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1 This document is downloaded from CityU Institutional Repository, Run Run Shaw Library, City University of Hong Kong. Title Channel polarization: A method for constructing capacity-achieving codes Author(s) Chuai, Jie ( 揣捷 ) Citation Chuai, J. (2011). Channel Polarization: A method for constructing capacity achieving codes (Outstanding Academic Papers by Students (OAPS)). Retrieved from City University of Hong Kong, CityU Institutional Repository. Issue Date 2011 URL Rights This work is protected by copyright. Reproduction or distribution of the work in any format is prohibited without written permission of the copyright owner. Access is unrestricted.

2 Department of Electronic Engineering FINAL YEAR PROJECT REPOR RT BEECE-2010/11-PL-05 Channel Polarization: A Method forr Constructing Capacity-Achieving Codes Student Name: CHUAI, Jie Student ID: Supervisor: : Prof. LI, Ping Assessor: Dr. DAI, Lin Bachelor of Engineering (Honours) in Electronic and Communication Engineering (Full-time)

3 Student Final Year Project Declaration I have read the student handbook and I understand the meaning of academic dishonesty, in particular plagiarism and collusion. I declare that the work submitted for the final year project does not involve academic dishonesty. I give permission for my final year project work to be electronically scanned and if found to involve academic dishonesty, I am aware of the consequences as stated in the Student Handbook. Project Title: Channel Polarization: A Method for Constructing Capacity-Achieving Codes Student Name: CHUAI Jie Student ID: Signature: Date: 10 April, 2011 i

4 No part of this report may be reproduced, stored in a retrieval system, or transcribed in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of City University of Hong Kong. ii

5 Contents LIST OF FIGURES... v LIST OF TABLES... vii ABSTRACT... viii 1. INTRODUCTION... 1 a) Objectives of the Project... 1 b) Scope of the Report MOTIVATION... 2 a) A Milestone in Coding ory... 2 b) Project Requirements THEORY... 3 a) Basic Concepts in Coding ory... 3 b) Channel Coding orem... 5 c) ory of Polar Codes... 6 i. Preliminary Definitions... 6 ii. Channel Construction Procedures... 8 iii. Equivalent Channel at the Input iv. Channel Polarization Effect v. Polar Coding by Channel Selection vi. Decoding Rule vii. Analysis of the Channel Polarization Phenomenon iii

6 4. IMPLEMENTATION a) Coding Algorithm b) Decoding Algorithm c) Evaluation of the Z Value RESULTS a) Set of Parameters i. BEC ii. BSC iii. AWGN DISCUSSIONS CONCLUSION REFERENCES iv

7 LIST OF FIGURES Figure 1 A Simple Model of the Communication System...4 Figure 2 Channel Coding orem [4]...5 Figure 3 N uses of the channel W...6 Figure 4 Channel Models Used in Simulation...7 Figure 5 Polar Code Construction with N=2 [1]...9 Figure 6 Polar Code Construction with N=4 [1]...9 Figure 7 Polar Code Construction with length N [1]...10 Figure 8 Equivalent Channel at the Input...11 Figure 9 Channel Polarization Effect for BEC...12 Figure 10 orem on Channel Polarization [1]...13 Figure 11 Effect of Block Length on Polarization...13 Figure 13 Equivalent Channels (N=2)...15 Figure 14 Channel Construction with N=4 (Step 1)...17 Figure 15 Channel Construction with N=4 (Step 2)...18 Figure 16 Flowchart of the Simulation Program...19 Figure 17 Coding Implementation Algorithm...20 Figure 18 Fundamental Structure in Decoding Process...20 Figure 19 Decoding Procedure (1)...22 Figure 20 Decoding Procedure (2)...23 Figure 21 Decoding Procedure (3)...24 Figure 22 Decoding Procedure (4) Figure 23 Decoding Procedure (5)...26 Figure 24 Decoding Procedure (6)...27 Figure 25 Flowchart for finding the Z value...30 Figure 26 BER vs. Erasure Probability (BEC, Code Rate=0.5)...34 Figure 27 BER vs. Code Rate (BEC, Bit Erasure Probability=0.5) 35 Figure 28 BER vs. Block Length (BEC, Bit Erasure Probability=0.5)..35 v

