digital communication systems - siit.tu.ac.th€¦ · c. e. shannon (1916-2001) 4 1938 mit...

Digital Communication SystemsECS 452

Asst. Prof. Dr. Prapun Suksompong(ผศ.ดร.ประพันธ ์สขุสมปอง)

prapun@siit.tu.ac.th

1. Intro to Digital Communication Systems

Office Hours: BKD, 6th floor of Sirindhralai building

Monday 10:00-10:40Tuesday 12:00-12:40Thursday 14:20-15:30

“The fundamental problem of

communication is that of

reproducing at one point either exactly or approximately a message selected at another point.”

Shannon, Claude. A Mathematical Theory Of

Communication. (1948)

Shannon: Father of the Info. Age

3 [http://www.uctv.tv/shows/Claude-Shannon-Father-of-the-Information-Age-6090][http://www.youtube.com/watch?v=z2Whj_nL-x8]

Documentary Co-produced by the

Jacobs School, UCSD-TV, and the California Institute for Telecommunications and Information Technology

Won a Gold award in the Biography category in the 2002 Aurora Awards.

C. E. Shannon (1916-2001)

1938 MIT master's thesis: A Symbolic Analysis of Relay and Switching Circuits

Insight: The binary nature of Booleanlogic was analogous to the ones and zeros used by digital circuits.

The thesis became the foundation of practical digital circuit design.

The first known use of the term bit to refer to a “binary digit.”

Possibly the most important, and also the most famous, master’s thesis of the century.

It was simple, elegant, and important.

C. E. Shannon: Master Thesis

Boole/Shannon Celebration

Events in 2015 and 2016 centered around the work of George Boole, who was born 200 years ago, and Claude E. Shannon, born 100 years ago.

Events were scheduled both at the University College

Cork (UCC), Ireland and the Massachusetts

Institute of Technology (MIT)

http://www.rle.mit.edu/booleshannon/

An Interesting Book

The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

by Paul J. Nahin

ISBN: 9780691151007

http://press.princeton.edu/titles/9819.html

C. E. Shannon (Con’t)

1948: A Mathematical Theory of Communication Bell System Technical Journal,

vol. 27, pp. 379-423, July-October, 1948.

September 1949: Book published. Include a new section by Warren Weaver that applied Shannon's theory to human communication.

Create the architecture and concepts governing digital communication.

Invent Information Theory: Simultaneously founded the subject, introduced all of the major concepts, and stated and proved all the fundamental theorems.

A Mathematical Theory of Communication

Link posted in the “references” section of the website.

[An offprint from the Bell System Technical Journal]

C. E. Shannon

10 …with some remarks by Toby Berger.

Claude E. Shannon Award

Claude E. Shannon (1972)

David S. Slepian (1974)

Robert M. Fano (1976)

Peter Elias (1977)

Mark S. Pinsker (1978)

Jacob Wolfowitz (1979)

W. Wesley Peterson (1981)

Irving S. Reed (1982)

Robert G. Gallager (1983)

Solomon W. Golomb (1985)

William L. Root (1986)

James L. Massey (1988)

Thomas M. Cover (1990)

Andrew J. Viterbi (1991)

Elwyn R. Berlekamp (1993)

Aaron D. Wyner (1994)

G. David Forney, Jr. (1995)

Imre Csiszár (1996)

Jacob Ziv (1997)

Neil J. A. Sloane (1998)

Tadao Kasami (1999)

Thomas Kailath (2000)

Jack KeilWolf (2001)

Toby Berger (2002)

Lloyd R. Welch (2003)

Robert J. McEliece (2004)

Richard Blahut (2005)

Rudolf Ahlswede (2006)

Sergio Verdu (2007)

Robert M. Gray (2008)

Jorma Rissanen (2009)

Te Sun Han (2010)

Shlomo Shamai (Shitz) (2011)

Abbas El Gamal (2012)

Katalin Marton (2013)

János Körner (2014)

Arthur Robert Calderbank (2015)

Alexander S. Holevo (2016)David Tse (2017)

[ http://www.itsoc.org/honors/claude-e-shannon-award ]

IEEE Richard W. Hamming Medal

1988 - Richard W. Hamming1989 - Irving S. Reed1990 - Dennis M. Ritchie and Kenneth L. Thompson1991 - Elwyn R. Berlekamp1992 - Lotfi A. Zadeh1993 - Jorma J. Rissanen1994 - Gottfried Ungerboeck1995 - Jacob Ziv1996 - Mark S. Pinsker1997 - Thomas M. Cover1998 - David D. Clark1999 - David A. Huffman2000 - Solomon W. Golomb2001 - A. G. Fraser2002 - Peter Elias2003 - Claude Berrou and Alain Glavieux2004 - Jack K. Wolf2005 - Neil J.A. Sloane

2006 -Vladimir I. Levenshtein2007 - Abraham Lempel2008 - Sergio Verdú2009 - Peter Franaszek2010 -Whitfield Diffie,

Martin Hellman, and Ralph Merkle

2011 - Toby Berger2012 - Michael Luby, Amin Shokrollahi2013 - Arthur Robert Calderbank2014 - Thomas Richardson

and Rüdiger L. Urbanke2015 - Imre Csiszar2016 - Abbas El Gamal2017 - Shlomo Shamai2018 - Erdal Arikan

“For contributions to Information Theory, including source coding and its applications.”

[http://www.ieee.org/documents/hamming_rl.pdf]

[http://www.cvaieee.org/html/toby_berger.html]

Information Theory

The science of information theory tackles the following questions [Berger]

1. What is information, i.e., how do we measure it quantitatively?

2. What factors limit the reliability with which information generated at one point can be reproduced at another, and what are the resulting limits?

3. How should communication systems be designed in order to achieve or at least to approach these limits?

Elements of communication sys.

Noise, Interference,Distortion

ReceiverTransmitterInformation Source DestinationChannel

ReceivedSignal

TransmittedSignalMessage Message

ModulationCoding

Analog (continuous)Digital (discrete)

+ Transmission loss (attenuation)

AmplificationDemodulationDecodingFiltering

(ECS 332)

[Shannon, 1948]

The Switch to Digital TV

Japan: Starting July 24, 2011, the analog broadcast has ceased and only digital broadcast is available.US: Since June 12, 2009, full-power television stations nationwide have been broadcasting exclusively in a digital format.Thailand: Use DVB-T2. Launched in 2014.

[https://upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Digital_broadcast_standards.svg/800px-Digital_broadcast_standards.svg.png]

News: The Switch to Digital Radio

Norway (the mountainous nation of 5 million) is the first country to shut down its national FM radio network in favor of digital radio.

Start on January 11, 2017 At which point, 99.5% of the population has access to DAB reception

with almost three million receivers sold. 70% of Norwegian households regularly tune in digitally

Take place over a 12-month period, conducting changes region by region.

December 13, 2017: All national networks are DAB-only. Local broadcasters have five years to phase out their FM stations.

New format: Digital Audio Broadcasting (DAB)http://gizmodo.com/norway-is-killing-fm-radio-tomorrow-1791019824http://www.worlddab.org/country-information/norwayhttp://www.smithsonianmag.com/smart-news/norway-killed-radio-star-180961761/http://www.latimes.com/world/la-fg-norway-radio-20170114-story.htmlhttps://www.newscientist.com/article/2117569-norway-is-first-country-to-turn-off-fm-radio-and-go-digital-only/http://fortune.com/2017/12/18/norway-fm-radio-digital-audio-broadcasting/

Digital Audio Broadcasting

Initiated as a European research project in the 1980s.

The Norwegian Broadcasting Corporation (NRK) launched the first DAB channel in the world on 1 June 1995 (NRK Klassisk)

The BBC and Swedish Radio (SR) launched their first DAB digital radio broadcasts in September 1995.

Audio quality varies depending on the bitrate used.

