfect CSI) ZF detector. 0. 10. 16-QAM,R c=1/2 16-QAM,R c=3/4 64-QAM,R c=2/3 64-QAM,R c=3/4 -1. W=SINR W=1. BER. 10. -2. 10. -3. 10. -4. 10. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. SNR_dB. Fig. 3-5: BER of bit metrics calculation with equal and weighted coefficients by ZF detector for 16-QAM and 64-QAM in channel B, 2x2. 3.3 MMSE Criterion. In this section, use MMSE approach to detect signal. It is similar to a ZF receiver. Assume the MIMO channel is multipath quasi-static Rayleigh fading channel. The received signal vector after FFT/remove-GI is defined in (3.14) and the output signal vector y. .k. of MMSE detector is defined as. y .k =G MMSE r .k k. (3.28). Now, base on the MMSE criterion to minimize the error of the detected signal vector y .k and a transmitter signal vector s .k. {. G kMMSE =arg min E y .k-s .k MMSE Gk. 2. } =arg min E { G G MMSE k. - 30 -. MMSE k. r .k-s .k. 2. }. (3.29).

(43) See Appendix A, assume the energy of signal is equal to 1. Then, the coefficients of an MMSE detector is G. MMSE k. ( ). = Hk. H. ( ). ⎡H H k ⎢⎣ k. H. +σ I ⎤ ⎥⎦. -1. 2 n NR. (3.30). 3.3.1 Approximation of Bit Metrics. Observe the p th sub-stream detected signal. ( ). y p.k = g kp. H. y p.k , ⎛. ( ) ⎜∑h s H. h kp s p.k + g kp. ⎝ j≠ p. j k. j .k. ⎞ p ⎟ + gk ⎠. ( ). H. n. (3.31). .k. where we define h kj is the j th column vector of H k h kj = ⎡⎣ H k1, j , Because. (g ) p k. H. ,H kN R , j ⎤⎦. T. (3.32). h kj ≠ 0, for any j , there are co-antenna interference μ p.k ⎛. ⎞. μ p.k = ( g kp ) ⎜ ∑ h kj s j.k ⎟ H. ⎝. (3.33). ⎠. j≠ p. In [21], H.V. Poor and S.Verdu show that the MMSE estimate approximates a Gaussian distribution. Hence, the co-antenna interference and noise are considered. z p.k. together as complex Gaussian noise. with Gaussian approximation.. ⎛. ( ) ⎜∑h s. z p.k = g kp. H. ⎝. j≠ p. σ. Due to s 1.k ,. { }. =E z. p 2 .k. ,s N.kT and n1.k ,. ⎧ ⎪ =E ⎨ g kp ⎩⎪. j .k. ⎞ p ⎟ + gk ⎠. ( ). H. n. (3.34). .k. z p.k is. The variance of complex Gaussian noise 2 z p.k. j k. ( ). H. ⎛ j j ⎞ p ⎜ ∑ h k s .k ⎟ + g k ⎝ j≠ p ⎠. ( ). 2 H. n. .k. ,n N.kR are statistically independent, - 31 -. ⎫ ⎪ ⎬ ⎭⎪. (3.35).

