A capacity-approaching coded modulation scheme for non-coherent fading channels

Transcription

1 Louisiana State University LSU Digita Commons LSU Master's Theses Graduate Schoo 008 A capacity-approaching coded moduation scheme for non-coherent fading channes Youngjeon Cho Louisiana State University and Agricutura and Mechanica Coege Foow this and additiona wors at: Part of the Eectrica and Computer Engineering Commons Recommended Citation Cho, Youngjeon, "A capacity-approaching coded moduation scheme for non-coherent fading channes" (008). LSU Master's Theses This Thesis is brought to you for free and open access by the Graduate Schoo at LSU Digita Commons. It has been accepted for incusion in LSU Master's Theses by an authorized graduate schoo editor of LSU Digita Commons. For more information, pease contact gradetd@su.edu.

2 A CAPACITY-APPROACHING CODED MODULATION SCHEME FOR NON-COHERENT FADING CHANNELS A Thesis Submitted to the Graduate Facuty of the Louisiana State University and Agricutura and Mechanica Coege in partia fufiment of the requirements for the degree of Master of Science in Eectrica Engineering In The Department of Eectrica and Computer Engineering By Youngjeon Cho Bacheor of Engineering, Korea Miitary Academy, 999 May 008

3 Acnowedgments It is a peasure to than the many peope who made this thesis possibe. First and foremost, I woud ie to than Professor Xue-Bin Liang for his supervision, mentorship, mora and technica support during my graduate research studies at Louisiana State University. I truy appreciate his guidance and wisdom in executing this study. I aso than Professor Guoxiang Gu and Professor Shuangqing Wei for their invauabe hep and wiingness to serve on my dissertation research committee. My wife, Eunhye, and my daughter, Soji, gave me encouragement and deight. I reay coud not have done this without them. Finay, I woud ie to than the Korean Army for sending me here to earn many things. ii

4 Tabe of Contents Acnowedgments ii List of Figures.. v Abstract..vii Chapter. Introduction.... Channe Coding.... Sma Scae Fading Frequency Fat Fading. 4.3 Unitary Space-Time Moduation. 5.4 Probem Statement..6.5 Thesis Organization....7 Chapter. Capacity for Non-Coherent Fading Channes Signa Mode Mutua Information. 8.3 Utimate Formua Obtaining G Capacity....4 Simuation Resut.. Chapter 3. Turbo Codes 4 3. Convoutiona Codes.4 3. Turbo Encoder Intereaver Turbo Decoder The MAP Decoding Agorithm Principe of Iterative Decoding Simuation Resut.. Chapter 4. LDPC Codes Fundamentas of Linear Boc Codes LDPC Encoder Parity Chec Matrix H Generator Matrix G LDPC Decoder Tanner Graph Soft Decision Simuation Resut.. 3 Chapter 5. Design of Coded Moduation for Non-Coherent Fading Channes.3 iii

5 5. Turbo Codes with Unitary Space-Time Moduation Encoding Decoding LDPC Codes with Unitary Space-Time Moduation Encoding Decoding Codewords Chapter 6. Simuation Resuts Simuation Resuts of Turbo Codes Performance Evauation of Turbo-Coded Moduation Scheme Simuation Resuts of LDPC Codes Performance Evauation of LDPC-Coded Moduation Scheme....4 Chapter 7. Concusion...43 Bibiography..45 Vita. 48 iv

6 List of Figures. Channe Coding at Transmitter and Receiver..... Types of Channe Codes Many Paths in Mobie Communication (Diffraction, Refection, Scattering)..3.4 Two Types of Wireess Channes Basic Spatia Mutipexing Scheme with Three Tx and Three Rx Antennas. Ai, Bi, and Ci Represent Symbo Consteations.5. Capacity versus SNR, Singe Antenna Used and Considered T 3. Two Types of Convoutiona Codes 4 3. Treis Diagram for 8 States Turbo Encoder Boc Diagram of Turbo Decoder State and Branch Metrics Dependencies in the MAP Agorithm Soft-Input Soft-Output Decoder Modue The BER Performance of the Turbo Codes over AWGN Channes 4. Boc Diagram of LDPC Codes Boc Diagram of Boc Code Encoder Parity Chec Matrix Tanner Graph Factor Graph at The LDPC Decoder The Iustration of (a) Update Chec Nodes (b) Update Bit Nodes Boc Diagram of LDPC Decoder The BER Performance of the LDPC Codes over AWGN Channes The Boc Diagram of the Channe Encoder and Unitary Space-Time Moduation..3 v

7 5. The Boc Diagram of the Receiver with Channe Decoder The Boc Diagram of the Receiver with LDPC Decoder Turbo Codes over Non-coherent Channe (T) with One Transmit Antenna and One Receive Antenna (L) Turbo Codes over Non-coherent Channe (T) with One Transmit Antenna and One Receive Antenna (L4) LDPC Codes over Non-coherent Channe (T) with One Transmit Antenna and One Receive Antenna (L) LDPC Codes over Non-coherent Channe (T) with One Transmit Antenna and One Receive Antenna (L4) LDPC Codes over Non-coherent Channe (T) with One Transmit Antenna and One Receive Antenna (L8) 4 7. New Receiver Structure of LDPC-Coded MIMO System...43 vi

8 Abstract Approaching the Shannon imit of the communication channes has been studied by many researchers to efficienty and reiaby transmit data through the channes. To sove this probem, various methods and schemes have been proposed for approaching the theoretica imit for Shannon s channe capacity. Among them, both ow-density parity chec (LDPC) codes and Turbo codes have been proposed to minimize the bit error rate (BER). Therefore, understanding of LDPC codes and Turbo codes is usefu for their appications in modern communication systems. The study about non-coherent channes, which do not require expicit nowedge or estimation of the channe state information, has become a major issue in mobie communication. Specificay, a new signaing scheme caed unitary space-time moduation has been invented which is suitabe for non-coherent channes. Combining channe coding with unitary space-time moduation is expected to mae good performance for non-coherent fading channes. In this thesis, non-coherent capacity of a mobie communication channe in Rayeigh fat fading is cacuated for the case of coherence time of ength two. Aso, LDPC codes and Turbo codes are combined with unitary space-time moduation to enhance the efficiency and reiabiity of communication over non-coherent fading channes. The performance resuts are compared to the cacuated channe capacity. Simuation resuts show that both LDPC codes and Turbo codes are we performed for non-coherent fading channes. The LDPC and Turbo coded unitary space-time moduation schemes have BER performance much better than the uncoded moduation schemes and the performance is cose to the cacuated channe capacity. vii

