The document is the outline for Hakim Alhussien's PhD final oral examination on April 6, 2009. It discusses channel matched iterative decoding for magnetic recording systems. The outline covers perpendicular magnetic recording channels, error correcting codes for recording channels, error pattern correction coding and its enhancements, tensor product parity codes, and EPC-LDPC tensor product codes. It also discusses Hakim's thesis contributions.
The document discusses analyzing and reducing errors in DNA computing algorithms. It focuses on a sensitivity analysis of Gloor's algorithm for the shortest common superstring problem. Key steps include encoding input strings, generating all possible solutions, and iteratively matching strings. The analysis finds that encoding and extraction steps are most sensitive to errors. Various techniques are proposed to make the algorithm more error-resistant, such as carefully designing the encoding to avoid mismatches, and using multiplexing to tolerate a certain number of erroneous operations. The goal is to tune the algorithm and underlying biological operations to be robust against real-world error rates in DNA computing.
Advances in coding for the fading channelwtyru1989
The document discusses coding techniques for fading channels. It notes that most coding wisdom is based on high SNR Gaussian channels, but fading channels have different properties like low and variable SNR and non-Gaussian statistics. For fading channels, the optimal coding criteria depends on the channel model. Techniques like trellis coded modulation (TCM) and bit-interleaved coded modulation (BICM) can provide more robust performance over fading channels compared to codes optimized for Gaussian channels. BICM in particular separates demodulation and decoding, improving performance when the exact channel model is unknown. The document also discusses coding for block fading channels, where a codeword experiences a limited number of independent fading gains rather than full interleaving.
This document discusses error control coding, also known as channel coding. It begins by introducing concepts such as bit error rate and methods to reduce it, including increasing transmission power, diversity techniques, automatic repeat request (ARQ), and forward error correction (FEC) codes. It then provides details on parity checks, cyclic redundancy checks (CRCs), and block error correction codes. Specific error control coding techniques like Reed-Solomon codes are also mentioned.
Nyquist criterion for distortion less baseband binary channelPriyangaKR1
binary transmission system
From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]
The document summarizes Lecture 7 which covered:
1) A review of Lecture 6 on PCM waveforms and the remaining portion of Chapter 2 on spectral densities of PCM waveforms and multi-level signaling.
2) An overview of Chapter 3 on baseband demodulation/detection including matched filters, correlators, Bayes' decision criterion, and maximum likelihood detection.
3) Key aspects of line codes including how pulse shaping can control the signal spectrum and ensure symbol transitions, comparisons of line codes based on power spectral density, DC component, and bandwidth.
Locally decodable codes allow recovery of individual data symbols even after data loss by accessing only a small number of codeword symbols. Reed-Muller codes provide locality but only up to a rate of 0.5, while multiplicity codes achieve higher rates but have weaker locality guarantees. Matching vector codes can match the best known locality bounds, constructing codes of length n with locality r for constant r, but the optimal tradeoff between rate, length and locality remains an open problem.
The document discusses locally decodable codes, which allow recovery of individual data symbols from a coded data set even after erasures. Reed-Muller codes and multiplicity codes were early constructions that provided locality but only up to a rate of 0.5. Matching vector codes were later introduced and can achieve locality r for codes of positive rate and length n=O(r^2). However, the optimal tradeoff between rate, length, and locality remains an open problem.
Performances des turbo codes parallèles pour un canal satellite non linéaireRachidz
1) The document analyzes the performance of parallel concatenated codes (turbo codes) with iterative decoding for error correction on nonlinear satellite channels.
2) It simulates a digital satellite transmission system using parallel turbo codes with QPSK modulation.
3) The simulation evaluates how varying parameters like constraint length, interleaver size, and number of iterations affects the bit error rate performance of turbo codes compared to Viterbi decoding.
The document discusses analyzing and reducing errors in DNA computing algorithms. It focuses on a sensitivity analysis of Gloor's algorithm for the shortest common superstring problem. Key steps include encoding input strings, generating all possible solutions, and iteratively matching strings. The analysis finds that encoding and extraction steps are most sensitive to errors. Various techniques are proposed to make the algorithm more error-resistant, such as carefully designing the encoding to avoid mismatches, and using multiplexing to tolerate a certain number of erroneous operations. The goal is to tune the algorithm and underlying biological operations to be robust against real-world error rates in DNA computing.
Advances in coding for the fading channelwtyru1989
The document discusses coding techniques for fading channels. It notes that most coding wisdom is based on high SNR Gaussian channels, but fading channels have different properties like low and variable SNR and non-Gaussian statistics. For fading channels, the optimal coding criteria depends on the channel model. Techniques like trellis coded modulation (TCM) and bit-interleaved coded modulation (BICM) can provide more robust performance over fading channels compared to codes optimized for Gaussian channels. BICM in particular separates demodulation and decoding, improving performance when the exact channel model is unknown. The document also discusses coding for block fading channels, where a codeword experiences a limited number of independent fading gains rather than full interleaving.
This document discusses error control coding, also known as channel coding. It begins by introducing concepts such as bit error rate and methods to reduce it, including increasing transmission power, diversity techniques, automatic repeat request (ARQ), and forward error correction (FEC) codes. It then provides details on parity checks, cyclic redundancy checks (CRCs), and block error correction codes. Specific error control coding techniques like Reed-Solomon codes are also mentioned.
Nyquist criterion for distortion less baseband binary channelPriyangaKR1
binary transmission system
From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]
The document summarizes Lecture 7 which covered:
1) A review of Lecture 6 on PCM waveforms and the remaining portion of Chapter 2 on spectral densities of PCM waveforms and multi-level signaling.
2) An overview of Chapter 3 on baseband demodulation/detection including matched filters, correlators, Bayes' decision criterion, and maximum likelihood detection.
3) Key aspects of line codes including how pulse shaping can control the signal spectrum and ensure symbol transitions, comparisons of line codes based on power spectral density, DC component, and bandwidth.
Locally decodable codes allow recovery of individual data symbols even after data loss by accessing only a small number of codeword symbols. Reed-Muller codes provide locality but only up to a rate of 0.5, while multiplicity codes achieve higher rates but have weaker locality guarantees. Matching vector codes can match the best known locality bounds, constructing codes of length n with locality r for constant r, but the optimal tradeoff between rate, length and locality remains an open problem.
The document discusses locally decodable codes, which allow recovery of individual data symbols from a coded data set even after erasures. Reed-Muller codes and multiplicity codes were early constructions that provided locality but only up to a rate of 0.5. Matching vector codes were later introduced and can achieve locality r for codes of positive rate and length n=O(r^2). However, the optimal tradeoff between rate, length, and locality remains an open problem.
Performances des turbo codes parallèles pour un canal satellite non linéaireRachidz
1) The document analyzes the performance of parallel concatenated codes (turbo codes) with iterative decoding for error correction on nonlinear satellite channels.
2) It simulates a digital satellite transmission system using parallel turbo codes with QPSK modulation.
3) The simulation evaluates how varying parameters like constraint length, interleaver size, and number of iterations affects the bit error rate performance of turbo codes compared to Viterbi decoding.
This document contains a resume for Can Bayrak, who has a Master's degree in Aviation Management & Safety and a Bachelor's degree in Aerospace Engineering. It summarizes his qualifications, which include over 8 years of experience in aviation and aerospace fields, as well as expertise in safety management systems and operational/managerial aviation aspects. His educational background and work experience are also outlined, along with descriptions of relevant projects, skills, honors/awards, and references.