(44) σ. 2 z p.k. =σ. ∑ (g ). 2 s. p H k. j≠ p. 2. h. j k. NR. +σ ⋅ ∑ g kp,q 2 n. 2. (3.36). q=1. Then the detect signal is shown as. ( ). y p.k = g kp. y p.k. The conditional pdf of. H. h kp s p.k +z p.k. (3.37). is a complex Gaussian distribution,. (. ). p y p.k s p.k =ψ =. (. ⎧ exp ⎨ σ-2 1 y p.k- g kp ⎩ z p.k. 1. π σz. p .k. ( ). H. h kpψ. ) ⎫⎬⎭ 2. (3.38). By the equation (3.10) and (3.38), the bit metrics c p.k ,m is equal to. (. Λ c. p .k , m. ). ( (. (. y p.k. To normalize. ) ). 2 H ⎛ ⎧ -1 p ⎫⎞ p p exp y - ψ g h 2 ⎨ ⎬⎟ ⎜ ψmax .k k k (1) σ p =Ψ m z .k ⎩ ⎭⎟ ⎜ =ln 2 ⎟ ⎜ H ⎧ ⎫ exp ⎨ σ-2 1 y p.k- g kp h kpψ ⎬ ⎟ ⎜ ψmax p =Ψ (m0 ) ⎩ z .k ⎭⎠ ⎝ 2 H 1 ⎧ = 2 ⎨- min(1) y p.k- g kp h kpψ + min( 0 ) y p.k- g kp ψ =Ψ m σ z p ⎩ ψ =Ψ m .k. ( ). ( ). (g ). p H k. by dividing. ). ( ). (. (3.39). ( ). H. h kpψ. ) ⎫⎬⎭ 2. h kp ,. ζ p.k =y p.k ⋅ ⎡⎢( g kp ) h kp ⎤⎥ =s p.k +z p.k ⋅ ⎡⎢( g kp ) h kp ⎤⎥ ⎣ ⎦ ⎣ ⎦ -1. H. H. -1. (3.40). Then, the bit metrics is. (. ). Λ c p.k ,m =. ( ) g kp. H. 2. h kp. σ z2. p .k. {. (. - min(1) ζ p.k-ψ ψ =Ψ m. ). 2. (. + min( 0 ) ζ p.k-ψ ψ =Ψ m. )} 2. (3.41). The coefficient of bit metrics c p.k ,m for BICM in MMSE detector is. p .k. W =. (g ) p k. H. σ z2. 2. h kp. p .k. ( ). = g. p H k. 2. h. p k. ⎡ 2 p ⎢σ s ∑ g k ⎣ j≠ p. ( ). H. 2. h. p j. By the way, the signal-to-interference-and-noise ratio of. - 32 -. NR. +σ ⋅ ∑ g 2 n. q=1. y p.k. is. p,q 2 k. ⎤ ⎥ ⎦. -1. (3.42).

(45) { }=. E s p.k SNR=. {. ( ). 2. E z p.k 2. }. g kp. σ. 2 s. ∑ (g ) j≠ p. p H k. H. 2. h kp σ s2 2. h. j k. NR. +σ ⋅ ∑ g 2 n. (3.43) p,q 2 k. q=1. p. So the coefficient of bit metrics c .k ,m for BICM is directly proportional to the signal-to-interference-and-noise ratio of detected signal y p.k .. 3.3.2 Simulation Results. Our simulation platform is based on the proposal of TGn Sync. The signal bandwidth (BW) is 20MHz. The transmitter and receiver use 128-points IFFT and FFT, respectively. The antenna spacing in the transmitter and receiver are equal to 0.5 wavelength. The decoder uses soft Viterbi algorithm to decide information bits with trace back length of 128. Assume there are perfect synchronization in the receiver, i.e. without frequency offset, clock offset, and phase rotation. The channel is well-kwon in the receiver. There are 8000 information bits per packet. There are at least 500 packet errors down to 1% packet error rate (PER) or a total of 10,000 packets in our simulation. The detector design in this section is based on the MMSE criterion. Compare the performance of equal and weighted coefficients of bit metrics calculation. The SNR is defined in chapter 2.. - 33 -.

(46) Case1: Orthogonal AWGN channel, 2x2 The signal is transmitted through the AWGN channel with orthogonal MIMO channel. H=. 1 2. ⎡1 1 ⎤ ⎢1 -1⎥ ⎣ ⎦. (3.44). From Fig. 3-6, we can find that the performances of bit metrics calculation with equal and weighted coefficients are almost the same. That’s because the frequency response of all subchannel are equal.. PER vs. SNR_dB PerfectCSI (AWGN). 0. PER. 10. -1. 10. BPSK,R c=1/2 QPSK,R c=1/2 QPSK,R c=3/4 16-QAM,R c =1/2. W=SINR W=1. 16-QAM,R c =3/4 64-QAM,R c =2/3 64-QAM,R c =3/4. -2. 10. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. SNR_dB. Fig. 3-6: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector in AWGN channel, 2x2. - 34 -.

