國 立 交 通 大 學
電信工程學系
碩 士 論 文
編碼多天線頻域正交多工信號
在時變通道下之偵測
On the Detection of Coded MIMO OFDM
Signals in Time-Varying Channels
研究生：洪佳君
指導教授：蘇育德 博士
編碼多天線頻域正交多工信號在時變通道下之偵測
On the Detection of Coded MIMO OFDM Signals in
Time-Varying Channels
研 究 生：洪佳君
Student : Chia-Chun Hung
指導教授：蘇育德 博士
Advisor
:
Dr.
Yu
T.
Su
國 立 交 通 大 學
電信工程學系碩士班
碩 士 論 文
A Thesis Submitted to
The 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
June 2005
Hsinchu, Taiwan, Repubic of China
編碼多天線頻域正交多工信號在時變通道下之偵測
研究生：洪佳君
指導教授：蘇育德 博士
國立交通大學電信工程學系碩士班
摘要
這 篇 論 文 主 要 是 針 對 使 用 低 密 度 同 位 校 正 (Low Density Parity Check, LDPC) 碼之頻域正交多工 (Orthogonal Frequency Division Multiplexing, OFDM) 系統，從理論上探討其接收端在時變通道下，相關的系統架構以及信號處理流程 等重要的設計課題。 我們提出一種利用結合低密度同位校正碼的解碼器以及頻域正交多工系統 之通道估測與解調的接收系統架構。透過所謂的渦輪原理(Turbo Principle)，我們 讓接收器反覆的在解碼器以及通道估測解調器之間交換軟輸出資訊。同時，我們 也提出了改良式的模型基礎(model-based)通道估測法，這種新型估測法可以有效 的改善先前類似的通道估測法之效能，並保留其追蹤時變通道頻率反應的能力。
上述反覆式接收器架構要從單收發天線(Single Input Single Output, SISO)系 統延伸到多收發天線(Multiple Input Multiple Output, MIMO)系統時，會有額外的 同天線(空間)干擾(Co-Antenna Interference, CAI)的問題要解決。我們討論了兩個 方案並建議一種適合在此環境下使用的反覆式架構。這種架構包含了兩個反覆迴 圈(Iterative Loop)，內迴圈(Inner Loop)結合了同天線干擾消除與解碼，外迴圈 (Outer Loop)之功能則類似單收發天線接收系統，從事反覆的通道估計與解碼工 作。透過電腦模擬，可以正確的評估我們所提出架構與方法之效能。種種數值模 擬的結果都顯示，我們的接收系統的確可以保證極低的位元錯誤率(Bit Error Rate, BER)以及訊框錯誤率(Frame Error Rate, FER)。
On the Detection of Coded MIMO OFDM Signals
in Time-Varying Channels
Student : Chia-Chun Hung Advisor : Yu T. Su
Institute of Communications Engineering National Chiao Tung University
Abstract
This thesis is mainly concerned about the developments of the receiver structure and the signal processing algorithms for LDPC-coded OFDM systems in a time-varying channel.
We suggest a joint channel estimation and decoding receiver structure using an iter-ative decoding approach that exchanges the information between the low density parity check (LDPC) code decoder output and the orthogonal frequency division multiplexing (OFDM) channel estimator input. Efficient new model-based algorithms for estimat-ing time-varyestimat-ing multi-carrier channel frequency responses are given. We consider both single-input single-output (SISO) and multiple-input multiple-output (MIMO) cases. Computer simulation is performed to estimate the performance of the proposed algo-rithms. Numerical results reveal that our algorithms do yield excellent bit error rate (BER) and frame error rate (FER) performance.
誌謝
本篇論文得以順利完成，首先我要感謝我的指導老師蘇育德教授
的悉心指導，在論文的研究以及撰寫過程中，都給予極大的支持與協
助，除了電信的專業領域外，老師更不時帶我們去郊外踏青，讓我們
在身心方面都可以得到適度的調適。
而我的爸爸、媽媽在我的求學生涯中，更是提供我精神上源源不
絕的鼓勵，在我遇到瓶頸的時候支持著我，提醒著我不能放棄，讓我
在低潮的時候能重新振作起來，若我能獲得任何一絲的榮耀，那也是
因為你們而發光的。
在實驗室共同為各自將來而打拼的呂子逸同學，不但給予我許多
研究上的建議，更是在研究過程中，培養出患難與共的精神。而我生
活中的好夥伴林裕欽同學，則是我壓力的出口，讓我可以保持著愉悅
的心情專注在研究上。
最後，我還要感謝所有關心我的老師、家人、朋友，和 Lab 811
的成員，以及承蒙聯發科技公司贊助此計劃，有你們的幫助，才能讓
我順利完成此篇論文。
Contents
Contents 2
List of Figures 4
List of Tables 7
1 Introduction 1
2 System Model Based on an Industrial Standard 4
2.1 802.11n Systems . . . 4
2.2 OFDM Subsystem . . . 8
2.3 Fading Channel Model . . . 10
2.4 LDPC Codes . . . 13
3 Joint Channel Estimation and Decoding in SISO-OFDM Systems 15 3.1 Model Based Channel Estimation . . . 16
3.2 Computer Experiments . . . 18
3.3 Improved Channel Estimates . . . 19
3.3.1 Piece-wise polynomial model . . . 19
3.3.2 Modified model-based channel estimation . . . 23
3.4 Joint Channel Estimation and Decoding in SISO-OFDM Systems . . . . 25
4 Joint Channel Estimation and Decoding in MIMO-OFDM Systems 33 4.1 Channel Estimation in MIMO-OFDM Systems . . . 35
4.2 MIMO Signal Detection . . . 36 4.2.1 MIMO MMSE detector . . . 36 4.2.2 MIMO MLD detector . . . 39 4.3 Joint Channel Estimation and Decoding in MIMO-OFDM Systems . . . 40
5 Conclusion 46
Appendix 47
A 802.11n Specification 47
A.1 Training symbols specification . . . 47 A.2 Pilot symbols specification . . . 48 A.3 LDPC code specification . . . 49
List of Figures
2.1 The 802.11n transmitter for 2 transmit antennas using direct map in 20M Hz. . . 5 2.2 Tone format for 20M Hz channelization. . . 6 2.3 PPDU format for NT x mandatory basic MIMO transmission. . . 7
2.4 A block diagram of a LDPC-coded OFDM system and its major building blocks. . . 9 2.5 An example of the channel impulse response with channel length CL= 16. 11
2.6 An example of the channel response with Doppler frequency fd = 500Hz. 11 2.7 The factor graph representation of a LDPC code with n = 4 and r = 3. . 13 3.1 The factor graph representation of the model-based channel estimate. . . 17 3.2 Performance comparison of the model-based channel estimates and the
LS channel estimate. . . 17 3.3 The coded and uncoded BER performance of the LS, M B(4, 56), M B(7, 56)
estimates, and the one with perfect CSI case. . . 20 3.4 The coded FER performance of the LS, M B(4, 56), M B(7, 56) estimates.