This document contains 57 questions ranging from math word problems to logical reasoning questions. Some key questions include:
- A question about the probability of getting the same musical sound 5 times consecutively from a toy train that makes 10 sounds.
- A question about the present age of Peter given information about his and Paul's ages.
- A question about the speed of a dog being chased by a horse over a certain distance and time period.
- Various math word problems involving ratios, averages, percentages, and other calculations.
- Logical reasoning questions involving statements made by people with different truth-telling tendencies on different days of the week.
The questions cover a wide range of topics and
1. The document outlines the plans and preparations for an international society discussion group.
2. Topics under consideration included Africa, Burma, Iraq, and North Korea, which were evaluated based on member interest, potential, and suitability.
3. Going forward, the group created a to-do list focusing on arranging discussions, preparing knowledge on topics, practicing English, and conducting fieldwork over the coming months in July, August, and September.
This document lists quantities of various items including 2 liters, 25 liters, 400 grams, and 10 kilograms. It also lists three plant diseases: Botrytis, Fusarium pseudograminaerum, and Pineapple Disease.
The document provides an overview of GSM and GPRS networks. It describes key components of the GSM access network including the BTS, BSC and MSC. It also explains the GSM core network elements such as the HLR, VLR, AuC and SMS centers. For GPRS, it outlines the new GPRS support nodes - SGSN and GGSN, and how they interface with existing GSM network elements.
Food 3.0. Bardzo ciekawy materiał o tym jak social media i aplikacje na iPhone mogą być źródłem wiedzy o tym co jemy, wpływając na przyzwyczajenia żywieniowe... tysięcy ludzi
This document provides an overview and summary of the STS-107 Columbia accident. It discusses the accident timeline, including the launch, events during orbit, issues noticed during re-entry, and the emergency response. It examines the investigation into contributing factors such as foam shedding from the external tank damaging the left wing. The document outlines recommendations to address issues with the thermal protection system, imaging, data collection, crew safety, and organizational culture. It also discusses deficiencies in NASA's safety culture identified by the investigation, including a lack of testing and overconfidence.
EDGE is an upgrade to GSM networks that allows higher data transmission speeds of up to 473 kbps. It uses more advanced modulation techniques like 8-PSK compared to GSM's GMSK, allowing more bits to be transmitted per symbol. EDGE is backwards compatible with GPRS networks and provides benefits like increased network capacity and data rates. Future evolutions of EDGE, such as EDGE Evolution, aim to achieve speeds up to 1 Mbps through techniques like dual antennas, advanced QAM modulation, and additional coding schemes. EDGE allows GSM networks to provide multimedia services at a lower cost than upgrading to 3G networks.
Strategic Analysis of Netjets Management. The final presentation for the Planning Systems module in Embry Riddle Aeronautical University\'s Masters program.
This document provides an overview of FedEx Corporation and its logistics operations. It discusses FedEx's founding in 1971 and expansion over the decades through acquisitions. It describes FedEx's various operating companies including FedEx Express, FedEx Ground, FedEx Freight, and FedEx Custom Critical. It also outlines FedEx's large global transportation network, use of technology, services offered to customers like transportation management and returns, and efforts around innovation.
The Gulfstream Aerospace Production Review and Analysis presentation for Embry-Riddle MSA 641 Production and Procurement Management course in graduate studies
Audio documents can be summarized in 3 sentences or less as follows:
The document discusses various audio coding models, techniques, and standards including entropy coding, differential coding, LPC and parametric coding, sub-band coding, and audio compression standards from ITU, ISO, and MPEG. It also covers topics like audio data rates and file sizes, psychoacoustic models, compression algorithm requirements, and structured audio coding techniques like MIDI. The models, techniques, and standards discussed aim to compress audio signals while maintaining sufficient quality for the intended application.
Error detection enhanced decoding of difference set codes for memory applicat...Sherin Deena Sam
This document discusses error detection techniques for memory applications using difference set codes. It begins by introducing difference set codes and the (21,11) difference set code. It then describes the conventional decoder for this code and issues with silent data corruption when decoding words with 3 or more errors. The document proposes an error detection majority logic decoding technique that can detect errors in 3 cycles to reduce latency and detect uncorrectable errors. This approach avoids the issues with the conventional decoder while enhancing error detection capabilities for memory applications.
This document discusses different channel coding techniques. It begins by introducing the concept of a noisy communication channel and the need for error correction. It then discusses early techniques like repetition coding and parity check codes. More advanced linear block codes and convolutional codes are presented. The document notes that while these codes help reduce errors, they do not achieve the theoretical maximum rate allowed by Shannon's channel capacity theorem. Finally, it introduces turbo codes as a breakthrough that can operate very close to channel capacity with reasonable complexity.
This document discusses using low density generator matrix (LDGM) codes for source coding and joint source-channel coding of correlated sources. Key points:
- LDGM codes can achieve near-Shannon limit performance for channel coding with lower complexity than LDPC codes. Serial and parallel concatenated LDGM codes are proposed.
- LDGM codes are applied to source coding of correlated sources modeled by a hidden Markov model. Simulation results show the codes achieve rates close to the theoretical limits.
- LDGM codes are also used for joint source-channel coding of correlated sources over independent noisy channels. Different decoding schedules are evaluated. Results show the codes perform well for both AWGN and Rayleigh fading channels.
This document contains a resume for Can Bayrak, who has a Master's degree in Aviation Management & Safety and a Bachelor's degree in Aerospace Engineering. It summarizes his qualifications, which include over 8 years of experience in aviation and aerospace fields, as well as expertise in safety management systems and operational/managerial aviation aspects. His educational background and work experience are also outlined, along with descriptions of relevant projects, skills, honors/awards, and references.
This document contains 57 questions ranging from math word problems to logical reasoning questions. Some key questions include:
- A question about the probability of getting the same musical sound 5 times consecutively from a toy train that makes 10 sounds.
- A question about the present age of Peter given information about his and Paul's ages.
- A question about the speed of a dog being chased by a horse over a certain distance and time period.
- Various math word problems involving ratios, averages, percentages, and other calculations.
- Logical reasoning questions involving statements made by people with different truth-telling tendencies on different days of the week.
The questions cover a wide range of topics and
1. The document outlines the plans and preparations for an international society discussion group.
2. Topics under consideration included Africa, Burma, Iraq, and North Korea, which were evaluated based on member interest, potential, and suitability.
3. Going forward, the group created a to-do list focusing on arranging discussions, preparing knowledge on topics, practicing English, and conducting fieldwork over the coming months in July, August, and September.
This document lists quantities of various items including 2 liters, 25 liters, 400 grams, and 10 kilograms. It also lists three plant diseases: Botrytis, Fusarium pseudograminaerum, and Pineapple Disease.
The document provides an overview of GSM and GPRS networks. It describes key components of the GSM access network including the BTS, BSC and MSC. It also explains the GSM core network elements such as the HLR, VLR, AuC and SMS centers. For GPRS, it outlines the new GPRS support nodes - SGSN and GGSN, and how they interface with existing GSM network elements.