9 Chapter Introduction The roe of communication has been changed. Peope are no onger satisfied with voice communication. They want to watch TV and to connect to the internet with their ce phones. In order to satisfy peope s desires, the technoogy in wireess communication has been deveoped rapidy and new terminoogy has been emerged in this fied. Remarabe breathrough has been made in channe coding area to minimize bit error rate (BER) which are turbo codes and ow-density parity chec (LDPC) codes. However, fading may be too fast to get the nowedge of the channe state information at the receiver. So, the case where the channe state information is unnown to the transmitter and the receiver has been studied. In this chapter, the basic bacground of a mobie communication system wi be presented.. Channe Coding Origina Data Source Coding e.g. MPEG4 Channe Coding e.g. boc code Moduation Received Data Source Decodin Channe Decoding Demoduation Figure. Channe coding at transmitter and receiver. In modern society, it is very important to get reiabiity of data. However, as the quantity of data increased, it has high possibiity to produce errors. So, error-correcting coding emerged and has been deveoped to minimize the errors. The principe for error detection and correction is simpe. Figure. shows the basic diagram of communication

10 and iustrates how channe coding changes bits. Channe encoder receives data or 0, but it adds redundant data and sends it to the moduator. For exampe, if there is an information bit,, which wi be sent to the receiver, channe encoder adds redundancy and produces. Then, the coded data go to the moduator. Receiver can detect as ong as error occurs just one time ie 0 or 0. Thus, we can assume that error detection/correction adds redundancy to the origina data so that receiver can detect the information error. Channe coding is an error-correcting coding by which codes can be constructed to detect and correct errors which may be caused by noise and interference. A mobie communication system woud be unreiabe without channe coding. The modern approach to error contro in digita communications originated from the wor of Shannon [8] and Hamming [9]. Since then, many different error correcting codes have emerged. Historicay, these codes have been incuded boc codes and convoutiona codes. The main difference for the two codes is the presence of memory in the encoders. Various approaches have been proposed for approaching the theoretica imit for Shannon s channe capacity. The we-nown channe coding schemes to approach Shannon s imit have LDPC codes and Turbo codes among others. Figure. Types of channe codes (Adopted from Ref. [3])

11 Figure. iustrates types of channe codes. It aso shows that Turbo codes use convoutiona codes and iterative decoding. Liewise, LDPC codes aso use iterative decoding to approach Shannon imit, but LDPC codes is boc codes. Sum-product agorithm which is used in LDPC codes has ow compexity as compared to the maximum posteriori probabiity (MAP) agorithm in Turbo codes. Richardson s research shows that LDPC codes can approach the Shannon imit coser than Turbo codes with the 6 boc ength 0 and code rate []. Both LDPC codes and Turbo codes are main streams in error-correction channe coding for digita communication system.. Sma Scae Fading Fading is the channe condition varying as time or frequency. It is caused by many reasons such as time variation of received signa, changes in transmission paths, and movement of antenna. Sma scae fading pays important roe in a mobie communication and is dominated by two factors, muti path propagation and Dopper shift. Figure.3 iustrates muti paths in mobie communication. Figure.3 Many paths in a mobie communication (diffraction, refection, scattering) (Adopted from Ref. [3]) 3

12 Signas can tae many different paths between transmitter and receiver due to refection, scattering, diffraction. If ine of sight (LOS) exists, LOS pays major important roe in communication. However, it is very difficut to get LOS in mobie communication. We need to consider the environment where LOS does not exist. In this case, diffraction and refection can be primary factors at the receiver. The muti path propagation maes signas arrive at the receiver at different times. It is caed deay spread. In mobie communication, deay spread is even worse if transmitter and receiver move and channe characteristics change over time and ocation. When mutipath components of one puse overap components of subsequent puse, it is caed intersymbo interference (ISI)... Frequency Fat Fading The wireess channe is said to be fat fading, when the channe has constant gain and inear phase and inear response is much arger than the bandwidth of the signa []. Let BS be the bandwidth of the signa and Bc be the coherence bandwidth, then for fat fading channe, B >> B. It means that when coherence bandwidth is arger than c S bandwidth of the signa, fat fading channe happens. This yieds another important fact that ISI is negigibe in fat fading channe [4]. Figure.4 iustrates how to divide wireess channe into two types of fading channes. Frequency seective fading channe T S >> τ S T S t Im p u s e tra in W ir e e s s C hanne T S < τ S B S τ S f Fat fading channe f Figure.4 Two types of wireess channe (Adopted from Ref. [3]) 4

13 .3 Unitary Space-Time Moduation A non-coherent communication system is a system where channe state information is not nown at the receiver end, i.e., fading coefficients are not nown at the receiver. Unitary space-time moduation is invented for mutipe antennas in non-coherent channe that operates in a Rayeigh fat-fading environment [0]. It is a signaing scheme which consteation comprising compex-vaued signas Φ,, L L with respect to time, among the transmitter antennas. Space-time signas Φ are T M matrix where M is the number of transmits antennas and T is the coherence time during which the fading is approximatey constant. Figure.5 shows when three transmitter and three receiver antennas are used. However, we ony consider a mobie communication system that empoys one transmitter antenna and one receiver antenna that operates in a Rayeigh fat fading environment. The mode at the receiver can be described as foows. Χ ρ SH + W (-) Figure.5 Basic spatia mutipexing scheme with three Tx and three Rx antennas. Ai, Bi, and Ci represent symbo consteations (Adopted from Ref. []) 5