The perfect CSI case is also given for comparison. . . 20 3.5 The factor graph representation of the piece-wise polynomial model based
channel estimator. . . 21 3.6 An example of the piece-wise polynomial model based estimate M B(3, 14)
3.7 The BER performance of the M B(3, 14), M B(4, 56), and M B(7, 56) es-timates. . . 22 3.8 The FER performance of the M B(3, 14), M B(4, 56), and M B(7, 56)
es-timates. . . 22 3.9 The factor graph representation of the modified model based channel
estimator. . . 24 3.10 An example of the channel estimates using the two-stage M B(3, 16, 6)
estimate. . . 24 3.11 The BER Performance for the M B(3, 14), M B(3, 16, 6), and M B(7, 56)
Channel Estimates. . . 26 3.12 The FER Performance for the M B(3, 14), M B(3, 16, 6), and M B(7, 56)
Channel Estimates. . . 26 3.13 An example of the channel estimates using the two-stage M B(3, 16, 6)
estimate when the average SNR is high. . . 27 3.14 The detail of the regional curve fitting. . . 27 3.15 The joint channel estimation and decoding receiver in SISO-OFDM systems. 28 3.16 The factor graph representation of joint channel estimation and decoding
in SISO-OFDM systems. . . 29 3.17 The BER performance of joint the M B(3, 16, 6) estimation and decoding
with kout = 0, 1, 3. . . 30 3.18 The FER performance of joint the M B(3, 16, 6) estimation and decoding
with kout = 0, 1, 3. . . 30
3.19 The BER performance of joint the M B(7, 56) estimation and decoding with kout = 0, 1, 3. . . 31
3.20 The FER performance of joint the M B(7, 56) estimation and decoding with kout = 0, 1, 3. . . 31
4.2 HT-LTF tone interleaving across 4 transmit antennas. . . 36 4.3 Influence of the inner loop iteration number kin on the BER performance
of 2× 2 MIMO-OFDM systems with MMSE detector. . . 38 4.4 Influence of kin on the FER performance of 2× 2 MIMO-OFDM systems
with MMSE detector. . . 38 4.5 Influence of kin on the BER performance of 2× 2 MIMO-OFDM systems
with MLD detector. . . 41 4.6 Influence of kin on the FER performance of 2× 2 MIMO-OFDM systems
with MLD detector. . . 41 4.7 An iterative receiver structure for LDPC-coded MIMO signals. . . 42 4.8 Influence of kin and the outer loop iteration number kout and with the
M B(3, 13, 6) estimation at the preamble and the M B(2, 11, 6) estimation at the data symbols on the BER performance of 2 × 2 MIMO-OFDM systems with MMSE detector. . . 44 4.9 Influence of kin and kout and with the M B(3, 13, 6) estimation at the
preamble and the M B(2, 11, 6) estimation at the data symbols on the FER performance of 2× 2 MIMO-OFDM systems with MMSE detector. 44 4.10 Influence of kin and kout and with the M B(3, 13, 6) estimation at the
preamble and the M B(2, 11, 6) estimation at the data symbols on the BER performance of 2× 2 MIMO-OFDM systems with MLD detector. . 45 4.11 Influence of kin and kout and with the M B(3, 13, 6) estimation at the
preamble and the M B(2, 11, 6) estimation at the data symbols on the FER performance of 2× 2 MIMO-OFDM systems with MLD detector. . 45
List of Tables
2.1 Timing related parameters for 802.11n in 20 MHz. . . 6 A.1 Tone partitioning into sets for 20M Hz (56 tones) . . . 48
Chapter 1
Introduction
The OFDM technique has enjoyed increasing popularity in a variety of wireless ap-plications in recent years. This widespread popularity is due to several key features of the OFDM scheme, namely, high data rate transmission capability with high bandwidth efficiency and robustness to frequency-selective fading. It has been used in wireless local area network (LAN) standards such as American IEEE802.11 and the European equivalent HIPERLAN/2, in multimedia wireless services such as Japanese Multime-dia Mobile Access Communications, and in various digital broadcasting standards such Eureka, DMB, etc.
Amongst the industrial standards mentioned above, the IEEE 802.11 is perhaps the most popular and the corresponding WLAN has been deployed worldwide. The IEEE 802.11 is a set of the wireless LAN standards developed by working group 11 of the IEEE LAN/MAN Standards Committee. The original version of the standard IEEE 802.11 released in 1997, which is now called “802.11 legacy”, specifies the data rates of 1 and 2M bps. The 802.11a amendment to the original standard was approved in 1999, and uses the same core protocol as the original standard but operates in 5GHz band uses a 52-sub-carrier OFDM and the maximum data rate of 54M bps.
IEEE announced to form a new 802.11 Task Group (TGn) to develop a new amend-ment to the 802.11 in January 2004, which services enhanceamend-ment of the higher through. The real data throughput will be least 100M bps (up to 200M bps). 802.11n builds upon
previous 802.11 standards by adding MIMO scheme and uses a 64-sub-carrier OFDM, the additional transmit and receive antennas allow for increasing the data throughput through spatial multiplexing.
There are two competing variants of the 802.11n standard: • WWiSE (Worldwide Spectrum Efficiency)
Backed by companies including Airgo Networks, Broadcom, Conexant Systems, Motorola, Nokia, Texas Instruments, and others.
• TGn Sync
Backed by Atheros, Intel, Sony, Matsushita, Toshiba, and others.
Recently, the IEEE working group dedicated to the next-generation 802.11n standard has settled on a signal proposal, TGn Sync over the competing WWiSE proposal.
After being transmitted over a multi-path fading channel, the received signal ex-periences the fluctuation of amplitude attenuation and phase distortion in addition to inter-path (or inter-symbol) interference. In practical systems, the receiver does not have perfect channel state information, and so it is thus necessary for the receiver to recover the unknown amplitude and phase variations. Besides initial synchronization, to coherently detect the received signal in a time-varying environment, accurate tracking of the channel response is also needed [2]. The optimum approach is to do joint channel estimation and decoding.
The channel response can be estimated by inserting the pilot tones into all of the sub-carriers in a block of OFDM symbols that form a period called preamble, or by inserting the pilot tones into some of the sub-carriers at each OFDM symbols. The first approach is more appropriate for a slow-fading channel. Least-squares (LS) estimation and minimum mean-square error (MMSE) estimation proposed in [3] are based on this kind of pilot arrangement. The second approach is proposed to meet the need of channel tracking and equalization when the channel changes from symbol to symbol. Such a channel
estimation scheme often first obtain the channel estimates at pilot frequencies and then derive the channel responses at other sub-carriers by a proper interpolation method. Well-known pilot-assisted channel estimation algorithms include LS, MMSE or least mean-square (LMS) methods. The interpolation approaches reported in the literature include linear interpolation, second order interpolation [5], piecewise-linear interpolation [6], low-pass interpolation, spline cubic interpolation, time domain interpolation, and regression function interpolation [9].
In the specification of 802.11n proposed by TGn Sync [1], these two kinds of pilot arrangements are used, and the channel can be accurately estimated by the combination of these two approaches.
This thesis is organized as following, in chapter 2, the system model including the 802.11n system, and the LDPC codes are introduced. Chapter 3 then proposes a method to estimate the channel in OFDM systems. By joint channel estimation and decoding will bring improvement in BER and FER performance. The extension for MIMO scenario using joint channel estimation, data detection, and decoding is shown in chapter 4. The numerical results of the algorithms are shown at the above chapter. Finally, the conclusion are summarizes in chapter 5.
Chapter 2
System Model Based on an
Industrial Standard
We use the IEEE 802.11n standard proposed by the TGn Sync group as an applica-tion example and a model for the MIMO-OFDM scheme. The figure shown in Fig. 2.1 is an 802.11n transmitter block diagram. The transmitted waveform is a LDPC-coded 64-tone OFDM signal. The aim of this chapter is to provide a brief description of this industrial standard to which subsequent discourse can be refer. It also serves to show the readers important elements and critical design considerations of a typical MIMO-OFDM system.
2.1
802.11n Systems
The TGn Sync complete proposal meets the IEEE 802.11n project authorization request (PAR), meets all functional requirements, and addresses all mandatory require-ments of the comparison criteria. In this thesis, we construct a joint channel estimation and decoding receiver based on this proposal in 20 MHz channel bandwidth, and the transmitter used here is shown in Fig. 2.1 with the parameter given in Table 2.1.
The forward error correction (FEC) architecture uses a single LDPC encoder. The LDPC code used in this proposal can be found in Section A.3 of Appendix A. Different code rates are produced within the encoder and external puncturing is not needed. The coded bits are then mapped to QAM constellation points. The mapping may be BPSK,
Frequency Interleaver across 52 data tones Constellation Mapper Insert GI window LD PC E nc ode r sp atia l pa rs er iFFT 64 tones in 20MHz 56 populated tones 52 data tones 4 pilot tones RF BW ~ 17MHz Frequency Interleaver across 52 data tones Constellation Mapper Insert GI window iFFT 64 tones in 20MHz 56 populated tones 52 data tones 4 pilot tones RF BW ~ 17MHz
Figure 2.1: The 802.11n transmitter for 2 transmit antennas using direct map in 20M Hz. QPSK, 16 QAM, 64 QAM or 256 QAM.