Food 3.0. Bardzo ciekawy materiał o tym jak social media i aplikacje na iPhone mogą być źródłem wiedzy o tym co jemy, wpływając na przyzwyczajenia żywieniowe... tysięcy ludzi
This document provides an overview and summary of the STS-107 Columbia accident. It discusses the accident timeline, including the launch, events during orbit, issues noticed during re-entry, and the emergency response. It examines the investigation into contributing factors such as foam shedding from the external tank damaging the left wing. The document outlines recommendations to address issues with the thermal protection system, imaging, data collection, crew safety, and organizational culture. It also discusses deficiencies in NASA's safety culture identified by the investigation, including a lack of testing and overconfidence.
EDGE is an upgrade to GSM networks that allows higher data transmission speeds of up to 473 kbps. It uses more advanced modulation techniques like 8-PSK compared to GSM's GMSK, allowing more bits to be transmitted per symbol. EDGE is backwards compatible with GPRS networks and provides benefits like increased network capacity and data rates. Future evolutions of EDGE, such as EDGE Evolution, aim to achieve speeds up to 1 Mbps through techniques like dual antennas, advanced QAM modulation, and additional coding schemes. EDGE allows GSM networks to provide multimedia services at a lower cost than upgrading to 3G networks.
Strategic Analysis of Netjets Management. The final presentation for the Planning Systems module in Embry Riddle Aeronautical University\'s Masters program.
This document provides an overview of FedEx Corporation and its logistics operations. It discusses FedEx's founding in 1971 and expansion over the decades through acquisitions. It describes FedEx's various operating companies including FedEx Express, FedEx Ground, FedEx Freight, and FedEx Custom Critical. It also outlines FedEx's large global transportation network, use of technology, services offered to customers like transportation management and returns, and efforts around innovation.
The Gulfstream Aerospace Production Review and Analysis presentation for Embry-Riddle MSA 641 Production and Procurement Management course in graduate studies
Audio documents can be summarized in 3 sentences or less as follows:
The document discusses various audio coding models, techniques, and standards including entropy coding, differential coding, LPC and parametric coding, sub-band coding, and audio compression standards from ITU, ISO, and MPEG. It also covers topics like audio data rates and file sizes, psychoacoustic models, compression algorithm requirements, and structured audio coding techniques like MIDI. The models, techniques, and standards discussed aim to compress audio signals while maintaining sufficient quality for the intended application.
Error detection enhanced decoding of difference set codes for memory applicat...Sherin Deena Sam
This document discusses error detection techniques for memory applications using difference set codes. It begins by introducing difference set codes and the (21,11) difference set code. It then describes the conventional decoder for this code and issues with silent data corruption when decoding words with 3 or more errors. The document proposes an error detection majority logic decoding technique that can detect errors in 3 cycles to reduce latency and detect uncorrectable errors. This approach avoids the issues with the conventional decoder while enhancing error detection capabilities for memory applications.
This document discusses different channel coding techniques. It begins by introducing the concept of a noisy communication channel and the need for error correction. It then discusses early techniques like repetition coding and parity check codes. More advanced linear block codes and convolutional codes are presented. The document notes that while these codes help reduce errors, they do not achieve the theoretical maximum rate allowed by Shannon's channel capacity theorem. Finally, it introduces turbo codes as a breakthrough that can operate very close to channel capacity with reasonable complexity.
This document discusses using low density generator matrix (LDGM) codes for source coding and joint source-channel coding of correlated sources. Key points:
- LDGM codes can achieve near-Shannon limit performance for channel coding with lower complexity than LDPC codes. Serial and parallel concatenated LDGM codes are proposed.
- LDGM codes are applied to source coding of correlated sources modeled by a hidden Markov model. Simulation results show the codes achieve rates close to the theoretical limits.
- LDGM codes are also used for joint source-channel coding of correlated sources over independent noisy channels. Different decoding schedules are evaluated. Results show the codes perform well for both AWGN and Rayleigh fading channels.
This document provides an overview of various channel coding techniques used in digital communication systems to combat noise and errors during transmission. It describes forward error correction methods like block codes, cyclic codes, convolutional codes and turbo codes. It also discusses error detection techniques like cyclic redundancy checks. Finally, it covers automatic repeat request protocols for retransmitting corrupted data packets.
This document discusses various techniques for speech coding used in digital communication systems. It covers fundamental concepts like sampling theory, quantization, predictive coding, and linear predictive coding (LPC). It then describes specific speech codecs including PCM, ADPCM, CELP, LD-CELP, ACELP, and LPC vocoders. It discusses characteristics of speech coding like being lossy and metrics like SNR and MOS. Finally, it provides details on widely used standards like G.711, G.729, G.723.1, and GSM.
Pulse code modulation (PCM) is a method of digitally representing sampled analog signals. In PCM, the instantaneous voltage of an analog signal is sampled regularly at uniform intervals, then quantized to a series of digital codes. This allows the analog signals to be transmitted over digital communication networks or stored in digital memory. Key aspects of PCM include sampling the analog signal, quantizing the samples to discrete levels, encoding the quantized levels into binary code words, and transmitting the encoded binary data as a serial bit stream. PCM provides advantages like noise immunity, easy processing and storage, and the ability to multiplex and transmit multiple signals over the same channel.
This document proposes a speaker-dependent WaveNet vocoder to generate high-quality speech from acoustic features. It uses a WaveNet model conditioned on mel-cepstral coefficients and fundamental frequency to directly model the relationship between acoustic features and speech waveforms. Evaluation shows the proposed method improves sound quality over traditional vocoders, as measured by objective metrics and subjective listening tests. Future work will apply this approach to other tasks and make the model independent of individual speakers.
This document discusses various techniques used to improve mobile radio link performance including equalization, diversity, and channel coding. It describes equalization techniques that compensate for intersymbol interference caused by multipath. It explains different types of diversity including spatial, time, and frequency diversity that are used to mitigate fading. Specifically, it outlines four common spatial diversity techniques: selection diversity, maximal ratio combining, equal gain diversity, and scanning diversity. The document also discusses time diversity and RAKE receivers used in code division multiple access systems to exploit multipath for additional time diversity gain.
Turbo codes provide reliable communication at high data rates by combining concepts from block and convolutional codes. They use two convolutional encoders separated by an interleaver, producing redundant parity bits. During iterative decoding, probabilistic information from the first decoder is used as a priori information for the second decoder, and vice versa, improving the estimates of the transmitted bits at each iteration. Turbo codes achieve performance close to the theoretical channel capacity limit with low error rates. They have applications in deep space communications, mobile wireless systems, and other areas requiring high reliability transmission.
Pulse code modulation (PCM) is an analog-to-digital conversion technique used to represent sampled analog signals as digital data. PCM involves sampling the analog signal at regular intervals, quantizing the amplitude of the signal at each point to a few discrete levels, and coding it as digital data. The sampling rate must be greater than twice the highest frequency of the analog signal as per the Nyquist sampling theorem. PCM was invented in 1937 but was not widely adopted until the 1940s. It became the standard method for digital telephony due to its robustness and ability to efficiently regenerate and transmit signals.
This document discusses OFDM and OFDMA technologies. It begins with an outline of topics including the need for multi-carrier transmission, how OFDM addresses this need using FFT and IFFT, guard time insertion using cyclic prefixes, drawbacks of OFDM including high PAPR, channel estimation techniques, and an OFDM block diagram. It then discusses OFDMA which allows simultaneous transmissions to multiple users using OFDM signaling. Diversity techniques including time, frequency, and spatial diversity are also summarized.