(47) BER vs. SNR_dB PerfectCSI (AWGN). -2. 10. -3. BER. 10. -4. 10. BPSK,R c=1/2 QPSK,R c=1/2. -5. 10. QPSK,R c=3/4 16-QAM,R c=1/2 16-QAM,R c=3/4. W=SINR W=1. 64-QAM,R c=2/3 64-QAM,R c=3/4. -6. 10. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. SNR_dB. Fig. 3-7: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector in AWGN channel, 2x2. Case2: Channel B of IEEE802.11n, 2x2 From Fig. 3-8 and Fig. 3-9, we can find that the performance of weighted gain for bit metrics computation is better than those of equal gain. There are about 1dB improvement for BPSK and QPSK, about 3dB improvement for 16-QAM and about 4dB improvement for 64-QAM under the PER=0.1. Compare to ZF detectors, the improvement of MMSE detects is smaller than those of ZF detector, especially for lower modulation. That is because we use the Gaussian approximation in MMSE detector. Then in the low modulation scheme and fewer sub-streams, the Gaussian approximation of interference is loose.. - 35 -.

(48) 10. P E R vs . S N R _d B C h -B 2 x2 (P erfec t C S I) M M S E D etecto r. 0. B P S K ,R c =1/2 QP S K ,R c =1/2 QP S K ,R c =3/4. PER. W=SINR W=1. 10. -1. 10. -2. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. S N R _d B. Fig. 3-8: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x2. PER vs. SNR_dB Ch-B2x2 (Perfect CSI) MMSE Detector. 0. PER. 10. -1. 10. W=SINR W=1 16-QAM,R c =1/2 16-QAM,R c =3/4 64-QAM,R c =2/3 64-QAM,R c =3/4 -2. 10. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. SNR_dB. Fig. 3-9: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x2. - 36 -.

(49) BER vs. SNR_dB Ch-B 2x2 (Perfect CSI) MMSE detector. -1. 10. BPSK,R c=1/2 QPSK,R c=1/2 QPSK,R c=3/4 -2. 10. W=SINR W=1. -3. BER. 10. -4. 10. -5. 10. -6. 10. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. SNR_dB. Fig. 3-10: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x2. BER vs. SNR_dB Ch-B 2x2 (Perfect CSI) MMSE detector. 0. 10. 16-QAM,R c=1/2 16-QAM,R c=3/4 64-QAM,R c=2/3 64-QAM,R c=3/4 -1. W=SINR W=1. BER. 10. -2. 10. -3. 10. -4. 10. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. SNR_dB. Fig. 3-11: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x2. - 37 -.

(50) Case3: Channel B of IEEE802.11n, 2x3 In this case, the receiver uses three antennas to receive signal. From Fig. 3-12 and Fig. 3-13, we can find that the performance of weighted gain for bit metrics computation is better than those of equal gain. There are smaller than 0.5dB improvement for BPSK and QPSK, about 1dB improvement for 16-QAM and about 1.5dB improvement for 64-QAM under the PER=0.1. Compare to case2, the receiver in the case3 uses more receiver antenna than those in case2, and then the receiver has more diversity gain. Therefore, the weight for bit metrics is close to. PER. equal.. P E R vs . S N R _d B C h -B 2x3 (P erfect C S I) M M S E D etecto r. 10. 0. 10. -1. W=SINR W=1 B P S K ,R c =1/2 QP S K ,R c =1/2 QP S K ,R c =3/4 10. -2. -5. -4. -3. -2. -1. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. S N R _d B. Fig. 3-12: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x3. - 38 -.

(51) PER. 10. 0. 10. -1. P E R v s. S N R _d B C h -B 2x3 (P erfe ct C S I) M M S E D etecto r. W=SINR W=1 16-QA M ,R c =1/2 16-QA M ,R c =3/4 64-QA M ,R c =2/3 64-QA M ,R c =3/4 10. -2. 5. 6. 7. 8. 9. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35. S N R _d B. Fig. 3-13: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x3. BER vs. SNR_dB Ch-B 2x3 (Perfect CSI) MMSE detector. 0. 10. BPSK,R c=1/2 QPSK,R c=1/2 QPSK,R c=3/4. -1. 10. W=SINR W=1 -2. BER. 10. -3. 10. -4. 10. -5. 10. -6. 10. -5. -4. -3. -2. -1. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. SNR_dB. Fig. 3-14: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x3. - 39 -.