14 where X is the T received signa, ρ is the signa to noise ratio(snr) at the receiver antenna, S is the T transmit signa, H is the compex-vaued fading coefficient which is constant for t,,t, and W is the T additive white Gaussian noise (AWGN) Fading coefficient H is constant for T symbo period (In this thesis, T wi be ony considered). Fading coefficient H and Gaussian noise W are independent and identicay distributed CN(0,). The transmitted signa power can be written as E, t,,t (-) s t If coherence interva T is changed, transmit signa s shoud be aso changed using the equation of (-) in order to get equa transmitted power. A consteation of L unitary space-time signas, {, } S KS L are defined to comprise of T M compex-vaued matrices such that S Tφ,,,L, where is an H H isotropicay distributed T M matrix with the constraints φ φ K φ φ I [0]. Φ L L M The coumns of Φ are orthonorma. Because of the maximum-ieihood (ML) agorithm for a consteation of unitary space-time signas, which is shown in [5], we see to the singuar vaues of φ H φ as sma as possibe to minimize pair wise probabiity error..4 Probem Statement Combining LDPC codes or turbo codes with non-coherent fading channes has been performed by some researchers [9][5][7]. The resuts of those papers show that coded moduation schemes outperform uncoded moduation. However, those papers did not use optimum codewords for unitary space-time moduation. Moreover, they did not compute the capacity for comparing the BER performance with the capacity of the non-coherent channes. One contribution of this thesis is to cacuate capacity for certain cases based on 6

15 Hassibi s paper [7]. With the cacuated capacity, either LDPC codes or Turbo codes wi be combined with unitary space time moduation to see if the resuts can approach the Shannon imit. In order to get the best resuts, optimum codewords wi be used for certain coherent time T..5 Thesis Organization The rest of this thesis is organized as foows. Chapter presents the capacity for the non-coherent fading channe in the case of coherence time T. Chapter 3 gives a brief review of Turbo codes and Chapter 4 presents LDPC codes. In Chapters 3 and 4, we wi see the simuation resuts for LDPC and Turbo codes over AWGN channes. Chapter 5 introduces the channe mode which is used for simuation of LDPC and Turbo coded moduation schemes for non-coherent fading channes. Chapter 6 presents the simuation resuts. The concusion of the thesis is given in Chapter 7. 7

16 Chapter Capacity for Non-Coherent Fading Channes We want to compute capacity for mobie communication with no channe state information avaiabe either to the transmitter or to the receiver. Hassabi and Marzetta computed the capacity using isotropicay random unitary inputs [7], but the paper ony introduced resuts for mutua information per symbo versus duration of coherence interva T. Since our goa of this thesis is to compare the capacity with coded performance, detaied capacity for certain cases are needed. In this chapter, We wi briefy describe how to obtain the cose form capacity which was introduced in [7]. Then, the resut wi be appied to the specia case which one transmit and one receive antenna are used with T.. Signa Mode The propagation coefficients are assumed to be constant for T symbo periods. If M transmit and N receive antennas are given, the signa mode can be given by X ρ / M SH + W (-) where ρ is the signa to noise ratio (SNR) The signa S is the product of two independent random matrices [] and can be written by S T Φ D (-) where Φ is a unitary matrix, and D is a nonnegative rea diagona matrix. In this thesis, D I M wi be considered in the paper [7].. Mutua Information The mutua information is given by P(X S) I(X;S) E og E[ ogp(x S) - ogp(x) ] (-3) P(X) 8

17 where X is the received signas and S is the transmit signas. Equation of (-) shows that the mutua information I(X;S) can be obtained provided that ogp(x S), and ogp(x) are cacuated. To get these probabiities, Marzetta s paper [5] is referred which offered properties of capacity for non-coherent fading channe. The conditiona probabiity density of the received signas for non-coherent channe is exp( tr p( X S) TN π det + { Λ XX } N Λ ) H where Λ + ( ρ / M )S S is covariance matrix of the coumns of X. I T (-4) From the equation of (-), we can rewrite the equation of (-4) as p exp + + [ trx (I + ρβφd Φ ) X ] T ( X S) TN π det(it + ρβφd π TN exp det(i + [ trx X ] M + ρβd ) N Φ + ) N exp tr( X + Φ(I M + D ρβ ) + Φ X (-5) where we define β T / M. Aso, we can get p(x) from P(X D), the detaied steps are suggested in [7]. p(x) is written as p( X ) ( ) ( T M )( T M ) / exp( tr( TN π ( + ρβ ) α ) X MN + X ) Γ( T ) Γ( M + ) det G Γ( T M ) Γ() (-6) ρβ where G is a (T-M) (T-M) Hane matrix, α, and G is given by + ρβ G mn exp( ασ ) Q ( ασ ) ( ασ + ασ ) Q- ( ασ ) ( exp( ασ )) q 0 q! q, when Q>0 where σ is nonzero eigenvaue of + XX. 9

18 Monte Caro estimate can be used to get mutua information I ( X; S) L og L p(x p Φ ( X ) 0 ) L L Γ( T - M) Γ( ) exp og Γ( T) Γ( M + ) + + ( αtrxx ( IT + Φ 0Φ 0 ) ( T-M)( T-M-) / ( ) det G (-7) Q- ( ασ ) ( exp( ασ )) q 0 q! exp where ( ασ ) G Q ( ασ ) ( ασ + ασ ) Equation of (-7) is the cosed form capacity for non-coherent channes which is shown in [7]. However, the cosed form invoves many matrixes to compute the capacity when T is arger, due to the (T-M) (T-M) G matrix. In this thesis, we ony consider specia case for capacity for non-coherent channes with a singe transmit and receive antenna for coherence interva T of. In order to get the resuts for this case, we need to do further steps..3 Utimate Formua.3. Obtaining G Because coherence time T is and one transmit and one receive antenna are considered, we can get K min (T, N), Q, Φ Therefore Hane matrix G is written as I M q. G exp( ασ ) Q ( ασ ) ( ασ + ασ ) Q- ( ασ ) ( exp( ασ )) q 0 q! q exp( ασ ) ασ (-8) 0