8 Figure 29 BER vs. Bit Error Probability of Channel (BSC, Code Rate=0.25).36 Figure 30 BER vs. Eb/N0 (AWGN, Code Rate=0.5).36 Figure 31 Block Error Rate vs. Channel Error (BEC)..38 Figure 32 Block Error Rate vs. Code Rate (BEC) 40 Figure 33 Block Error Rate vs. Block Length (BEC) 40 Figure 34 Block Error Rate vs. Channel Error (BSC)..41 Figure 35 Block Error Rate vs. Eb/N0 (AWGN) 43 vi

9 LIST OF TABLES Table 1 Simulation Parameters (BEC, Code Rate=0.5, Various ε).31 Table 2 Simulation Parameters (BEC, Bit Erasure Probability=0.5, Various Rates)...32 Table 3 Simulation Parameter (BEC, Bit Error Probability=0.5, Various N)..32 Table 4 Simulation Parameters (BSC, Code Rate=0.5, Various ε ) 33 Table 5 Simulation Parameter (AWGN, Code Rate=0.5, Various sigma) 33 vii

10 ABSTRACT According to Shannon's orem, the possible code rate of a noisy channel is limited by the channel capacity; and there exist coding methods that could achieve the capacity of noisy channels. A method, called channel polarization (or polar coding), proposed by Erdal Arikan, was the first one mathematically proved to be capacity-achieving for symmetric binary-input discrete memoryless channels (B-DMC). In this project, the algorithm of this polar coding method was studied, and a simulation package was produced to evaluate its performance. Channels including Binary Erasure Channel (BEC), Binary Symmetric Channel (BSC) and Additive White Gaussian Noise channel (AWGN) were assumed in the simulation. As shown by the simulation results, for a give coding block length N=2 n the polar coding method re-allocated the capacities over the channels, so that the equivalent channel capacities at different bit locations polarized as the coding block length becomes large; and the block error rate obtained in the simulation agreed with the bound given by Erdal Arikan in his paper. se results supported the capacity-achieving performance of the polar coding method. viii

11 1. INTRODUCTION a) Objectives of the Project This final year project involved an investigation into the polar coding method proposed by Erdal Arikan in his paper [1]. main objectives of this project were: To study the working principles of the polar coding method; To evaluate the performance of the polar codes under specified channel conditions. Based on these two above objectives, the working progress of this final year project was divided into the following three phases: i. Study the paper [1] by Erdal Arikan and other relevant materials. ii. iii. Implement a simulation package for the polar coding method. Test the simulation program with different channel specifications and evaluate the coding performance. b) Scope of the Report This report outlines the motivation of the project, the theoretical analysis of the polar coding method, the implementation algorithms and analyses of the simulation results. 1

12 2. MOTIVATION a) A Milestone in Coding ory In 1948, Shannon proved the existence of coding schemes that could achieve the capacity of noisy channels in his influential paper [2]; however, no specific coding sequence was shown explicitly. Thus, it has been a goal for communication engineers in the past decades to construct a coding scheme that could reach the limit stated in Shannon s paper. Erdal Arikan introduced a phenomenon called channel polarization in his paper [1] in 2008, and the coding scheme based on this phenomenon was called polar coding. This code construction could achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels (B-DMC). This is the first coding scheme that is provable to be capacity-achieving thus fills the theoretical gap in coding theory [3]. At the same time, this code construction algorithm has low complexities in encoder and decoder implementations. refore, it is worthy of investigating this family of codes. b) Project Requirements project requires a verification of the polar codes performance by comparing the simulation results with the theoretical values, and an evaluation on its noise-resistance capabilities based on the simulation. 2