(52) BER vs. SNR_dB Ch-B 2x3 (Perfect CSI) MMSE detector. 0. 10. 16-QAM,R c=1/2 16-QAM,R c=3/4 64-QAM,R c=2/3 64-QAM,R c=3/4. -1. 10. -2. BER. 10. -3. 10. -4. 10. -5. 10. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. SNR_dB. Fig. 3-15: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x3. - 40 -.

(53) Case4: Channel B of IEEE802.11n, 3x3 PER vs. SNR_dB Ch-B 3x3 (PerfectCSI) MMSE Detector. 0. PER. 10. -1. 10. W=SINR W=1 BPSK,R c=1/2 QPSK,R c=1/2 QPSK,R c=3/4 -2. 10. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30. SNR_dB. Fig. 3-16: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 3x3 PER vs. SNR_dB Ch-B 3x3 (PerfectCSI) MMSE Detector. 0. PER. 10. -1. 10. W=SINR W=1 16-QAM,R c =1/2 16-QAM,R c =3/4 64-QAM,R c =2/3 64-QAM,R c =3/4 -2. 10. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. SNR_dB. Fig. 3-17: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 3x3. - 41 -.

(54) BER vs. SNR_dB Ch-B 3x3 (PerfectCSI) MMSE Detector. 0. 10. BPSK,R c=1/2 QPSK,R c=1/2 QPSK,R c=3/4. -1. 10. W=SINR W=1 -2. BER. 10. -3. 10. -4. 10. -5. 10. -6. 10. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30. SNR_dB. Fig. 3-18: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 3x3. BER vs. SNR_dB Ch-B 3x3 (PerfectCSI) MMSE Detector. 0. 10. 16-QAM,R c=1/2 16-QAM,R c=3/4 64-QAM,R c=2/3 64-QAM,R c=3/4 -1. W=SINR W=1. BER. 10. -2. 10. -3. 10. -4. 10. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45. SNR_dB. Fig. 3-19: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 3x3 - 42 -.

(55) Case5: Compare MMSE and ZF detector in channel B of IEEE802.11n, 2x2 10. P E R v s. S N R _d B C h -B 2x 2 (p erfect C S I) W =S IN R. 0. B P S K ,R c =1/2 QP S K ,R c =1/2 QP S K ,R c =3/4. PER. MMSE Detector ZF Detector. 10. -1. 10. -2. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. S N R _d B. Fig. 3-20: PER of bit metrics calculation with weighted coefficients by MMSE detector and ZF detector for BPSK and QPSK in channel B, 2x2. 10. P E R v s. S N R _d B (p e rfe ct C S I) W =S IN R. 0. 16-QA M ,R c =1/2. MMSE Detector ZF Detector. 16-QA M ,R c =3/4 64-QA M ,R c =2/3. PER. 64-QA M ,R c =3/4. 10. -1. 10. -2. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. S N R _d B. Fig. 3-21: PER of bit metrics calculation with weighted coefficients by MMSE detector and ZF detector for 16-QAM and 64-QAM in channel B, 2x2 - 43 -.

(56) 3.4 Conclusions. In this chapter, we derived the approximation of bit metric for MMSE detector and ZF detector, respectively. We analyze the performance of bit metric calculation with equal and weighted coefficients for the MMSE detector and the ZF detector There are about 3~4dB improvement by using weighted coefficients compared to equal coefficients. But in the lower modulation scheme, the Gaussian approximation of the interference would be loose. Hence, the improvement for BPSK and QPSK is only about 1dB in the MMSE detector. By the way, the ZF detector has noise enhancement so the performance of MMSE detector is better than those of ZF detector about 1~4dB, especially at lower SNR. At high SNR, the performance of the ZF detector is close to those of the MMSE detector.. - 44 -.