19 Now, we want to compute σ which is nonzero eigenvaue of + XX. a a0 + ai Let X, where, b are compex numbers. b a b0 + bi + X X can be written as a + X X b [ a b ] a * a b a b. b * Thus, we can get the nonzero eigenvaue of.3. Capacity + XX is σ a + b. Our utimate goa is to simpify the equation of (-7). The gamma function, which is shown in equation of (-7), defines Γ (x) (x-)!, if x is positive. Since coherence time T is and the number of transmit antenna is, the gamma function of (-7) is cacuated by Γ Γ ( T - M) Γ( ) ( T) Γ( M + ) Γ() 0!. We aso have Tr (X X + ) Γ()! a + b. Using these resuts, the exponentia part of I ( X; S ) can be simpified by * + exp( α trx X ( I + Φ Φ + T 0 0 ) a 0 0 ab exp α tr exp (-α b * ) a 0 b b Thus, we can get the capacity I ( X; S) L L exp( α b og det G.4 Simuation Resut L L ) α exp( ασ b og L ασ ) L (-9) To simuate the equation of (-9), a and b shoud be determined. Since a and b are distributed according to the equation of (-4), the conditiona probabiity of X is written as

20 a 0 exp tr p ( X S) [ ] 0 b a exp tr a b + [ 0] π ( + ρβ ) 0 ρβ b exp a0 a b0 b π + ρ + ρ + ρ (-0) With equations of (-9) and (-0), the graph of capacity for non coherent fading channe is obtained. Matab is used to mae this simuation. Figure. iustrates the capacity for the case of T. L30000 is used for Monte Caro estimate. The graph shows that when the SNR is increased, capacity can aso be increased. The resuts wi be used in Chapter 6, where LDPC codes and Turbo codes are depicted and they wi be compared to this capacity. The graph ony considered coherence time T of. The graph for different Ts can be obtained by the same idea that 3.5 C db Figure. Capacity versus SNR, singe antenna used and ony considered T.

21 we proceeded in this Chapter. However, it is not surprising that increasing T causes more compexity to cacuate capacity. 3

22 Chapter 3 Turbo Codes Turbo codes, which was proposed in993 by C. Berrou et a [3], achieves amost reiabe data communication at signa-to-noise ratio that is very cose to the Shannonimit despite of its compexity. Since turbo encoder consists of two constituent convoutiona codes separated by an intereaver, inear convoutiona codes wi be briefy expained to understand Turbo codes in this Chapter. Aso, the MAP agorithm, which is used at the turbo decoder, wi be discussed. 3. Convoutiona Codes Convoutiona codes generate coded symbos by passing the information bits through a inear finite-state shift register with tap [3]. Figure 3. shows two types of convoutiona codes in which one is recursive and the other is non-recursive. Tapped outputs of shift register are added by moduo- addition and the outputs of moduo- adders are mutipexing. { u } { d } { u } { d } d d d a a a { v } { v } (a) Non-recursive systematic (b) Recursive systematic Figure 3. Two types of convoutiona codes. 4

23 Let n be the number of output bits, be the number of input bits and m be the number of memory registers. The quantity /n is caed the code rate. The constraint ength L represents the number of bits in the encoder memory that affects the generation of the n output bits. It is defined as L (m-). Current output at the encoder depends on the current input and m previous inputs of the encoder. Encoder generator is used to mae convoution encoder. Assuming that bits of the encoder, convoutiona encoder can be written as Y is the output Y g d i i mod. (3-) L i 0 where G : { g i } is the encoder generator which is one or zero. It can be represented in octa form. Graphicay, there are three ways in which we can oo at the encoder or decoder to gain better understanding of its operation. These are State Diagram, Tree Diagram, and Treis Diagram. Figure 3. shows treis diagram which is a powerfu way of describing the decoding. In Figure 3., 8 different bits combinations (8 states) represent 3 memory registers. There are two outputs for each state. Since memory registers have no previous data before data is encoded, the treis begins at state 000. Figure 3. Treis diagram for 8 states. (Adopted from Ref. [6]) 5

24 3. Turbo Encoder Figure 3.3: Turbo encoder. The common form of convoutiona encoder is the non-recursive systematic convoutiona encoder which is shown in Figure 3.3. However, the turbo encoder consists of two recursive systematic convoutiona encoders separated by an intereaver. As turbo encoder is systematic, the message bit d produces x without processing encoding. The message bit d is aso encoded by the first encoder to produce parity bits Y. And intereaved message bit d is encoded by the second encoder to mae another parity bits Y. Figure 3.3 aso shows how to combine two systematic convoutiona encoders with one intereaver. The rate of the Turbo codes can be varied by the symbo size of input and output. If punctured code, which produces codes of many different rates, is used, different code rate can be obtained by deeting output bits according to a chosen puncturing pattern [3]. 3.. Intereaver An intereaver is an device that permutes the ordering of a sequence of symbos from input, i.e., the symbos at the input are ordered by intereaver [8]. This intereaver can eiminate the correation among bits so that burst errors can be avoided. There are many 6

25 inds of intereavers such as boc intereaver, heica intereaver, and random intereaver. Turbo codes uses random intereaver which is we nown to the best intereaver. It is hard to find the correation among input bits, when random intereaver is used. Because the intereaver puts the input bits randomy. 3.3 Turbo Decoder Maximum a Posteriori (MAP) agorithm was modified to be used for turbo codes in 993 [3]. MAP agorithm for turbo decoder uses two decoding stages. It is caed BCJR agorithm which is named after the inventor s name (Bah, Coce, Jeine, and Raviv). This agorithm minimizes the bit error rate at the turbo decoder. The main fow of BCJR agorithm is that the decoder receives a the bits (noise added) of one frame and computes the probabiities for a paths, and outputs the soft-output. Two types of parity bits are added noise and faded through channes. And then, each parity bits go to different decoders. Figure 3.4 shows turbo decoder in which two decoders are used. Each of the two decoding stages uses a BCJR agorithm to sove MAP detection probem. In Figure 3.4, Decoder receives information sequence x and parity sequence () y which is obtained from the first convoutiona encoder. Figure 3.4 Boc diagram of turbo decoder. 7