The next step is the antenna map transformation. The antenna map transforma-tion maps NSS spatial streams to NT x transmit antenna streams. Note that MIMO
transmission requires where NSS ≤ min{NT x, NRx} is the number of active receiving
antennas. In particular, we must always have NSS ≤ NT x in the data transmit path. This transformation is generically represented by the matrix Q. In general, there may be a different NT x× NSS transformation for each sub-carrier. There are three standard
typesQ transformations, direct map, spatial spreading, and transmit beamforming. Di-rect map MIMO is the simplest antenna mapping transformation. The Q matrix is simply the identity mapping, i.e. each spatial stream maps to one antenna. Hence, there is a one-to-one correspondence between spatial streams and transmit antennas. In this thesis, we use direct map and NSS = NT x.
After the antenna map, antenna streams are transformed into the time domain se-quence via the inverse fast Fourier transform (IFFT), with a cyclically extended guard interval (GI) inserted. Finally, time domain windowing is applied to create a continuous time-domain signal, and analog and RF processing yields the signals that are applied to transmit antennas.
con-Parameter For 20M Hz Channel NSD Number of data subcarriers 52
NSP Number of pilot subcarriers 4
NSN Number of center null subcarriers 1 (tone = 0)
NSR Subcarrier index range 28(-28 ... +28)
∆F Subcarrier frequency spacing 0.3125 MHz
TF F T IFFT/FFT period 3.2µsec
TGI GI duration 0.8µsec
TShortGI Short GI duration 0.4µsec
TGI2 Legacy LongTraining symbol GI duration 1.6µsec
TSym Symbol interval 4µsec
TL−LT F Legacy Long training field duration 8µsec
THT−LT F HT Long training field duration 7.2µsec
TL−ST F Legacy Short training field duration 8µsec
THT−ST F HT Short training field duration 2.4µsec
TS Nyquist sampling interval 50nsec
Table 2.1: Timing related parameters for 802.11n in 20 MHz.
structed from 56 tones, of which 52 are data tones and 4 are pilot tones. Tone assigned during 20M Hz for DATA symbols can be summarized as Fig. 2.2:
-21 +7 +21 Tone Fill
20MHz
-7 +31
-32 -1+1
Figure 2.2: Tone format for 20M Hz channelization.
The PHY protocol data unit (PPDU) format for transmission using NT x antennas
in a 20M Hz channelization is shown in Fig. 2.3. The PPDU is formed by the preamble and the data units. The preamble is a concatenation of the legacy preamble (identical to 802.11a/g) and a high throughput (HT)-specific preamble.
The functions performed by the preamble include: 1. Start-of-Packet (SoP) detection
L-STF 8 us RATE 4 bits Reserve 1 bit Legacy PLCP Header TxAnt.1 TxAnt.2 L -LTF 8 us Coded OFDM BPSK, r = 1/2 LENGTH 12 bits Parity 1 bit Tail 6 bits PSDU Pad Bits L -SIG 4 us Coded OFDM BPSK, r = 1/2 DATA.X1 Variable Number of OFDM Symbols
DATA.X2 Variable Number of OFDM Symbols
Tail 6 bits
HT-SIG 8 us
DATA.XN Variable Number of OFDM Symbols HT-STF 2.4 us HT-STF 2.4 us HT-LTF 7.2 us HT-LTF 7.2 us HT-STF 2.4 us HT-LTF 7.2us TxAnt.N HT-LTF 4 us HT-LTF 4 us HT-LTF 4 us
Coded MIMO - OFDM Using RATE.X indicated in SIGNAL.X LENGTH 12 bits MCS 6 bits Advanced coding 1 bit Sounding packet 1 bit Number HT-LTF 2 bits Short GI 1 bit Scrambler Init 2 bits CRC 8 bits 1 bit Tail 6 bits 20/40 Tx LENGTH 12 bits MCS 6 bits Advanced coding 1 bit Sounding packet 1 bit Number HT-LTF 2 bits Short GI 1 bit Scrambler Init 2 bits CRC 8 bits 1 bit Tail 6 bits 20/40
Figure 2.3: PPDU format for NT x mandatory basic MIMO transmission.
2. Automatic Gain Control (AGC) 3. Coarse Frequency Offset Estimation 4. Coarse Timing Offset Estimation 5. Fine Timing Offset Estimation 6. Fine Frequency Offset Estimation 7. Channel Estimation
The legacy part of the HT preamble allows a legacy receiver to properly decode the L-SIG. A HT receiver shall also perform functions 1 through 7 (SoP to Channel Esti-mation) to successfully decode the HT-SIG. It is anticipated that functions 1 through 4 (and perhaps function 5) are performed by the legacy short training field (L-STF), and functions 5 through 7 are performed by the legacy long training field (L-LTF).
The HT short training symbol uses 24 tones (L-STF uses 12 tones in a 20 MHz channel). The 24 tones are interleaved across the antennas, i.e., 6 tones are used per antenna if there are 4 transmitter antennas, and likewise 12 are used per antenna if there are 2 transmitter antennas. The length of the HT-STS is 1.6µs. Together with the 0.8µs
GI, the total length of the HT-STF is 2.4µs. The GI of the HT-STF, i.e. the first 0.8µs, would contain echoes from the previous symbol. Hence, a clean measurement of power is made in the 1.6us portion of the HT-STF. It is anticipated that the RF transient settling after the AGC switch would occur in the GI of the HT-LTF that follows the HT-STF.
Channel estimation and fine frequency offset estimation are performed using the HT Long Training Symbols. The number of HT-LTFs is equal to the number of antennas in the basic MIMO mode. For example, with two transmit antennas, two HT-LTFs are sent. Therefore, one HT-LTF would cover half of the tones in the two antenna case. Remaining tones are covered in the second HT-LTF. The HT-LTF consists of two Long Training Symbols (LTS) as in 802.11a/g and a regular GI of 0.8µs. The total length of one HT-LTF is 7.2µs. Details of the HT-LTF symbols can be found in Section A.1 of Appendix A.
2.2
OFDM Subsystem
OFDM is a multi-carrier transmission technique, which splits the available band-width into many narrow band channels, and each sub-carrier being modulated by dif-ferent data. The sub-carriers for each channel are made orthogonal to one another in order to allow them to be spaced very close together.
The orthogonality of the sub-carriers means that each sub-carrier has an integer number of cycles over one symbol period. Due to this, the spectrum of each sub-carrier has a null at the center frequency of each of the other sub-carriers in the system. This results in no interference between the sub-carriers, allowing then to be spaced as close as theoretically possible.
A multi-path fading channel is often modeled by a linear finite impulse response filter with a sample complex channel impulse response (CIR) as that depicted in Fig. 2.5. The corresponding synchronous discrete-time system model represents a transmitter that
sends information bit stream in discrete sampling time intervals nTs, n= 0, 1, 2,· · · and
the receiver operates with a symbol timing clock synchronized at a rate of symbols per second with the incoming samples. A critical measure concerning a multi-path-channel is the channel length τmax of the longest path with respect to the shortest path. A
received symbol is likely to be interfered by τmax
Ts previous symbols where the channel
length is CL = τmaxTs . This interference has to be estimated and compensated for in
the receiver, a task which may become very challenging.
For simplicity, we assume that the baseband pulse used for transmission in an OFDM system is a rectangular pulse. This choice has the advantage that the task of pulse forming and modulation can be performed by a simple inverse discrete Fourier transform (IDFT) which can be implemented very efficiently as an Inverse fast Fourier transform (IFFT). Further pulse-shaping can be accomplished by using a window during the IFFT operation (transmit windowing). Accordingly in the receiver we need either a FFT or a windowed FFT (receive windowing) to reverse this operation.
LDPC
Enc oder Serial/Parallel . . OFDM Modulator (IFFT ) Parallel/Serial and GI Ins ertion Channel OFDM Dem odulator (FFT ) GI Rem oval and Serial/Parallel LDPC
Dec oder Parallel/Serial
Channel Equalizer Nois e ˆ mn X mn X mn Y = XmnHmn +Nmn . . . . . . . .