The Switch to DAB in Norway

Co-exist with FM since 1995. Provide a clearer and more reliable network that can better

cut through the country's sparsely populated rocky terrain. FM has always been problematic in Norway since the nation’s

mountains and fjords makes getting clear FM signals difficult.

Offer more channels at a fraction of the cost. Allow 8 times as many radio stations Norway currently has five national FM radio stations. With DAB, it will be able to have around 40.

The FM radio infrastructure was coming to the end of its life, Need to either replace it or fully commit to DAB anyway

Can run at lower power levels the infrastructure electricity bills are lower

The Switch to Digital Radio

Switzerland and Denmark are also interested in phasing out FM Great Britain says it will look at making the switch

once 50 percent of listeners use digital formats currently at 35 percent Unlikely to happen before 2020.

and when the DAB signal reaches 90 percent of the population. Germany had set a 2015 date for dumping FM many years ago, but

lawmakers reversed that decision in 2011. In North America,

FM radio, which has been active since the 1940s, shows no sign of being replaced any time soon, either in the United States or Canada.

There are around 4,000 stations using HD radio technology in the United States, and HD radio receivers are now common fixtures in new cars.

In Thailand, NBTC planed to start digital radio trial within 2018.

http://thaidigitalradio.com/ความคบืหน้าล่าสดุ-วิทยุ/

Selected by the U.S. FCC in 2002 as a digital audio broadcasting method for the United States.

Embed digital signal “on-frequency” immediately above and below a station’s standard analog signal

Provide the means to listen to the same program in either HD (digital radio with less noise) or as a standard broadcast (analog radio with standard sound quality).

Spectrum of FM broadcast station

without HD Radio with HD Radio

[ https://en.wikipedia.org/w

iki/HD

_Radio

Countries using DAB/DMB

21 https://en.wikipedia.org/wiki/Digital_audio_broadcasting

2017 hurricane season in the US

22 http://edition.cnn.com/2017/10/10/weather/hurricane-nate-maria-irma-harvey-impact-look-back-trnd/index.htmlhttps://www.vox.com/energy-and-environment/2017/9/28/16362522/hurricane-maria-2017-irma-harvey-rain-flooding-climate-change

Radio broadcasts are critical during a disaster

In areas hit hardest by things like hurricanes, earthquakes, fires, or even shootings:

Vulnerabilities of mobile phone infrastructure Cell phone infrastructure is often knocked out. Overwhelmed from everyone trying to access information. Three weeks after Hurricane Maria pummeled Puerto Rico,

more than 76 percent of cell sites still aren’t functioning.

Radio broadcast signals, which use low frequencies and can travel much further distances and penetrate through obstacles, usually remain up.

FM capability in modern cellphone

FM capability is baked into the Qualcomm LTE modem inside nearly every cellphone. You can easily turn your phone into an FM radio if it has an embedded chipset and the

proper circuitry to connect that chip to an FM antenna. Need

an app like NextRadio something to act as an antenna, such as headphones or nonwireless speakers.

Until a few years ago, device manufacturers disabled the function. Wireless carriers wanted customers to stream music and podcasts, and consume more

data. Broadcasters and public safety officials have long urged handset manufacturers and

wireless carriers to universally activate the FM chip. ITU (International Telecommunications Union) issued an opinion in March 2017 urging all

mobile phone makers to include and turn on FM radios on their devices. Major US carriers now allow FM chips to be turned on. Manufacturers like Samsung,

LG, HTC and Motorola have activated FM radio on their phones. September 28, 2017: FCC blasted Apple for not activating FM receivers built into

iPhones. Apple responded that iPhone 7 and iPhone 8, and iPhone X don’t use a chipset with an embedded FM radio.

https://www.cnet.com/news/everything-you-need-to-know-about-fm-radio-on-your-phone/https://spectrum.ieee.org/tech-talk/consumer-electronics/gadgets/fcc-wants-apple-to-turn-on-iphone-fm-receivers-that-may-not-existhttps://www.wired.com/2016/07/phones-fm-chips-radio-smartphone/

Pokémon Communications

Pikachu's language

Some of Pikachu's speech is consistent enough that it seems that some phrases actually mean something.

Pikachu always uses "Pikapi" when referring to Ash (notice that it sounds somewhat similar to "Satoshi").

Pi-Kachu: He says this during the sponsor spots in the original Japanese, Pochama (Piplup)

Pikachu-Pi: Kasumi (Misty)

Pika-Chu: Takeshi (Brock), Kibago (Axew)

Pikaka: Hikari (Dawn)

PiPiPi: Togepy (Togepi)

PikakaPika: Fushigidane (Bulbasaur)

PikaPika: Zenigame (Squirtle), Mukuhawk (Staraptor), Goukazaru (Infernape) or Gamagaru (Palpitoad)

PiPi-kachu: Rocket-dan (Team Rocket)

Pi-Pikachu: Get da ze! (He says this after Ash wins a Badge, catches a new Pokémon or anything similar.)

Pikachu may not be the only one to use this phrase, as other Pokémon do this as well. For example, when Iris caught Emolga, Axew said Ax-Axew (Ki-Kibago in the Japanese).

Pika-Pikachu: He says this when referring to himself.

Four-symbol variable-length code?

[https://www.youtube.com/watch?v=XumQrRkGXck]

Rate-Distortion Theory

The theory of lossy source coding

Asst. Prof. Dr. Prapun Suksompongprapun@siit.tu.ac.th

2. Source Coding

Elements of digital commu. sys.

Noise & In

terferen

Information Source

Destination

Channel

ReceivedSignal

TransmittedSignal

Message

Recovered Message

Source Encoder

Channel Encoder

DigitalModulator

Source Decoder

Channel Decoder

DigitalDemodulator

Transmitter

Receiver

Remove redundancy

Add systematic redundancy

System Under Consideration

Noise & In

terferen

Information Source

Destination

Channel

ReceivedSignal

TransmittedSignal

Message

Recovered Message

Source Encoder

Channel Encoder

DigitalModulator

Source Decoder

Channel Decoder

DigitalDemodulator

Transmitter

Receiver

Remove redundancy

Main Reference

Elements of Information Theory

2006, 2nd Edition

Chapters 2, 4 and 5

‘the jewel in Stanford's crown’

One of the greatest information theorists since Claude Shannon (and the one most like Shannon in approach, clarity, and taste).

English Alphabet (Non-Technical Use)

The ASCII Coded Character Set

6[The ARRL Handbook for Radio Communications 2013]

0 16 32 48 64 80 96 112

(American Standard Code for Information Interchange)

Example: ASCII Encoder

Characterx

Codewordc(x)

⋮E 1000101

⋮L 1001100

⋮O 1001111

⋮V 1010110

SourceEncoder

Information Source

“LOVE”“1001100100111110101101000101”

>> M = 'LOVE';>> X = dec2bin(M,7);>> X = reshape(X',1,numel(X))X =1001100100111110101101000101

MATLAB:

Remark:numel(A) = prod(size(A))(the number of elements in matrix A)

English Redundancy: Ex. 1

J-st tr- t- r--d th-s s-nt-nc-.

yxx cxn xndxrstxndwhxt x xm wrxtxngxvxn xf x rxplxcx xllthx vxwxls wxth xn 'x' (t gts lttl hrdr f y dn'tvn kn whr th vwls r).