26 () With x and parity bits y, the Decoder produces extrinsic information. The L e Decoder receives intereaved extrinsic information L e from Decoder. Decoder aso receives parity sequence () y which is intereaved data from the second convoutiona code at the transmitter in order to maes the extrinsic information L e. Deintereaver changes intereaved L e into origina bits sequence and this extrinsic information goes bac to the Decoder. This procedure iterates severa times unti no errors can be found The MAP Decoding Agorithm At the receiver, og-ieihood can be represented as foows L( d Pr ( d observation) ) og (3-) P ( d 0 observation) r P where r ( d i observation) is the APP of the data bit. If L( ) 0, the decoded bit is, otherwise, the decoded bit is 0. N Let ( ) Pr{ S m R } λ, the APP of a decoded data bit is equa to: m N i P { d i R } λ ( m) m d i 0, (3-3) where N N + R { R, R, R + } P( A, B, C, D) P( B A, C, D) P( A, C, D) Bayes rue is P ( A B, C, D) P( B, C, D) P( B, C, D) P( B A, C, D) P( D A, C) P( A, C) P( B, C, D) Using Bayes rue, i λ (m) can be rewritten by K N N λ m ) P( R d i, S m, R ) P( R d i, S m, R i ( K + N ) N P ( d i, S m, R ) / P( R N ) (3-4) 8

27 To simpify the equation of (3-4), et us introduce three auxiiary metrics α { N Pr m, R }, β (m) { R S m} (m) S Pr and (m) + γ Pr{ S m, R S m' }, where α (m) is a forward state metric, β (m) is a bacward state metric and γ (m) is a branch metric. These metrics can be rewritten as this: M α (m) m' 0 Pr( S m, R S m', R ) Pr( S m', R ) M Pr( S m, R ) Pr( S m, R S m' ) α ( m'). γ ( m', m) (3-5) m' 0 m' 0 M M β (m) m' 0 Pr( S N + m', R + S m) M N Pr( S + m', R +, S Pr( S m) m ' 0 m) M N Pr( S + m', R + S m) Pr( R + S + m' ) β ( m'). γ ( m, m' ) (3-6) m' 0 m Figure 3.5 shows how the above three metrics wor in treis codes. These are essentia to i compute λ (m) which can be used to get L d ). Thus, L( d ) can be written using metrics ( M ' 0 L N Pr ( d R ) m ( d ) og og N 0 Pr ( d 0 R ) λ m λ ( m) ( m) og m m' m m' α α ( m'). γ ( m, m'). β ( m) ( m'). γ ( m, m'). β ( m) 0 (3-7) Figure 3.5 State and branch metrics dependencies in the MAP agorithm. (Adopted from Ref. [6]) 9

28 In the case of RSC encoders, the received vaues are spit into two uncorreated s p s p components and, where is the received systematic component and is the y y coded parity component. Thus, og-ieihood can be rewritten as y R y L( d ) Pr ( y og P ( y r s s d d ) 0) m m m' m' α α ( m'). γ ( y ( m'). γ ( y 0 p p, m, m'). β ( m), m, m'). β ( m) Pr ( y og P ( y r d d ) 0) + og m m' m m' α α ( m'). γ ( m, m'). β ( m) ( m'). γ ( m, m'). β ( m) 0 (3-8) The right side of the equation of (3-8) is extrinsic information Principes of Iterative Decoding The second term in the right side of the equation of (3-8) is the redundant information which caed extrinsic information. This information improves the LLR for bit d A priori vaues in L (d) Detector a posteriori LLR vaues ^ L '( d) (x) L c + L (d) L c ( x Channe vaues in ) Soft-in Soft-out Decoder (d) L e Extrinsic vaues out ^ L '( d) A posteriori vaues ^ L d) L '( d) + (d) ( ^ L e Figure 3.6 Soft-input soft-output decoder modue. 0

29 Log-ieihood Ratio of the information bit conditioned by the received symbo is L( d x Pr ( d x ) ) og. Using Bayes rue, it can be changed ie this: P ( d x ) r Pr ( x d ) Pr ( d ) L ( d x ) og + og (3-9) P ( x d ) P ( d ) r If AWGN channes are considered, the conditiona pdf can be written as r p( x d ) exp( ( x ) ), p( x exp( ( ) ) d ) x + πσ σ πσ σ We can now compute P ( x d ) ( x ) ( x + ) r og + x Pr ( x d ) σ σ σ E 4 N S 0 x Thus, we obtain ES Pr ( d ) L ( d x ) 4 x + og L c ( x ) + L a ( d ) (3-0) N P ( d ) 0 r where L x ) c ( E 4 S x, L a ( d ) N0 Pr ( d ) og P ( d ) r : a priori vaue for bit d If we consider the Log-ieihood Ratio of the information bit conditioned by the whoe observation sequence, we can now rewrite L ( d ) L ( d x ) + L e ( d ) L c ( x ) + L a ( d ) + L e ( d ) (3-) where L e d ) is the extrinsic information derived from equation of (3-7). ( Decoder sends the new reiabity information to Decoder, and Decoder revises the first information by new information which came from Decoder. The two decoders aternatey update their probabiity measures by iterations.

30 3.4 Simuation Resuts The BER of the turbo coded system is presented in Figure 3.7. Matab is used to mae simuations. Generating poynomias are ( 37,) octa, which is the same Turbo codes used in the origina paper [3], but 000 of the number of information bits N are used which is different to the origina paper. 0 - BER 0 - uncoded BER 0-3 iter iter 0-4 iter 5 iter 3 iter SNR (db) Figure 3.7 The BER performance of the Turbo over AWGN (N 000, Rate ). Figure 3. 7 shows that Turbo codes outperforms over the uncoded scheme. The resut for 5 iterations is cose to the db of SNR. It is remaraby improved and cose to the Shannon imit. We aso observe that as iterations are increased, the performance of the Turbo codes is improved. However, the gain is smaer as the iteration is arger.