Figure 2.4: A block diagram of a LDPC-coded OFDM system and its major building blocks.
A diagram showing the basic building blocks of a LDPC-coded OFDM system is given in Fig.2.4. As long as the length of the GI Tg is longer than the maximum channel
preserved. Transmission of OFDM signals over a multi-path fading channel will then render a simple model: The transmitted symbols Xmnat the mth subcarrier and the nth
time-slot are only disturbed by a complex factor–the channel response Hmn (the Fourier transform of the CIR) at the mth subcarrier frequency and the nth time-slot–and by additional complex white Gaussian noise Nmn.
Ymn = HmnXmn+ Nmn, (2.1)
where Nmn = NI,mn+ jNQ,mn, with independent in-phase and quadrature phase
com-ponents and identical variance var(NI) = var(NQ) = N20 σn2.
(2.1) implies that, if Hmn is perfectly known, the maximum likelihood (ML) receiver would make its decision based on the statistic ˆXmn = HYmnmn for it is the ML solution to
|Ymn− HmnXmn|2 = 0.
The preamble used in our SISO-OFDM system is the sequence 1 given in Section A.1 of Appendix A. At the beginning of a packet, pilot tones Xmn are inserted at all
the sub-carriers. The initial channel estimation can thus be obtained by applying the simple least square (LS) method
HLS,mn =
Ymn
Xmn
. (2.2)
The LS channel estimation which computes a tentative channel estimate at all NSD+NSP
tones through the use of the preamble needs 5· (NSD+ NSP) = 280 real multiplications
per OFDM symbol.
Given the initial channel estimate available, the scattered pilots at some selected sub-carriers in every data symbol can be of help in updating the channel estimate. The location and the value of these pilots are indicated in A.2.
2.3
Fading Channel Model
Jakes model [7] is a very popular channel model for mobile multi-path fading channels that takes into account the Doppler frequency and is based on the assumptions that the
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -0.45 -0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Channel Impulse Response hn
Channel Length Index (T S)
real part image part
Figure 2.5: An example of the channel impulse response with channel length CL= 16.
0 10 20 30 40 50 60 70 0 20 40 60 80 −1 −0.5 0 0.5 sub−carrier index m time index n
channel response (real part)
receiver is moving at speed v while the arrival angles of multi-path components are uniformly distributed. The time correlation of the channel is then given by
R(t) = J0(2π· t · fd), (2.3)
where the Doppler frequency fd= v·fcc, fc is the center frequency and c is the speed of
light. J0(x) is the zero-th order Bessel function of the first kind.
The multi-path fading channel is modelled by an exponentially decaying power profile whose tap weights are independent Rayleigh processes. That is, the complex envelope associated with each path is a correlated Rayleigh process generated by Jakes model while the Rayleigh processes associated with different paths are independent. Taking into account the Doppler frequency and the terminal speed, a multi-path fading channel with a channel impulse response (CIR) consists of CL taps over the CP length, we model
the CIR h(τ ) by
h(τ ) =
CL−1 i=0
αiδ(τ − iTs) (2.4)
where Tsis the sampling time and αiis a Jakes model attenuation factor of the ith
multi-path component with an exponentially decaying power-delay profile over CL paths. The
exponentially decaying power-delay profiles are as (2.5). σ02 = 1− e−Ts/TRMS
σk2 = σ20e−kTs/TRMS. (2.5)
Fig. 2.5 shows a period time of channel impulse response for the multi-path channels in time domain with channel length CL = 16,the moving speed v = 108km/hour, Ts =
50ns, TRM S = 30ns, and fd = 500Hz where Fig. 2.6 shows in frequency domain that
characterize either broadcasting, or mobile communication environments, and the change of the channel response between each sub-carrier is very smooth. The characteristic of the channel response is helpful for the estimation of channel response.
2.4
LDPC Codes
Low density parity check (LDPC) codes were invented by Gallager in early 1960’s [8]. The name comes from the characteristic of their parity-check matrix which contains only sparse 1’s in comparison to the amount of 0’s. Their main advantage is that they provide a performance which is very close to the capacity for a lot of different channels and linear time complex algorithms for decoding.
LDPC codes are linear codes obtained from sparse bipartite graphs. Suppose that Gis a factor graph with n upper node (bit nodes) and r lower node (check nodes). The graph gives rise to a linear code of block length n and dimension at least n− r in the following way: The n coordinates of the codeword are associated with the n bit nodes. The codeword are those vectors (X0, ..., Xn−1) such that for all check nodes fc0, ..., fcr−1
the sum of the neighboring positions among the message nodes is zero; see Fig. 2.7 for an example. LDP C Bit No de LDP C Che c k N o de 0 X 0 C f 1 C f 2 C f 1 X X2 X3 0 X +X3=0 X0+ X1+X2=0 X1+ X2+X3=0
Figure 2.7: The factor graph representation of a LDPC code with n = 4 and r = 3. The parity check matrix H is a binary r × n matrix in which the entry (i, j) is 1 if and only if the ith check node is connected to the jth bit node in the graph. Then the LDPC code defined by the graph is the set of vectors X = (X0, ..., Xn−1) such that
HTX = 0.
A general class of decoding algorithms for LDPC codes is called message passing algorithms, and which is iterative decoding algorithms. The reason for their name is that at each round of the algorithms messages are passed from bit nodes to check nodes, and from check nodes back to bit nodes. The messages from bit nodes to check nodes are computed based on the observed value of the bit node and some of the messages passed from the neighboring check nodes to that bit node. An important aspect is that the message that is sent from a bit node X to a check node f must not take into account the message sent in the previous round from f to X. The same is true for messages passed from check nodes to bit nodes.
The details of the LDPC code specification are given in Appendix A.3. The codeword length used in our simulation is equal to 1248 bits, with code rate = 1/2.
Chapter 3
Joint Channel Estimation and
Decoding in SISO-OFDM Systems
Transmitting a radio signal over a multi-path fading channel, the received signal will encounter unknown amplitude and phase distortions. In order to coherently detect the received signals, accurate channel estimation is essential. In this chapter, we introduce a channel estimation method used in SISO-OFDM systems. At the beginning of the transmission, the preamble is first transmitted, and then is the data symbols transmis-sion. At the preamble, the pilot tones are inserted into all of the sub-carriers. But at the data symbols, the pilot tones are inserted into only some of the sub-carriers.
Firstly, the model based channel estimation is introduced by the combination of the LS estimation and the regression function interpolator at the preamble. In order to reduce the complexity of the channel estimation, the piecewise polynomial model based channel estimation is proposed, and the Hermite curve can be applied to smooth the disconnection caused by the piecewise scheme. Since the channel response changes rapidly, to re-estimate the channel response at the present OFDM symbol is needed. After estimating the initial channel, the data symbols can be detected and decoded by the current channel estimate, and then, the channel can be re-estimated by the information computed from the LDPC decoder. The joint channel estimation and decoding algorithm is obtained.
3.1
Model Based Channel Estimation
As in [9], the channel response Hm is viewed as a sampled version of a continuous
complex fading process. The receiver can model the true sampled fading process by a regression function
F[m] =
MO−1 k=0
akmk=qTma = Hm+ g[m], (3.1)
where g[m] represents the modelling error and MO is the (polynomial) model order.
The ML estimate of the coefficients a is the solution to min a ML−1 m=0 |Ym− F [m]Xm|2 = min a ML−1 m=0 |Xm|2 ˆHLS,m− F [m] 2 , (3.2) where the error between the LS estimation and the regression function is minimized, and ML is the number of frequency-domain units within our modeling range and will be
referred to as the modeling block length.
