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EEE8099 - INFORMATION THEORY & CODING
创建日期：2024-09-06 15:37:44
所属分类：数学mathematics
标签： EEE8099
INFORMATION THEORY & CODING

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EEE8099 - INFORMATION THEORY & CODING

RESIT

PROBLEM-SOLVING EXERCISES

Specific instructions:

(a)  This assignment is open book, which means that you are allowed to use your lecture notes to answer questions.

(b) Be concise in your answers. What is important when answering a question is to demonstrate your understanding of a particular topic by focusing your explanations on its key aspects only.

Therefore, as a rough guideline, the length of each answer should not exceed  half  of an A4  page,  including  any  drawing,  diagram,  and equation.

(c) Please make sure that your submitted PDF file is very clear and easy to read.

(d) You can use drawing(s) and/or equation(s) in your answers, if you wish to do so. However, remember that the length of your answer for any specific question should ideally not exceed half of an A4 page, including any drawing, diagram, and equation.

Question 1

All linear block codes can be defined using a parity-check matrix. In other words, the operation of any linear block code can be described using a Tanner graph that connects a set of bit nodes to another set of check nodes.

In this question, assume transmission over a binary symmetric channel (BSC) with an error probability p.

Also assume that the bit-flipping algorithm is used to decode a particular linear block code over this BSC.

We have seen in class that the bit-flipping algorithm is going to work well provided that each check node in the Tanner graph of this linear block code is fed with, at most, one wrong bit.

(a) Consider a particular check node connected to l bit nodes c0 , c1 , c2 , … , and cl−1 .

Find the expression of the probability that this check node receives two or more wrong bits.

Simplify the equation thus obtained by assuming that p  ≪  1.     [10 marks]

(b)  By using the result obtained in (a), determine a condition that must be satisfied for successful decoding of a linear block code using the bit-flipping algorithm.    [10 marks]

Question 2

Consider the convolutional encoder depicted in Figure 1. At each time n . T, where T and n denote the encoder period and an integer, respectively, the encoder is fed with an information bit dn and generates two coded bits xn  and yn .

(a)  Draw the trellis diagram for this encoder.

Use this trellis diagram to determine the free distance, dfree ,  for this encoder.

Show  the  path(s)  in  the  trellis  that  correspond(s)  to  this  free distance.   [10 marks]

(b)  Find the expression of the  bit error probability obtained at high SNR using this convolutional code over a BPSK, AWGN channel.

What is the value of the asymptotic coding gain over uncoded BPSK?   [5 marks]

(c)  We  now use the convolutional encoder depicted in Figure  1 to design a turbo encoder, as shown in Figure 2.

The turbo encoder input is a k-bit message M. This message is encoded  into  three  k-bit  binary  words.  The  first  one   is  the message M itself.

The second binary word is the sequence P1 of parity bits produced by the first convolutional encoder. Finally, the third binary word is the   sequence   P2      of   parity   bits   produced   by   the   second convolutional encoder which is fed by an interleaved version of M.

Find the expression of the first term of the union bound for this turbo code.

To save time and space on your answer sheet, you are allowed to use some results given in the lecture notes, without having to demonstrate them again.  [10 marks]

(d)  The turbo encoder studied in (c) is designed using convolutional encoders that are non-recursive.

Based on the analysis performed in (c), would you recommend using non-recursive convolutional codes for the design of turbo codes?

Justify your answer.   [10 marks]

Question 3

Consider the linear block code defined by the generator matrix

(a)  Determine the minimum distance, dmin , between codewords for this code.   [10 marks]

(b)  Find the expression of the  bit error probability obtained at high SNR with this code. Assume transmission over a BPSK, AWGN channel and the use of maximum-likelihood (ML) decoding.

What is the value of the asymptotic coding gain over uncoded BPSK?   [5 marks]

(c)  Find the parity-check matrix H for this code.                   [5 marks]

(d)  Draw the Tanner graph for this code.                              [5 marks]

(e)  Assume transmission over a binary symmetric channel (BSC),

and that the received word is R  =  (111110). Use the bit-flipping algorithm to decode R.

Comment on your result.   [5 marks]

(f)   Perform a slight modification of the Tanner graph determined in

(d) in order to improve the error-correction capability of the code when bit-flipping decoding is used.

As an illustration, use the modified Tanner graph to decode the received word R = (111110).

Comment on your result.    [5 marks]

Question 4

We propose to design a channel code, called C1,2 , by combining two linear block codes C1  and C2 .

The parameters of C1  are k1 , n1 , and dmin, 1 , whereas the parameters of C2  are k2 , n2 , and dmin,2 , where k1   and k2   denote the message lengths, n1  and n2  designate the codeword lengths, and dmin, 1  and dmin,2  are the minimum Hamming distances between codewords.

The encoding algorithm is explained below.

Each message composed of k1 ×k2  information bits is stored in a two- dimensional (2-D) array of size k1 ×k2   bits.  This array thus  has k1 columns and k2  rows.

At the first step of encoding, each row of this array is encoded into a codeword of length n1  bits using C1 . The row encoding results in a 2- D array composed of n1  columns and k2  rows.

At the second and final step of encoding, each of the n1   columns is encoded into a codeword of length n2  using C2 . The column encoding results in a 2-D array composed of n1 columns and n2  rows. This array is the codeword associated with the k1 ×k2 -bit array present at the encoder output.

(a) The minimum Hamming distance, dmin , between codewords for

C1,2  is the  product of dmin, 1  and dmin,2 . In other words, we can write dmin  = dmin, 1  ∙ dmin,2 .

How would you demonstrate this result?   [5 marks]

(b)  How would you decode the code C1,2  in practice?

To  illustrate  your  explanations,  feel free  to  draw  the  decoder structure that you propose.     [5 marks]

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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 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