To be, or xxx xx xx, xxxx xx xxx xxxxxxxx

Entropy Rate of Thai Text

Introduction to Data Compression

12 [ https://www.khanacademy.org/computing/computer-science/informationtheory/moderninfotheory/v/compressioncodes ]

Introduction to Data Compression

ASCII: Source Alphabet of Size = 128

0 16 32 48 64 80 96 112

(American Standard Code for Information Interchange)

Ex. Source alphabet of size = 4

Ex. DMS (1)

, , , ,X a b c d e

a c a c e c d b c ed a e e d a b b b db b a a b e b e d cc e d b c e c a a ca a e a c c a a d cd e e a a c a a a bb c a e b b e d b cd e b c a e e d d cd a b c a b c d d ed c e a b a a c a d

Information Source

1 , , , , ,50, otherwise

x a b c d ep x

Approximately 20% are letter ‘a’s[GenRV_Discrete_datasample_Ex1.m]

Ex. DMS (1)

17 [GenRV_Discrete_datasample_Ex1.m]

clear all; close all;

S_X = 'abcde'; p_X = [1/5 1/5 1/5 1/5 1/5];

n = 100;MessageSequence = datasample(S_X,n,'Weights',p_X)MessageSequence = reshape(MessageSequence,10,10)

>> GenRV_Discrete_datasample_Ex1

MessageSequence =

eebbedddeceacdbcbedeecacaecedcaedabecccabbcccebdbbbeccbadeaaaecceccdaccedadabceddaceadacdaededcdcade

MessageSequence =

eeeabbacdeeacebeeeadbcadcccdcebdcacccaedebabcbedacdceeeacadddbccbdcbacdeecdedccaeddcbaaeddcecabacdae

Ex. DMS (2)

1,2,3,4X

1 , 1,21 , 2,41 , 3,480, otherwise

Information Source

Approximately 50% are number ‘1’s

2 1 1 2 1 4 1 1 1 11 1 4 1 1 2 4 2 2 13 1 1 2 3 2 4 1 2 42 1 1 2 1 1 3 3 1 11 3 4 1 4 1 1 2 4 14 1 4 1 2 2 1 4 2 14 1 1 1 1 2 1 4 2 42 1 1 1 2 1 2 1 3 22 1 1 1 1 1 1 2 3 22 1 1 2 1 4 2 1 2 1

[GenRV_Discrete_datasample_Ex2.m]

Ex. DMS (2)

19 [GenRV_Discrete_datasample_Ex2.m]

clear all; close all;

S_X = [1 2 3 4]; p_X = [1/2 1/4 1/8 1/8];

n = 20;

MessageSequence = randsrc(1,n,[S_X;p_X]);%MessageSequence = datasample(S_X,n,'Weights',p_X);

rf = hist(MessageSequence,S_X)/n; % Ref. Freq. calc.stem(S_X,rf,'rx','LineWidth',2) % Plot Rel. Freq.hold onstem(S_X,p_X,'bo','LineWidth',2) % Plot pmfxlim([min(S_X)-1,max(S_X)+1])legend('Rel. freq. from sim.','pmf p_X(x)')xlabel('x')grid on

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

Rel. freq. from sim.pmf pX(x)

DMS in MATLABclear all; close all;

S_X = [1 2 3 4]; p_X = [1/2 1/4 1/8 1/8]; n = 1e6;

SourceString = randsrc(1,n,[S_X;p_X]);

rf = hist(SourceString,S_X)/n; % Ref. Freq. calc.stem(S_X,rf,'rx','LineWidth',2) % Plot Rel. Freq.hold onstem(S_X,p_X,'bo','LineWidth',2) % Plot pmfxlim([min(S_X)-1,max(S_X)+1])legend('Rel. freq. from sim.','pmf p_X(x)')xlabel('x')grid on

SourceString = datasample(S_X,n,'Weights',p_X);

Alternatively, we can also use

[GenRV_Discrete_datasample_Ex.m]

A more realistic example of pmf:

21 [http://en.wikipedia.org/wiki/Letter_frequency]

Relative freq. of letters in the English language

A more realistic example of pmf:

Relative freq. of letters in the English languageordered by frequency

[http://en.wikipedia.org/wiki/Letter_frequency]

Characterx

Codewordc(x)

⋮E 1000101

⋮L 1001100

⋮O 1001111

⋮V 1010110

SourceEncoder

Information Source

“LOVE”“1001100100111110101101000101”

MATLAB:

c(“L”) c(“O”) c(“V”) c(“E”)

Remark:numel(A) = prod(size(A))(the number of elements in matrix A)

0 16 32 48 64 80 96 112

A Byte (8 bits) vs. 7 bits

>> dec2bin('I Love ECS452',7)ans =1001001010000010011001101111111011011001010100000100010110000111010011011010001101010110010

>> dec2bin('I Love ECS452',8)ans =01001001001000000100110001101111011101100110010100100000010001010100001101010011001101000011010100110010

>> dec2bin('I Love You',8)ans =01001001001000000100110001101111011101100110010100100000010110010110111101110101

Geeky ways to express your love

>> dec2bin('i love you',8)ans =01101001001000000110110001101111011101100110010100100000011110010110111101110101

https://www.etsy.com/listing/91473057/binary-i-love-you-printable-for-your?ref=sr_gallery_9&ga_search_query=binary&ga_filters=holidays+-supplies+valentine&ga_search_type=all&ga_view_type=galleryhttp://mentalfloss.com/article/29979/14-geeky-valentines-day-cardshttps://www.etsy.com/listing/174002615/binary-love-geeky-romantic-pdf-cross?ref=sr_gallery_26&ga_search_query=binary&ga_filters=holidays+-supplies+valentine&ga_search_type=all&ga_view_type=galleryhttps://www.etsy.com/listing/185919057/i-love-you-binary-925-silver-dog-tag-can?ref=sc_3&plkey=cdf3741cf5c63291bbc127f1fa7fb03e641daafd%3A185919057&ga_search_query=binary&ga_filters=holidays+-supplies+valentine&ga_search_type=all&ga_view_type=gallery http://www.cafepress.com/+binary-code+long_sleeve_tees

Summary: Source Encoder

SourceEncoder

Information Source

“LOVE”“1001100100111110101101000101”

c(“L”) c(“O”) c(“V”) c(“E”)

• An encoder · is a function that maps each of the symbol in the source alphabet into a corresponding (binary) codeword.

• The list for such mapping is called the codebook.

Source Symbolx

Codewordc(x)

⋮E 1000101

⋮L 1001100

⋮O 1001111

⋮V 1010110

Discrete Memoryless Source (DMS)

• The codewordcorresponding to a source symbol is denoted by

.• the length of

• Each codeword is constructed from a code alphabet.

• For binary codeword, the code alphabet is 0,1

source string encoded string

• The source alphabet is the collection of all possible source symbols.

• Each symbol that the source generates is assumed to be randomly selected from the source alphabet.

w/o extension

Morse code

Telegraph network

Samuel Morse, 1838

A sequence of on-off tones (or , lights, or clicks)

(wired and wireless)

Example

29 [http://www.wolframalpha.com/input/?i=%22I+love+you.%22+in+Morse+code]

Example

Morse code: Key Idea

Frequently-used characters are mapped to short codewords.

Relative frequencies of letters in the English language

Frequently-used characters (e,t) are mapped to short codewords.

Relative frequencies of letters in the English language

Frequently-used characters (e,t) are mapped to short codewords.

Basic form of compression.

รหสัมอร์สภาษาไทย

Character Codeword

⋮E 1000101

⋮L 1001100

⋮O 1001111

⋮V 1010110

SourceEncoder

Information Source

“LOVE”“1001100100111110101101000101”

MATLAB:

Another Example of non-UD code

Suppose we want to convey the sequence of outcomes from rolling a dice.

x c(x)

A sequence of throws such as 53214 is encoded as 10111101100

Suppose we want to convey the sequence of outcomes from rolling a dice.

x c(x)

The encoded string 11 could be interpreted as 11: 1 1 3: 11

The encoded string 110 could be interpreted as 12: 1 10 6: 110

x c(x)

C 01110

D 1110

E 10011

x c(x)

C 01110

D 1110

E 10011

Consider the encoded string 011101110011.

It can be interpreted as CDB: 01110 1110 011 BABE: 011 1 011 10011

[ https://en.wikipedia.org/wiki/Sardinas%E2%80%93Patterson_algorithm ]

Game: 20 Questions

20 Questions is a classic game that has been played since the 19th century.

One person thinks of something (an object, a person, an animal, etc.)