(57) Chapter 4: Low-Complexity Iterative Detection Under the condition that the transmitter architecture is of no change and the receiver only uses available received signals, this chapter utilizes an iterative method to improve the performance of MIMO BICM systems. The receiver joints signal detection and soft decoding with turbo principles to suppress the strong co-antenna interference in MIMO systems. The receiver returns soft information of the MAP decoder back to the multistage detector to enhance the ability of detecting signals. The subchannel, i.e. subcarrier, of MIMO-OFDM system has constant channel gain on the multipath Rayleigh fading channel. The MIMO-OFDM receiver detects signals per subcarrier. It is similar to the receiver of MIMO Single-Carrier system on the flat fading channel. Here, our proposed algorithm can be used for general MIMO systems. It is more convenient to me to depict our proposed algorithm for MIMO BICM systems. The block diagram of MIMO transmitter structure is shown in Fig. 4-1.. { } cn1. {bn }nb=0. L -1. {c } n. Lc -1. {c }. Lc -1 p n n =0. Lc -1. {s }. L -1 1 s t t =0. n =0. {s } p. t. Ls -1. t =0. n =0. {c } NT n. Fig. 4-1: A MIMO transmitter. - 45 -. Lc -1 n =0. { } stNT. Ls -1. t =0.

(58) where i p ∈ {1,. , NT } transmitter antenna index. i n : bit sequence index i t : symbol sequence (time) index i NT : the number of transmitter antennas i Lb : the number of information bits bn i Lc =. Lb. : the number of coded bits cn , where R c = nk00 is code rate. Rc. i Lc = ⎡. Lc ⎤ : the number of interleaved bits cnp per tx antenna NT. i Ls = ⎡. Lc ⎤ : the number of symbols stp per tx antenna for M -QAM log 2 M. The MIMO channel is shown in Fig. 4-2.. {s }. {r }. H t1,1. L -1 1 s t t =0. 1 Ls -1 t t =0. H tN R ,1. {s } p. t. {n }. L -1 1 s t t =0. Ls -1. t =0. q. t. H t1,NT. {s } NT t. {r }. H Ls -1. t =0. {r } NR. N R ,NT t. t. {n } NR t. t =0. Ls -1. Ls -1. t =0. Ls -1. t =0. Fig. 4-2: The MIMO channel. The received signal is NT -1. rt = ∑ H tq,p ⋅ stp +ntp ⇒ rt =H t st +nt q. p =0. where ⎡ H t1,1 ⎢ i assume H t = ⎢ ⎢ H tN R ,1 ⎣. H t1,NT ⎤ ⎥ ⎥ ∈ N R ,NT ⎥ Ht ⎦. N R x NT. - 46 -. is a flat Rayleigh fading channel. (3.45).

(59) The block diagram of MIMO iterative receiver architecture is shown in Fig. 4-3.. { } rt1. {r } q. t. Ls -1. t =0. {y }. Ls -1 p t t =0. Ls -1 SISO DeMUD. t =0. { ( )} {λ ( c )} Λ i cnp. e i. Lc -1. n=0. p n. +. Bit Metrics. Lc -1. {λ ( c )}. n =0. a o. Deinterleaver. -. {r } NR. t. Ls -1. t =0. q ,p t. {λ ( c )} a i. Channel Estimation. p n. Lc -1. Interleaver. {λ ( c )} e o. n =0. n. n =0. +. Lc -1. {Λ ( c )} o. { } bˆn. MAP Decoder (BCJR). -. {H }. n. Lc -1. n. Lc -1 n =0. n=0. Fig. 4-3: A MIMO iterative receiver. where. ( ). ( ). i Λ i cnp , Λ o cn : a posteriori log likelihood ratio. ( ) ( ) ( ) ( ). i λia cnp ,λie cnp ,λoa cn ,λoe cn : log likelihood ratio. (. ). (. i λia ( cnp ) =π λoe ( cn ) and λoa ( cn ) =π λie ( cnp ). ). The index i and o denote the log likelihood ratio (LLR) associated with the inner detector and outer decoder, respectively. And the superscripts a and e denote a priori (intrinsic) information and extrinsic information, respectively. π ( • ) is an interleaver function.. This chapter is organized as follows: In the section 4.1, to describe the optimal detector based on MAP algorithm and MAP (BCJR) decoder. In the section 4.2, to provide the suboptimal low-complexity linear detector based on MMSE algorithm, and we propose four approximations to reduce the computation complexity of iterative MMSE receiver. Finally, in the section 4.3, the performances of various - 47 -. Lb -1 n =0.

