Lecture 4: Linear Codes

Home , Block code, Generator matrix, Linear code

TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Firma xy Systems Herrn Mustermann Beispielstraße 11 12345 Musterstadt Information and Communication Theory

Prof. Dr. Giuseppe Caire

Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin

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Text…& Sekretariat HFT6 Patrycja Chudzik

Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] Lecture 4: Linear Codes

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Prof. Dr. Giuseppe Caire

Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin

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Text…& = q q =2 SekretariatWe HFT6 let q for some prime power . Most important case: (binary Patrycja Chudzik F

• Telefon +49 (0)30 314-28459 X Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] codes).

Without loss of generality, we may represent the information message as a • sequence of k symbols from Fq.

We have = qk, and R = k log q bits/symbol. • |C| n 2

www.mk.tu-berlin.de k Deﬁnition 22. A (q ,n) block code over = Fq is called a linear (n, k) code if X n its codewords form a k-dimensional vector subspace of the vector space Fq . ⌃

Prof. Dr. Giuseppe Caire

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The code is an additive group, in particular, if c, c then c + c and Telefon +49 (0)30 314-29668 0 0 Subject: Telefax +49 (0)30 314-28320 [email protected] • C 2 C 2 C

Text…& Sekretariat HFT6 Patrycja Chudzikc .

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The all-zero vector is a codeword: 0 . • 2 C Linear combination of codewords are codewords: c ,...,c and • 1 ` 2 C a1,...,a` Fq, then 2 a c + a c 1 1 ··· ` ` 2 C

www.mk.tu-berlin.de There exist (non-unique) sets of k linearly independent codewords that • generate the whole code, i.e.,

k 1 = u`g` : u0,...,uk 1 Fq C ( 2 ) X`=0 where g0,...,gk 1 are codewords that form a basis for the code . C

Prof. Dr. Giuseppe Caire

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Text…& SekretariatWe HFT6 can arrange the basis as rows of a k n matrix Patrycja Chudzik

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g0,0 g0,1 g0,2 g0,n 1 ··· g1,0 g1,1 g1,2 g1,n 1 G = 2 ··· 3 . . 6 gk 1,0 gk 1,1 gk 1,2 gk 1,n 1 7 6 ··· 7 4 5 This is called a generator matrix for the code. Letting u =(u ,...,u ) we 0 k 1

www.mk.tu-berlin.de can write the encoding function as

c = uG

Prof. Dr. Giuseppe Caire

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Text…& Sekretariat HFT6 Patrycja Chudzik

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1101000 0110100 G = 2 11100103 6 10100017 6 7 4 5

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Prof. Dr. Giuseppe Caire

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Text…& Sekretariat HFT6 Patrycja Chudzik

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110 G = 101 

G = 111

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Prof. Dr. Giuseppe Caire

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Text…& SekretariatSometimes, HFT6 a desirable property of the encoding function is that the vector Patrycja Chudzik

• Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] of information symbols appears explicitly as part of the codeword.

For linear codes, a generator matrix in systematic form is given by • G =[P I ] | k k (n k)

where P ⇥ and I denotes the k k identity. Fq k www.mk.tu-berlin.de 2 ⇥ In this way, we have c = uG =[uP u]. • |

Prof. Dr. Giuseppe Caire

Berlin, 1. Month 2014 Einsteinufer 25 10587Since Berlin is a vector subspace of of dimension k, then it can be seen as the Fq Telefon +49 (0)30 314-29668 Subject: Telefax +49 (0)30 314-28320 • [email protected] C n n k

Text…& SekretariatKernel HFT6 of some linear transformation Fq Fq . Patrycja Chudzik

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The matrix of such linear transformation is called parity-check matrix and it is • T (n k) n denoted by H , where H Fq ⇥ : 2 n T = c Fq : cH = 0 C 2 n