The others playing can ask 20 questions in an effort to guess what it is.

20 Questions: Example

Shannon–Fano coding

Proposed in Shannon’s “A Mathematical Theory of Communication” in 1948

The method was attributed to Fano, who later published it as a technical report. Fano, R.M. (1949). “The transmission of information”.

Technical Report No. 65. Cambridge (Mass.), USA: Research Laboratory of Electronics at MIT.

Should not be confused with Shannon coding, the coding method used to prove Shannon's

noiseless coding theorem, or with Shannon–Fano–Elias coding (also known as Elias coding), the

precursor to arithmetic coding.

Prof. Robert Fano (1917-2016)Shannon Award (1976 )

Huffman Code

MIT, 1951 Information theory class taught by Professor Fano. Huffman and his classmates were given the choice of

a term paper on the problem of finding the most efficient binary code.

or a final exam.

Huffman, unable to prove any codes were the most efficient, was about to give up and start studying for the final when he hit upon the idea of using a frequency-sorted binary tree and quickly proved this method the most efficient.

Huffman avoided the major flaw of the suboptimal Shannon-Fanocoding by building the tree from the bottom up instead of from the top down.

David Huffman (1925–1999)Hamming Medal (1999)

Huffman’s paper (1952)

44[D. A. Huffman, "A Method for the Construction of Minimum-Redundancy Codes," in Proceedings of the IRE, vol. 40, no. 9, pp. 1098-1101, Sept. 1952.][ http://ieeexplore.ieee.org/document/4051119/ ]

Summary:

[2.16-17] A good code must be uniquely decodable (UD). Difficult to check.

[2.24] Consider a special family of codes: prefix(-free) code. Always UD. Same as being instantaneous.

All codes

Nonsingular codes

UD codes

Prefix-free

Huffman

[Defn 2.30] Huffman’s recipe Repeatedly combine the two least-likely (combined) symbols. Automatically give prefix-free code.

[2.37] For a given source’s pmf, Huffman codes are optimalamong all UD codes for that source.

[Defn 2.36]

Each source symbol can be decoded as soon as we come to the end of the codeword corresponding to it

No codeword is a prefix of any other codeword.

[Defn 2.18]

Huffman coding

Ex. Huffman Coding in MATLAB

47 [Huffman_Demo_Ex1]

Observe that MATLAB automatically give the expected length of the codewords

pX = [0.5 0.25 0.125 0.125]; % pmf of XSX = [1:length(pX)]; % Source Alphabet[dict,EL] = huffmandict(SX,pX); % Create codebook

%% Pretty print the codebook.codebook = dict;for i = 1:length(codebook)

codebook{i,2} = num2str(codebook{i,2});endcodebook

%% Try to encode some random source stringn = 5; % Number of source symbols to be generatedsourceString = randsrc(1,10,[SX; pX]) % Create data using pXencodedString = huffmanenco(sourceString,dict) % Encode the data

[Ex. 2.31]

codebook =

[1] '0' [2] '1 0' [3] '1 1 1'[4] '1 1 0'

sourceString =

1 4 4 1 3 1 1 4 3 4

encodedString =

0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0

[Huffman_Demo_Ex1]

49 [Huffman_Demo_Ex2]

pX = [0.4 0.3 0.1 0.1 0.06 0.04]; % pmf of XSX = [1:length(pX)]; % Source Alphabet[dict,EL] = huffmandict(SX,pX); % Create codebook

[Ex. 2.32]

The codewords can be different from our answers found earlier.

The expected length is the same.

>> Huffman_Demo_Ex2

codebook =

[1] '1' [2] '0 1' [3] '0 0 0 0' [4] '0 0 1' [5] '0 0 0 1 0'[6] '0 0 0 1 1'

2.2000

pX = [1/8, 5/24, 7/24, 3/8]; % pmf of XSX = [1:length(pX)]; % Source Alphabet[dict,EL] = huffmandict(SX,pX); % Create codebook

[Exercise]

codebook = [1] '0 0 1'[2] '0 0 0'[3] '0 1' [4] '1'

EL =1.9583

>> -pX*(log2(pX)).'ans =

1.8956

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

Entropy and Description of RV

57 [ https://www.khanacademy.org/computing/computer-science/informationtheory/moderninfotheory/v/information-entropy ]

Entropy and Description of RV

58 [ https://www.khanacademy.org/computing/computer-science/informationtheory/moderninfotheory/v/information-entropy ]

Summary: Optimality of Huffman Codes

Consider a given DMS withknown pmf … [Defn 2.36] A code is optimal

if it is UD and its corresponding expected length is the shortestamong all possible UD codes for that source.

[2.37] Huffman codes are optimal.

All codes

Nonsingular codes

UD codes

Prefix-free

Huffman

Expected length (per source symbol) of an optimal code

Expected length (per source symbol) of a Huffman code

[2.49-2.54] Bounds on expected lengths:

Summary: Entropy

Entropy measures the amount of uncertainty (randomness) in a RV.

Three formulas for calculating entropy: [Defn 2.41] Given a pmf of a RV , ≡ ∑ log .

[2.44] Given a probability vector ,

≡ ∑ log .

[Defn 2.47] Given a number , ≡ log 1 log 1

[2.56] Operational meaning: Entropy of a random variable is the average length of its shortest description.

Set 0log 0 0.

binary entropy function

Examples

Example 2.31

Example 2.32

Huffman

1.75H X X

Huffman

2.14 2.2H X X

Efficiency = 100%

Efficiency 97%

Examples

Example 2.33

Example 2.34

Huffman

2.29 2.3H X X

Huffman

1.86 2H X X Efficiency 93%

Efficiency 99%

Summary: Entropy

Important Bounds

deterministic uniform The entropy of a uniform (discrete) random variable:

The entropy of a Bernoulli random variable:

binary entropy function

Huffman Coding: Source Extension

1 2 3 4 5 6 7 80.4

n: order of extension

i.i.d.

BernoullikX p0.1p

[Ex.2.40]

Huffman Coding: Source Extension

1 2 3 4 5 6 7 80.4

Order of source extension

i.i.d.

BernoullikX p0.1p

[Ex.2.40]

Summary: Source Extension

The encoder operates on the blocks rather than on individual symbols.

[Defn 2.39] -th extension coding: 1 block = successive source symbols

= expected (average) codeword length per source symbol when Huffman coding is used with -th extension

1 2 3 4 5 6 7 80.4

Order of source extension

3 Discrete Memoryless Channel (DMC)

3.1 DMC Models

Summary: DMC

Channel Input Alphabet

matrix

matrixrow vector

row vector “AND”

Put together All values at corresponding positions in the matrix.

,X Y i ji

y 1 j ny y y

jq y q pQ

j iQ y x

Review: Evaluation of Probability from the Joint PMF Matrix

Consider two random variables X and Y.

Suppose their joint pmf matrix is

0.1 0.1 0 0 0 0.1 0 0 0.1 00 0.1 0.2 0 00 0 0 0 0.3

2 3 4 5 6xy

3 4 5 6 75 6 7 8 96 7 8 9 108 9 10 11 12

2 3 4 5 6xy

Step 1: Find the pairs (x,y) that satisfy the condition“x+y < 7”

One way to do this is to first construct the matrix of x+y.

0.1 0.1 0 0 0 0.1 0 0 0.1 00 0.1 0.2 0 00 0 0 0 0.3

2 3 4 5 6xy

3 4 5 6 75 6 7 8 96 7 8 9 108 9 10 11 12

2 3 4 5 6xy

Step 2: Add the corresponding probabilities from the joint pmf (matrix)

x y7 0.1 0.1 0.1

0.1 0.1 0 0 0 0.1 0 0 0.1 00 0.1 0.2 0 00 0 0 0 0.3

2 3 4 5 6xy

Review: Sum of two discrete RVs

0.1 0.1 0 0 00.1 0 0 0.1 00 0.1 0.2 0 00 0 0 0 0.3

2 3 4 5 6xy

3 4 5 6 75 6 7 8 96 7 8 9 108 9 10 11 12

2 3 4 5 6xy

x y7 0.1

3.2 Decoder and

Recipe for finding of any decoder

Use the matrix. If unavailable, can be found by scaling each row of the matrix by its

corresponding . Write values on top of the values for the matrix. For column in the matrix, circle the element whose corresponding

value is the same as . the sum of the circled probabilities. 1 .

