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EE 6331, Spring, 2009 Advanced Telecommunication. Zhu Han Department of Electrical and Computer Engineering Class 21 Apr. 14 th , 2009. Outline. Shrividya has a baby girl!!! Review Scrambler Cyclic Code Encryption Basics Convolutional code Encoder Decoder: Viterbi decoding
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EE 6331, Spring, 2009Advanced Telecommunication Zhu Han Department of Electrical and Computer Engineering Class 21 Apr. 14th, 2009
Outline Shrividya has a baby girl!!! Review Scrambler Cyclic Code Encryption Basics Convolutional code Encoder Decoder: Viterbi decoding Turbo Code LDPC Code TCM modulation Elementary field theory and RS codes (1 - 4pm, SR 140). ECE6331
Scrambling • Example 7.2 • Exercise: 100000000000 ECE6331
Cyclic code • Cyclic codes are of interest and importance because • They posses rich algebraic structure that can be utilized in a variety of ways. • They have extremely concise specifications. • They can be efficiently implemented using simple shift register • Many practically important codes are cyclic • In practice, cyclic codes are often used for error detection (Cyclic redundancy check, CRC) • Used for packet networks • When an error is detected by the receiver, it requests retransmission • ARQ ECE6331
Cyclic Code Decoder • Divider • Similar structure as multiplier for encoder ECE6331
Cyclic Redundancy Checks (CRC) ECE6331
Example of CRC ECE6331
Checking for errors ECE6331
BCH Code • Bose, Ray-Chaudhuri, Hocquenghem • Multiple error correcting ability • Ease of encoding and decoding • Page 653 • Most powerful cyclic code • For any positive integer m and t<2^(m-1), there exists a t-error correcting (n,k) code with n=2^m-1 and n-k<=mt. • Industry standards • (511, 493) BCH code in ITU-T. Rec. H.261 “video codec for audiovisual service at kbit/s” a video coding a standard used for video conferencing and video phone. • (40, 32) BCH code in ATM (Asynchronous Transfer Mode) ECE6331
Reed-Solomon Codes • An important subclass of non-binary BCH • Page 654 • Wide range of applications • Storage devices (tape, CD, DVD…) • Wireless or mobile communication • Satellite communication • Digital television/Digital Video Broadcast(DVB) • High-speed modems (ADSL, xDSL…) ECE6331
Encryption • Encryption is a translation of data into a secret code. Encryption is the most effective way to achieve data security. To read an encrypted file, you must have access to a secret key that enables you to decrypt it. Unencrypted data is called plain text; encrypted data is referred to as cipher (text). • Encryption can be used to ensure secrecy, but other techniques are still needed to make communications secure: authentication, authorization, and message integrity. • Message integrity - both parties will always wish to be confident that a message has not been altered during transmission. The encryption makes it difficult for a third party to read a message, but that third party may still be able to alter it in a useful way. • Authentication is a way to ensure users are who they say they are - that the user who attempts to perform functions in a system is in fact the user who is authorized to do so. • Authorization protects computer resources (data, files, programs, devices) by allowing those resources to be used by resource consumers having been granted authority to use them. • Digital rights management etc. ECE6331
Encryption – cipher taxonomy CIPHERS ROTORMACHINES CLASSICALCIPHERS MODERN CIPHERS PRIVATE KEY SUPERPOSITION PUBLIC KEY Quantum CIPHERS TRANSPOSITION ECE6331
Convolutional Code Introduction • Convolutional codes map information to code bits sequentially by convolving a sequence of information bits with “generator” sequences • A convolutional encoder encodes K information bits to N>K code bits at one time step • Convolutional codes can be regarded as block codes for which the encoder has a certain structure such that we can express the encoding operation as convolution ECE6331
Encoder • Convolutional codes are applied in applications that require good performance with low implementation cost. They operate on code streams (not in blocks) • Convolution codes have memory that utilizes previous bits to encode or decode following bits (block codes are memoryless) • Convolutional codes achieve good performance by expanding their memory depth • Convolutional codes are denoted by (n,k,L), where L is code (or encoder) Memory depth (number of register stages) • Constraint lengthC=n(L+1) is defined as the number of encoded bits a message bit can influence to ECE6331
Example • Convolutional encoder, k = 1, n = 2, L=2 • Convolutional encoder is a finite state machine (FSM) processing information bits in a serial manner • Thus the generated code is a function of input and the state of the FSM • In this (n,k,L) = (2,1,2) encoder each message bit influences a span of C=n(L+1)=6 successive output bits = constraint length C • Thus, for generation of n-bit output, we require n shift registers in k = 1 convolutional encoders ECE6331