Since for QPSK|Xm|2 is constant, the above optimization problem can be simplified
to ˆ a = arg min a ML−1 m=0 ˆHLS,m− qTma 2 . (3.3)
Taking the derivative of the argument on the right hand side of (3.3) with respect toa, we obtain ˆ a = ML−1 m=0 ( ML−1 r=0 qT rqr)−1qTmHˆLS,m, (3.4)
The (regression) model-based channel estimate is given by ˆ
Hm=qTmˆa. (3.5)
The factor graph representation of this model-based channel estimate is shown in Fig. 3.1, with the function nodes defined by
fM B( ˆHLS) = NSD+NSP−1 m=0 N SD+NSP−1 r=0 qT rqr −1 qT mHˆLS,m, (3.6)
,0 ˆ LS H HˆLS, . . . MB f 0 ˆa ˆ o M a . . . MB f . . . M odel Based C hannel Estimation
with model order Mo
C ompute the soft infromation for the LD PC decoder
SD N +NSP -1 -1 ˆ LS H ( ) ˆa ( ) ,0 ˆ MB H HˆMB ,NSD+NSP-1 ,h
Figure 3.1: The factor graph representation of the model-based channel estimate.
0 10 20 30 40 50 60 -0.6 -0.4 -0.2 0.0 0.2 0.4 Real Part of the Channel Response sub-carrier index
True Channel Response LS
MB(7,56) MB(4,56)
Figure 3.2: Performance comparison of the model-based channel estimates and the LS channel estimate.
fM B,h(ˆa) = qTma.ˆ (3.7)
Fig. 3.2 plots the channel estimates obtained by various model-based estimates and the LS channel estimate. The label M B(MO, ML) denotes the channel estimate that
uses an MO-order polynomial model with ML samples in the modelling block. In the
preamble period, the model-based approach is used as a refinement of the initial LS channel estimate. Unlike the crude LS estimate, the model-based estimate has a much smoother channel estimate and follows the real channel response much closer. It is also observed that as the model-order MO increases, the channel estimate becomes closer to the true channel response. The data unit can apply the channel estimate obtained in the preamble unit to detect the received samples, providing the LDPC decoder a soft input.
3.2
Computer Experiments
The simulation results reported in this section assume a data length of 624 bits. With 1/2 code rate, the codeword length is 1248 bits. The coded bit stream is then mapped by a QPSK modulator to form a complex symbol sequence. The channel response is generated with a channel length of CL = 16, terminal speed v = 108km/hour = 30m/sec, resulting in a Doppler frequency of fd = 500 Hz. The maximum iteration
number of the LDPC decoder A.3 is assumed to be 100. We define the average Eb/N0
as the average signal energy per data bit/noise power density per Hz.
Based on the channel estimate obtained by the M B(4, 56) or M B(7, 56) model-based scheme in the preamble unit, we demodulate the frequency domain data vectors and produce tentative decision vectors for soft LDPC decoding. The coded and uncoded BER and FER performance with different channel estimators are shown in Figs. 3.3-3.4, where the curves labelled by “Perfect CSI” represent those obtained with perfect channel state information (CSI), i.e., perfect channel estimate. As expected, the model-based estimate is much better than the LS estimate. There are more than 1 dB performance gain
between the LS estimate and the M B(4, 56) estimate. For a constant ML, the
model-based estimate with larger MO gives slightly better performance but at the expense of
large computational complexity increase.
The model-based channel estimate, M B(MO, ML), is computed through three steps:
(i) LS channel estimate (2.2), (ii) the computation of regression coefficients (3.4), and (iii) channel response estimate using the regression model(3.5). To compute the regression coefficients, ML× MO × (MO + 2) real multiplications are needed while the last step
needs ML× MO× 2 real multiplications per OFDM symbol.
For the M B(7, 56) estimate, the total computational complexity is 5× 56 + 56 × 7 × 11 = 4592 real multiplications in one OFDM symbol but the M B(4, 56) estimate only uses 2072 real multiplications in one OFDM symbol.
3.3
Improved Channel Estimates
Since the computational complexity of the model-based estimate tends to increase with the required estimation accuracy and the modelling order MO dominates the
com-plexity. To have a low complexity channel response model, we need to reduce the order of the model MO. Without compromising the estimation accuracy, ML, the number of
sub-carriers within one regression modelling block has to be decreased.
3.3.1
Piece-wise polynomial model
As shown in Fig. 3.5, partition the set of the sub-carriers into several parts, and use the regression function for each independent part. The example, Fig. 3.6, for channel estimate using 4 regression functions with MO = 3 and ML = 14, can also accurately
model the channel response.
As the piece-wise polynomial model-based estimate uses small model order, its com-putational complexity is very small in comparison with the regular model-based estima-tion, e.g., M B(3, 14) uses only 1456 real multiplications in one OFDM symbol.
8 10 12 14 16 1E-4 1E-3 0.01 BER avg{ Eb/N0} Perfect CSI
Perfect CSI UnCoded LS
MB(4,56) UnCoded MB(4,56)
MB(7,56) UnCoded MB(7,56)
Figure 3.3: The coded and uncoded BER performance of the LS, M B(4, 56), M B(7, 56) estimates, and the one with perfect CSI case.
8 10 12 14 16 1E-3 0.01 0.1 FER avg{ Eb/N0} Perfect CSI LS MB(4,56) MB(7,56)
Figure 3.4: The coded FER performance of the LS, M B(4, 56), M B(7, 56) estimates. The perfect CSI case is also given for comparison.
,0 LS H HLS,
. . .
MB f 0 a aMo. . .
MB f. . .
M odel Based C hannel Estimationwith model order Mo
C ompute the soft infromation for the LD PC decoder
-1 -1 LS H ( ) a ( ) ,0 MB H HMB,ML-1 ,h , LS H NSD+NSP-1 ,NSD MB H SP N + -1 L M
. . .
0 a aMo. . .
. . .
-1. . .
. . .
Figure 3.5: The factor graph representation of the piece-wise polynomial model based channel estimator. 0 10 20 30 40 50 60 -0.6 -0.4 -0.2 0.0 0.2 0.4 Real Part of the Channel Response sub-carrier index
True Channel Response MB(7,56)
MB(3,14)
Figure 3.6: An example of the piece-wise polynomial model based estimate M B(3, 14) compare with the model based estimate M B(7, 56).
8 10 12 14 16 1E-4 1E-3 0.01 BER avg{ Eb/N0} Perfect CSI LS MB(4,56) MB(7,56) MB(3,14)
Figure 3.7: The BER performance of the M B(3, 14), M B(4, 56), and M B(7, 56) esti-mates. 8 10 12 14 16 1E-3 0.01 0.1 FER avg{ Eb/N0} Perfect CSI LS MB(4,56) MB(7,56) MB(3,14)
Figure 3.8: The FER performance of the M B(3, 14), M B(4, 56), and M B(7, 56) esti-mates.
The BER and FER performance in Fig. 3.7 and Fig. 3.8 show that the piece-wise polynomial channel estimate M B(3, 14) performs slightly worse than the M B(4, 56) estimate, especially at low SNRs while achieving almost the same performance of the M B(7, 56) estimate at high SNRs. As can be seen from these two figures, most esti-mation errors occur in junctions that connect two modelling blocks where a block is a region that is modelled by a single regression function. To improve the performance we need to remove the discontinuities in the junction areas so that the estimated channel response is a smooth function across the whole range of interest.
3.3.2
Modified model-based channel estimation
Piece-wise polynomial model based channel estimation reduces the complexity at the cost of reduced accuracy when the average SNR is low. As just mentioned, it is necessary to smooth the discontinuities between two regression-modelled blocks. We use a interpolation function called Hermite curve to solve this problem. Hermite curves are very easy to calculate but are also very powerful. They are often used to provide smooth interpolation between 2 points.
Given two end points P0 P1 and the associated end differentials R0 R1, we can use a simple function to find the curve with order three between these two end points and with these two end differentials. In other words, given x(0) = P0, x(1) = P1, x(0) = R0, and x(1) = R1, for t∈ (0, 1), we have
x(t) = P0· (2t3− 3t2+ 1) + P1· (−2t3+ 3t2) + R0· (t3− 2t2+ t) + R1· (t3− t2). (3.8) As the Hermite curve is defined in the closed interval t∈ (0, 1), to apply the Hermite curve-fitting to our problem we have to normalize the length of the selected overlapped regions between two neighboring model block to one. If the duration of the overlap region is equal to L sampling intervals, then the original sampling instants are normalized to t0 = 0, t1 = ∆t,· · · , ti = i∆t,· · · , tL−1 = 1. Substituting the corresponding end-point
,0 ˆ LS H HˆLS, . . . MB f 0 ˆa ˆ o M a . . . MB f . . .