(60) iterative MMSE receiver schemes proposed in this chapter are examined. Equation Section 4. 4.1 Optimal Receiver Based on MAP Algorithm. Assume MIMO channel is an flat quasi-static Rayleigh fading channel matrix H. The received signal rt q at the q th receiver antenna at time t is N T -1. rt = ∑ H tq,p ⋅ stp +ntp q. (4.1). p =0. Then the received signal vector rt is defined as rt =H t st +nt ∈. ⎡ rt1 ⎤ ⎢ ⎥ where rt = ⎢ ⎥ , st = ⎢ rt N R ⎥ ⎣ ⎦. NT ×1. (4.2). ⎡ H t1,1 ⎡ st1 ⎤ ⎡ nt1 ⎤ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ , nt = ⎢ ⎥ and H t = ⎢ ⎢ H tN R ,1 ⎢ stNT ⎥ ⎢ ntNT ⎥ ⎣ ⎦ ⎣ ⎦ ⎣. H t1,NT ⎤ ⎥ ⎥ ∈ H tN R ,NT ⎥⎦. N R × NT. How to design an optimal receiver for MIMO system is to maximize a posteriori probability of information bit bn with all received signal vectors.. {. }. Ls -1 ⎤ bn =arg max p ⎡⎢bn {rt }t =0 bn ∈(0,1) ⎣ ⎦⎥. (4.3). Define a posteriori log likelihood ratio of bn as a posterior LLR: Λ optimal ( bn ). ⎛ p ⎡b =1 {r }Ls -1 ⎤ ⎞ t t =0 ⎥ ⎟ ⎜ ⎢ n ⎦ ln ⎜ ⎣ Ls -1 ⎟ ⎜ p ⎡⎢bn =0 {rt }t =0 ⎤⎥ ⎟ ⎦⎠ ⎝ ⎣. (4.4). Detect information bit bn , ⎪⎧bn =1 , if Λ optimal ( bn ) ≥ 0 ⎨ ⎪⎩bn =0 , if Λ optimal ( bn ) < 0. (4.5). By the total probability theorem, the a posteriori probability of bn can be shown as s p ⎡⎢bn {rt }t =0 ⎣. L -1. ⎤= ⎥⎦ ∑ {s }Ls -1. s s p ⎡⎢bn {rt }t =0 , {sˆ t }t =0 ⎣. L -1. t t =0. - 48 -. L -1. ⎤ ⋅ p ⎡{sˆ }Ls -1 {r }Ls -1 ⎤ ⎥⎦ ⎢⎣ t t =0 t t =0 ⎥⎦. (4.6).