31 Chapter 4 Low-Density Parity Chec (LDPC) Codes Low-Density Parity Chec (LDPC) codes is a inear boc code which is we nown to correct errors in digita communication channes. LDPC codes was invented by Robert Gaager in the eary 960s []. However, it had been ignored for a ong time due to the requirements of high compexity computations unti it was rediscovered by Macay in 999 [3]. It has shown that LDPC codes is the best error correction code to cose Shannon imit, which is the imitation in communication channes. Figure 4. shows the boc diagram of LDPC codes. The mechanism of LDPC encoder is the same as the inear boc encoder, but the main difference between LDPC codes and inear boc codes are decoding agorithm. LDPC decoder uses sum-product agorithm which is a ey technique for LDPC codes. Therefore, inear boc code wi be briefy described to understand LDPC encoder. Aso, sum-product decoding agorithm wi be discussed. Encoder T G Moduation Channe Sum-Product agorithm Demoduation Figure 4. Boc diagram of LDPC codes. 4. Fundamentas of Linear Boc Codes Generator matrix G is used to produce coded bits as depicted in Figure 4.. Generator matrix G is reated to parity chec matrix H which is used at the receiver to find errors. Let m0, m, m,... constitute a boc of message bits and b0, b, b,... denote m b n 3

32 m[ m 0, m, m,... m ] T G c[ c 0, c, c,... c n ] Figure 4. Boc diagram of boc code encoder. the (n-) parity bits in the code word and parity. Parity bits can be obtained by b ( n ) m P ( n ) where P are one s if b depends on m, otherwise P are zero s. We may therefore write codeword c as foow: c [ b : m] m[ P ( n ) : I ] mg (4-) where I is the by identity matrix and G is the generator matrix. Let parity chec matrix H [ : ], based on the eqation of (4-) and parity chec matrix H, H G T I n P ( n ) T 0 can be obtained. Using this resut, we can cacuate c H to find errors which is caed syndrome. For binary symmetric channe (BSC), the received T codeword is c added with an error vector e. If we have the equation of c H 0, a errors are detected. 4. LDPC Encoder 4.. Parity Chec Matrix H In order to mae LDPC codes, first, we need to mae parity chec matrix H. And then, generator matrix G can be made by parity chec matrix H. Let us denote that M is number of row, N is number of coumn. H is represented as M N matrix. Parity chec matrix H of LDPC codes is different from H of inear boc codes. One 4

33 Figure 4.3 Parity chec matrix (Adopted from Ref. [3]) remarabe characteristic of LDPC codes is that LDPC codes uses parity chec matrix H which does not have identity matrix and contains very few s in each row and coumn. Define an (N, j, ) parity chec matrix that has j ones in each coumn, and ones in each row, and zeros esewhere. Figure 4.3 shows a method to mae H (0, 3, 4) which was introduced by Gaager []. Parity chec matrix H is sub-divided into smaer submatrices (bocs) to ensure s are we dispersed. The first boc of Figure 4.3 creates a singe in each coumn and the other bocs are merey a coumn permutated version of the first boc. This is Gaager s method to mae H, but there are many ways to mae parity chec matrix H [][][3]. If the number of s per coumn or row is constant, the code is caed a reguar LDPC codes, otherwise it is a irreguar LDPC codes. Usuay the irreguar LDPC codes outperforms over other codes []. The ey to mae parity chec matrix H is that the number of in the matrix shoud be sma. 4.. Generator Matrix G Given H, we can get generator matrix G to mae encoder. Let H [ Z : X ], where Z is a non singuar matrix, H shoud be satisfied with T ch 0. Using this eqaution, parity bit b 5

34 can be rewritten b m( A B) T mp. And generator matrix G is G [ ( Z X ) T MI ] (4 ) K Therefore, LDPC encoder can be made by generator matrix G. 4.3 LDPC Decoder LDPC decoding agorithm has severa names such as beief propagation agorithm, the message passing agorithm, and the sum-product agorithm. In this thesis, the name of sum-product agorithm wi be used. Tanner graph [9] wi be introduced first in order to expain sum-product agorithm,. And sum-product agorithm wi be extended to wor with soft decision Tanner Graph Tanner graph is a bipartite graph which is an undirected graph whose nodes are divided into two casses, where edges ony connect two nodes [30]. To understand Tanner graph, Let us introduce a parity chec matrix H H (4 3) The parity chec matrix H of (4-3) has 7 rows and 3 coumns, and it can be represented by Tanner graph which is shown in Figure 4.4. The nodes of the tanner graph are separated into two distinctive sets and edges are ony connecting nodes of two different types. The two types of nodes in a Tanner graph are caed bit nodes and chec nodes. Edges between bit nodes and chec nodes indicate the participation of bit (variabe) parity chec n. m in 6

35 v v f v v3 v5 v7 0 v 3 v 4 f v v3 v6 v7 0 v 5 v 6 f 3 v 4 v5 v6 v7 0 v 7 bit nodes Chec nodes Figure 4.4 Tanner graph. Tanner graph consists of m chec nodes (the number of parity bits) and n variabe nodes (the number of bits in a codeword). Chec node c is connected to bit nodes b, provided that the eement of H is one. As Tanner graph represents parity chec matrix H, we can mae decoder as simpe as we can. For exampe, v which is shown at the right side of Figure 4.4 shoud be the same with sum of moduo- of, and v. It means v3 v5 7 that we can cacuate using sum of moduo- of, and v. Liewise, sum of v v3 v5 7 v3 v5 v7 moduo- of, and can be used to get v. Each bit node updates their vaues using the probabiity of other bit nodes. The decoding is accompished by iterating these steps. Tanner graph enabes us to observe the agorithm as a good graphica view point. Now we are ready to oo at the sum-product agorithm mathematicay Soft Decision At the receiver, the received bits are decoded 0 or. If the majority is, decoder chooses, otherwise 0 is chosen. It is caed hard decision. Soft decision which is 7

36 based on the probabiity yieds in a better decoding performance and it is concerned as the preferred method. Thus, the sum-product agorithm cacuates approximations for ogieihood ratios. The brief expanation for sum-product agorithm is ie this. Messages of probabiities of or 0 are transmitted. Based on these probabiities, probabiities of bit nodes and probabiities of chec nodes are updated. This procedure can be divided into three steps. First step is that bit nodes receive a message. In this step, a bit node has ony a posteriori probabiity p ( x y). Second step is that bit nodes send a message to their chec nodes which are connected each other. Chec nodes send message bac to the bit nodes after comparing incoming bits. For exampe, Tanner graph in Figure 4.4 shows that chec node f receives 4 bits from bits nodes. If the received bits are [ ], chec node sends [0 ] bac to the bit nodes according to v v v v Third step is to use additiona information which came from chec node. If the number of iteration is arger, bit nodes get more information from chec nodes. Figure 4.5 depicts how messages are processed in tanner graph. Beow are steps for soft decision whose idea is the same as the above expanation. Input of the LDPC decoder is a posterior probabiities p p c x y ) n ( n n f Figure 4.5 Factor graph at the LDPC decoder. 8