In particular, the rows of H are n-vectors in that are orthogonal to all Fq www.mk.tu-berlin.de • codewords. Indeed, they generate the orthogonal subspace ?, also known C as the dual code of . C T If G =[P Ik], then H =[In k P ], in fact, we have • | |

T In k GH =[P I ] = P P = 0 | k P 

Prof. Dr. Giuseppe Caire

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Telefon +49 (0)30 314-29668 Subject: Telefax +49 (0)30 314-28320 [email protected] A q-ary symmetric channel can always be represented as an additive noise

Text…& Sekretariat HFT6 • Patrycja Chudzik

Telefon +49 (0)30 314-28459 Telefaxchannel +49 (0)30 314-28320 over , such that Prof. Dr. Giuseppe Caire [email protected] Fq y = c + z n where z Fq . 2 The “noise” pmf is given by • 1 for z =0

PZ(z)=

/(q 1) for z =0 www.mk.tu-berlin.de ⇢ 6

The syndrome of the error vector is given by • s = zHT

Prof. Dr. Giuseppe Caire

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Text…& SekretariatNotice HFT6 that the decoder can compute the syndrome even though it does not Patrycja Chudzik

• Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] know z, in fact,

yHT =(c + z)HT = cHT + zHT = zHT = s

Therefore, the syndrome of the error vector is an index that can be used by

www.mk.tu-berlin.de • the decoder to “undo” the bit-ﬂips.

If s = 0, then y . In this case, we let c = y. • 2 C If s = 0, then y / . In this case theb decoder knows that an error has • 6 2 C occurred (detectable error).

Copyright G. Caire 97 TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Firma xy Systems Herrn Mustermann Beispielstraße 11 12345 Musterstadt Information and Communication Theory

Prof. Dr. Giuseppe Caire

Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin

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Text…& SekretariatIn HFT6 order to correct the error, the decoder needs to ﬁnd z, an estimate of z, Patrycja Chudzik

• Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] and correct the errors as c = y z. b Unfortunately, the systemb of equationsb • s = zHT

where s is known (the syndrome) and z is unknown, is underdetermined (n

www.mk.tu-berlin.de unknowns and n k equations).

n k k For every syndrome s Fq , we have q possible error vectors. • 2 We have to solve this problem in a probabilistic sense ... for each set of qk • possible error vectors corresponding to a given syndrome, we shall pick the most likely.

Prof. Dr. Giuseppe Caire n n k T Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin y s = yH The linear map such that has Kernel Fq Fq Telefon +49 (0)30 314-29668 Subject: Telefax +49 (0)30 314-28320 • [email protected] ! 7!

Text…& Sekretariat HFT6 Patrycja Chudzik T Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Ker(H )= Prof. Dr. Giuseppe Caire [email protected] C

Linear maps are group homomorphisms (they preserve the group operation, • that in this case is componentwise addition in Fq).

n n

A coset of in is given by the translate v + for some v . Fq Fq www.mk.tu-berlin.de • C C 2 The factor group (group of cosets), with respect to the coset addition, satisﬁes • (canonical isomorphism): n n k Fq / Fq C ⌘

n k The standard array is the correspondence between the syndromes s Fq • n 2 and the cosets of in Fq , induced by this isomorphism. C

s = 0 c = 0c c k Prof. Dr. Giuseppe Caire 0 0 1 q 1

Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin ···

Telefon +49 (0)30 314-29668 Subject: Telefax +49 (0)30 314-28320 s1 v1 v1 + c1 v1 + c k [email protected] q 1

Text…& Sekretariat HFT6 ··· Patrycja Chudzik k Telefon +49 (0)30 314-28459 s v c + c v + c 2 2 2 1 2 q 1 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] ··· . . . sqn k 1 vqn k 1 vqn k 1 + c1 vqn k 1 + cqk 1 ···

n For every new row, ﬁnd the vector in Fq with minimum Hamming weight that • not yet appeared in the array.

www.mk.tu-berlin.de The corresponding row is obtained by adding this vector to all codewords. • All row are distinct, and yield the same syndrome. • These vectors of minimum weight are called coset leaders. •

