Transcript Systematic Design of Space-Time Trellis Codes for Wireless

ECE 6332, Spring, 2014
Wireless Communication
Zhu Han
Department of Electrical and Computer Engineering
Class 20
Apr. 02th, 2014
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
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 length C=n(L+1) is defined as the number of encoded bits a
message bit can influence to
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
Example

(3,2,1) Convolutional encoder
x ' j  mj3  mj2  mj
x '' j  mj3  mj1  mj
x ''' j  mj2  mj
Here each message bit influences
a span of C = n(L+1)=3(1+1)=6
successive output bits
Generator sequences
Convolution point of view in encoding and
generator matrix
3
Example: Using generator matrix
 g  [1 0 11] 
 g ( 2 )  [111 1] 


(1)
Representing convolutional codes: Code tree
(n,k,L) = (2,1,2) encoder
 x ' j  m j2  m j1  m j

 x '' j  m j2  m j
xout  x '1 x ''1 x '2 x ''2 x '3 x ''3 ...
This tells how one input bit
is transformed into two output bits
(initially register is all zero)
 x ' j  0  1  0

 x '' j  0  0
m j 2 m j 1  0 1
 x ' j  0  1  1

 x '' j  0  1
Representing convolutional codes
compactly: code trellis and state diagram
Input state ‘1’
indicated by dashed line
Code trellis
State diagram
Shift register states
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

Lower the coding rate, larger the L, then larger the distance
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.
Decoding of convolutional codes
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
d
decoder outputs
if this path is selected
correct:1+1+2+2+2=8;8  (0.11)  0.88
false:1+1+0+0+0=2;2  (2.30)  4.6
total path metric:  5.48
The largest metric, verify
that you get the same result!
Note also the Hamming distances!
The Viterbi algorithm

Problem of optimum decoding is to find the minimum distance
path from the initial state back to initial state (below from S0 to
S0). The minimum distance is the sum of all path metrics
ln p(y, xm )  j0 ln p( y j | xmj )
Received code
sequence
Decoder’s output sequence
for the m:th path
that is maximized by the correct path

Exhaustive maximum likelihood
method must search all the paths
in phase trellis (2k paths emerging/
entering from 2 L+1 states for
an (n,k,L) code)

The Viterbi algorithm gets its
efficiency via concentrating into
survivor paths of the trellis
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 can
be discarded -> all path alternatives need not
to be considered

Note that in principle whole transmitted
sequence must be received before decision. 2L nodes, determined
However, in practice storing of states for by memory depth
2k branches enter each node
input length of 5L is quite adequate
Example of using the Viterbi algorithm

Assume the received sequence is
y  01101111010001
and the (n,k,L)=(2,1,2) encoder shown below. Determine the
Viterbi decoded output sequence!
states
(Note that for this encoder code rate is 1/2 and memory depth L = 2)
The maximum likelihood path
After register length L+1=3
branch pattern begins to repeat
(1)
Smaller accumulated
metric selected
(1)
1
(1)
(1)
(2)
1
(0)
(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?

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 point
at a memory depth J
– J is a random variable whose applicable
magnitude shown in the figure (5L)
has been experimentally tested for
negligible error rate increase
– Note that this also introduces the
delay of 5L!
J  5L stages of the trellis
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
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).
Interleaving to get time diversity
Combat burst error. Block and convolutional interleaver
Block interleaver where source bits are read into columns and out as n-bit rows
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
Motivation: Performance of Turbo Codes

Comparison:
– Rate 1/2 Codes.
Theoretical Limit!
– 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
Gain of almost 2 dB!
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
Deinterleaver
Inner
Decoder
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.
Interleaver
Input
Encoder
#1
Systematic Output
MUX
Encoder
#2
Parity
Output
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.
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.
xi( 0 )
mi
D
xi
D
xi(1)
Constraint Length K= 3
xi( 0 )
mi
An arbitrary input will cause
a “good” (high weight)
output with high probability.
Some inputs will cause “bad”
(low weight) outputs.
xi
ri
D
D
xi(1)
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.
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
Decoder
#1
systematic
data
parity
data
DeMUX
Interleaver
Interleaver
Decoder
#2
hard bit
decisions
The Turbo-Principle

Turbo codes get their name because the decoder uses feedback,
like a turbo engine.
Performance as a Function of Number of Iterations
K=5, r=1/2, L=65,536
0
10
-1
10
1 iteration
-2
10
2 iterations
-3
10
BER

-4
10
6 iterations
3 iterations
-5
10
10 iterations
-6
10
18 iterations
-7
10
0.5
1.5
1
Eb/No in dB
2
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.
Influence of Interleaver Size
Constraint Length 5. Rate r = 1/2. Log-MAP decoding. 18 iterations.
AWGN Channel.
-1
10
L=
L=
L=
L=
-2
10
1,024
4,096
16,384
65,536
Voice
-3
10
BER

Video
Conferencing
-4
10
Replayed
Video
-5
10
-6
10
Data
-7
10
0.5
1
1.5
Eb/No in dB
2
2.5
Power Efficiency of Existing Standards
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.
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
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
s0
H=
s1
s2
s3
s4
s5
11100000
00011100
10010010
01001001
s6
s7
Symbol nodes
Check nodes
c0
c1
c2
c3
Decoding of LDPC Codes

For linear block codes
– If c is a valid codeword, we have

c HT = 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

ADVANTAGES
– 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.

DISADVANTAGES
– 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

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.
Transmission
Encoded packets
Encoding
Engine
Original
content
Users reconstruct
Original content as
soon as they receive
enough packets