13 Simulation results in the following types of channels were required: i. Binary Erasure Channel (BEC) ii. iii. Binary Symmetric Channel (BSC) Additive White Gaussian Noise (AWGN) Channel following curves were needed for the evaluation purpose: i. Bit Error Rate (BER) vs. Channel Noise Level ii. iii. BER vs. Code Rate BER vs. Coding Block Length block error rate should also be obtained at the same time for comparing with the theoretical values in paper [1]. 3. THEORY a) Basic Concepts in Coding ory Coding theory studies the properties of different codes and their applications in various fields. As a branch of coding theory, channel coding aims at improving the noise-resistant capability of the transmitted information. By introducing extra bits and correlations between bits, the communication system thus has the error-detection and correction capability. 3

14 A simple model of the communicationn system is shown in the followingg figure: Figure 1 A Simple Model of the Communica tion Systemm As shown in Figure 1, information too be transmitted is divided into blocks with length K. encoder then transforms the K-bit information into N-bit codeword (N> >K). modulatedd signal experiences the noise in the channel and finally reached the receiving end. Decoding scheme s is applied to obtain the K-bit K information from the demodulated N-bit codeword. code rate is defined ass R K N (1) where N is the code block length andd K is the number of information per block. This parameter measures the bandwidth efficiency of the coding scheme. noise level in the channel influences the Signal-to-Noise Ratioo (SNR) off the system. Apparently, a higher tolerable noise level is more desirable d since it increases the power efficiency. 4

15 maximum information transmission rate (unit: bits perr use) through a channel is limited by its capacity. Forr a discrete memoryless channel, the t capacityy is defined as max I X; Y C p x (2) where X is the input alphabet of the channel and Y is the output o alphabet. p(x) iss the distribution of the input random variable [4]. capacity of a channel representss the number of information (in bits) thatt could be effectively transmittedd per use off the channel. b) Channel Coding orem Figure 2 Channel Coding orem [4] theorem, stated and proved byy Shannon in his paper in 1948, indicates that arbitrarily small BER could be achieved with a code rate R lowerr than C and a sufficiently long block length. Whenn N tends to infinity, the maximumm code rate for reliable transmission tendss to the capacity of the channel. 5

16 i. c) ory of Polar Codes Preliminary Definitions B-DMC system B-DMC refers to Binary-input Discrete Memoryless Channel. We usee W to denote a B-DMC in this paper, and its input alphabet is X and output o alphabet is Y. transition probabilities is W(y x) where x and y are elements of X and Y. Since we are discussing binary-input channels in this paper, the input alphabet X only has two elements {0, 1} }. When transmitting N bits through thee channel W, it is said to t be N usess of the channel W. This is equivalent to use N channels to send one bit each e at the same time.. We denote this group of N channels as W N ; therefore, the system could bee written as W N : X N Y N, with a transition probability W N (y x ). Figure 3 N usess of the channel W Channels discussed in thiss paper Three types of channels are considered in this report. 6 y are Binary Erasure

17 Channel (BEC), Binary Symmetric Channel (BSC) and Additive White Gaussian Noise (AWGN) Channel. following figure shows the graphic model of these three t types. (a) BEC Model (b) BSC B Modell (c) AWGN Model Figure 4 Channel Models Used in Simulation Symmetric Capacity of W symmetric capacity off B-DMC W is definedd as [1] I W 1 2 W y x W y x log 1 1 W y W y 1 (3) 7