This document discusses speech compression using linear predictive coding (LPC). It begins with the objectives of developing low bit-rate speech coders for cellular networks. It then introduces LPC and how it models the human vocal tract. The key aspects of LPC encoding and decoding are described, including analysis, synthesis, and the Levinson-Durbin algorithm. Simulation results on compressing male and female speech are presented, showing compression ratios and signal-to-noise ratios. The document concludes that LPC is well-suited for secure telephone systems by preserving the meaning of speech at low bit rates.
1. The document discusses various waveform coding techniques including PCM, DPCM, ADPCM, DM, and LPC. PCM involves sampling and quantizing an analog signal. DPCM and ADPCM encode differences between signal values to reduce bitrate. DM encodes changes in signal amplitude. LPC models speech as a linear system to remove redundancy.
2. LPC uses linear prediction to estimate future signal values from previous samples. The prediction error is minimized to find predictor coefficients. Different algorithms like autocorrelation, covariance, and reflection solve the LPC equations.
3. LPC parameters like reflection coefficients, line spectral frequencies, and impulse response relate to each other mathematically. The roots of the
Space time coding is used in MIMO wireless systems to improve communication performance by exploiting spatial diversity. It uses multiple transmit and receive antennas. The Alamouti code is a simple and effective space time block code that achieves full transmit diversity without requiring channel state information at the transmitter. It transmits symbols from two transmit antennas in two time slots so that the receiver can recover the symbols with low complexity decoding. MIMO combined with space time coding can provide high data rates, minimize errors, and increase capacity for wireless applications such as 4G networks.
Non-Binary LDPC codes are LDPC codes where parity check equations are performed over a Galois Field GF(q) of cardinality greater than 2. This allows operations such as addition and multiplication to be defined for symbols in the field. Decoding of NB-LDPC codes can be done using belief propagation on a Tanner graph by passing messages in the form of LLRs between variable and check nodes. NB-LDPC codes have better performance than binary LDPC codes for low code rates and lengths due to their higher mutual information and lack of need for bit marginalization in decoding. However, they also have increased complexity compared to binary LDPC codes.
Digital image compression techniques aim to reduce the number of bits required to represent an image by minimizing redundancy. There are two main categories: lossless compression preserves all image information while lossy compression provides higher data reduction but less than perfect image reproduction. Common methods include removing coding, interpixel, and psychovisual redundancies through techniques like variable length coding, discrete cosine transform, and quantization.
1) The document discusses various topics related to digital communication including sampling theory, analog to digital conversion, pulse code modulation, quantization, coding, and time division multiplexing.
2) In analog to digital conversion, an analog signal is sampled, quantized by assigning it to discrete amplitude levels, and coded by mapping each level to a binary sequence.
3) The Nyquist sampling theorem states that a signal must be sampled at a rate at least twice its highest frequency to avoid aliasing when reconstructing the original signal.
1. The document discusses different compression techniques for text, audio, images, and video.
2. It provides examples of compression ratios achieved using lossy and lossless compression methods. For example, text compression can achieve 3:1 ratios using Lempel-Ziv coding while audio compression can achieve ratios between 3:1 to 24:1 using MP3.
3. The techniques discussed include entropy encoding, run-length encoding, Huffman coding, discrete cosine transforms, and differential encoding which takes advantage of redundancies in the data. The best approach depends on the type of data and acceptable quality.
The document discusses turbo codes, a type of forward error correction code. Turbo codes use parallel concatenated convolutional encoders with pseudorandom interleaving and iterative decoding. This structure allows turbo codes to achieve error correction performance very close to the theoretical limit defined by the Shannon capacity. Some key advantages of turbo codes are their remarkable power efficiency and ability to support delivery of multimedia services through design tradeoffs. However, turbo codes also have long latency and poor performance at very low bit error rates.
1. April 2009
Channel Matched Iterative Decoding for
Magnetic Recording Systems
Final Oral Examination
Hakim Alhussien, PhD Candidate
Adviser: Jae Moon
Communications and Data Storage (CDS) Laboratory
Department of Electrical and Computer Engineering
University of Minnesota
April 06, 2009
1
2. Hakim, April 2009
Outline
Perpendicular magnetic recording channel.
• ECC for recording channels.
• Error Pattern Correction Coding (EPCC).
EPCC enhanced TE (TE-EPCC).
• Error rate analysis of TE-EPCC.
• TE-EPCC and TP-EPCC for PMRC.
Tensor product parity codes (TPPC).
• Linear-time Encoding of tensor product codes.
• Hard decoding of EPC-RS tensor product codes.
• Error rate analysis of EPC-RS tensor product codes.
EPC-LDPC tensor product codes.
• Soft-syndrome decoding of EPC-LDPC tensor product code.
• Simulation study of EPC-LDPC.
Thesis contributions.
2
3. Hakim, April 2009
Perpendicular Magnetic Recording (PMR) Channel
Recording channel is “transition-response fixed”
• To achieve the same normalized user density at a lower coding rate, the
( )
SNR is degrader by ∼ 10 × log10 1 R 2 use high rate codes.
Saturated-level recording (binary-constrained input)
• Optimal-precoding or SNR water-filling not possible.
Channel impaired by long error bursts.
• Due to ISI, disk defects, and thermal asperities.
• Symbol-correcting codes effective in burst correction,
such as RS, LDPC over GF(q).
Data reread is expensive in terms of latency
• Standard frame error rate is very low 10−13 ∼ 10−14
Fixed ISI channel with dominant odd and even error events
• Utilize ECC targeting dominant errors events after ML detection.
DC full PRML target
• A DC wandering compensation loop is required.
Transition-dependent medium noise due to zigzag domain boundaries
• Channel detector trellis incorporates PDNP.
3
4. Hakim, April 2009
ECC for PMR Read Channel
Reed Solomon (RS)
• Minimum distance of RS > LDPC for same block length and rate.
• ML decoding of RS outperforms ML decoding LDPC.
• Iterative Belief propagation decoding approaches ML performance.
• RS parity check matrix very dense – large number of 4-cycles.
• Iterative decoding of LDPC significantly outperforms RS iterative decoding.
RS with inner LDPC or turbo codes
• Error behavior of LDPC is catastrophic for strong codes.
• Requires high-rate low-column weight LDPC – weak family of codes.
• Convolutional based Turbo: long tail in symbol-error distribution.
Stand-alone LDPC
• Extensive research on lowering the SER error floor.
• LDPC with sector-wide codeword has low minimum distance.
• Sparse LDPC: improved iterative decoding – larger girth.
• Dense or large-block length LDPC: better Hamming weight spectrum
• Consider: Sparse non-binary LDPC of sector-length codeword!
4
6. Hakim, April 2009
The Channel Matched ECC Paradigm
Premise: for a given ISI channel all dominant error patterns are known a
priori.
• Hyperbolic tangent transition response
at a channel density of 1.4, High density
• 10% AWGN and 90% jitter noise.,
perpendicular
• Target response: 1+0.9D,
• Bit error rates: 2.3276×10-3 (1 PDNP) ,
recording channel
• Captured # of error patterns: 223,676,
• Edt/N90 = 13.5 dB
Strong general
Strong general Channel-matched
Channel-matched Write head/medium/read head
Write head/medium/read head
ECC encoder
ECC encoder EPC encoder
EPC encoder
corrects focuses on correcting
a few dominant error
remaining errors patterns
Strong general Channel-matched
Channel-matched
Strong general Equalizer/Detector
EPC encoder Equalizer/Detector
ECC decoder
ECC decoder EPC encoder
6
7. Hakim, April 2009
EPCC Design: Target List =5 most dominant errors
Target the 5 most dominant errors,
which account for 92.04% of possible errors.