37 The first step : Initiaization No other information other than a posteriori probabiity is given in this step. The probabiity of the first step is written as q o mn p( x 0 y) and q mn p( x y) (4-4) The second step : Chec node update p n Now we have a posteriori probabiities q mn. The probabiities are sent to the chec nodes. Chec nodes cacuate r which is the conditiona probabiity of chec z being mn satisfied given by a bit c. Thus, r is represented as P zm cn x. n mn r ( ) o Let δ q mn - qmn, and δ be the product of δ q matrix eements aong with row, ij qmn rmn mn m excuding the (m, n ) position. δ r can be written as mn rij δ δ q { n' N } m, n mn', where we define the set of bits that participate in chec m, except for bit n is { N N n},. The conditiona m n m \ probabiity of r mn can be written as r 0 mn ( + δ ) / and r ( δ ) / (4-5) r mn mn r mn q mn r mn q mn r mn q mn r mn bit nodes Chec nodes bit nodes Chec nodes (a) (b) Figure 4.6 The iustration of (a) update chec nodes (b) update bit nodes. 9

38 Figure 4.6 shows the iustration of updating chec nodes and bit nodes. The third step : Bit node update o Now we have other information to update the vaues of the probabiity of and. o 0 0 q mn α mn pn rm ' n and q mn α mn pn rm ' n (4-6) { m' } { m' } M n, m M n, m q mn q mn 0 where constants are seected to ensure +. However, these probabiities are α n q mn q mn just used to do iterations. The pseudo posterior probabiity q n is needed to get utimate probabiities for bit nodes. If q >0.5, a decision is made that x. q is written as n n n q P c x y,{ z 0, m M }) (4-7) n ( n n m n The equation of (4-7) shows that a chec nodes invoving represented as c n are satisfied. It is aso 0 0 q α p r and α p r (4-8) 0 n n n { m' M } n m' n q n n n { m' M } n m' n 0 where constants are seected to ensure +. α n q n Figure 4.7 shows the boc diagram of LDPC decoder. Updating chec nodes and bit nodes are iterated severa times unti a errors are detected. q n Start Initiaizatio n Update chec node Update bit node No M ax.iteration? No End Yes H ˆxˆ 0 E s tim a te tran sm itted vector x Figure 4.7 The boc diagram of LDPC decoder. 30

39 4.4 Simuation Resuts BER 0-3 iter iter iter SNR(dB) Figure 4.8 The BER performance of the LDPC over AWGN (N 000, Rate ) Figure 4.8 iustrates the BER of the LDPC coded system over AWGN channes. Reguar parity chec matrix is used that has 3 ones in each coumn, and 3 ones in each row. Sum-product agorithm is performed at the decoder with 8 iterations. And the numbers of bits, which are used in the simuation, are 000. Despite of sma ength of bits sequences, the graph shows that LDPC codes performs very we over AWGN channes. As the numbers of iterations were increased, we can see the fact that the resut is cosed to the zero. 3

40 Chapter 5 Design of Coded Moduation for Non-Coherent Fading Channes In this Chapter, we wi discuss the methods for coded moduations for non-coherent fading channes. Unitary space-time moduation, which is suitabe for MIMO, is used to communicate over non-coherent fading channes. Since we have cacuated capacity for singe antenna and T of, we ony consider one transmit and one receive antenna, and coherence time T wi be restricted to. The resuts of this Chapter are shown in Chapter Turbo Codes with Unitary Space-Time Moduation 5.. Encoding Figure 5. iustrates how to combine turbo codes with unitary space-time moduation. Message bits are divided into bocs of N bits and are encoded by Turbo codes. As we discussed in Chapter 4, the turbo encoder consists of two recursive systematic convoutiona encoders and it produces three types of outputs. Then, the encoded bits are intereaved by pseudorandom intereaver which is used to spread burst errors. However, the intereaver which is used at transmitter is different from the intereaver which is used in turbo encoder. And the next step is to moduate the intereaved data. Unitary space- Message bits Encoder Intereaver Unitary Space-Time Moduation Figure 5. The boc diagram of the channe encoder and unitary space-time moduation. 3

41 time moduation divides the intereaved bits into a consteation of L signas [0]. Given the coherence time of the channe of T, one transmit and one receive antenna, the transmitted signa matrix S from a unitary space-time consteation is ie this S T Φ,...,L, (5-) where Φ is an isotropicay distributed T matrix and obeys Φ + Φ I. The channe gain is constant for a period of T symbos when the unitary space-time moduation is used over a Rayeigh fat-fading channe. 5.. Decoding Lieihood Computation Decoder Deintereaver Figure 5. The boc diagram of the receiver with channe decoder. Hochwad and Marzetta s paper [0] shows maximum-ieihood (ML) agorithm and its performance when H is unnown. In their paper, maximum-ieihood decoding becomes + + Φ arg max tr{ X Φ Φ X } (5- ) Φ Φ,, { } Φ L However, instead of using the equation of (5-), we wi use a suboptima decoding agorithm which computes the og-ieihoods of the transmitted bits, and uses them as if they are the og-ieihoods of the observations from a BPSK moduation over an AWGN [4][5][6]. This approach enabes turbo codes to cacuate the received data easiy. It is aso used in LDPC codes. 33

42 Notice that the received signa X corresponds to TR coded bits, where R is bit rate. We shoud compute the og-ieihoods of TR coded bits using received set of signas. Let us denote the TR bits b ( b, L, b T, Lb TR ) that construct S, then LLR can be written as [ b X, L, X m ] [ b 0 X, L, X ] Pr Pr b, X, L, X Λ ( b ) og og Pr Pr b 0, X, L, X m [ m ] [ ] m og b: b b: b 0 Pr Pr [ b, X, L, X ] [ b 0, X, L, X ] m m (5-3) Assuming f ( ) is the mapping from b to C and that consteation. Signas are equiprobabe, equation of (5-3) can be written as S: S f ( b ), b Pr S: S f ( b ), b 0 [ X S ] Λ( b ) og (5-4) Pr [ X S ] This equation expains how we use the LLR as if they are the LLR of the observations from a BPSK moduation over an AWGN channe. In Chapter, the equation of (-3) is the conditiona probabiity density of the received signas given the transmitted signa. Thus, we can rewrite the LLR of (5-4) as LLR( d ) Φ S / Φ S / T : S f ( d ), d T : S f ( d ), d 0 exp( tr X + / Tρ exp( tr X + / Tρ + + ΦΦ ΦΦ + + X ) X ) (5-5) Now, we have LLR for non-coherent fading channes. The LLR is considered as LLR of BPSK over AWGN channe. Therefore, the received bits are computed using (5-5) and are fed to turbo decoder. The turbo decoder iterates severa times unti it gets good resuts 34