0.10 0.04 0.060.24 0.

.5 0.21

0.30.3 0.4 0.3 32 0.24

1 2 3 1 2 3x y xy1 1 0 x̂ y

3.3 Optimal Decoder

Review: ECS315 (2017)

Guessing Game 1

There are 15 cards. Each have a number on it. Here are the 15 cards:

1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 One card is randomly selected from the 15 cards.

You need to guess the number on the card.

Have to pay 1 Baht for incorrect guess.

The game is to be repeated n = 10,000 times.

What should be your guess value?

close all; clear all;

n = 5; % number of time to play this game

D = [1 2 2 3 3 3 4 4 4 4 5 5 5 5 5];X = D(randi(length(D),1,n));

if n <= 10X

g = 1cost = sum(X ~= g)

if n > 1averageCostPerGame = cost/nend

>> GuessingGame_4_1_1X =

3 5 1 2 5g =

1cost =

4averageCostPerGame =

0.8000

n = 5; % number of time to play this game

if n <= 10X

g = 3.3cost = sum(X ~= g)

>> GuessingGame_4_1_1X =

5 3 2 4 1g =

3.3000cost =

5averageCostPerGame =

n = 1e4; % number of time to play this game

if n <= 10X

g = ?cost = sum(X ~= g)

Guessing Game 1

341 1.5 2 2.5 3 3.5 4 4.5 5

Guess value

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60.65

Guess value

Guessing Game 1

Optimal Guess: The most-likely value

Summary: MAP vs. ML Decoders

Decoder is derived from the matrix Select (by circling) the maximum value

in each column (for each value of ) in the matrix.

The corresponding value is the value of .

optimal

ˆ argmax ,

argmax

x y p x y

P X x Y y

Q y p xx

MLˆ argmaxx

x y Q y x

Decoder is derived from the matrix Select (by circling) the maximum value

in each column (for each value of ) in the matrix.

The corresponding value is the value of .

Once the decoder (the decoding table) is derived and are calculated from adding the corresponding probabilities in the matrix.

Optimal at least when is uniform (the channel inputs are equally likely)

a posteriori probability

likelihood funtion

Can be derived without knowing the channel input probabilities.

prior probability

Example: MAP Decoder [Ex. 3.36]

0.10 0.04 0.060.24 0.

.5 0.21

0.30.3 0.4 0.3 32 0.24

1 2 3 1 2 3x y xy1 1 1 MAPx̂ y

( ) 0.24 0.32 0.( ) 1 21 24 0.P P

Example: ML Decoder [Ex. 3.47]

( ) 0.10 0.( 32 0.06) 1 1 0.52PP

( ) 0.10 0.( 32 0.24) 1 1 0.34PP

0.10 0.04 0.060.24 0.

.5 0.21

0.30.3 0.4 0.3 32 0.24

1 2 3 1 2 3x y xy0 1 0 MLx̂ y

0.10 0.04 0.060.24 0.

.5 0.21

0.30.3 0.4 0.3 32 0.24

1 2 3 1 2 3x y xy

MLx̂ y0 1 1

Sol 1:

Sol 2:

yy x[Same as Ex. 3.27]

[Agree with Ex. 3.33]

MAP Decoder

%% MAP DecoderP = diag(p_X)*Q; % Weight the channel transition probability by the

% corresponding prior probability.[V I] = max(P); % For I, the default MATLAB behavior is that when there are

% multiple max, the index of the first one is returned.Decoder_Table = S_X(I) % The decoded values corresponding to the received Y

%% Decode according to the decoder tablex_hat = y; % preallocationfor k = 1:length(S_Y)

I = (y==S_Y(k));x_hat(I) = Decoder_Table(k);

PE_sim = 1-sum(x==x_hat)/n % Error probability from the simulation

%% Calculation of the theoretical error probabilityPC = 0;for k = 1:length(S_X)

I = (Decoder_Table == S_X(k));Q_row = Q(k,:); PC = PC+ p_X(k)*sum(Q_row(I));

endPE_theretical = 1-PC

[DMC_decoder_MAP_demo.m]

ML Decoder

%% ML Decoder[V I] = max(Q); % For I, the default MATLAB behavior is that when there are

% multiple max, the index of the first one is returned.Decoder_Table = S_X(I) % The decoded values corresponding to the received Y

%% Decode according to the decoder tablex_hat = y; % preallocationfor k = 1:length(S_Y)

I = (y==S_Y(k));x_hat(I) = Decoder_Table(k);

PE_sim = 1-sum(x==x_hat)/n % Error probability from the simulation

%% Calculation of the theoretical error probabilityPC = 0;for k = 1:length(S_X)

I = (Decoder_Table == S_X(k));Q_row = Q(k,:); PC = PC+ p_X(k)*sum(Q_row(I));

endPE_theretical = 1-PC

[DMC_decoder_ML_demo.m]

3.5 Repetition Code in Communications Over BSC

Hypercube

48 https://www.youtube.com/watch?v=Q_B5GpsbSQw

Hypercube

49 https://www.youtube.com/watch?v=Q_B5GpsbSQw

n-bit space

Channel Encoder and Decoder

Noise & In

terferen

Information Source

Destination

Channel

ReceivedSignal

TransmittedSignal

Message

Recovered Message

Source Encoder

Channel Encoder

DigitalModulator

Source Decoder

Channel Decoder

DigitalDemodulator

Transmitter

Receiver

X: channel input

Y: channel output

Better Equivalent Channel

Noise & In

terferen

Information Source

Destination

Channel

ReceivedSignal

TransmittedSignal

Message

Recovered Message

Source Encoder

Channel Encoder

DigitalModulator

Source Decoder

Channel Decoder

DigitalDemodulator

Transmitter

Receiver

Remove redundancy

Add systematic redundancy Better

EquivalentChannel

Example: Repetition Code

Original Equivalent Channel:

BSC with crossover probability p = 0.01

New (and Better) Equivalent Channel:

Use repetition code with n = 5 at the transmitter Use majority vote at the receiver New BSC with

Repetition Code with

Majority Vote

1053 1 5

4 1 55 1

Example: ASCII Encoder and BSC

Noise & In

terferen

Information Source

Destination

Channel

ReceivedSignal

TransmittedSignal

Message

Recovered Message

Source Encoder

Channel Encoder

DigitalModulator

Source Decoder

Channel Decoder

DigitalDemodulator

Transmitter

Receiver

X: channel input

Y: channel output

“LOVE”

“1001100100111110101101000101”

0 16 32 48 64 80 96 112

Character Codeword

⋮E 1000101

⋮L 1001100

⋮O 1001111

⋮V 1010110

SourceEncoder

Information Source

“LOVE”“1001100100111110101101000101”

MATLAB:

c(“L”) c(“O”) c(“V”) c(“E”)

System considered

Noise & In

terferen

Information Source

Destination

Channel

ReceivedSignal

TransmittedSignal

Message

Recovered Message

Source Encoder

Channel Encoder

DigitalModulator

Source Decoder

Channel Decoder

DigitalDemodulator

Transmitter

Receiver

X: channel input

Y: channel output

THE WIZARD OF OZ (1900)written by L. Frank Baum

Introduction

Folklore, legends, myths and fairy tales have followed childhoodthrough the ages, for every healthy youngster has a wholesome andinstinctive love for stories fantastic, marvelous and manifestlyunreal. The winged fairies of Grimm and Andersen have brought morehappiness to childish hearts than all other human creations.Yet the old time fairy tale, having served for generations, may