Syndrome set produced by g(x) = 1 + x +x3 + x5 + x6 Target Error Polynomial Syndrome
• Order of g(x)=12. Period
• Total number of distinct syndrome sets: 5. 1 12
• 5 distinct, non-overlapping syndrome sets are
utilized to distinguish 5 target error. 1+ x 12
Cyclic generator polynomial used to design a cyclic
(12,6) code of rate=0.5, and code cord length 12. 1+ x + x 2 6
Single occurrences of error types {1,2,4,5} decoded 1+ x + x 2 + x 3 12
without ambiguity.
Via channel reliability information and the polarity of 1 + x + x 2 + x 3 + x 4 12
data support, error type 3 can be decoded reliably.
Unique syndrome-error mapping via channel side
information.
7
8. Hakim, April 2009
EPCC Design: Target List =10 most dominant errors
Target the 10 most dominant errors, Target Error Polynomial Syndrome Syndrome
Period g1(x) Period g2(x)
account for 99.67% of possible errors.
g1(x) = 1 + x2 +x3 + x5 + x6 +x8 1 18 30
• Order of g1(x)=18.
1+x 9 15
• 10 distinct syndrome sets.
• Cyclic generator polynomial used to 1+x +x 2 18 10
design a cyclic (18,10) code of
1+x +x 2 +x3 9 15
rate=0.56, and codeword length 18.
g2(x) = 1 +x3 + x5 + x8 1+x +x 2 +x3 +x 4 18 6
• Order of g2(x)=30. 1+x +x 2 +x3 +x4 +x5 9 5
• 10 distinct syndrome sets.
1+x +x 2 +x3 +x4 +x5 +x6 18 30
• Cyclic generator polynomial used to
design a cyclic (30,22) code of 1+x +x 2 +x3 +x4 +x5 +x6 +x7 9 15
rate=0.73, and codeword length 30.
1+x +x 2 +x3 +x4 +x5 +x6 +x7 +x8 2 10
Unique syndrome-error mapping via
channel side information. 1+x +x 2 +x3 +x4 +x5 +x6 +x7 +x8 +x9 9 3
8
9. Hakim, April 2009
Approaches to Increase Code Rate of EPCC
Syndrome sets produced by g(x) = 1 + x3 + x5 + x8
• Order of g(x): 30 → (30, 22) base cyclic code
• 10+3 extra distinct, non-overlapping syndrome sets are utilized to distinguish 13
target error patterns.
Multiply g(x) by a degree 6 primitive polynomial which is not a factor of
any target error polynomials :
• The extended code is a (630,616) code of rate 0.98.
Extended periods of syndrome
sets produced by g ′(x )
Tensor product coding paradigm.
• Short codeword length (outer ECC symbol length), very high total code rate.
9
11. Hakim, April 2009
WER ML Bound
Word Error Probability:
x2
1 M
x − x m′
PW ≤ ∑∑ Q m
M m =1 m ′ ≠ m 2σ
T1,dmin
2 2
d E = d min
1 M ∞ d
PW ≤ ∑∑
M m =1 d E =1
Tm ,d E Q E
2σ
x3 x1
∞
d
= ∑
d E = d m in
T ( d E )Q E
2σ
Decrease number of codewords at Increase Euclidean minimum
Euclidean minimum distance distance
x4
(Turbo codes) (Trellis coded modulation)
11
12. Hakim, April 2009
BER ML Bound
∞
T (d E ) w(d E ) d E
• Bit Error Probability: Pb ≤ ∑
d E = d min K
Q
2σ
• Average number of codeword sequences of channel noiseless outputs
separated by dE:
N
T (d E , C ) = ∑ A ( d )Pr( d
d =1
E | d,C )
• Average Hamming distance between information words that generate codewords
of channel noiseless outputs separated by dE2:
N
1
w (d E , C ) =
T (d E , C )
∑ A ( d ) A ( d )Pr ( d
d =1
E | d,C)
Average input Hamming weight
# of codeword sequences of weight d of codewords of weight d
∞ N
A ( d ) A ( d )Pr( d E | d , C ) d E
Pb ≤ ∑ ∑
d E = d min d =1 K
Q
2σ
12
14. Hakim, April 2009
Partial Response Class-1 (PR1) Channel (1+D)
0 0/0 Trellis of Dicode channel
1/1 0 /1
1 1/ 2
2 … non-dominant error pattern
d = 2 + 4 × bcr
E
2
2
dE = 1 dE = 1
2
dE = 4
dominant error pattern
2 …
d =2
E
2 2
dE = 1 dE = 1
2
dE = 0
14
15. Hakim, April 2009
Dicode Channel (1-D)
0 0/0 Trellis of Dicode channel
1/1 0 / −1
1 1/ 0
2
dE = 2 … dominant error pattern
2
2
dE = 1 dE = 1
2
dE = 0
non-dominant error pattern
2
d = 2 + 4 × bcr
E
…
2 2
dE = 1 dE = 1
2
dE = 4
15
16. Hakim, April 2009
A Dicode multiple error occurrence
m: # of error patterns in EPCC sub-code
… … … …
4
1 1 1 1 1 1 1
• Merging branches correspond to
∑
zero error Hamming weight 2
dE
16
17. Hakim, April 2009
Distribution of dE given d and m
# of ways we can have # of crossing branches
crossing branches
d − m d −m
1
2
2 d E − 2m
, > 0 integer, mdom < m
d E − 2m 2
4
4
d − mdom
1 2
Pr(d E | d , m) = , d E = 2mdom , mdom = m
2
0 , otherwsie
d: Hamming weight of multiple error,
m: # of error patterns, mdom: # of dominant error patterns
17
18. Hakim, April 2009
Enumerators for error Hamming weights
d H (e) = i K
A(d , i) Information sequence
d H ( e) = d N
RSCC codeword
1
Π
N
d
d H ( e) = d N
Interleaved RSCC codeword
L
d H ( e) = ∑ d i L × Nc
i =1
Nc
di
d H (e) = di N c + Pc EPCC codeword
N − di di − 1 N − di d i − 1
m m − 1
i i mi − 1 mi − 1
Nc closed Nc
closed
d + d
i error patterns
open i
error patterns
18
19. Hakim, April 2009
Enumerators for error Hamming weights
Distribution of Euclidean distance given Distribution of sub-code Hamming weights
Hamming weight of sub-codes given Hamming weight of outer code
Pr( d E | d ) = Pr(d E | d , d1 ,..., d L ) × Pr( d1 ,..., d L | d )
= Pr( d E | d , d1 ,..., d L , m, m1 ,..., mL )
×Pr(m, m1 ,..., mL | d1 ,..., d L , d ) × Pr( d1 ,..., d L | d )
Distribution of sub-code multiple error patterns
given Hamming weight of sub-codes
d d
Pr(d E | d ) = ∑ ... ∑ Pr ( d ,..., d
d1 = 0 d L =0
1 L | d)
∑ i=1 di
L
d=
d d1 dL L
× ∑ ∑ ...∑ Pr ( d
m =1 m1 = 0 mL
E | d , m ) ∏ Pr(mi | di )
i =1
∑ i=1
L
m= mi
19
20. Hakim, April 2009
Enumerators for error Hamming weights
Joint distribution of the sub-code Hamming weights:
Nc Nc Nc # of sub-code words
× ... × of Hamming weight d
Pr(d1 ,..., d L | d ) = d1 d 2 dL L
N # of interleaved RSCC words
of Hamming weight d
d
Distribution of the number of error patterns per sub-code:
# of ways mi error patterns are N c − di d i − 1 # of ways di is decomposed
arranged in sub-code i. × into mi error patterns.