Prof. Dr. Giuseppe Caire

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Text…& (3, 2) SekretariatConsider HFT6 the SPC , with parity-check matrix Patrycja Chudzik

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H =[ 111]

Exercise: build the standard array for the Hamming (7, 4) code. •

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Text…& s = yH 1. SekretariatCompute HFT6 the syndrome . Patrycja Chudzik

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2. Use the standard array and ﬁnd the most likely error vector z compatible with s (coset leader). b 3. The minimum Hamming distance decision rule is given by

c = y z

www.mk.tu-berlin.de b b

n k 4. A (n, k) linear code is able to correct q error patterns (error vectors)

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(n, k) = c : cH = 0 Text…& Consider a linear block code . Sekretariat HFT6 Patrycja Chudzik

• Telefon +49 (0)30 314-28459 C { }

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It is deﬁned by the set of parity-check equations • h x + h x + + h x =0 1,1 1 1,2 2 ··· 1,n n h x + h x + + h x =0 2,1 1 2,2 2 ··· 2,n n . .

www.mk.tu-berlin.de hn k,1x1 + hn k,2x2 + + hn k,nxn =0 ···

The Tanner graph of the code is a bipartite graph with n “bit-nodes” and n k • “check-nodes”, such that an edge (i, j) exists if hi,j =1, that is, if bit xj participate in the parity-check equation i.

Prof. Dr. Giuseppe Caire

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Text…& SekretariatParity-check HFT6 equations of the Hamming (7, 4) code: Patrycja Chudzik

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x1 + x2 + x4 + x5 =0

x1 + x3 + x4 + x6 =0

x2 + x3 + x4 + x7 =0

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+ + +

Prof. Dr. Giuseppe Caire

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TelLow-Densityefon +49 (0)30 314-29668 Parity-Check (LDPC) codes are linear binary codes with the Subject: Telefax +49 (0)30 314-28320 [email protected]

• Text…& Sekretariat HFT6 Patrycjacharacteristic Chudzik that their parity-check matrix is sparse: the number of “ones”

Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] in the matrix is proportional to the block length n.

Notice that a randomly generated binary matrix H with dimensions n(1 R) • ⇥ n has an average number of ones equal to n2(1 R)/2, i.e., quadratic with n. A regular (d ,d ) LDPC code has Tanner graph with constant left and right • ` r degrees d and d , respectively. ` r

www.mk.tu-berlin.de Example: a (3, 6) regular LDPC code of length n = 10, given by the parity- • check matrix 0111110001 1001010111 2 3 H = 0010111011 6 11101011007 6 7 6 11010011107 6 7 4 5

Fig. 2.11 shows an instance of the regular ensemble , also known as -Gallager code of length 10. We notice that all bitnodes have degree 3 and all checknodes have degree 6. The coding rate is . The corresponding parity-check matrix is

(3, 6) LDPC code with n = 10

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Text…& Sekretariat HFT6 Patrycja Chudzik 10 Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected]

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−5

−10

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Figure 2.11: Tanner graph for an instance of the Gallager code of length 10.

2.5.2 Iterative decoding for the BEC We consider transmission of a linear binary code over the BEC. Let be the parity-check matrix of the code and let be the transmitted codeword. The received codeword is , that coincides with over the positions and contains erasures in positions . The set of erasure positions is called erasure pattern. Introducing memory

Prof. Dr. Giuseppe Caire

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Text…& SekretariatA HFT6 binary linear block encoder is a linear transformation 2 2 . Patrycja Chudzik F F

• Telefon +49 (0)30 314-28459 ! Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected]

What about using a Linear time-invariant linear system for encoding? • Convolutional codes consider small k and n, but introduce memory into the • encoding process of a sequence of consecutive blocks.