Example • (3,2,1) Convolutional encoder Here each message bit influences a span of C = n(L+1)=3(1+1)=6successive output bits ECE6331
Generator sequences ECE6331
Convolution point of view in encoding and generator matrix 3 ECE6331
Example: Using generator matrix ECE6331
Representing convolutional codes: Code tree (n,k,L) = (2,1,2) encoder This tells how oneinput bit is transformed into two output bits (initially register is all zero) ECE6331
Representing convolutional codes compactly: code trellis and state diagram Input state ‘1’ indicated by dashed line State diagram Code trellis Shift register states ECE6331
Inspecting state diagram: Structural properties of convolutional codes • Each new block of k input bits causes a transition into new state • Hence there are 2k branches leaving each state • Assuming encoder zero initial state, encoded word for any input of k bits can thus be obtained. For instance, below for u=(1 1 1 0 1), encoded word v=(1 1, 1 0, 0 1, 0 1, 1 1, 1 0, 1 1, 1 1) is produced: -encoder state diagram for (n,k,L)=(2,1,2) code - note that the number of states is 2L+1 = 8 ECE6331
Distance for some convolutional codes • Lower the coding rate, larger the L, then larger the distance ECE6331
Puncture Code • A sequence of coded bits is punctured by deleting some of the bits in the sequence according to some fixed rule. • The resulting coding rate is increased. So a lower rate code can be extended to a sequence of higher rate codes. ECE6331
Decoding of convolutional codes ECE6331
Example of exhaustive maximal likelihood detection • Assume a three bit message is transmitted [and encoded by (2,1,2) convolutional encoder]. To clear the decoder, two zero-bits are appended after message. Thus 5 bits are encoded resulting 10 bits of code. Assume channel error probability is p = 0.1. After the channel 10,01,10,11,00 is produced (including some errors). What comes after the decoder, e.g. what was most likely the transmitted code and what were the respective message bits? a b states c decoder outputsif this path is selected d ECE6331
The largest metric, verify that you get the same result! Note also the Hamming distances! ECE6331
The Viterbi algorithm • Problem of optimum decoding is to find the minimum distance path from the initial state back to initial state (below from S0to S0). The minimum distance is the sum of all path metricsthat is maximized by the correct path • Exhaustive maximum likelihood method must search all the pathsin phase trellis (2k paths emerging/entering from 2 L+1 states for an (n,k,L) code) • The Viterbi algorithm gets itsefficiency via concentrating intosurvivor paths of the trellis Decoder’s output sequence for the m:th path Received code sequence ECE6331
The survivor path • Assume for simplicity a convolutional code with k=1, and up to 2k = 2 branches can enter each state in trellis diagram • Assume optimal path passes S. Metric comparison is done by adding the metric of S into S1 and S2. At the survivor path the accumulated metric is naturally smaller (otherwise it could not be the optimum path) • For this reason the non-survived path canbe discarded -> all path alternatives need notto be considered • Note that in principle whole transmittedsequence must be received before decision.However, in practice storing of states for input length of 5L is quite adequate ECE6331
Example of using the Viterbi algorithm • Assume the received sequence is and the (n,k,L)=(2,1,2) encoder shown below. Determine the Viterbi decoded output sequence! (Note that for this encoder code rate is 1/2 and memory depth L = 2) ECE6331
1 1 The maximum likelihood path Smaller accumulated metric selected After register length L+1=3 branch pattern begins to repeat (Branch Hamming distances in parenthesis) First depth with two entries to the node The decoded ML code sequence is 11 10 10 11 00 00 00 whose Hamming distance to the received sequence is 4 and the respective decoded sequence is 1 1 0 0 0 0 0 (why?). Note that this is the minimum distance path. (Black circles denote the deleted branches, dashed lines: '1' was applied) ECE6331
How to end-up decoding? • In the previous example it was assumed that the register was finally filled with zeros thus finding the minimum distance path • In practice with long code words zeroing requires feeding of long sequence of zeros to the end of the message bits: this wastes channel capacity & introduces delay • To avoid this path memory truncation is applied: • Trace all the surviving paths to the depth where they merge • Figure right shows a common pointat a memory depth J • J is a random variable whose applicablemagnitude shown in the figure (5L) has been experimentally tested fornegligible error rate increase • Note that this also introduces thedelay of 5L! ECE6331