M odel Based C hannel Estimation
with model order Mo
C ompute the soft infromation for the LD PC decoder
-1 -1 ˆ LS H ( ) ˆa ( ) ,0 ˆ MB H ,h L M . . . . . . . . . . . . . . . . . . 0 P R0 P1 R1 Hermite f (P0, R0, P1, R1) . . .
{
O verlap lengthFigure 3.9: The factor graph representation of the modified model based channel esti-mator. 0 10 20 30 40 50 60 -0.6 -0.4 -0.2 0.0 0.2 0.4 Real Part of the Channel Response sub-carrier index
True Channel Response Step1: Regression Function Step2: Hermite Curve
Figure 3.10: An example of the channel estimates using the two-stage M B(3, 16, 6) estimate.
estimates at the selected positions of the resulting Hermite curve. Combining with the original regression curves obtained by a three-step M B(MO, ML) estimate, we obtain
an overall smooth curve. The factor graph representation of the modified model-based channel estimate is plotted in Fig. 3.9. Fig. 3.10 shows a typical M B(3, 16, 6) two-stage regression fitting process, where the three parameters MO = 3, ML = 16, L = 6 denote
respectively the model order, the modelling block length and the overlap block length. The discontinuous curve marked by empty circles represents the first-stage block-wise regression fitting result while the one marked by stars is resulted from Hermite curve at the overlap regions so that the empty circles in the non-overlapped regions and the stars in the overlapped regions constitute the final estimate (smooth curve). We also give the true channel response (the one marked by filled squares) for convenience of comparison. By adding an additional interpolation stage, the M B(3, 16, 6) estimate outperforms the M B(7, 56) estimate except when the average SNR is relatively high, as indicated by the BER and FER performance curves of Fig. 3.11 and Fig. 3.12. Note that when the average SNR is high, a regression model-based estimate has already achieved excellent MSEE performance even at the block junctions whence no further smoothing is needed; see Figs. 3.13 and 3.14 . It is therefore suggested that a test stage be added before the Hermite-fitting stage so that at the end of the first piece(block)-wise fitting stage, one can check if it is necessary to proceed to perform the smoothing operation.
The computational complexity for the additional Hermite curve for the overlap length Lis 2× (ML− 1) × 2 + (L − 2) × 4 × 2 real multiplications per overlap block. The total
multiplication complexity of M B(3, 16, 6) is 2496 per OFDM symbol.
3.4
Joint Channel Estimation and Decoding in
SISO-OFDM Systems
Since the communication devices become smaller, the devices will be moved more easily and the channel response will change through time more quickly. The receivers
8 10 12 14 16 1E-4 1E-3 0.01 BER avg{ Eb/N0} Perfect CSI MB(7,56) MB(3,14) MB(3,16,6)
Figure 3.11: The BER Performance for the M B(3, 14), M B(3, 16, 6), and M B(7, 56) Channel Estimates. 8 10 12 14 16 1E-3 0.01 0.1 FER avg{ Eb/N0} Perfect CSI MB(7,56) MB(3,14) MB(3,16,6)
Figure 3.12: The FER Performance for the M B(3, 14), M B(3, 16, 6), and M B(7, 56) Channel Estimates.
0 10 20 30 40 50 60 -0.2 0.0 0.2 0.4 0.6 Real Part of the Channel Response sub-carrier index
True Channel Response Step1: Regression Function Step2: Hermite Curve
Figure 3.13: An example of the channel estimates using the two-stage M B(3, 16, 6) estimate when the average SNR is high.
20 30 40 0.3 0.4 Real Part of the Channel Response sub-carrier index
True Channel Response Regression Function Regional Curve Fitting
need to use the information from not only the training signals but all the transmitted signals in order to estimate the channel response at present. The optimal receiver per-forms to detect the transmitted data given the all received signals (based on whatever information of the channel statistics that can be considered known). If the data bits are coded, after decoding the data, we can get the data part information, and this informa-tion can help estimating the channel response at different symbols, so the receiver can operate iteratively by joint channel estimation and data decoding.
Channel Equalizer
Channel Es tim ation
LDP C Dec oder
Data Sym bol Inform ation
P ream ble S ym bols
ˆ
mX
ˆ
mX
ˆ
mH
mY
Dec oded Data
Outer loop
Inner loop
Figure 3.15: The joint channel estimation and decoding receiver in SISO-OFDM systems. The receiver operation structure is shown as Fig. 3.15, and the corresponding factor graph is shown as Fig. 3.16. After the channel estimation at the preamble, we can get the channel estimate ˆH, and use this channel estimate to detect the data symbols
ˆ
X, and after several times of the LDPC decoding (here we use 25 iterations) the more
reliable data symbols can be seen as the pilot symbols, the channel response at each symbol can be re-estimated by these pilot symbols. The factor graph representation of the joint channel estimation and decoding is shown in Fig. 3.16.
LD PC Bit N ode
LD PC C heck N ode LS C hannel Estimator
0 X Y0 Received Signals LDPC f fLS Y . . . . . . . . .
Pilots Bit N ode
. . . X ,0 ˆ LS H HˆLS, = P {P0 P . . . MB f 0 ˆa ˆ o M a . . . MB f . . .
M odel Based C hannel Estimation
with model order Mo
C ompute the soft infromation for the LD PC decoder
codeword length
X
. . .
. . .
C hannel Estimation for one O FD M symbol
SD N ( SD N +NSP X) ,..., } SP N NSD+NSP -1 -1 -1 -1 -1 -1 (X,Y ) ˆ LS H ( ) ˆa ( ) ,0 ˆ MB H HˆMB ,NSD+NSP-1 ,h
Figure 3.16: The factor graph representation of joint channel estimation and decoding in SISO-OFDM systems.
8 10 12 14 16 1E-4 1E-3 0.01 BER avg{ Eb/N0} Perfect CSI LS MB(3,16,6) kout=0 MB(3,16,6) kout=1 MB(3,16,6) kout=3
Figure 3.17: The BER performance of joint the M B(3, 16, 6) estimation and decoding with kout= 0, 1, 3. 8 10 12 14 16 1E-3 0.01 0.1 FER avg{ Eb/N0} Perfect CSI LS MB(3,16,6) kout=0 MB(3,16,6) kout=1 MB(3,16,6) kout=3
Figure 3.18: The FER performance of joint the M B(3, 16, 6) estimation and decoding with kout= 0, 1, 3.
8 10 12 14 16 1E-4 1E-3 0.01 BER avg{ Eb/N0} Perfect CSI LS MB(7,56) kout=0 MB(7,56) kout=1 MB(7,56) kout=3
Figure 3.19: The BER performance of joint the M B(7, 56) estimation and decoding with kout = 0, 1, 3. 8 10 12 14 16 1E-3 0.01 0.1 FER avg{ Eb/N0} Perfect CSI LS MB(7,56) kout=0 MB(7,56) kout=1 MB(7,56) kout=3
Figure 3.20: The FER performance of joint the M B(7, 56) estimation and decoding with kout = 0, 1, 3.
times of the outer loop, and at each time of the outer loop, there are 25 inner loop iterations. The channel estimate used by the outer loop are M B(3, 16, 6) and M B(7, 56), respectively, show that at each iteration of the channel re-estimation will bring some performance improvement. kout = 0 means the channel response only estimated at the
preamble but not re-estimated after decoding. kout= 1, 3 denotes the iterative number of
channel re-estimating, and the total iterations of the LDPC decoding is equal to 50, 100, respectively.
Finally, we observe that re-estimating channel once after 25 decoding iterations can bring about 1.5dB gain in BER and FER performance. If there are 3 times of channel re-estimation, as the performance curves labelled by M B(3, 16, 6) 3 and M B(7, 56) 3 have indicated, we then come very close to the theoretical lower bound–that achieved by a receiver with perfect channel estimate.