101010010010001000101010000010101111001001101101010000011010010100010001000001001111100011001000001001111101101001000000100000010000001010000110001011100101100000110000010100100010100001001111011111100101101001111010011101001100101110111001000001100010111100101000001001100010111001000001000110111001011000011101110110101101000001000010110000111101011101101000101000010100001010100100111011101110100111001011011111100100111010111000111110100110100111011111101110000101000010100100000010000010001101101111110110011010111101100110111111100101100101010110001000001101100110010111001111100101110111011001001110011010110001000001101101111100111101001101000111001101000001100001110111011001000100000110011011000011101001111001011110010100000111010011000011101100110010111100110100000110100011000011110110110010101000001100110110111111011001101100110111111101111100101110010001000001100011110100011010011101100110010011010001101111110111111001000001010111010011010001110010110111111101011100111110100001000001110100110100011001010100000110000111001111100101111001101011000100000110011011011111110010010000011001011110110110010111100101111001010000011010001100101110000111011001110100110100011110010100000111100111011111110101110111011001111110011111010011001011110010010000011010001100001111001101000001100001010000011101111101000110111111011001100101111001111011111101101110010101000001100001110111011001000001010110100111011101110011111010011010011101110110001111101001101001111011011001010100000110110011011111110110110010101000001100110110111111100100100000111001111101001101111111001011010011100101111001101000001100110110000111011101110100110000111100111110100110100111000110101100010000011011011100001111001011101101100101110110011011111110101111001101000001100001110111011001000100000110110111000011101110110100111001101100101111001111101001101100111100100010101110101110111011100101100101110000111011000101110010000010101001101000110010101000001110111110100111011101100111110010111001000100000110011011000011101001111001011010011100101111001101000001101111110011001000001000111111001011010011101101110110101000001100001110111011001000100000100000111011101100100110010111100101110011110010111011100100000110100011000011110110110010101000001100010111001011011111110101110011111010001110100010000011011011101111111001011001010001010110100011000011110000111000011010011101110110010111100111110011010000011101001101111010000011000111101000110100111011001100100110100111100111101000010000011010001100101110000111100101110100111001101000001110100110100011000011101110010000011000011101100110110001000001101111111010011010001100101111001001000001101000111010111011011100001110111001000001100011111001011001011100001111010011010011101111110111011100110101110000101001000000100000101100111001011110100010000011101001101000110010101000001101111110110011001000100000111010011010011101101110010101000001100110110000111010011110010111100101000001110100110000111011001100101010110001000001101000110000111101101101001110111011001110100000111001111001011110010111011011001011100100010000011001101101111111001001000001100111110010111011101100101111001011000011110100110100111011111101110111001101011000100000110110111000011111001

[ErrorProbabilityoverBSC.m]

Results

101010010010001000101010000010101111001001101101010000011010010100010001000001011111100011001000001001111101101001000000100000010000001010000110001011100101100000110000011100100110100001001111011111100101101001111010011101001100101110111001000001100010111100101000001001100010111001000001000110111001011000011101110110101101100001000010110000111101011101101000101000010100001010100100111011101110100111001011011111100100111010111000111110100110100111011111101110000101000010100100000011000010001101101111110110011010111101100110111111100101100101010111000000001101100110010111001111100101110111011001001010011000110001000001101101111100111101001101000111001101000001100001110111011001000100000110011011000011101001110001011110010100000111010011000011101100110011111100110100000110100011000011110110110010101000001100110110110111011001101100110111111101111100101110010001000001100011110100011010011101100110010011010001101111110111111001000001010111010011010001110010110111111101011100111110100001000001110100110100011001010100000110000111001111100101111001101011000000000110011011011111110010010010011001011110110110010111100101111001010000011011101100101110000111011001110100110100011110010100000111100111011111110101110111011001111110011111010011001011110010010000011010001100001111001101000001100001010000011101111101000110111111011001100101111001111011111101101110010101000001100001110111011001000001010110100111011001110011111000011110011101110110001111101001101001111001011001010100000110110011011111110110110010101000001100110110111111100100100000111001111101001101111111001011010011100101111000101000001100110110000101011101110100110000111100111110100110100111000110101100010000011011011100001111001011101101100101110110011011111110101011001101000001100101110111011001000100000110110111000011101110110100111001101100101111001111101001101100111100100010101110101110111011100101100101110000111011000101110010000010101001101000110010101000001110111110100111011101100111110010111001000100000110011011000011101001111001011010011100101111001101000001101111110011001000001000111111001011010011101101110110101000001100001110111011001000101010100000111011101100100110010111100101110011110010111011100100000110100011000011110110100010101000001100010111001011011111110101110011111010001110100010000011011011101111111001011001010001010110100011000011110000111000011010011101110110010111100111110011000000011101001101111010000011000111101000110100111010001100100110100111100111101000010000011010001100101110000111100101110100111001101000001110100110100011000011101110010000011000011101100110110001000001101111110010011010001110101111001001000001101000111010111011011100001110111001000001100011111001011001011100001111010011010011101111110111011100110101111000101001000000100000101100111001011110100010000011101001101000110010101100001101111110110011001000100010111010011010011101101110110101000001100110110000111010010110010111100101000001110100110000111011001100101010110001000001101000110000111101101101001110111011001110100000111000111001011110010111011011001011100100010000011001101101111111001001000001100111110010111011101100101111001011000011110100110100111011111101110111000101011000100000110110111000011111001

101010010010001000101010000010101111001001101101010000011010010100010001000001001111100011001000001001111101101001000000100000010000001010000110001011100101100000110000010100100010100001001111011111100101101001111010011101001100101110111001000001100010111100101000001001100010111001000001000110111001011000011101110110101101000001000010110000111101011101101000101000010100001010100100111011101110100111001011011111100100111010111000111110100110100111011111101110000101000010100100000010000010001101101111110110011010111101100110111111100101100101010110001000001101100110010111001111100101110111011001001110011010110001000001101101111100111101001101000111001101000001100001110111011001000100000110011011000011101001111001011110010100000111010011000011101100110010111100110100000110100011000011110110110010101000001100110110111111011001101100110111111101111100101110010001000001100011110100011010011101100110010011010001101111110111111001000001010111010011010001110010110111111101011100111110100001000001110100110100011001010100000110000111001111100101111001101011000100000110011011011111110010010000011001011110110110010111100101111001010000011010001100101110000111011001110100110100011110010100000111100111011111110101110111011001111110011111010011001011110010010000011010001100001111001101000001100001010000011101111101000110111111011001100101111001111011111101101110010101000001100001110111011001000001010110100111011101110011111010011010011101110110001111101001101001111011011001010100000110110011011111110110110010101000001100110110111111100100100000111001111101001101111111001011010011100101111001101000001100110110000111011101110100110000111100111110100110100111000110101100010000011011011100001111001011101101100101110110011011111110101111001101000001100001110111011001000100000110110111000011101110110100111001101100101111001111101001101100111100100010101110101110111011100101100101110000111011000101110010000010101001101000110010101000001110111110100111011101100111110010111001000100000110011011000011101001111001011010011100101111001101000001101111110011001000001000111111001011010011101101110110101000001100001110111011001000100000100000111011101100100110010111100101110011110010111011100100000110100011000011110110110010101000001100010111001011011111110101110011111010001110100010000011011011101111111001011001010001010110100011000011110000111000011010011101110110010111100111110011010000011101001101111010000011000111101000110100111011001100100110100111100111101000010000011010001100101110000111100101110100111001101000001110100110100011000011101110010000011000011101100110110001000001101111111010011010001100101111001001000001101000111010111011011100001110111001000001100011111001011001011100001111010011010011101111110111011100110101110000101001000000100000101100111001011110100010000011101001101000110010101000001101111110110011001000100000111010011010011101101110010101000001100110110000111010011110010111100101000001110100110000111011001100101010110001000001101000110000111101101101001110111011001110100000111001111001011110010111011011001011100100010000011001101101111111001001000001100111110010111011101100101111001011000011110100110100111011111101110111001101011000100000110110111000011111001