Concepts to Learn • You understand the differences between cyclic codes and convolutional codes • You can create state diagram for a convolutional encoder • You know how to construct convolutional encoder circuits based on knowing the generator sequences • You can analyze code strengths based on known code generation circuits / state diagrams or generator sequences • You understand how to realize maximum likelihood convolutional decoding by using exhaustive search • You understand the principle of Viterbi decoding ECE6331
Viterbi Algorithm • As a youth, Life Fellow Andrew Viterbi never envisioned that he’d create an algorithm used in every cellphone or that he would cofound Qualcomm, a Fortune 500 company that is a worldwide leader in wireless technology. • Viterbi came up with the idea for that algorithm while he was an engineering professor at the University of California at Los Angeles (UCLA) and then at the University of California at San Diego (UCSD), in the 1960s. Today, the algorithm is used in digital cellphones and satellite receivers to transmit messages so they won’t be lost in noise. The result is a clear undamaged message thanks to a process called error correction coding. This algorithm is currently used in most cellphones. • “The algorithm was originally created for improving communication from space by being able to operate with a weak signal but today it has a multitude of applications,” Viterbi says. • For the algorithm, which carries his name, he was awarded this year’s Benjamin Franklin Medal in electrical engineering by the Franklin Institute in Philadelphia, one of the United States’ oldest centers of science education and development. The institute serves the public through its museum, outreach programs, and curatorial work. The medal, which Viterbi received in April, recognizes individuals who have benefited humanity, advanced science, and deepened the understanding of the universe. It also honors contributions in life sciences, physics, earth and environmental sciences, and computer and cognitive sciences. • Qualcomm wasn’t the first company Viterbi started. In the late 1960s, he and some professors from UCLA and UCSD founded Linkabit, which developed a video scrambling system called Videocipher for the fledgling cable network Home Box Office. The Videocipher encrypts a video signal so hackers who haven’t paid for the HBO service can’t obtain it. • Viterbi, who immigrated to the United States as a four-year-old refugee from facist Italy, left Linkabit to help start Qualcomm in 1985. One of the company’s first successes was OmniTracs, a two-way satellite communication system used by truckers to communicate from the road with their home offices. The system involves signal processing and an antenna with a directional control that moves as the truck moves so the antenna always faces the satellite. OmniTracs today is the transportation industry’s largest satellite-based commercial mobile system. • Another successful venture for the company was the creation of code-division multiple access (CDMA), which was introduced commercially in 1995 in cellphones and is still big today. CDMA is a “spread-spectrum” technology—which means it allows many users to occupy the same time and frequency allocations in a band or space. It assigns unique codes to each communication to differentiate it from others in the same spectrum. • Although Viterbi retired from Qualcomm as vice chairman and chief technical officer in 2000, he still keeps busy as the president of the Viterbi Group, a private investment company specializing in imaging technologies and biotechnology. He’s also professor emeritus of electrical engineering systems at UCSD and distinguished visiting professor at Technion-Israel Institute of Technology in Technion City, Haifa. In March he and his wife donated US $52 million to the University of Southern California in Los Angeles, the largest amount the school ever received from a single donor. • To honor his generosity, USC renamed its engineering school the Andrew and Erna Viterbi School of Engineering. It is one of four in the nation to house two active National Science Foundation–supported engineering research centers: the Integrated Media Systems Center (which focuses on multimedia and Internet research) and the Biomimetic Research Center (which studies the use of technology to mimic biological systems). ECE6331
Turbo Codes • Backgound • Turbo codes were proposed by Berrou and Glavieux in the 1993 International Conference in Communications. • Performance within 0.5 dB of the channel capacity limit for BPSK was demonstrated. • Features of turbo codes • Parallel concatenated coding • Recursive convolutional encoders • Pseudo-random interleaving • Iterative decoding ECE6331
Motivation: Performance of Turbo Codes • Comparison: • Rate 1/2 Codes. • K=5 turbo code. • K=14 convolutional code. • Plot is from: • L. Perez, “Turbo Codes”, chapter 8 of Trellis Coding by C. Schlegel. IEEE Press, 1997 Theoretical Limit! Gain of almost 2 dB! ECE6331