(61) s Due to information bit bn depending on detected signal vector sequences {st }t =0 , then. L -1. s s p ⎡⎢bn {rt }t =0 , {sˆ t }t =0 ⎣. L -1. L -1. ⎤ =p ⎡b {sˆ }Ls -1 ⎤ ⎥⎦ ⎢⎣ n t t =0 ⎥⎦. (4.7). The channel is a flat fading and discrete memoryless channel so the detected signal vector st at time t only depends on the received signal vector rt at time t. Then, Ls -1. Ls -1 L -1 p ⎢⎡{sˆ t }t =0 {rt }t =0s ⎦⎥⎤ =∏ p ⎡⎣sˆ t rt ⎤⎦ ⎣ t =0. (4.8). Finally, the optimal receiver is able to calculate a posteriori LLR of information bit bn . ⎛ p ⎡b =1 {r }Ls -1 ⎤ ⎞ t t =0 ⎥ ⎟ ⎜ ⎢ n ⎦ Λ ( bn ) =ln ⎜ ⎣ Ls -1 ⎟ ⎜ p ⎡b =0 {rt }t =0 ⎥⎤ ⎟ ⎦⎠ ⎝ ⎢⎣ n. Ls -1 ⎛ ⎛ ⎡ ⎞⎞ Ls -1 ⎤ ˆ ⎜ ∑ ⎜ p ⎢bn =1 {st }t =0 ⎥ ⋅ ∏ p ⎡⎣sˆ t rt ⎤⎦ ⎟ ⎟ ⎦ t =0 ⎠⎟ ⎜ {sˆ }Ls -1 ⎝ ⎣ =ln ⎜ t t =0 ⎟ (4.9) Ls -1 ⎛ ⎞ Ls -1 ⎡ ⎤ ⎜ ⎜ p ⎣⎢bn =0 {sˆ t }t =0 ⎦⎥ ⋅ ∏ p ⎣⎡sˆ t rt ⎦⎤ ⎟ ⎟ ⎜ {sˆ∑ Ls -1 t =0 ⎠ ⎠⎟ ⎝ t }t =0 ⎝. But the computation complexity of the optimal receiver is too high. It is impossible to realize an optimal receiver. In order to reduce the computation complexity, we divide the receiver into two parts: inner detector and outer decoder, as Fig. 4-4 .. {rt }t =0. Ls -1. {rt }t =0. Optimal receiver (MAP). {st }t =0. {bn }n=0. Lb -1. L s -1. Ls -1. Inner detector (DeMUD). Outer decoder. {bn }n=0. Lb -1. Fig. 4-4: A inner detector and a outer decoder. 4.1.1 MAP Detector. The optimal detector for iterative receiver is an a posteriori probability (APP) detector. - 49 -.

(62) {. }. sˆ t =arg max p ⎡⎣st rt ⎤⎦ st ∈Ψ. (4.10). p ⎡⎣rt st ⎤⎦ ⋅ p [ st ] ∝ p ⎣⎡rt st ⎦⎤ ⋅ p [ st ] p [rt ]. (4.11). By the Bayes rule, p ⎣⎡st rt ⎦⎤ =. At the first iteration, there is no soft information about transmitted signal vector. st .. It means that p [st ] are equal. Then, the MAP detector is a maximum-likelihood (ML) detector.. {. }. {. }. st =arg max p ⎡⎣st rt ⎤⎦ =arg max p ⎡⎣rt st ⎤⎦ st ∈Ψ st ∈Ψ. (4.12). The computation complexity of MAP detector (ML detector) is order of M NT . MAP detector is not feasible for larger number of transmit antennas or higher modulation schemes. The suboptimal detector is a linear detector based on MMSE criterion.. 4.1.2 MAP (BCJR) Decoder. In this section, we describe how to use a MAP decoder as an optimal decoder and how to calculate the soft information pass to inner detector. Because the transmitter uses a bit interleaver after a convolutional encoder to overcome Rayleigh fading channel, the receiver needs to calculate the bit metrics before a bit de-interleaver for soft Viterbi decoding or MAP decoding. The de-interleaved codeword is denoted by c n . It is an encoder output tuple by encoding information bit bn . Assume the code rate of a convolutional encoder is Rc = 1/2 .. cn = ( cn,0 ,cn,1 ). bn → encoder → c n - 50 -. (4.13).