Introduction

THE WIZARD _F OZ (19009 written by L. Frank0Baum

Introduction

0Folklore. legendS myths and faiby talgs have fmllowed childhoodthrough the ages, for$every nealthy youngster has a wholesome andilspynctire love for storieq fa.tastic, marvelou3 end manifestlyunreal. The winged fairies of Grimm and*Andersen havE brought morehappiness to chihdish hearts than all odhur human creations/Yet the0old"timm fai2y tale, having qerved for generationq, may

Results

Introduction

THE WIZARD _F OZ (19009 written by L. Frank0Baum

Introduction

0Folklore. legendS myths and faiby talgs have fmllowed childhoodthrough the ages, for$every nealthy youngster has a wholesome andilspynctire love for storieq fa.tastic, marvelou3 end manifestlyunreal. The winged fairies of Grimm and*Andersen havE brought morehappiness to chihdish hearts than all odhur human creations/Yet the0old"timm fai2y tale, having qerved for generationq, may

The whole book which is saved in the file “OZ.txt” has 207760 characters (symbols).

The ASCII encoded string has 207760×7 = 1454320 bits.

The channel corrupts 14545 bits.

This corresponds to 14108 erroneous characters.

Results

The file “OZ.txt” has 207760 characters (symbols).

This corresponds to 14108 erroneous characters.

>> ErrorProbabilityoverBSCbiterror =

14545BER =

0.010001237691842theoretical_BER =

0.010000000000000characterErrror =

14108CER =

0.067905275317674theoretical_CER =

0.067934652093010

Results

The file “OZ.txt” has 207760 characters (symbols).

This corresponds to 14108 erroneous characters (symbols).

CER A character (symbol) is successfully recovered if and only if none of its bits are corrupted.

BSC’s crossover probability

Crossover probability and readability

When the first novel of the series, Harry Potter and the Philosopher's Stone (published in some countries as Harry Potter and the Sorcerer's Stone), opens, it is apparent that some significant event has taken place in the wizarding world--an event so very remarkable, even the Muggles notice signs of it. The full background to this event and to the person of Harry Potter is only revealed gradually through the series. After the introductory chapter, the book leaps forward to a time shortly before Harry Potter's eleventh birthday, and it is at this point that his magical background begins to be revealed.

When the first novel of the series, Harry Pottez and the Philosopher's Stone (p5blished in some countries as Harry Potter cnd the Sorcerep's Stone), opens, it i3 apparent that soMecignifacant event!haS taken0place in the wi~arding 7orld--ao event so`very!bemark!blu, even the Mufgles nodice signs"of it. The fuld background to this event and to the person of Harry P/tTer is only revealed gradually through th series. After the introfuctory chapter, the boo+ leaps forward to a time shortly before Harpy Potteb7s eleventh`birthday, and )t is at this poi~t that his -agikal bac{ground begins to be revealed.

0.01 CER 0.07

Original

When the first novel of the series, Harry Pottez and the Philosopher's Stone (p5blished in some countries as Harry Potter cnd the Sorcerep's Stone), opens, it i3 apparent that soMecignifacant event!haS taken0place in the wi~arding 7orld--ao event so`very!bemark!blu, even the Mufgles nodice signs"of it. The fuld background to this event and to the person of Harry P/tTer is only revealed gradually through th series. After the introfuctory chapter, the boo+ leaps forward to a time shortly before Harpy Potteb7s eleventh`birthday, and )t is at this poi~t that his -agikal bac{ground begins to be revealed.

0.01 CER 0.07

Human may be able to correct some (or even all) of these errors.

w(en th% birst .ovo,òf the serieq, Hcrry Potter(and the Phidosopèr's Suone (xub|ishe$(in some!Countries as @arry Potter and tè Snr#erer's S|ong)- opens, it is apparent thatsoeesmgfificant erent ha3 taieN place in the wizardino world-,an event!so very remarkable, even thEEugglds notiae qigns of it. Tledfull back'tound to this event ane to the perron of Harry Popter is onl{ reveqned gsadeally thro}gH th%$serias. After the int2oducpory chcptur, the0jook deapsforward to"a!tmme shmrtly befosE Harry"Potter's eleventh jirthdiy cnd ht is a| thi3 po{nt tHat@is mAgiial background begijs to rm rerealed.

Whethe first nOwgl nf the susi-Q-@Harr} PoutEr(and |he PhilosoxhEr's Ctonepuclyshed in som% coultries as Harrx @ tter and the S_rcermr7s Spone), opdns, id is apparent that {omg`signifikant evmnt ias taKen!Placa in tHe 7ijardIng world--an event so Very remaroqble, eve.!thE MugglaC fotice signc of"it. Uhe full backf2ound`to thas even| ant`0o the pEssoj of @arry Qotteb iw only revealed gradu!lly vhvoug` the rerier. Afte2 the IndRoductori chaptar,t`ebook leats ForwaRf tc a 4imE shostl= before!Hcssy potter's u|Eveoth$firthdA{, and iT is ad this pomNt uhav `ir magica, back'bound cegins to bE 2evealed.

0.02 CER 0.13

0.03 CER 0.19

When phe fir v okval ov"th% serie3, @`rry0Pntter efdxtxeThil soph%rs Stone0(p}blisjed!in{ooe c un|pye{ agav0y Potter aj`(the sorcerer"s S4o|e)< opdns- mt"Is!apParEnt 4hat somusiwnidiga.v evant iAs take."plhge in(uhe w)zard)ng wo{|d--An event so very Rumar{ableleteN0Dhe %ugcles$n t)ce signs of$At. Tje!&ul|!backep/und Dk thkw`event ajt(to vhd per{On of8Ikxry P_Pter is oN,y rereAeud gredualli 4hroufh5ie qeriesn Af|ir the )~trofUctkry!ciapter,$tler%ok lE`ps for erd8to!a d)hg 3Hostly redobd HArry(Potter/r elaventI(birpl%ay,))nd(iD i3 1t tlishohlt vhat iis$iagical bac+gropnd bedans to bg rEve!ied/

Whef th% &i2sv nkvdl"On(txE"serm-s< HaRtY Qo|p%R$anlthe Phi$)qop8gb'r YtoNe (puclirhedin23/ee c uNpr9es aZ Harby!PovDdZ qnd0THA!Uorojev's Qpof'), pegsL iT is0aqazenP Tiet`{nlesau*!fICQ~t eve.t`xA# raken pOqb%%)D }Hm`wizprdYjv"wOrnd--a~%W%Jv s' tury2maskABdd$`eden(tl| LuxGxec`nOtike c)gzq of ktTiu!f5mm"cackG@ Ud(to"vhhQ a~aNd alt tn0vid veRckn of HaRvq$Xntter#isxohk{ regea,ed@&saduadLy u(2otGh"tau griEs."AfTex0T`g mntr DUCt ry kh `ter,$thd(fomN0j`apv ngrwarTt-0c t,me"1xortly bEemsL |ar2q Pnfter'3 aMen-n5i@Fipth$`q, aoh It i3d1t piac0pmhnP d*if Zas mafibin"je#k7poUndpb%dins tk`be qe6e!lgd.

0.05 CER 0.30

0.10 CER 0.52

BER vs. CER

660 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Bit Error Rate (BER)

rror r

[BERCER.m]

BER vs. CER

67 10-2 10-1 10010-2

Bit Error Rate (BER)

rror r

[BERCER.m]

4. Mutual Information and Channel Capacity

Reference for this chapter

Elements of Information Theory

By Thomas M. Cover and Joy A. Thomas

2nd Edition (Wiley)

Chapters 2, 7, and 8

1st Edition available at SIIT library: Q360 C68 1991

Information-Theoretic Quantities

Conditional Entropies

≡ log∈

| ≡ | ≡ | log |∈

| ≡ |∈

Amount of randomness in

Amount of randomness still remained in when we know that .