18 This parameter measures the capacity of the channel W when using the two inputs 0 and 1 with the same probability. For the symmetric channels discussed in this paper, the symmetric capacity I (W) is equal to the capacity C of W as defined in equation (2). Bhattacharyya Parameter parameter is defined as Z W W y 0 W y 1 (4) parameter will be called Z parameter in the rest of the report. This Z parameter is the upper bound of the error probability when transmitting 0 or 1 through W with the maximum-likelihood (ML) decision rule [1]. It should be expected that when I (W) tends to one, Z (W) will tend to zero; Z (W) approaches one when I (W) approaches to zero. Z parameter is used instead of I (W) in the polar coding method, since it fits better with the decoding algorithm of polar codes as will be seen in this report. ii. Channel Construction Procedures polar coding method synthesizes the group of N channels W N into a new channel W N with an input vector [u 1 u 2 u N ] (denoted as u ) and an output vector [y 1 y 2 y N ] (denoted as y ). construction is an iterative process. following are two 8

19 examples of the code construction. Example 1: N= =2 Figure 5 Polar Code Construction n with N=2 [1] Example 2: N= =4 Figure 6 Polar Code Construction n with N=4 [1] Generally, a polar code sequence withh a block length N is constructed as follows: 9

20 Figure 7 Polar Code Construction with w length N [1] As could be seen from the above figure, the channel W N iss synthesized by using two identical channels W N/2. input vector u first performss mod-2 addition in groups of two to obtain the vector s ; a permutation operator R N is then used to sort all the odd-indexed elements in s in ascending order as the vector v /, and the even-indexed elements as the vector v ; the two vectorss are then inputted into the two channels W N/2 to get the t finally output y. block length N of polar codes is always 2 n, which could be seen directly from the construction figure. iii. Equivalent Channel at the Input equivalent channel seen by the input u i is defined to have h an output of y and u. channel is denoted as W. 10

21 u i W y,u Figure 8 Equivalent Channel at the Input refore, the transition probability of this equivalent channel is W y,u u 1 2 W yy u (5) definition of the equivalent channel is closely related too the decoding rule of polar codes, which will be explained in section vi. iv. Channel Polarization Effect symmetric capacity equation (5) and (3). An of the equivalent channel couldd be calculated based on efficient algorithm for finding the symmetric capacities of BECs was given in Arikan s paper [1]. following figuree shows a plot of the result. 11

22 Figure 9 Channel Polarization Effect for BEC figure plots the Symmetric Capacity vs. the Channel Index for BEC with a bit erasure probability ε block length is N=1024. Itt could be clearly seen that symmetric capacities of the equivalent channels polarized; and channels with higher indices are more probable to have I W close to 1; while w channels with lower indices are more probable to havee capacities close to zero. However, it is not necessarily the case that a higher-indexed channel has a higher capacity, as could be seen from the figure. Effect of Block Length N on Channel Polarization theorem related to channel polarization phenomenon in Arikan s paper is shown in the following: 12

23 Figure 10 orem on Channel Polarization [1] That is, when N tends to infinity through powers of two, capacitiess of part off the channels will tend to one; ; while the rest tend to have capacities of zero. number of the full-capacity-approaching channels is equal to N*I (W). Figure 11 Effect off Block Length on Polarization above figure is a demonstration of block length s effect on channel polarization. BEC with ε 0.4 is used as the channel condition. T horizontal axis is the threshold capacity; while the verticall axis showss the percentage of channels that have symmetric capacities under the threshold for various block b length. It could be expected from the figure that, t when N is infinity, half of the channels have capacities 13

24 equal to one; while the rest have capacities equal to zero. Thus the capacity of this BEC (i.e., C 1 ε 0.6) could be achieved. v. Polar Coding by Channel Selection Composition of the Input Vector In polar coding, the input vector u to the channel consists of two parts: u and u. u is the sub-vector used to convey the information bits; while u is called frozen vector of which the bits are known and will be used in the decoding stage. subscript A refers to the set of which its elements are the indices of channels selected for transmitting u. Channel Selection In the coding stage, the Z parameter Z W for each channel is first calculated. Given the code rate R, a number of N R channels with the lowest Z W are selected for information transmission, and their indices are the elements of set A. 14