mi mi − 1
Pr(mi | di ) =
Nc # of sub-code words
of Hamming weight di
di
20
21. Hakim, April 2009
Euclidean distance Enumerator of TE-EPCC, when EPCC is tuned off:
d d d d1 dL
1
Pr( d E | d ) = ∑0 ... d∑0 ∑1
N d1 =
∑0 ...m∑0
L= m= m1 = L=
d =∑ iL=1 di L
m: d E − 2 m = 0 mod 4, m = ∑ mi
d
2
i =1
d −m d −m L
2 1 Nc − d j d j − 1
× d E − 2m ∏
2
j =1 m j m j − 1
4
Euclidean distance Enumerator of all correctable TE-EPCC codewords:
1 min( d ,dc ) min( d ,dc ) d min( d1 , mc ) min( d L , mc )
Pr(d E | d , C ) =
∑ ... d∑0
N d1 =0 ∑ ∑ ... ∑
L= m =1 m1 = 0 mL = 0
d= d
d ∑ i=1 i
L L
2
m: m = d E 2, m = ∑ mi
i =1
d −m L
1 Nc − d j d j − 1
× ∏
2 j =1 m j m j − 1
Euclidean distance Enumerator of non-correctable TE-EPCC codewords:
Pr( d E | d , C ) = Pr( d E | d ) − Pr( d E | d , C )
21
22. Hakim, April 2009
Interleaver Gain Exponent of TE
Approximations:
d
N ( N − d + 1) Nd
>
d d! d!
m − µ +1
N − d m − µ + 1 N − d + 1 N − d + 1 ( N − d + 1) N m − µ +1
= <
m − µ N − d + 1 m − µ + 1 m − µ + 1 (m − µ + 1)! (m − µ + 1)!
d2
d E 1 − 4σE2
Q ≤ e .
2σ 2
Modified TE bound: 2
∞ dT 1 d dE
1 −
Pb <
2K
∑ ∑∑
µ
d E =1 d = 2 = 0
∑
m =1
B d E ,d ,m,µ N m − µ −d
e 4σ 2
2
m: d E − 2 m + µ = 0mod 4
d −m d −m
B d E ,d ,m ,µ = A ( d ) A (d )
d! 1 2 d −1
( m − µ )! 2 d E − 2m + µ
m − 1
4
22
23. Hakim, April 2009
Interleaver Gain Exponent of
TE-EPCC(dc = 10, mc = 3, L = 1)
2
∞ dE 1
1 −
Pb <
2K
∑e
d E =1
4σ 2
∑
µ
=0
dT d min( dT , d c ) min( d , mc )
∑
d =2
∑m =1
B d E ,d ,m, µ N m − µ −d
− ∑
d =2
∑
m =1
B d E ,d ,m, µ N m − µ − d
2 2
m: d E − 2 m + µ = 0mod 4 m: d E = 2 m − µ
23
24. Hakim, April 2009
Interleaver Gain Exponent of
TE-EPCC(dc = 10, mc = 3, L = 1)
24
25. Hakim, April 2009
Interleaver Gain Exponent of
TE-EPCC(dc = 10, mc = 3, L = 1)
25
26. Hakim, April 2009
Interleaver Gain Exponent of
TE-EPCC(dc = 10, mc = 3, L = 1)
26
27. Hakim, April 2009
Interleaver Gain Exponent of
TE-EPCC(dc = 10, mc = 3, L = 1)
27
28. Hakim, April 2009
Interleaver Gain Exponent of
TE-EPCC(dc = 10, mc = 3, L = 1)
Asymptotic BER bound of conventional TE:
1 1 3
A(2) A(2) − 4σ 2 A (2) A(2) − 2σ 2 A(2) A(2) − 4σ 2
Pb < e + e + e
2 KN 2 2 KN KN
1 5 3
A (2) A(2) −σ 2 3A(3) A(3) − 4σ 2 A(3) A (3) − 2σ 2
+ e + e + e
2K 2 KN 2K
7
2A (4) A(4) − 4σ 2
+ e +O
KN
Asymptotic BER bound of TE-EPCC(dc = 10, mc = 3, L = 1):
1 1 3
155925A (10) A (10) − 4σ 2 155925A (10) A (10) − 2σ 2 779625A (10) A (10) − 4σ 2
Pb < 11
e + 10
e + 10
e
8 KN 8KN 2 KN
1 5 3
779625A(10) A(10) − σ 2 A (2) A (2) − 4σ 2 A(2) A (2) − 2σ 2
+ e + e + e
4 KN 9 2 KN 2 2 KN
7
2 A (4) A (4) − 4σ 2
+ e +O
KN
28
29. Hakim, April 2009
“Spectral Thinning” of TE-EPCC
30
20
10
0
log T(dE)
-10
-20
precoded Dicode, TE
-30 unprecoded Dicode, TE
unprecoded Dicode, TE-EPCC
-40
-50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
d2
E
• TE: K = 4096, punctured R=8/9, (31, 33) RSCC.
• TE-EPCC: (L = 7)EPCC, mc = 3, dc = 10.
• EPCC sub-code: (630, 616), R = 0.98.
29
30. Hakim, April 2009
Precoded TE
0 0/ 0 Precoded TE
Convolutional
1/1 1/ −1 Encoder Dicode Channel
(RSCC) ∏ 1 1⊕ D (1-D)
1 0/ 0
Unprecoded Dicode: trellis paths corresponding to different code bits
are at 0 Euclidean distance → long error events have a high
probability of generating low Euclidean distance errors.
Precoded Dicode: trellis paths corresponding to different code bits
accumulate Euclidean distance → ONLY low Hamming weight
errors generate low Euclidean distance errors.
The average number of Hamming weight 2 errors that generate dE2 =2
is more for precoded compared to unprecoded Dicode.
• Unprecoded TE achieves a lower error floor compared to precoded TE.
30
38. Hakim, April 2009
Perpendicular Magnetic Recording (PMR) channel
Hyperbolic tangent transition response for perpendicular recording
2t (H. Sawaguchi et al., “Performance analysis of modified
h(t ) = tanh PRML channels for perpendicular recording systems,” J.
0.5795 ⋅ π ⋅ pw50 Magn. Magn. Mater., 2001.)