Chapter 4
Joint Channel Estimation and
Decoding in MIMO-OFDM Systems
The combination of MIMO signal processing with OFDM waveform offers a practical solution to achieve very high spectral efficiencies and unprecedented data rates. With a rich scattered multi-path propagation environment and appropriate signal processing at the receiver side, the received streams can be separated so that a MIMO wireless channel can be viewed as virtual parallel independent channels.
LDPC Enc oder T rains m itter 1 IFFT T rain s mitter NT x IFFT . . data 1 X Tx N X 11 H Tx N H 1 Y Rx N Y 1 Tx N H 1NRx H MIMO Detec tor LDPC Enc oder . . . . 1 ˆ X ˆNTx X ˆ H
Es tim ated Channel
i X Hi1 +N1 Rx N Tx N 1 i= =
Figure 4.1: A LDPC-coded MIMO-OFDM system model.
A MIMO-OFDM system model is depicted in Fig. 4.1 in which the data stream is first encoded by a single LDPC encoder, then serial-to-parallel converted into NT x
parallel data sub-streams, where NT x is the number of transmit antennas. Each data
sub-stream is mapped onto a stream of symbols and an IFFT is performed on each sub-stream of symbols and a cyclic prefix is inserted in front of each OFDM symbol as the GI. Denoting the signal vector transmitted from NT x transmit antennas on the mth
sub-carrier by
Xm = [Xm1, Xm2, ..., X NT x
m ]
T, (4.1)
and assuming the channel delay spread is less than the GI so that the ISI can be fully removed, we can express the received signal vector Ym at the mth sub-carrier as
Ym =HmXm+Nm, (4.2)
where Ym ∈ CNRx×1, Hm ∈ CNRx×NT x, Xm ∈ CNT x×1, and Nm ∈ CNRx×1, and Hmij
denotes the channel response from the ith transmit antenna to the jth receive antenna at the mth sub-carrier, and NT x,NRx is the number of transmit and receive antennas,
respectively.
Separating the desired signal Xi
m from the interference from other transmit antennas Xi m, we have Ym =Him· X i m+H i mX i m+Nm, where Hi m = [H i1 m, H i2 m, ..., H iNRx m ] T Hi m = [H1m,H2k, ...,H i−1 m ,H i+1 m , ...,H NT x m ] T Xi m = [Xm1, Xm2, ..., X i−1 m , X i+1 m , ..., X NT x m ] T , and Hi mXi
m is called the co-antenna interference (CAI).
Because of the presence of CAI in a MIMO-OFDM system, to estimate the channel response between each transmit antenna and each receive antenna based on Yj
m, j =
In the remaining part of this chapter, we first deal with the (initial) channel estima-tion problem using the special preamble symbol structure of the 802.11n specificaestima-tion. In the ensuing section we discuss the MIMO signal detection issue, i.e., the problem of extracting soft information for the LDPC decoder in the presence of CAI. We close this chapter with a presentation of an iterative joint channel estimation/tracking and decoding algorithm and its performance.
4.1
Channel Estimation in MIMO-OFDM Systems
As shown in Fig. 2.1, the coded bit stream is de-multiplexed (spatially parsed) into substreams that are to be transmitted via individual antennas. Each frequency domain data stream is interleaved before being modulated by the associated constellation mapper. As mentioned in Chapter 2, the packet format of the IEEE 802.11n standard, PPDU, is divided into two concatenated units: the preamble unit and the data unit. Such a signal structure is a common practice in almost all wireless protocols. In other words, it calls for the transmission of a preamble prior to the payload that contains the information block.
The preamble structure and the contents of the HT-LTF for the different number of transmit antennas can be found at A.1. An example of tone interleaving across 4 transmit antennas is shown in Fig. 4.2. The HT-LTFs set 0, set 1, set 2, and set 3, as given in Table A.1, form a partition of the 56 tones in the HT-LTF sequence 4. At the OFDM symbol interval, each set of tones maps to single transmit antenna. Only after the overall preamble is transmitted, all sets get mapped to each transmit antenna. By using the preamble unit in which each tone is assigned to a single antenna, one can easily separate the signals transmitted from different antenna, channel estimation can thus be carried out by using the same algorithm as that for the SISO-OFDM systems.
HT-LTF 0 HT-LTF 1 HT-LTF 2 HT-LTF 3
Tx 0 Set 0 Set 3 Set 2 Set 1
Tx 1 Set 1 Set 0 Set 3 Set 2
Tx 2 Set 2 Set 1 Set 0 Set 3
Tx 3 Set 3 Set 2 Set 1 Set 0
time
0 1 2
-Transmit
antenna
Figure 4.2: HT-LTF tone interleaving across 4 transmit antennas.
4.2
MIMO Signal Detection
Unlike the received signals in SISO-OFDM systems, the baseband samples received by the jth receive antenna at the mth sub-carrier Yj
mis a combination of the transmitted
signals (Xm1, Xm2, ..., XNT x
m ) from all transmit antennas. Before decoding, one needs to
suppress CAI and estimate the transmitted data Xm from different transmit antennas
based on Ym and the channel estimate obtained during the preamble period.
4.2.1
MIMO MMSE detector
The MIMO MMSE detection scheme is proposed by Haykin [10] for Turbo-BLAST. The estimator for Xi
m is given by
ˆ
Xmi =wHi Ym− ui,
wherewi represents tap weights of a linear filter, and ui is the corresponding weighted
Invoking the MMSE principle to minimize the cost function ( ˆwi,uˆi) = arg min (wi,ui)E[|X i m− ˆXmi |2], (4.3) we obtain ˆ wi = (P + Q + ΣNRx)−1Him (4.4) ˆ ui = wˆiz (4.5) where P = Hi mH i m H ∈ CNRx Q = Hi m[INT x−1− Diag(E[Xi m]E[X i m] H)]Hi m H ∈ CNRx ΣNRx = σN2INRx ∈ CNRx z = Hi mE[X i m]∈ C NRx×1
Note that the expectations E[Xi
m] are the LDPC decoder soft outputs and z
repre-sents our estimate of CAI. A simplified sub-optimum solution is obtained by assuming E[Xi
m]E[Xim]H = 1 ∀i, m so that Q = 0NRx×NRx and the desired signal becomes
ˆ Xmi = (HimHHim+ σ2N)−1HimH(Ym− Hi mE[X i m]), (4.6)
with the initialization E[Xi
m] = 0. This data estimator will be referred to as MMSE
detector henceforth. Given the channel estimate obtained by applying the M B(3, 16, 6) estimate at the preamble, one can detect the data by first use the MIMO MMSE detec-tion algorithm and then proceed to perform the LDPC decoding. An iterative MMSE detection and LDPC decoding process that feedback the soft LDPC decoder output for MMSE detection is expected to enhance the BER (or FER) performance. Since the LDPC decoder converges rather slow, to feedback reliable and substantially improved soft values to the MMSE detector, many LDPC decoding iterations need to be carried out. We compare the FER and BER performance of the above three scenarios in Figs.
8 9 10 11 1E-5 1E-4 1E-3 0.01 BER avg{ Eb/N0} MMSE Detector Perfect CSI, kin=5 MB(3,16,6), k in=0 MB(3,16,6), kin=1 MB(3,16,6), kin=5
Figure 4.3: Influence of the inner loop iteration number kin on the BER performance of
2× 2 MIMO-OFDM systems with MMSE detector.
8 9 10 11 1E-3 0.01 FER avg{ Eb/N0} MMSE Detector Perfect CSI, kin=5 MB(3,16,6), kin=0 MB(3,16,6), kin=1 MB(3,16,6), kin=5
Figure 4.4: Influence of kin on the FER performance of 2× 2 MIMO-OFDM systems with MMSE detector.
4.4 and 4.3, respectively. In these figures, the curve labelled by M B(3, 16, 6), kin
rep-resents the scenario that soft output is obtained after kin LDPC decoding iterations,
kin = 0 represents the no-feedback case. We notice that the system performance
im-proves as the number of LDPC decoding iteration increase. For kin >5, the performance
gain becomes negligible and, for the sake of brevity, the corresponding curves are not shown here.