43 and maes a decision whether received bits are zero or one. Then the decoded data are deintereaved as the same order of intereaver to get origina information bits. 5. LDPC Codes with Unitary Space-Time Moduation 5.. Encoding The basic concept of LDPC encoder with unitary space-time moduation is the same as the turbo encoder. Information bits are encoded by LDPC encoder and then intereaved by pseudo-random intereaver. Unitary space-time moduation moduates the intereaved bits using reevant codeword which wi be introduced in Chapter 5.3. We consider ony coherence time of. Thus, if a consteation of unitary space-time signas with L is used at the transmitter, code rate for moduation wi be ½. Generator matrix G, which is obtained by parity chec matrix H, is needed to encode the information bits. Reguar parity chec matrix H in which the number of s in each row and coumn are the same wi be used. In this thesis, Radford M. Nea parity chec matrix wi be foowed [8]. We wi construct matrix which has a 0 eement according to code rate and size of information bits. And then, we wi exchange 0 for randomy. After that, generator matrix G can be made by parity chec matrix H. 5.. Decoding Figure 5-3 is the decoding agorithm for LDPC for non-coherent fading channes. Suboptima decoding agorithm, which is used in turbo decoding, wi compute the ogieihood ratio. The outputs of LLR are fed into the LDPC decoder. LDPC decoder, which uses sum-product agorithm, considers the inputs as if they are the og-ieihoods of the observations from a BPSK moduation over an AWGN channes. So, the sumproduct agorithm, which Chapter 4 introduced, performs at the decoder by updating 35

44 Lieihood Computation Bit nodes Deintereaver Chec nodes Figure 5.3 The boc diagram of the receiver with LDPC decoder. chec nodes and bit nodes. Since information bits are intereaved at the transmitter, deintereaver is needed at the receiver to reorder the output bits of the decoder. 5.3 Codewords We ist the code design that wi be used in Chapter 6. These are nown as optimum codewords for unitary space-time moduation, when coherence time T is. Turbo codes and LDPC codes wi use these codewords for non-coherent channe to get the best resuts. These are provided by Professor Xue-Bin Liang. When L, the codeword is constructed as 0 0 When L 4, the codeword is given as i i i i i i i i When L 8, the codeword is constructed as i i 36

45 i i i i i i i i i i i i i i 37

46 Chapter 6 Simuation Resuts Based on Chapter 5, Turbo codes and LDPC codes wi be appied for non-coherent channes. Coherence interva T wi be considered in this thesis. Matab is used to mae a simuations in this Chapter. And severa web pages are referred in order to mae LDPC codes and Turbo codes [][9][0]. 6. Simuation Resuts of Turbo Codes 0 0 turbo(l) uncoded BER SNR (db) Figure 6. Turbo codes in non-coherent channe (T) for one transmit antenna and one receive antenna (L). 38

47 0 0 turbo(l4) uncoded BER SNR (db) Figure 6. Turbo codes in non-coherent channe (T) for one transmit antenna and one receive antenna (L4). 6.. Performance Evauation of Turbo-Coded Moduation Scheme The above graphs iustrate simuations of Turbo codes in terms of bit error rate versus signa-to-noise ratio (SNR) for one transmit antenna and one receive antenna. ( ) n g d ( ) octa g, 37, generating poynomias are used which is the same turbo codes used in the origina turbo coding paper [3]. The numbers of information bits which are used in the simuation are 000 and the rate of the turbo codes is. Based on Chapter, capacity for non-coherent channes is depicted as a dashed ine. Figures 6. and 6. demonstrate channe capacity, Turbo codes and uncoded moduation for non-coherent 39

48 fading channe. Ceary, simuation resuts demonstrate that the Turbo codes have good performance in non-coherent fading channes. However, when comparing the resuts with the capacity, more than 5 db between capacity and simuation resut are found. The graph shows approximatey 8 db, when consteation L is. When consteation L is 4, we can find ess than 0 db. This gap is arge compared to the performance over AWGN channes. 6. Simuation Resuts of LDPC Codes LDPC(L) uncoded 0 - BER SNR (db) Figure 6. 3 LDPC codes in non-coherent channe (T) for one transmit antenna and one receive antenna (L). 40

49 LDPC(L4) uncoded 0 - BER SNR (db) Figure 6. 4 LDPC codes in non-coherent channe (T) for one transmit antenna and one receive antenna (L4) LDPC(L8) uncoded 0 - BER SNR(dB) Figure 6. 5 LDPC codes in non-coherent channe (T) for one transmit antenna and one receive antenna (L8). 4

50 6.. Performance Evauation of LDPC-Coded Moduation Scheme Despite communication toobox for matab gives LDPC codes function, we foow severa websites which have matab based LDPC source codes to understand LDPC codes [9] [0]. The reguar ow density parity chec matrix H, which is based on Radford M. Nea s programs coection [8], is used in this paper. The number of s in each row and coumn of parity chec matrix H are 3. And parity chec matrix H for our simuations maes ony rates. At the receiver, soft decision is used which is better than hard decision for BER versus SNR performance. 8 iterations are executed to obtain better resuts. Three graphs of Figure 6.3, 6.4, and 6.5 show LDPC codes performed over noncoherent fading channes for singe antenna and coherence time T of. As can be expected, LDPC codes demonstrate the great resuts compared to uncoded non-coherent channe. LDPC codes for non-coherent channe is coser to the Shannon imit. However, the resuts are not cose to the capacity as we expected. It has approximatey db gaps. Both Turbo codes and LDPC codes perform we in non-coherent fading channes, but they are not cose to the capacity which we computed. We can assume that it is not sufficient to just combine channe coding with non-coherent channes. To reduce the gap, other factors may be considered or encoding and decoding agorithm shoud be modified. 4