Channel density Ds ≡ pw50 / T
• pw50 : −50% to 50% width of the transition response
• T : symbol period
38
39. Hakim, April 2009
PMR Continuous-time Channel Model
Continuous-time channel model
• h(t) : Hyperbolic tangent transition response, i.e., h(t ) = tanh(λt )
1
• s(t) : dibit response, i.e., s (t ) = [ h (t ) − h (t −T )]
2 2 2
• h'(t) : First-order time derivative of h(t), i.e., h′(t ) = λ sech (λt )
• p(t) : Front-end band-limiting filter (7th-order butterworth filter)
• n(t) : Additive white Gaussian noise
• jk : Random transition position jitter
• Definition of energy Edt: 2
E dt = ∫−∞ [ h ′(t )] dt
∞
39
40. Hakim, April 2009
PMR Discrete-time Channel Model
Discrete-time channel model
• s k ≡ [s (t ) ∗ p (t )]t =kT , h j ≡ [ h′(t ) ∗ p (t )] , h n ∗ h n = [ p ( t ) ∗ p ( − t ) ]
s
k t =kT k −k t = kT
• Variance of the additive white Gaussian noise (AWGN) sequence nk : σ n2 = N o
2
• Variance of the jitter noise jk : σ 2j = M o
2
• Spectral height for the mixed noise: Nα = No + Mo
− Nα signifies α % jitter noise, i.e., α = M o × 100
No + M o
Edt
• SNR can be defined as SNR ≡
Nα
40
41. Hakim, April 2009
Partial Response Maximum Likelihood System
Channel density: 1.1
• Mixed noise: 10% AWGN and 90% jitter noise, DC full dibit response.
• Target response: 1+0.85D , optimized to whiten noise for the all-transition input.
Discrete Time Dibit Response at Ds=1.1 15-tap RLS Equlizer
0.4 3
0.3 2
1
0.2
0
0.1
-1
0
0 2 4 6 8 0 5 10 15
k k
Dibit Vs Taregt (Frequency) Target Vs Dibit*Equalizer (Frequency)
10 10
0 0
-10 -10
dB
dB
-20 Dibit -20
Target Equalized Dibit
-30 -30 Target
-40 -40
0 0.5 1 0 0.5 1
fT fT
41
42. Hakim, April 2009
EPCC-TE
Encoder
(630,616) Write head/medium/read head
RS Encoder (11,10)
Convolutional ∏ EPCC 1+0.9D PR,
xk t = 20 Encoder (RSCC) encoder 90% media noise + 10% electronic noise
(11,10) RSCC ∏-1 e EPCC SISO λke SISO Equalizer
RS Decoder
SISO decoder λ k List decoder 4 state BCJR,
t = 20 4 state BCJR
ˆ
xk rate ≈ 1 1 PDNP tap
∏
Decoder EPCC enhanced Turbo Equalizer
42
47. Hakim, April 2009
An EPC- Tensor Product Code
Chaichanavong and Siegel (2006) proposed a tensor product code based on a
single parity code + BCH as an inner code for outer RS ECC.
• Suitable for low density longitudinal recording channels were dominant errors
have odd weight of the form [+2] , [+2, −2, +2] .
• Code combined with MTR for perpendicular recording channels.
• Tensor product code has much higher rate than a short parity code.
• Parity code on the symbol-level – less multiple error occurrences.
To achieve performance gains with respect to QLDPC we will investigate a
tensor product code based on a short inner multiparty code (EPCC) and
outer QLDPC ECC.
• The EPC multiparty code corrects any single occurrence of a dominant targeted
error in a tensor symbol.
• An EPCC sequence of syndromes forms a codeword for QLDPC.
• EPCC is decoded jointly with the channel using post processing techniques that
generate a soft “syndrome-codeword” to be decoded by the QLDPC non-binary
message-passing decoder.
• Via channel side information, EPCC has a unique syndrome per dominant error
single occurrence. A list decoding scheme increases the decoding sphere radius
of EPCC to target multiple error occurrences.
47
48. Hakim, April 2009
Introduction to Tensor Product Codes
Jack K. Wolf, “On Codes Derivable form the Tensor Product of check Matrices,” IT 1965.
Constituent Codes: 1 0 1
• Binary (3,1) single error correcting code, H = = 1 α α2
0 1 1
1 0 1 α α 2
• Doubly-extended t=1 (5,3) RS on GF(22), H =
0 1 1 α2 α
The tensor product code parity check matrix in GF(22) is
1 α α 2 0 0 0 1 α α 2 α α 2 1 α 2 1 α
H GF (22 ) =
0 0 0 1 α α 2 1 α α 2 α 2 1 α α α 2 1
1. This binary (15,11) tensor product code 101 000 101 011 110
corrects any single tensor symbol error 011 000 011 110 101
=
provided it contains a single bit error.
2. Binary constituent code rate is 0.34 and H GF (2)
codeword length is 3 bits. 000 101 101 110 011
3. Tensor product code rate is 0.74
and codeword length is 15 bits. 000 011 011 101 110
Tensor Symbol
48
49. Hakim, April 2009
Encoding of Tensor Product Codes
Encoding of a tensor product code of binary code C1: (n1,k1), and non-binary
code C2: (n2,k2)
• Divide n1k2 information bits into k2 columns.
• Encode each column using C1 .
• Convert to GF ( 2 p ) .
1
• Encode intermediate non-binary syndromes using C2 .
• Convert back to GF ( 2 ) .
transmitted codeword
1 0 1 1 0
• Use remaining p2k1 information bits
and the calculated syndromes 1 1 1 1 1
bits to calculate p1p2 parity bits using
back substitution and systematic H1. 0 1 1 0 1
• Result : If C1 and C2 are linear time 1 1 0 0 1
Intermediate
syndromes
encodable, then C 1 ⊗ C 2 1 0 0 1 1
is linear time encodable! 2 2
α 1 0 α α
49
50. Hakim, April 2009
An EPC-RS Tensor Product code
EPC-RS constituent codes
• (18,10) EPCC over GF(2), Rate=0.556, 8 parity bits.
H = 1 α α 2 α 3 α 4 α 5 α 6 α 7 α 133 α 134 α 96 α 90 α 82 α 236 α 234 α 217 α 92 α 93
• (255,195) RS over GF(28), Rate=0.765, t=30, 60 parity symbols.
EPC-RS tensor product code is a binary (4590,4110) code, Rate=0.895, 480
parity bits.
• Codeword length = 18×255 bits, parity = 8×60 bits.
Tensor symbol (1) Tensor symbol (2) Tensor symbol (3) … Tensor symbol (255)
18 bits
Tensor code can correct any combination of 30 or less tensor symbol errors,
given that each 18-bit tensor symbol has a single occurrence of a dominant
error that is correctable by EPCC.
50
51. Hakim, April 2009
Hard Decoding of RS-EPC tensor product code
18 bits
Tensor symbol (1) Tensor symbol (2) Tensor symbol (3) … Tensor symbol (255)
Compute EPCC binary
syndromes and convert to GF(28)
EPCC Syndrome (1) EPCC Syndrome (2) EPCC Syndrome (3) … EPCC Syndrome (255)
RS hard decoding in GF(28)
8 bits or GF(28) symbol
(or any list soft decoding algorithm ).
RS symbol (1) RS symbol (2) RS symbol (3) … RS symbol (255)
Convert back to corrected binary
EPCC syndromes.
EPCC Error Synd (1) EPCC Error Synd (2) EPCC Error Synd (3) … EPCC Error Synd (255)
Find most likely single and
8 bits double errors.
Likely dominant Likely dominant Likely dominant Likely dominant Error
Error (1) Error (2) Error (3) … (255)
Add to ML word
18 bits
51
52. Hakim, April 2009
RS-EPC TPPC Residual Errors
Non-targeted single error occurrences.
e13 (x ) …
13 bits
18 bits
More than double multiple error occurrences.
…
2 bits 1 bit 3 bits
18 bits
Double error occurrences that have a zero EPCC syndrome, since RS generates
syndromes of errors as input to EPCC.
Residual errors can be corrected by an outer RS code of small correction power,
since the number of residual tensor symbols in error is small.
