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ECE 4371, Fall, 2013 Introduction to Telecommunication Engineering/Telecommunication Laboratory. Zhu Han Department of Electrical and Computer Engineering Class 19 Nov. 6 th , 2013. Outline. Convolutional Code Encoder Decoder Viterbi Interleaver Turbo Code LDPC Code Fountain Code.

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ECE 4371, Fall, 2013 Introduction to Telecommunication Engineering/Telecommunication Laboratory

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Ece 4371 fall 2013 introduction to telecommunication engineering telecommunication laboratory
ECE 4371, Fall, 2013Introduction to Telecommunication Engineering/Telecommunication Laboratory

Zhu Han

Department of Electrical and Computer Engineering

Class 19

Nov. 6th, 2013


  • Convolutional Code

    • Encoder

    • Decoder

    • Viterbi

    • Interleaver

  • Turbo Code

  • LDPC Code

  • Fountain Code

Convolutional code introduction
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


  • 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


  • 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


  • (3,2,1) Convolutional encoder

Here each message bit influences

a span of C = n(L+1)=3(1+1)=6successive output bits

Representing convolutional codes code tree
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)

Representing convolutional codes compactly code trellis and state diagram
Representing convolutional codes compactly: code trellis and state diagram

Input state ‘1’

indicated by dashed line

State diagram

Code trellis

Shift register states

Inspecting state diagram structural properties of convolutional codes
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

Distance for some convolutional codes
Distance for some convolutional codes convolutional codes

  • Lower the coding rate, larger the L, then larger the distance

Puncture code
Puncture Code convolutional codes

  • 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.

Decoding of convolutional codes
Decoding of convolutional codes convolutional codes

Example of exhaustive maximal likelihood detection
Example of exhaustive maximal likelihood detection convolutional codes

  • 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?





decoder outputsif this path is selected


Ece 4371 fall 2013 introduction to telecommunication engineering telecommunication laboratory

The largest metric, verify convolutional codes

that you get the same result!

Note also the Hamming distances!

The viterbi algorithm
The Viterbi algorithm convolutional codes

  • 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


The survivor path
The survivor path convolutional codes

  • 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

Example of using the viterbi algorithm
Example of using the Viterbi algorithm convolutional codes

  • 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)

The maximum likelihood path

1 convolutional codes


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)

How to end up decoding
How to end-up decoding? convolutional codes

  • 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!

Concepts to learn
Concepts to Learn convolutional codes

  • 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

Viterbi algorithm
Viterbi Algorithm convolutional codes

  • 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).

Interleaving to get time diversity
Interleaving to get time diversity convolutional codes

Combat burst error. Block and convolutional interleaver

Block interleaver where source bits are read into columns and out as n-bit rows

Turbo codes
Turbo Codes convolutional 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

Motivation performance of turbo codes
Motivation: Performance of Turbo Codes convolutional 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!

Concatenated coding
Concatenated Coding convolutional codes

  • 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)














Parallel concatenated codes
Parallel Concatenated Codes convolutional 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










Pseudo random interleaving
Pseudo-random Interleaving convolutional codes

  • 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.

Recursive systematic convolutional encoding
Recursive Systematic Convolutional Encoding convolutional codes

  • 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.



Constraint Length K= 3



Why interleaving and recursive encoding
Why Interleaving and Recursive Encoding? convolutional codes

  • 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.

Iterative decoding
Iterative Decoding convolutional codes

  • 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.











hard bit






The turbo principle
The Turbo-Principle convolutional codes

  • Turbo codes get their name because the decoder uses feedback, like a turbo engine.

Performance as a function of number of iterations
Performance as a Function of Number of Iterations convolutional codes

  • K=5, r=1/2, L=65,536

Performance factors and tradeoffs
Performance Factors and Tradeoffs convolutional codes

  • 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.

Influence of interleaver size
Influence of Interleaver Size convolutional codes

  • Constraint Length 5. Rate r = 1/2. Log-MAP decoding. 18 iterations. AWGN Channel.







Turbo code summary
Turbo Code Summary convolutional codes

  • 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.

Ldpc introduction
LDPC Introduction convolutional codes

  • 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

Tanner graph 1 2
Tanner Graph (1/2) convolutional codes

  • 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

Decoding of ldpc codes
Decoding of LDPC Codes convolutional codes

  • For linear block codes

    • If c is a valid codeword, we have

      • cHT = 0

    • Else the decoder needs to find out error vector e

  • Graph-based algorithms

    • Sum-product algorithm for general graph-based codes

    • MAP (BCJR) algorithm for trellis graph-based codes

    • Message passing algorithm for bipartite graph-based codes

Pro and con
Pro and Con convolutional codes


    • Near Capacity Performance: Shannon’s Limit

    • Some LDPC Codes perform better than Turbo Codes

    • Trellis diagrams for Long Turbo Codes become very complex and computationally elaborate

    • Low Floor Error

    • Decoding in the Log Domain is quite fast.


    • Long time to Converge to Good Solution

    • Very Long Code Word Lengths for good Decoding Efficiency

    • Iterative Convergence is SLOW

      • Takes ~ 1000 iterations to converge under standard conditions.

    • Due to the above reason

      transmission time increases

      • i.e. encoding, transmission and decoding

    • Hence Large Initial Latency

      • (4086,4608) LPDC codeword has a latency of almost 2 hours

Fountain code

Users reconstruct convolutional codes

Original content as

soon as they receive

enough packets



Encoded packets




Fountain Code

  • Sender sends a potentially limitless stream of encoded bits.

  • Receivers collect bits until they are reasonably sure that they can recover the content from the received bits, and send STOP feedback to sender.

  • Automatic adaptation: Receivers with larger loss rate need longer to receive the required information.

  • Want that each receiver is able to recover from the minimum possible amount of received data, and do this efficiently.

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