We may see this a the convolution of a sequence of information blocks ui www.mk.tu-berlin.de • { } with a matrix G of impulse responses, in order to generate a sequence of coded blocks c . { i}

Copyright G. Caire 107 “MA” k n linear systems ⇥ TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Firma xy Systems Herrn Mustermann Beispielstraße 11 12345 Musterstadt Information and Communication Theory

Prof. Dr. Giuseppe Caire

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Text…& k n SekretariatA HFT6 Moving Average (MA) system is deﬁned by: Patrycja Chudzik

• Telefon +49 (0)30 314-28459⇥ Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected]

m1,1 mk,1 (1) (1,1) (1) (k,1) (k) ci = g` ui ` + + g` ui ` ··· X`=0 X`=0 m1,2 mk,2 (2) (1,2) (1) (k,2) (k) ci = g` ui ` + + g` ui ` ··· `=0 `=0

X X www.mk.tu-berlin.de . m1,n mk,n (n) (1,n) (1) (k,n) (k) ci = g` ui ` + + g` ui ` ··· X`=0 X`=0

Copyright G. Caire 108 TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Firma xy Systems Herrn Mustermann Beispielstraße 11 12345 Musterstadt Information and Communication Theory

Prof. Dr. Giuseppe Caire (1) (n) Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin Deﬁning a vector output sequence c , c , c ,...such that c =(c ,...,c ) 0 1 2 i Telefon +49 (0)30 314-29668 i i Subject: Telefax +49 (0)30 314-28320 [email protected] • (1) (k)

Text…& Sekretariat HFT6 Patrycjaand Chudzik a vector input sequence u , u , u ,...such that u =(u ,...,u ),we 0 1 2 i i i Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] can write m

ci = ui `G` `=0 where we let m = max m . X { (i,j)} We obtain a block-Toeplitz notation •

www.mk.tu-berlin.de G G G 0 0 1 ··· m ··· 0 G0 Gm 1 Gm 2 . . . . 3 (c0, c1, c2, c3,...)=(u0, u1, u2, u3,...) . 0 .. . Gm 1 .. 6 G . 7 6 0 7 6 0 G ... 7 6 0 7 4 5

The impulse responses g(i,j) are called the code generators. •

Prof. Dr. Giuseppe Caire

Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin

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Text…& • Sekretariat HFT6 Patrycja Chudzik

Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] i u u(D)= u D (Laurent series) i ! i i X

Convolutional encoding in the D-transform domain: •

c(D)=u(D)G(D)

www.mk.tu-berlin.de or, equivalently,

g (D) g (D) g (D) 1,1 1,2 ··· 1,n g2,1(D) g2,2(D) g2,n(D) (c1(D),...,cn(D)) = (u1(D),...,uk(D)) 2 ··· 3 . . 6 g (D) g (D) g (D) 7 6 k,1 k,2 ··· k,n 7 4 5

Prof. Dr. Giuseppe Caire

Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin

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Text…& Sekretariat HFT6 Patrycja Chudzik

Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] (1) + ci = ui + ui 2 ui

(2) + ci = ui + ui 1 + ui 2

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Prof. Dr. Giuseppe Caire

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Text…& SekretariatA HFT6 code is deﬁned as the set of all output sequences (code sequences). Patrycja Chudzik

• Telefon +49 (0)30 314-28459 C Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected]

As for block codes, a convolutional code may have several input-ouput • C encoder implementations.

We seek encoders in canonical form: a general problem in system theory is • for a given system, deﬁned as the ensemble of all its output sequences, what

is the minimal canonical realization?

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State-space representation (ABCD): •

si+1 = siA + uiB, ci = siC + uiD

a minimal representation is a representation with the minimum number of state variables.

