Exam Schedule and Topics for Coding Theory Course, Slides of Digital Communication Systems

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Announcements Quiz # 7 will be held on Friday 19th December in the class and the Makeup Quiz # 1 will take place on the same day in the evening time 5:00pm Quiz # 8 will be held on Friday 26th December in the class and the Makeup Quiz # 2 will take place on the same day in the evening time 5:00pm docsity.com Agenda Structured Sequences Linear Block Codes Examples Error Detecting and Correcting Capabilities Hamming Code Example Convolutional Codes Quiz docsity.com Automatic Repeat Request » ARQ vs. FEC a ARQ is much simpler than FEC and need no redundancy. a ARQ is sometimes not possible if = Areverse channel is not available or the delay with ARQ would be excessive = The retransmission strategy is not conveniently implemented = The expected number of errors, without corrections, would require excessive retransmissions ® docsity.com  Transmitter 1 2 3 3 4 5 5 ; ee u* a 7 ors <F Transmission & & = oy = x Receiver 1 2 34] 3 4 Ps 5 Error Error (a) Transmitter 1,;2/3)/4/5/)6)/7/8/4)/5/]e6]7;8]3 }10 8/9 )10 _— ~~ AOC SIE III IVES “ek Transmission OS PPE LESEEEEEEE a e CEE Receiver 1f2}3bkadsfel[7[ela]s]efrqe nuj7is 1 1 Error Error (b} Transmitter T)}2)/3/4/)5/6)/7 [8] 4] 9 }10}11 11/16/17) 18 . * ee wo O wm eH a ~~ Ae ete Moke he Fe ke He, - ae Transmission wo ee eS oF ey ee oe we oe ee + Receiver 4 2/3 bay 5/6/7])s8s])4]9 14)15/)911/16 Error Figure 6.7: Automatic Repeat Request (ARQ) (a) Stop and wait ARQ (b) tinuous ARQ with pullback (c) Continuous ARQ with selective repeat docsity.com  6.3 Structured Sequences = Block codes = Convolutional codes = Turbo codes 6.3.2 Code Rate and Redundancy = Incase of block codes, encoder transforms each k-bit data block into a larger block of n-bits called code bits or or channel symbol = The (n-k)-bits added to each data block are called redundant bits, parity bits or check bits = Ratio of redundant bits to data bits: (n-k)/k is called redundancy of code = Ratio of data bits to total bits, k/n is called code rate Channel Data block Codeword encoder SS eSSSSSSSSSSSSSSSSsSA SSeS k bits n bits ® docsity.com  6.4 Linear block Codes 6.4.1 Vector Spaces = The set {0,1}, under modulo 2 binary addition and multiplication forms a field. Addition Multiplication 0®80=0 0-0=0 0@l=1 0-1=0 180=1 1-0=0 1@1=0 ll=1 ® docsity.com  Some definitions — cont’d = Examples of vector spaces a The set of binary n-tuples, denoted by V, = {(0000), (0001), (0010), (0011), (0100), (0101),(0111), (1000), (1001), (1010), (101 1),(1100),(1101),(1111)} = Vector subspace: a A subset S of the vector space V’,is called a subspace if: un The all-zero vector is in S. = The sum of any two vectors in S is also in S (Closure Property). Example: {(0000), (0101), (L010), (1 111)} isasubspaceof V,. a These are fundamental properties of Linear Block Codes ® docsity.com  = The subset chosen for the code should include as many as elements to reduce the redundancy but they should be as apart as possible to maintain good error performances 2" n-tuples constitute the entire space V,, TS 2% n-tuples constitute the subspace of codewords Linear block-code structure ® docsity.com  Example: = Let the generator matrix be: V, 1 1 0 1 0 0 G=|V,/=|}0 1 1 0 1 0 V; 1 01 00 1 =n Where V,,V,and V, are linearly independent vectors that can generate all fhe code