25 Figure 12 Polar Coding by Channel Selection (Red dots representt channels with lower Z values) vi. Decoding Rule Polar code uses a successive cancellation (SC) decoding algorithm, which usess the output vector y and the decoded input u to estimate the current bit u i. following decoding rule applies: if i A (cal if i A, cal led frozen bits), u u ; culate the following Likelihood Ratio (LR) W LR u y,u u 0 W y,u u 1 (6) 0 if LR u u 1 1 if LR u 1 transition probabilities in equation (6) use the estimated value u instead of 15

26 the real input u in equation (5). This does not affect the polarization behavior of polar codes, since u tends to coincide with u when N tends to infinity. vii. Analysis of the Channel Polarization Phenomenon reason for the channel polarization phenomenon comes c fromm the iterative construction of the channel W N. e two-channel structure in Figuree 5 is actually a capacity-reallocation structure. By using it repeatedly, capacities are e removed from f some of the channels and added to the rest continuously, resulting in the polarization phenomenon. Example: N=2 equivalent channels seen at the input when N=2 is shown in the following figure, and a lighter color represents a lowerr capacity. Figure 13 Equivalent Channels (N=2) (W1 and W2 in this figure are equivalent to W and W ) It can be proved that the symmetric s capacities of the two channels have the following relations: I W1 I W2 2I W 16

27 I W1 I W2 (7) the equality is valid when I (W) is zero or one. above relation holds as longg as two transformed in the same way in Figure 5 [1]. identical binary-inputb t channels are Example: N=4 By using the above analysis, the channel construction withh N=4 is first transformed into the following form: Figure 14 Channel Construction with N=4 (Step 1) capacity re-allocation is then performed the second time, and the resulted capacity distribution is shown in the following figure. 17

28 y, v &v y, v, v,u Figure 15 Channel Construction with N= =4 (Step 2) As shown in Figure 6, v u u and v u. Thus thee output of W3 and W4 are the same as y,u and y,u, whichh agrees with our previous definition in iii. above two examples are used to facilitate understandin ng the channel polarization phenomenon as stated in the theorem in iv. 4. IMPLEMENTAT TION simulation package of the polar coding is implementedd by using MATLAB. following figure shows the flowchart of the whole program.. 18

29 Figure 16 Flowchart of the Simulation Program P a) Coding Algorithm iterative constructionn of the polar code shown in Figure 7 could be transformed into an algorithm with a computational complexity of O(NlogN) [1].. following figure shows the coding procedure with N=8. node in the t above figure represents the mod-2 addition. It could be noticed that the output vector of the encoder x is arranged in a bit-reversal order. 19

30 Figure 17 Codingg Implementation Algorithm (Note: the x vector is in bit-reversal order) b) Decoding Algorithm decoding process is just the reverse of the above coding algorithm. By knowing the corrupted version of x, the decoding process tries too restore the input u. As can be seen from the above, the figure is an iterative combination of the following fundamental structure: Figure 18 Fundamental Structure in Decoding Process decoding rule of the above structure is as following: 20

31 LR a LR b LR b 1 LR b LR b (8) LR b LR b if a 0 LR a LR b if a LR b 1 proof of equation (8) is shown in paper [1]. refore, again the decoding procedure of the polar codes becomes an iterative process. following shows a decoding example for N=8 with a code rate R=0.25. Example: Decoding Procedures for N=8 and R=0.25 channels used for transmitting information bits are Channel 7 and Channel 8. program proceeds decoding by estimating every input bit one by one from u to u according to the rule in equation (6). Since the first six inputs are frozen, the program gets their estimated values without calculating any LR value. For u 7, its LR needs to be obtained (Figure 19). 21

32 Figure 19 Decoding Procedure (1) Notes on the Use of Symbols: A circle is used to represent the estimated binary valuee (or hard information); while the triangle is used to representt the LR value (or soft information); An empty shape means the information, either softt or hard, needs to be calculated. A solid shape means the information is already obtained. According to equation (8), the LR values for s 7 and s 8 should be found.. To find LR(s 7 ), we should know LR(v 5 ), LR(v 7 ) and s ; similarly, three values needs to be knownn for 22