Concatenated Coding • A single error correction code does not always provide enough error protection with reasonable complexity. • Solution: Concatenate two (or more) codes • This creates a much more powerful code. • Serial Concatenation (Forney, 1966) Outer Encoder Block Interleaver Inner Encoder Channel Outer Decoder De- interleaver Inner Decoder ECE6331
Parallel Concatenated Codes • Instead of concatenating in serial, codes can also be concatenated in parallel. • The original turbo code is a parallel concatenation of two recursive systematic convolutional (RSC) codes. • systematic: one of the outputs is the input. Systematic Output Input Encoder #1 MUX Parity Output Interleaver Encoder #2 ECE6331
Pseudo-random Interleaving • The coding dilemma: • Shannon showed that large block-length random codes achieve channel capacity. • However, codes must have structure that permits decoding with reasonable complexity. • Codes with structure don’t perform as well as random codes. • “Almost all codes are good, except those that we can think of.” • Solution: • Make the code appear random, while maintaining enough structure to permit decoding. • This is the purpose of the pseudo-random interleaver. • Turbo codes possess random-like properties. • However, since the interleaving pattern is known, decoding is possible. ECE6331
Recursive Systematic Convolutional Encoding • An RSC encoder can be constructed from a standard convolutional encoder by feeding back one of the outputs. • An RSC encoder has an infinite impulse response. • An arbitrary input will cause a “good” (high weight) output with high probability. • Some inputs will cause “bad” (low weight) outputs. D D Constraint Length K= 3 D D ECE6331
Why Interleaving and Recursive Encoding? • In a coded systems: • Performance is dominated by low weight code words. • A “good” code: • will produce low weight outputs with very low probability. • An RSC code: • Produces low weight outputs with fairly low probability. • However, some inputs still cause low weight outputs. • Because of the interleaver: • The probability that both encoders have inputs that cause low weight outputs is very low. • Therefore the parallel concatenation of both encoders will produce a “good” code. ECE6331
Iterative Decoding • There is one decoder for each elementary encoder. • Each decoder estimates the a posteriori probability (APP) of each data bit. • The APP’s are used as a priori information by the other decoder. • Decoding continues for a set number of iterations. • Performance generally improves from iteration to iteration, but follows a law of diminishing returns. Deinterleaver APP APP Interleaver Decoder #1 Decoder #2 systematic data hard bit decisions parity data DeMUX Interleaver ECE6331
The Turbo-Principle • Turbo codes get their name because the decoder uses feedback, like a turbo engine. ECE6331
Performance as a Function of Number of Iterations • K=5, r=1/2, L=65,536 ECE6331
Performance Factors and Tradeoffs • Complexity vs. performance • Decoding algorithm. • Number of iterations. • Encoder constraint length • Latency vs. performance • Frame size. • Spectral efficiency vs. performance • Overall code rate • Other factors • Interleaver design. • Puncture pattern. • Trellis termination. ECE6331
Influence of Interleaver Size • Constraint Length 5. Rate r = 1/2. Log-MAP decoding. 18 iterations. AWGN Channel. Voice Video Conferencing Replayed Video Data ECE6331
Turbo Code Summary • Turbo code advantages: • Remarkable power efficiency in AWGN and flat-fading channels for moderately low BER. • Deign tradeoffs suitable for delivery of multimedia services. • Turbo code disadvantages: • Long latency. • Poor performance at very low BER. • Because turbo codes operate at very low SNR, channel estimation and tracking is a critical issue. • The principle of iterative or “turbo” processing can be applied to other problems. • Turbo-multiuser detection can improve performance of coded multiple-access systems. ECE6331
LDPC Introduction • Low Density Parity Check (LDPC) • History of LDPC codes • Proposed by Gallager in his 1960 MIT Ph. D. dissertation • Rediscovered by MacKay and Richardson/Urbanke in 1999 • Features of LDPC codes • Performance approaching Shannon limit • Good block error correcting performance • Suitable for parallel implementation • Advantages over turbo codes • LDPC do not require a long interleaver • LDPC’s error floor occurs at a lower BER • LDPC decoding is not trellis based ECE6331
Tanner Graph (1/2) • Tanner Graph (A kind of bipartite graph) • LDPC codes can be represented by a sparse bipartite graph • Si: the message nodes (or called symbol nodes) • Ci: the check nodes • Because G is the null space of H, H.xT = 0 • According to the equation above, we can define some relation between the message bits • Example • n = 7, k=3, J = 4, λ=1 ECE6331