vectors = The sum of any two generating vectors does not yield any of the other generating vectors = Generate Codeword U4 for the fourth message vector 1 1 0 in Table 6.1 U, =[1 1 0] V=V,+ V,+ 0*V, = 110100 + 011010 + 000000 =101110 ( Codeword for the message vector 110) ® docsity.com  6.4.5 Systematic Linear Block codes = Asystematic (n,k) linear block code is a mapping a k-dimensional message vector to an n-dimensional code word such that part of the sequence has k message digits and remaining (n-k) are parity digits = Asystematic linear code will have a generator matrix Pll PI2 - Pl(w-ky) 1 O 1 0 ; P21 P22 Pr(n-k) O 1 ov 0 G-|P : & |= 21 P22 2,(n-k) Pel Pk2 °* Pk(n-k) 9 Ov 1 nxk I, =kxk identity matrix P, =kx(n—k) matrix = Combining Pu Po 7 Praew 1 0 _ Px Px "Pawar U,,U,,...U, =[M,,m,,...m,]x]-, Pu Pr 7 Pra-r 00 -- 1 ® docsity.com  Where u, =mM,p,+ M,p,;+ ....M,p,; for i=1,...(n-k) = Max for i=(n-k+1)....n =u The systematic code vector can be expressed as: U = Py; Poseees Pn, M5 5--+ My parity — bits message _ bits Py = MM, Py FM, Py Fe FM; P Hy Py =M Pix TM, Py Te FM, Py Prk = Pyne) FM Po (nny Ft FIM, Penk) & docsity.com  6.4.7 Syndrome Testing It is easy to verify from here: UH" = py + Diy Py + Pasee-Pn¢ + Pre = 0 Where U is a code word generated by matrix G iff UH™=0 Let r be received vector (one of 2" n-tuples) where U vector (one of 2« n-tuples) was transmitted : r=Ut+e =m The syndrome of r is defined as: S =rH™=(U+e)H™T=UH™+eH" S=eH™ ‘Daca souree| + Format m | Channel U, Modulation encoding channel Channel |,___Demodulation, Data sink Format anne eme “a 70 _Y m decoding r Detection [ docsity.com  = Requirements of the parity-check matrix a Nocolumn of H can be all zeros, or else an error in the corresponding codeword position would not affect the syndrome and would be undetectable a All columns of H must be unique. If two columns of H were identical, errors in these two corresponding codeword positions would be indistinguishable Example = CodewordU=101110,andr=001 110 Find S=rH" S=rH 7 l 0 0 =fo01110] l 0 l i] = = fl, 141, 1+ ® docsity.com S=eH'’ =[1 0 0 0 0 ojA’ =[1 0 ol] mMm™ OFF OH OO ia a)  6.4.8 Error Correction = Arranging 2" n-tuples; representing possible received vectors, in an array is called standard array. Standard array for (n,k) code is: Uy op U; vs Uy. zero codeword ] ©2 Uz +e) U; +e wee Uy +e74 coset &3 Un + 23 U; +63 occ Un. +83 C7 U> +e; Uj; +e; ot Us. +e; et leaders Conk U2 + Conk U; + Can-k Uns + i) aa = Each row, called a coset consists of an error pattern in the first column called coset leader = If error pattern is not a coset leader, erroneous decoding will result & docsity.com  U =(101110) transmitted. Syndrome r=(001110) is received. 000000 sagncc cons gecc asses procnncna 0007 The syndrome of ris computed : 000010 S =rH’ = (001110)H* =(100) 000700 Error pattern corresponding to this syndrome is _ é = (100000) 010000 100000 The corrected vector is estimated o_o U =r+é = (001110) +(100000) = (101110) Syndrome lookup Table ® docsity.com  Decoder implementation = The received signal is multiplied with the parity check matrix: S=rH! l 0 / 0 S = [77,73 1%, 15 ¥o | ; 0 l and s, =["r + 7, + 76) s, =[r, +7, + ¥5 ] s, =[r; +r, + re] or FE Oo Fr Oo eS Fe Ore COC SF docsity.com  Implementation of the (6,3) decoder Received — r Pr; — vector r ' 2 "3 my ra a | 6 Exclusive-OR ates Syndrome S 33 g + —+ Error AND gates patterne @5 eg Received vector r rs rg