EPCC can work as an error locating code: Erasure decoding of outer RS.
52
53. Hakim, April 2009
EPC-RS Hard Decoder
Tensor Product Hard Decoder
EPCC Syndrome
Generator
RS Decoder t=27 GF(28)
rk Binary
Viterbi
Modulo 2
ˆ
ck
hk
qk EPCC list ˆ
bk
− RS Decoder t=3 GF(210)
+
decoder
25 test words
53
54. Hakim, April 2009
Semi-Analytic & Fully-Analytic Multinomial SER estimations
Step 1: Estimate P1, …, Pm
• Simulation:
1. slide a window of size m symbols over the channel detector’s simulated hard output
and count occurrences of 1 to m consecutive symbol errors.
2. divide the m cumulative sums by the number of simulated symbols.
• Analytic:
1. P1=∑ (probability of 1 dominant error-pattern that spans 1 symbol).
2. P2=∑ (probability of 1 dominant error-pattern that spans 2 symbols)
+ ∑(probability of 2 dominant error-patterns encapsulated in two separate
consecutive symbols).
3. P3=∑ (probability of 1 dominant error-pattern that spans 3 symbols)
+ ∑ (probability of 1 dominant error-pattern that spans 2 consecutive symbols)
×(probability of 1 dominant error-pattern that spans a 3rd succeeding
symbol )
+ ∑ (probability of 3 dominant error-patterns encapsulated in 3 separate
consecutive symbols).
Step 2: n!
PW ≥ 1 − ∑ ∑ …∑ P0s0 P1s1 … Pmsm
s0 s1 sm s 0 ! s1 !… s m !
m m m
∀ si : ∑ is
i=0
i ≤ t, ∑s i =0
i = n ; P0 = 1 − ∑ Pi .
i =1
54
55. Hakim, April 2009
Symbol Error Event Probabilities
Single-level RS Vs EPC-RS
-1
10
-2
10
• ISI channel 5+6D-D3, AWGN.
• Shortened (450, 450-2T) RS -3
Symbol Error Event Probability
10
over GF(210).
• (18, 10) EPCC + shortened
-4
(250,250-2Ttp) RS over GF(28). 10
-5
10
P1, 10 bit sybmol
-6
10 P2, 10 bit sybmol
P3, 10 bit sybmol
-7
P1, 18 bit sybmol
10 P2, 18 bit sybmol
P3, 18 bit sybmol
-8
10
6.5 7 7.5 8 8.5 9 9.5
SNR (dB)
55
57. Hakim, April 2009
Difference of Minimum SNR Required for SFR=10-13
Single-level RS Vs EPC-RS
0.9
0.85
• ISI channel 5+6D-D3, AWGN.
minSNR(RS) - minSNR(TPRS)
0.8
• Shortened (450, 450-2T) RS
over GF(210).
• (18, 10) EPCC + shortened 0.75
(250,250-2Ttp) RS over GF(28).
0.7
0.65
0.6
0.55
0.5
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Rate, R
57
58. Hakim, April 2009
Minimum SNR Required for SFR=10-13
Single-level RS Vs EPC-RS
9
8.8
• ISI channel 5+6D-D3, AWGN.
• Shortened RS, GF(212), R=0.89. 8.6
• (24, 14) EPCC + shortened RS
over GF(210), total R=0.89. 8.4
S N R (d B ) 8.2
8
7.8
7.6
7.4
7.2
1/2 K 1K 3/2 K 2K 5/2 K 3K 7/2 K 4K
Sector size
58
60. Hakim, April 2009
Non-binary LDPC: Complexity and Performance
Davey and MacKay (1998) have shown that the near Shannon limit
performance of binary LDPC codes in AWGN can be significantly enhanced
by a move to fields of higher order.
For monotonic improvement in performance with field order the parity
check matrix for short blocks has to be very sparse
• Column weight 3 codes over GF(q) exhibit worse BER as q increases.
• Column weight 2 codes over GF(q) exhibit monotonically lower BER as q
increases.
• Results confirmed by Hu, Eleftheriou, and Arnold (2005): optimum degree
sequence favors a regular graph of degree-2 in all symbol nodes.
Chang and Cruz (2008) studied the decoding time complexity of non-binary
LDPC for PR channels
• Moving from binary to non-binary LDPC results in a gain of around 1 dB.
• Size of the Galois field does not affect the decoding complexity.
• The decoding complexity ratios of non-binary to binary LDPC-coded system
can be as high as 7.42 (in the number of FLP ops).
• Time complexity ratios are always smaller than the ratios of FLP ops.
60
61. Hakim, April 2009
Soft Decoding of EPC-LDPC tensor product code
LDPC
iteration
Convolve
Map to LDPC FFT-
EPCC list γ ( Syni e = j )
bit-level based
a priori info decoder j ∈ GF (26 ) SPA
GF (26 ) over GF(26)
λk λk
rk Global
iteration 1 ≤ i ≤ 390
Binary
Viterbi ˆ
ck
Tensor symbol i
hk :
Correlator(e1) :
qk 0 ≤ j ≤ α 63
− Correlator(e2) γ ( Syni ch = j )
+ Syndrome j j ∈ GF (26 )
Correlator(elmax) 1 ≤ i ≤ 390
List of likely
0 ≤ j ≤ α 63
errors and
RS decoder reliabilities
t=6
:
ˆ
bk :
61
62. Hakim, April 2009
p.m.f. of Tensor Symbol i
0.8
j=233
Pr[Syndrome(i)]=j
0.6
How to generate Syndrome p.m.f. for
0.4 j=66
each Tensor Symbol?
0.2 j=127
0
0 50 100 150 200 250
j
62
63. Hakim, April 2009
SFR Comparison of Single-level LDPC Systems
0
10
CU.I.D
• (4550, 4095) GF(2)-LDPC,
GF(256) LDPC, Col wt 2.
col. wt.= 5, cycle size= 91, and GF(64) LDPC, Col wt 2.
binary BCJR. 10×50 TE. -1 GF(64) LDPC, Col wt 3.
10
GF(2) LDPC, Col wt 5.
• (570, 510) GF(28)-LDPC, 4560
bits, col. wt.= 2, cycle size= 15
Sector Error Rate
symbols, and GF(28)-BCJR. -2
0×50 TE. 10
•(760, 684) GF(26)-LDPC, 4560
bits, col. wt.= 2, cycle size= 19
-3
symbols, and GF(26)-BCJR. 10
0×50 TE.
• (775, 700) GF(26)-LDPC, 4650
-4
bits, col. wt.= 3, cycle size= 25 10
symbols, and GF(26)-BCJR.
0×50 TE.
-5
10
3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8
E / N dB
b o
63
67. Hakim, April 2009
Thesis Contributions
Proposed a channel matched turbo equalization scheme based on the SISO list
decoder of EPCC, termed TE-EPCC.
Demonstrated the “Spectral Thinning” effect achieved by incorporating EPCC in
TE of the Dicode channel.
Derived an upper bound on the ML BER of TE-EPCC.
Proposed a turbo-product code based on EPCC.
Proposed an error-pattern correcting tensor product code that is linear time
encodable.
Derived a fully analytic multinomial method to estimate the SER of RS in ISI.
Designed a two-level coding scheme based on the tensor product of EPCC and
QLDPC that achieves a better complexity-performance trade-off compared to
single-level QLDPC.
Designed a soft iterative decoder of T-EPCC-QLDPC.
67