33 calculating LR(s 8 ) (Figure 20). Figure 20 Decoding Procedure (2) By continuously using rule in equation (8), all the required values could be labeled (Figure 21). 23

34 Figure 21 Decoding Procedure (3) LR value at the last column couldd be found by LR x W y 0 W y 1 where y is the corrupted output of x through the channell W. binary value of v should s be confirmed by using thee estimatedd inputs already obtained (i.e., u ). Thus, the t figure changes to the followingg (Figure 22): 24

35 Figure 22 Decoding Procedure (4) LR values from LR(v 5 ) to LR(vv 8 ) are then calculated. s and s s are found by using u and u (Figure 23) 25

36 Figure 23 Decoding Procedure (5) LR( (s 7 ) and LR(s 8 ) are then computedd using equation (8). After A that, LR(u 7 ) could be found. Thus, the hard information of u could be decided. Finally, LR(u 8 ) and u could be confirmed. 26

37 Figure 24 Decoding Procedure (6) As shown in the above figure, the hard information (i.e., the t estimated values off the inputs) flows from the leftt to the right to help relieve the influence of noise; while the soft information (i.e., the LR value) flows from right to decision could be made. the left soo that the final A Practical Issue A practical issue encountered when implementing the decoder is aboutt the calculation of LRs. oretically, LR has the opportunity to increase to infinityy or decrease to zero. However, these values are not suitable for numerical calculations of the computer. refore, an upper bound and a lower bound are set in the program to avoid the existence of infinity and zero. upper boundd is set to be and the 27

38 lower bound in the simulation since MATLAB has a double precision. c) Evaluation of the Z Value BEC For BECs, the Z parameter could be found using the recursive formulae [1]: Z W 2Z W Z W / Z W Z W / (9) BSC and AWGN No efficient algorithm is found to calculate the Z parameter for BSCs and AWGN channels in paper [1]. However, it was proved that Z W E W Y,U U 1 W Y,U U where E( ) is the expectation operator; Y and U are the random variables representing the output vector and the i th input respectively [1]. Thus a Monte Carlo approach was used to estimate the Z value of each equivalent channel. value under the square root operator is just the output value of the SC decoder by setting a code rate of zero. By sampling combinations of inputs and outputs repeatedly, the value of Z W is calculated again and again until it is considered to be stabilized. (10) 28

39 criteria used to decide whether the sampling process could be terminated are: Minimum samples. Required consecutive times of tolerable difference. flowchart of the Z estimation process is shown in the following: 29

40 Figure 25 Flowchart for finding the Z value 30

41 5. RESULTS a) Set of Parameters following tables show the number of trials for each set of channel condition parameter in the simulation. BEC Table 1 Simulation Parameters (BEC, Code Rate=0.5, Various ε, N=2 8, 2 10, 2 12 ) N= N= N= N= N= N= N=

42 Table 2 Simulation Parameters (BEC, Bit Erasure Probability=0.5, Various Rates, N=2 8, 2 10, 2 12 ) Rate N= N= N= Rate N= N= N= Table 3 Simulation Parameter (BEC, Bit Error Probability=0.5, Various N, Rate=0.25, 0.4) Block Length Rate= Rate= Block Length Rate= Rate= Block Length Rate= Rate=

43 BSC Table 4 Simulation Parameters (BSC, Code Rate=0.5, N=2 7, 2 8, 2 10, 2 12, Various ε ) N= N= N= N= N= N= N= N= N= AWGN Table 5 Simulation Parameter (AWGN, Code Rate=0.5, N=2 8, 2 10, 2 12, Various sigma) Sigma N= N= N= Sigma N= N= N=

44 i. b) Simulation Results BEC BER vs. Bit Erasure Probability (Code Rate=0.5) Figure 26 BER vs. Erasure Probability (BEC, Code Rate=0.5) 34