The average amount of randomness still remained in when we know

given a particular value

Apply the entropy calculation to a row from the matrix

Diagrams [Figure 16]

∩ \A

Venn Diagram

Information Diagram

∩ \A

Probability Diagram

Diagrams [Figure 16]

Information Diagram

∩ \A

Probability Diagram

Diagrams

Information Diagram

∩ \A

Probability Diagram

Operational Channel Capacity

Use repetition code with n = 5 at the transmitter Use majority vote at the receiver New BSC with

Repetition Code with

Majority Vote

1053 1 5

4 1 55 1

[Figure 14]

Use repetition code at the transmitter Use majority vote at the receiver

New BSC with new crossover probability

Repetition Code

Majority Vote

Reminder [From ECS315]

[From ECS315]

Repetition Code

Majority Vote

3 32 1 3

3 0.0280

5 53 1 5

4 1 55 0.0086

7 0.00279 8.9092 10

11 2.9571 10

Information Channel Capacity

Channel Capacity

“Operational”: max rate at which reliablecommunication is possible

“Information”: [bpcu]

Arbitrarily small error probability can be achieved.

Shannon [1948] shows that these two quantities are actually the same.

MATLAB

function H = entropy2s(p)% ENTROPY2 accepts probability mass function % as a row vector, calculate the corresponding % entropy in bits.p=p(find(abs(sort(p)-1)>1e-8)); % Eliminate 1p=p(find(abs(p)>1e-8)); % Eliminate 0if length(p)==0

H = 0;else

H = simplify(-sum(p.*log(p))/log(sym(2)));end

function I = informations(p,Q)X = length(p);q = p*Q;HY = entropy2s(q);temp = [];for i = 1:X

temp = [temp entropy2s(Q(i,:))];endHYgX = sum(p.*temp);I = HY-HYgX;

Capacity calculation for BAC

Capacity of 0.0918 bits is achieved by 0.5380, 0.4620p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1 0.90.4 0.6

Capacity calculation for BAC

close all; clear all;syms p0p = [p0 1-p0];Q = [1 9; 4 6]/sym(10);

I = simplify(informations(p,Q))

p0o = simplify(solve(diff(I)==0))

po = eval([p0o 1-p0o])

C = simplify(subs(I,p0,p0o))

eval(C)

>> Capacity_Ex_BACI =(log(2/5 - (3*p0)/10)*((3*p0)/10 - 2/5) - log((3*p0)/10 + 3/5)*((3*p0)/10 +

3/5))/log(2) + (log((5*2^(3/5)*3^(2/5))/6)*(p0 - 1))/log(2) +

(p0*log((3*3^(4/5))/10))/log(2)

p0o =(27648*2^(1/3))/109565 - (69984*2^(2/3))/109565 + 135164/109565

po =0.5376 0.4624

C =(log((3*3^(4/5))/10)*((27648*2^(1/3))/109565 - (69984*2^(2/3))/109565 + 135164/109565))/log(2) - (log((104976*2^(2/3))/547825 - (41472*2^(1/3))/547825 + 16384/547825)*((104976*2^(2/3))/547825 - (41472*2^(1/3))/547825 + 16384/547825) + log((41472*2^(1/3))/547825 - (104976*2^(2/3))/547825 + 531441/547825)*((41472*2^(1/3))/547825 - (104976*2^(2/3))/547825 + 531441/547825))/log(2) + (log((5*2^(3/5)*3^(2/5))/6)*((27648*2^(1/3))/109565 -(69984*2^(2/3))/109565 + 25599/109565))/log(2)

ans =0.0918

X Y0.1 0.90.4 0.6

Same procedure applied to BSC

close all; clear all;syms p0p = [p0 1-p0];Q = [6 4; 4 6]/sym(10);

I = simplify(informations(p,Q))

p0o = simplify(solve(diff(I)==0))

po = eval([p0o 1-p0o])

C = simplify(subs(I,p0,p0o))

eval(C)

>> Capacity_Ex_BSCI =(log((5*2^(3/5)*3^(2/5))/6)*(p0 - 1))/log(2) -(p0*log((5*2^(3/5)*3^(2/5))/6))/log(2) - (log(p0/5 + 2/5)*(p0/5 + 2/5) - log(3/5 - p0/5)*(p0/5 -3/5))/log(2)p0o =1/2po =

0.5000 0.5000C =log((2*2^(2/5)*3^(3/5))/5)/log(2)ans =

0.0290

X Y0.6 0.40.4 0.6

Blahut–Arimoto algorithm

function [ps C] = capacity_blahut(Q)% Input: Q = channel transition probability matrix% Output: C = channel capacity% ps = row vector containing pmf that achieves capacity

tl = 1e-8; % tolerance (for the stopping condition)n = 1000; % max number of iterations (in case the stopping condition

% is "never" reached") nx = size(Q,1); pT = ones(1,nx)/nx; % First, guess uniform X.for k = 1:n

qT = pT*Q;% Eliminate the case with 0% Column-division by qTtemp = Q.*(ones(nx,1)*(1./qT));%Eliminate the case of 0/0l2 = log2(temp); l2(find(isnan(l2) | (l2==-inf) | (l2==inf)))=0;logc = (sum(Q.*(l2),2))';CT = 2.^(logc);A = log2(sum(pT.*CT)); B = log2(max(CT));if((B-A)<tl)

breakend% For the next looppT = pT.*CT; % un-normalizedpT = pT/sum(pT); % normalizedif(k == n)

fprintf('\nNot converge within n loops\n')end

endps = pT;C = (A+B)/2; [capacity_blahut.m]

Capacity calculation for BAC: a revisit

Q = [1 9; 4 6]/10;

[ps C] = capacity_blahut(Q)

>> Capacity_Ex_BAC_blahutps =

0.5376 0.4624C =

0.0918

X Y0.1 0.90.4 0.6

Richard Blahut

Former chair of the Electrical and Computer Engineering Department at the University of Illinois at Urbana-Champaign

Best known for Blahut–Arimotoalgorithm (Iterative Calculation of C)

Claude E. Shannon Award

Claude E. Shannon (1972)

David S. Slepian (1974)

Robert M. Fano (1976)

Peter Elias (1977)

Mark S. Pinsker (1978)

Jacob Wolfowitz (1979)

W. Wesley Peterson (1981)

Irving S. Reed (1982)

Robert G. Gallager (1983)

Solomon W. Golomb (1985)

William L. Root (1986)

James L. Massey (1988)

Thomas M. Cover (1990)

Andrew J. Viterbi (1991)

Elwyn R. Berlekamp (1993)

Aaron D. Wyner (1994)

G. David Forney, Jr. (1995)

Imre Csiszár (1996)

Jacob Ziv (1997)

Neil J. A. Sloane (1998)

Tadao Kasami (1999)

Thomas Kailath (2000)

Jack KeilWolf (2001)

Toby Berger (2002)

Lloyd R. Welch (2003)

Robert J. McEliece (2004)

Richard Blahut (2005)

Rudolf Ahlswede (2006)

Sergio Verdu (2007)

Robert M. Gray (2008)

Jorma Rissanen (2009)

Te Sun Han (2010)

Shlomo Shamai (Shitz) (2011)

Abbas El Gamal (2012)

Katalin Marton (2013)

János Körner (2014)

Arthur Robert Calderbank (2015)

Alexander S. Holevo (2016)David Tse (2017)

[ http://www.itsoc.org/honors/claude-e-shannon-award ]

Berger plaque

Raymond Yeung

BS, MEng and PhD degrees in electrical engineering from Cornell University in 1984, 1985, and 1988, respectively.

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