Corrected us us output U docsity.com  6.8 Well-Anown Block Codes 6.8.1 Hamming Codes Simple class of block codes characterized by the structure: (n,k) = (2” -1,2” -1-m) Where m=2, 3, ...... These codes have a minimum distance of 3 and are capable of correcting single errors docsity.com  Example: Hamming Codes = Parameters of (n,k) linear block codes: a Block length: n=2™-1 a Number of message bits: k=2™-m-1 a Number of parity bits = Consider Hamming code with n=7 and k=4 (7,4) corresponding to m=3 = Generator Matrix: QD ll —_ — CG —_— ® docsity.com  Message | Code Weight of |Message | Code Weight of Word Word Code word | Word Word Code word 0000 0000000 |0 1000 1101000 |3 0001 1010001 |3 1001 0111001 |4 0010 1110010 |4 1010 0011010 |3 0011 0100011 |3 1011 1001011 | 4 0100 0110100 |3 1100 1011100 | 4 0101 1100101 |4 1101 0001101 |3 0110 1000110 |3 1110 0101110 |4 0111 0010111 |4 1111 1111111 |7 = Corresponding parity check Matrix 10 0; 1011 A7=|0 1 0] 1 1 1 £0 00 1], 011 1 —-—~ —_——~’ Lk pe ® docsity.com  7.1 CONVOLUTIONAL ENCODING = A convolutional code is described by three integers, n, k, and K where the ratio k/n is called the rate of the code = The integer K is constraint length; it represents number of k-tuple stages in the encoding shift register. =» Encoder has memory—the n-tuple emitted by the convolutional encoding procedure is not only a function of an input k-tuple, but is also a function of the previous K-7 input k-tuples ®) docsity.com  Block Diagram of a Typical Communication Link Information Convolutional Modulate source 4 encode 4 mM =m}, M2, ... , Mj, «.. U = G(m) {s;(t)} Input sequence = Uj, U2, ... , Uj, ... Codeword sequence AWGN where Uj = u4j, ... , Uji, ++. Uni channel Information Convolutional sink FT decode FP Demodulate aN M = 774, M2, ..., Mj, ... Z=2Z4, Zo, ...,Zj, ... ($:(£)} where Z; =24;, ... »Zjis os Eni and 2; is the jth demodulator output symbol of branch word Z; re 7.1: Encode/decode and modulate/demodulate portions of a communication link docsity.com  7.2 Convolutional Encoder Representation mM =m, MQ, ... , Mj; «.. 12 3... RK wee Inputsequence = —> a ~stage (shifted in & at a time) shit register n modulo-2 Mag 2M ote adders Codeword sequence U = Uj, Uo, ... , Uj, ... where U; = Uis very Ujiy ore Uni = ith codeword branch uj, = jth binary code symbol of branch word U; (Be 7.2: Convolutional encoder with constraint length K and rate k/n docsity.com  7.2.1.2 Polynomial Representation = Convolutional encoder maybe represented with a set of n-generator polynomials, one for each modulo-2 adders =» Continuing with the same example, we can write the generator polynomial for upper connections g,(X) and g,(X) for lower connections: e(X)=14X4X? ge (X)=14+X° = U(X) is the output sequence U(X)=m(X)g,(X) interlaced with m(X¥)g,(X) = Where m = 107,encoder can be found as: ® docsity.com  m(X)g(X) = (1+ X° 4+ X 4+. X°) = 14 X + XP 4X7 m(x)g,(X) = (1+ X7)\d+X°) =14+X"7 mX)g(X)=l+ X+0X7 + X° +X" m(X)g,(X) =1+0X +0X*+0X° +X" U(X) =(L) + (,0)X + (0,0).X° + (,0).X° + (LDX* U=l1!l1 10 00 +10 411 docsity.com  Encoder representation = Impulse response representaiton: a The response of encoder to a single “one” bit that goes through it. = Example: . Branch word Register =| contents uy Us 100 1 1 Inputsequence: | 0 O O10 10 Output sequence: 11 10 11 ~ oo nee eee eee -- 001 4d Input m Output l 11 10 Il 0 | 00 00 00 1 | Il 10 11 Modulo-2 sum: 11 10 00 10 II ® docsity.com