45 BER vs. Code Rate (Bit Erasure Probability=0.5) Figure 27 BER vs. Code Rate (BEC, Bit Erasure Probability= =0.5) BER vs. Block Length (Code Rate=0.5, Bit Erasure Probability=0.5) Figure 28 BER vs. Block Length (BEC, Bit Erasuree Probability=0.5) 35

46 ii. BSC BER vs. Bit Error Probability of Channel (Code Rate=0.25) ) Figure 29 BER vs. Bit Error Probability of Channel (BSC, Code Rate=0.25) iii. AWGN Figure 30 BER vs. Eb/N0 (AWGN, Code Rate=0.5) R 36

47 6. DISCUSSIONS block error rate of the t simulation results were compared with the theoretical values. theoretical upper bound for the block error rate is obtained by summing up all the Z values of the information i n channels, i.e., Upper Bound Z W (11)( lower bound is obtained by Lower Bound max Z W (12)( comparison results are shown in the following: BEC Block Error Rate vs. Bit Erasure Probability 37

48 Figure 31 Block Error Rate vs. Channel Error (BEC) 38

49 Block Error Rate vs. Code Rate 39

50 Figure 32 Block Error Rate vs. Code Rate (BEC) Block Error Rate vs. Block Length Figure 333 Block Error Rate vs. Block Length (BEC) 40

51 BSC Figure 34 Block Error Rate vs. Channel Error (BSC) 41

52 AWGN 42

53 Figure 35 Block Error Rate vs. Eb/N0 (AWGN) Lower Bound One phenomenon was found that, when N is short and the code rate iss low, the block error rate will be smaller than t the theoretical lower bound (e.g.,( Figuree 32 (1), Figure 33, Figure 34). This is because the cases where LR(u i ) =1 are treated as errors in the calculation of the Z value and lower bound (equation 12). However, in practice, u i is estimated to be 0 in these cases. refore, the estimationss are correctt for half of the time, resulting in a decrease in the BER. For example, whenn N=2 and R=0.25 in BEC (i.e., only one bit is the informationi n), both of the upper bound b and lower bound is This could be verified by trying all of the possible inputs and outputs, and finding out that 6.25% of the decodedd information bits havee an LR value of 1 with all of the rest being zero. Ass N goes longer, the influence due to these cases becomes little, and the lower bound is valid. 43

54 Upper Bound theoretical upper bound given in [1] sometimes exceeds 1, which is apparently not possible. Other methods have been developed by researchers to give a more accurate estimation of the upper bound. An example for estimating a tighter upper bound could be found in paper [5]. Inaccuracy in Simulation In Figure 35 (2), the simulated result at 3.74dB is higher than the upper bound, which may due to the variance of the small sampling size. 7. CONCLUSION Polar coding is the first coding scheme that could be mathematically proved to be capacity-achieving. project aimed at investigating the working principles of the polar coding method, and evaluated its performance by implementing a simulation package. report presented the motivation and requirement of this project, and theories related to the polar coding method. Implementation algorithms of the simulation package were included. simulation results were then demonstrated and compared with the theoretical results. Difference between the simulation results and the theoretical values were finally discussed. simulation results agreed with the theoretical values, and supported the capacity-achieving performance of polar coding. 44

55 8. REFERENCES [1] Arikan, E., "Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels," Information ory, IEEE Transactions on, vol.55, no.7, pp , July [2] Shannon, C. E., A Mathematical ory of Communications, Bell Syst. Tech. J., vol. 27, pp , , Jul-Oct, [3] Performance of Short Polar Codes under ML Decoding [4] Cover, T. M. and Thomas, J. A., Elements of Information ory, Second Edition, Chapter 7, pp [5] Mori, R.; Tanaka, T.;, "Performance and Construction of Polar Codes on Symmetric Binary-input Memoryless Channels," Information ory, ISIT IEEE International Symposium on, vol., no., pp , June July