(63) The a posteriori log likelihood ratio of cn, j for MAP decoder is denoted as ⎛ ⎡ a ⎜ p ⎢⎣ cn , j =1 λo ( ci ) ln ⎜ ⎜⎜ p ⎡ cn , j =0 λoa ( ci ) ⎝ ⎣⎢. {. ( ). Λ o cn , j. }. {. ⎞ ; decoding ⎤ ⎟ ⎥⎦ ⎟ Lb -1 ; decoding ⎤ ⎟⎟ ⎥⎦ ⎠ i=0. Lb -1. i=0. (4.14). }. where. { ( ). ( )}. i λoa ( c n ) = λoa cn ,0 , λoa cn, 1. ( ). i λoa cn , j : a priori log likelihood ratio (soft information). The a posteriori probability can be written as. {. }. p ⎡cn , j =k λoa ( ci ) ⎢⎣. Lb -1. i=0. ; decoding ⎤ = ⎥⎦. ∑ p ⎡⎣⎢ S. {. }. =S', Sn =S, λoa ( ci ). n-1. S(k) j. {. }. p ⎡ λoa ( ci ) ⎢⎣. Lb -1. i=0. ⎤ ⎥⎦. Lb -1. i=0. ⎤ ⎦⎥. (4.15). where i Sn :the state of information bit at time n th i S(k) bit of output j : the set of state transition from S' to S and the j. tuple c n is k ∈ (0,1). Define the forward metrics denoted by α n ( S) is. α n (S). {. }. p ⎡ Sn = S, λoa ( ci ) ⎢⎣. i=n i=0. ⎤ ⎥⎦. (4.16). ⎤ ⎥⎦. (4.17). Define the backward metrics denoted by α n ( S) is. β n (S). {. }. p ⎡ S n = S, λoa ( ci ) ⎢⎣. Lb -1. i=n+1. And define the branch metrics γ n ( S',S) from the state S' to the state S is. γ n ( S',S). p ⎡⎣ Sn = S, λoa ( c n ) Sn-1 =S'⎤⎦ - 51 -. (4.18).

(64) By [19], the authors tell us,. α n ( S) = ∑ S' α n -1 ( S') γ n ( S',S). (4.19). β n -1 ( S) = ∑ S' β n ( S') γ n (S',S). (4.20). and. ( ). ( ). Calculate the branch metrics γ n ( S',S) by a priori information λoa cn ,0 and λoa cn, 1 ,. γ n ( S',S) =p ⎣⎡λoa ( cn ) Sn =S, Sn-1 =S'⎦⎤ p ⎡⎣ Sn =S Sn-1 =S'⎤⎦ (4.21). j=1. =p ⎡⎣S S'⎤⎦ ∏ p ⎡⎣ cn, j ( S',S ) ⎤⎦ j=0. where. (. i p ⎡⎣ cn, j =1⎤⎦ = exp λoa ( cn, j ). (. ) {1+exp ( λ ( c ))}. i p ⎡⎣cn, j = 0 ⎤⎦ = exp -λoa ( cn, j ). a o. n, j. ) {1+exp ( λ ( c ))} a o. n, j. By the equations(4.14) and(4.15), the a posteriori LLR is. {. }. ⎛ p ⎡ Sn-1 =S', S n =S, λoa ( ci ) ⎜∑ ⎣⎢ (1) S ln ⎜ j ⎜ p ⎡ Sn-1 =S', S n =S, λoa ( ci ) ⎢⎣ ⎜∑ (0) ⎝ Sj. ( ). Λ o cn , j. {. ⎤⎞ ⎦⎥ ⎟ ⎟ Lb -1 ⎤⎟ i=0 ⎥ ⎦⎟ ⎠ Lb -1. i=0. (4.22). }. Because,. {. }. p ⎡ S n-1 =S', S n =S, λoa ( ci ) ⎢⎣. {. }. = p ⎡ S n-1 =S, λoa ( ci ) ⎣⎢. i=n-1 i=0. Lb -1. i=0. ⎤ ⎥⎦. {. }. ⎤ p ⎡ S =S, λ a ( c ) S =S'⎤ p ⎡ S =S, λ a ( c ) o n n -1 o i ⎦ ⎣⎢ n ⎦⎥ ⎣ n. = α n -1 ( S' ) β n ( S ) ⋅ γ n ( S',S ). Lb -1. i=n+1. ⎤ ⎦⎥. (4.23). then ⎛ ∑ S(1) α n -1 ( S') β n ( S) ⋅ γ n ( S',S ) ⎞ j ⎟ Λ o ( cn , j ) = ln ⎜ ⎜ ∑ S(0) α n -1 ( S') β n ( S) ⋅ γ n ( S',S) ⎟ j ⎝ ⎠ And - 52 -. (4.24).

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