4.2.2
MIMO MLD detector
Another MIMO detector called the soft-input/soft-output maximum likelihood de-coder (MLD) is used in the Zelst’s Turbo-BLAST [11] as an ideal BLAST de-mapper. To produce soft-decision outputs with the MLD, the log-likelihood ratio is used as an in-dication for the reliability of a bit. If Xi
m is the transmitted signal form the ith transmit
antenna at the m sub-carrier and with QPSK mapping, then we have to derive estimates (soft values) of the corresponding bits, bi
m,0 and bim,1, from the received samplesYm. For
example, the a posteriori log value of the estimated bit b1m,0 can be expressed as LM LD,p(bim,0|Ym) = ln P(b1m,0 = 0|Ym) P(b1m,0 = 1|Ym) = lnP(b = [0, 0, .., 0]|Ym) + ... + P (b = [0, 1, .., 1]|Ym) P(b = [1, 0, .., 0]|Ym) + ... + P (b = [1, 1, .., 1]|Ym) (4.7) whereb represents the transmitted coded bits [b1m,0, b1m,1, b2m,0, ..., bNT x
m,1].
Applying the Bayes rule, the above expression render the decomposition LM LD,p(bim,0|Ym) = lnP(Ym|b = [0, .., 0])P (b = [0, .., 0]) + ... + P (Ym|b = [0, .., 1])P (b = [0, .., 1]) P(Ym|b = [1, .., 0])P (b = [1, .., 0]) + ... + P (Ym|b = [1, .., 1])P (b = [1, .., 1]) = lnP(Ym|b = [0, .., 0])P (b 1 m,0 = 0)P (b1m,1 = 0)· · · P (b NT x m,1 = 0) + ... P(Ym|b = [1, .., 0])P (b1m,0 = 1)P (b1m,1 = 0)· · · P (b NT x m,1 = 0) + ... = lnP(b 1 m,0 = 0) P(b1m,0 = 1) + ln P(Ym|b = [0, .., 0])P (b1m,1 = 0)· · · P (b NT x m,1 = 0) + ... P(Ym|b = [1, .., 0])P (b1m,1 = 0)· · · P (b NT x m,1 = 0) + ... = La(bim,0) + LM LD,e(bim,0|Ym) (4.8)
where we have assumed that the coded bits are independent so that the joint probabil-ity is equal to the product of the corresponding marginal probabilities, e.g., P (bi
m,0 =
0, bi
m,1 = 0) = P (bim,0 = 0)· P (bim,1 = 0).
The above equation says that LM LD,e(bim,0|Ym) is the difference between the a priori
and the a posteriori log value at the MIMO MLD detector and consists of channel information and extrinsic information. The LM LD,e are taken as the a priori values to
the LDPC decoder for further iterative decoding steps. And after several iterations of LDPC decoding, a posteriori log values output from the LDPC decoding can feedback to the MIMO MLD detector as a priori log values. After estimating the channel response at the preamble by applying the M B(3, 16, 6) estimator, the MIMO MLD detector can detect the data and then proceed to perform the LDPC decoding.
An iterative MLD detection and LDPC decoding process that feedback the soft LDPC decoder output for MLD detection can also be expected to enhance the BER (or FER) performance. Since as the result as the previous subsection, after 5 LDPC decoding iterations, to feedback reliable and substantially will improve soft values to the MLD detector. We compares the BER and FER performance of the above three scenarios in Figs. 4.5 and 4.6, respectively.
4.3
Joint Channel Estimation and Decoding in
MIMO-OFDM Systems
We have shown that excellent performance is attainable with joint channel estimation and decoding in SISO-OFDM systems. It is thus natural apply the same approach to MIMO-OFDM systems. The proposed receiver structure is shown in Fig.4.7. The initial channel estimate ˆH derived from the preamble is sent to the MIMO detector to help estimating data symbols ˆX from different transmit antennas. The output from the MIMO detector is forwarded to the LDPC decoder whose soft output is feedback to both the MIMO detector and the channel estimator.
8 9 10 11 1E-5 1E-4 1E-3 0.01 BER avg{ Eb/N0} MLD Detector Perfect CSI, kin=5 MB(3,16,6), kin=0 MB(3,16,6), kin=1 MB(3,16,6), kin=5
Figure 4.5: Influence of kin on the BER performance of 2× 2 MIMO-OFDM systems with MLD detector. 8 9 10 11 1E-3 0.01 0.1 FER avg{ Eb/N0} MLD Detector Perfect CSI, kin=5 MB(3,16,6), kin=0 MB(3,16,6), kin=1 MB(3,16,6), kin=5
Figure 4.6: Influence of kin on the FER performance of 2× 2 MIMO-OFDM systems
1
Y
Rx NY
Channel Es tim ation Parallel to Serial MIMO Detec tor SISO LDPC Dec oder Serial to parallel P ilot S ym bolsT entative Data S ym bols Dec is ions from LDPC Dec oder Output
X
ˆ
ˆ
b
. . . . . . 1ˆ
X
ˆ
NTxX
1ˆ
X
X
ˆ
NTxˆ
H
ˆ =
X
Inner Loop Outer LoopFigure 4.7: An iterative receiver structure for LDPC-coded MIMO signals. Grant [12] proposed a simple method for joint decoding and channel estimation for MIMO channels. The decoder obtains a tentative data sequence estimate by using the current channel estimate. The initial channel estimation can be derived from the preamble. Let ˆXi
m be the decoder’s tentative output for the data from the ith transmit
antenna and denote by ˆHi
m the ith column of ˆHm. One then original compute an
the tentative data vectors associated with other transmit antennas ˆ Hi m[t] = Ym− k=i ˆ Xk mHˆkm[t− 1] ˆ Xi m . (4.9)
Removing the estimated CAI, one then go on to compute a new channel estimate by our SISO-OFDM solution proposed before.
Using the M B(3, 16, 6) channel estimate obtained during the preamble period, we perform iterative joint MMSE detection (or MLD detection) and decoding to obtain a tentative data vector estimate. Regarding this tentative decision as pilots, we can then re-estimate the channel response via the M B(2, 11, 6) estimate. We use a lower-order channel model for fear of severe CAI-induced estimation error. Our choice of MO = 2 means a straight line is used as a local approximation. Additional Hermite
curve fitting will smooth the piece-wise linear estimate and result in a much better approximation of the true channel response. Such an iterative detection scheme consists of two iteration loops: the inner loop performs iterative MMSE detection (or MLD detection) and LDPC decoding while the outer loop represents the iterative process between the channel estimator and the inner loop.
Figs. 4.8 and 4.9 are the BER and FER performance of the above double-loop iterative receiver by using MMSE detector, where Figs. 4.10 and 4.11 are using MLD detector. The curves labelled by M B(3, 16, 6) kin, M B(2, 11, 6) koutrepresent the
perfor-mance of the detection strategy that performs kout outer loop iterations where an outer
loop iteration mean a renewed channel estimation after a M B(3, 16, 6) kin round. The
performance of the M B(3, 16, 6) 5 iterative scheme, which re-estimate the received data for every 5 LDPC decoding iterations without channel re-estimation, is also included for convenience of comparison.
8 9 10 11 1E-5 1E-4 1E-3 0.01 BER avg{ Eb/N0} MMSE Detector Perfect CSI, kin=5 MB(3,16,6), kin=5 MB(3,16,6), kin=5; MB(2,11,6), kout=1 MB(3,16,6), kin=5; MB(2,11,6), kout=3
Figure 4.8: Influence of kin and the outer loop iteration number kout and with the
M B(3, 13, 6) estimation at the preamble and the M B(2, 11, 6) estimation at the data symbols on the BER performance of 2× 2 MIMO-OFDM systems with MMSE detector.
8 9 10 11 1E-3 0.01 FER avg{ Eb/N0} MMSE Detector Perfect CSI, kin=5 MB(3,16,6), kin=5 MB(3,16,6), kin=5; MB(2,11,6), kout=1 MB(3,16,6), kin=5; MB(2,11,6), kout=3
Figure 4.9: Influence of kin and kout and with the M B(3, 13, 6) estimation at the
pream-ble and the M B(2, 11, 6) estimation at the data symbols on the FER performance of 2× 2 MIMO-OFDM systems with MMSE detector.
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