Prof. Dr. Giuseppe Caire

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Text…& SekretariatIn HFT6 the (2, 1) example of before, the state is deﬁned as the content of the Patrycja Chudzik

• Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] memory elements, si =(ui 1,ui 2) therefore we have

01 s = s + u 10 i+1 i 00 i  ⇥ ⇤

www.mk.tu-berlin.de and 01 c = s + u 11 i i 11 i  ⇥ ⇤

Prof. Dr. Giuseppe Caire

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Text…& SekretariatA HFT6 convolutional code can be seen as a block code deﬁned on the ﬁeld q(D) Patrycja Chudzik F

• Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] of rational functions over Fq. Roughly speaking: rational functions are to polynomials as rationals Q to the • integers Z. Generalizing what seen before, we can consider G(D) with rational elements • gi,j(D).

In system theory, this corresponds to AR-MA linear systems. www.mk.tu-berlin.de • The code is preserved by elementary row operations. • It follows that for any G(D), we can ﬁnd a systematic generator matrix in the • form G(D)=[I P(D)] | where I is the k k identity, and P(D) is a k (n k) matrix of rational ⇥ ⇥ functions.

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Text…& SekretariatA HFT6 state-space realization with m binary state variables is a ﬁnite-state Patrycja Chudzik

• Telefon +49 (0)30 314-28459 m Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] machine (FSM) with a state space ⌃ = F2 .

In general, a FSM is described by its state transition diagram, i.e., by a • graph with ⌃ vertices, corresponding to all possible state conﬁgurations, | | and edges connecting those states for which a transition is possible.

Each edge (s, s0) ⌃ ⌃ is labeled by input and output vectors b and F2 www.mk.tu-berlin.de • n 2 ⇥ 2 c F2 , corresponding to the state transition between s and s0. 2

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Text…& Sekretariat HFT6 Patrycja Chudzik

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1/11 (1, 0) 1/10

0/00 0/01

(0, 0) (1, 1) www.mk.tu-berlin.de 1/00 1/01

0/11 0/10 (0, 1)

Prof. Dr. Giuseppe Caire

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Telefon +49 (0)30 314-29668 Subject: Telefax +49 (0)30 314-28320 [email protected] An alternative representation consists of the trellis section, i.e., by a bipartite

Text…& Sekretariat HFT6 • Patrycja Chudzik

Telefon +49 (0)30 314-28459 Telefaxgraph +49 (0)30 314-28320 with ⌃ state vertices on the left and ⌃ state vertices on the right. Prof. Dr. Giuseppe Caire [email protected] | | | | Left vertices represent the possible states at time i, and right vertices • represent the possible states at time i +1. Edges represent the possible state transitions corresponding to input ui and output ci.

The trellis representation follows from the state transition diagram by

• www.mk.tu-berlin.de introducing the time axis.

A trellis diagram for a convolutional code consists of the concatenation of an • inﬁnite number of trellis sections.

Given an initial state at time i =0, an input sequence u(D) determines an • output sequence c(D) and a state sequence s(D) that correspond to a path in the trellis.

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Text…& Sekretariat HFT6 Patrycja Chudzik

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(0, 0) 0/00 (0, 0) 1/11

0/11 (1, 0) (1, 0) 1/00

0/01

www.mk.tu-berlin.de 1/10 (0, 1) (0, 1)

0/10 (1, 1) (1, 1) 1/01

Prof. Dr. Giuseppe Caire

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Text…& SekretariatThree HFT6 types of variable nodes: information bitnodes ui, coded bitnodes ci Patrycja Chudzik

• Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected] and states nodes si.

The function nodes correspond to the state and output mappings •

si+1 = siA + uiB, ci = siC + uiD

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Copyright G. Caire 119 TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Firma xy Systems Herrn Mustermann Beispielstraße 11 12345 Musterstadt Information and Communication Theory

Prof. Dr. Giuseppe Caire

Berlin, 1. Month 2014 Einsteinufer 25 10587 Berlin

Telefon +49 (0)30 314-29668 Subject: Telefax +49 (0)30 314-28320 [email protected]

Text…& Sekretariat HFT6 Patrycja Chudzik

Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 Prof. Dr. Giuseppe Caire [email protected]

End of Lecture 4