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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

Tuesday 14:20-15:20Wednesday 14:20-15:20Friday 9:15-10:15

“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)

[ 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 Elias

2003 - Claude Berrou and Alain Glavieux2004 - Jack K. Wolf2005 - Neil J.A. Sloane2006 -Vladimir I. Levenshtein2007 - Abraham Lempel2008 - Sergio Verdú2009 - Peter Franaszek2010 -Whitfield Diffie, Martin Hellman and Ralph Merkle2011 -Toby Berger2012 - Michael Luby, Amin Shokrollahi2013 - Arthur Robert Calderbank2014 -Thomas Richardson and Rüdiger L. Urbanke2015 - Imre Csiszar2016 - Abbas El Gamal

“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)

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 start shutting 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. By the end of the year all national networks will be DAB-only, while

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/

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.

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

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

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).

The ASCII Coded Character Set

4[The ARRL Handbook for Radio Communications 2013]

0 16 32 48 64 80 96 112

(American Standard Code for Information Interchange)

Introduction to Data Compression

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

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

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)

12 [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)

14 [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:

16 [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]

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:

c(“L”) c(“O”) c(“V”) c(“E”)

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

The ASCII Coded Character Set

19[The ARRL Handbook for Radio Communications 2013]

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

Morse code

Telegraph network

Samuel Morse, 1838

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

(wired and wireless)

Example

23 [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

Example: ASCII Encoder

Character Codeword

⋮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:

Another Example of non-UD code

x c(x)

C 01110

D 1110

E 10011

Consider the string 011101110011.

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

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)

35[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/ ]

Huffman coding

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

Ex. Huffman Coding in MATLAB

37 [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]

39 [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

Let’s talk about TV series on HBO

Silicon Valley (TV series)

Focus around six young men who found a startup company in Silicon Valley.

In the first season, the company develop a “revolutionary” data compression algorithm: The “middle-out” algorithm

Behind the Scene

When Mike Judge set out to write Silicon Valley, he wanted to conceive a simple, believable widget for his characters to invent.

He teamed with Stanford electrical engineering professorTsachyWeissman and a PhD student Vinith Misra

They came up with a fictional breakthrough compression algorithm. “We had to come up with an approach that isn’t possible today,

but it isn’t immediately obvious that it isn’t possible,” says Misra.

The writers also coined a metric, the “Weissman Score,” for characters to use when comparing compression codes.

[May 2014 issue of Popular Science]

Middle-Out Algorithm: Into the Real World

Something like it can be found in Lepton A new lossless image compressor created by Dropbox.

Lepton reduces the file size of JPEG-encoded images by as much as 22 percent, yet without losing a single bit of the original.

How is this possible? Middle-out. Well, it’s much more complicated than that, actually. The middle-out bit comes toward the end of the decompression

Lepton is open source, and Dropbox has put the code for it on GitHub.

[ https://techcrunch.com/2016/07/14/dropboxs-lepton-lossless-image-compression-really-uses-a-middle-out-algorithm/ ]

Weissman Score: Into the Real World

It’s hard to convey to a lay audience that one compression algorithm is better than another.

Created by Misra (Prof. Weissman’s PhD student) for the show.

Metrics for compression algorithms that rate not only the amount of compression but the processing speed, are hard to find.

http://spectrum.ieee.org/view-from-the-valley/computing/software/a-madefortv-compression-metric-moves-to-the-real-worldhttp://spectrum.ieee.org/view-from-the-valley/computing/software/a-madefortv-compression-algorithm

Summary

A good code must be uniquely decodable (UD). Difficult to check.

A special family of codes called prefix(-free) code is always UD. They are also instantaneous.

Huffman’s recipe Repeatedly combine the two least-likely (combined) symbols Automatically give prefix code

For a given source’s pmf, Huffman codes are optimal among all UD codes for that source.

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

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

Entropy and Description of RV

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

Kronecker Product

An operation on two matrices of arbitrary size

Named after German mathematician Leopold Kronecker.

If A is an m-by-n matrix and B is a p-by-q matrix, then the Kronecker product AB is the mp-by-nq matrix

Example

a B a B

· · 2·0 2·5 0 5 0 102 · · 2·6 2·7 6 7 12 14

3 4 3·0 3·5 4·0 4·5 0 15 0 203·6 3·7 4·6 4·7 18

0 50 5 6

Use kron(A,B)in MATLAB.

Kronecker Product

>> p = [0.9 0.1]p =

0.9000 0.1000>> p2 = kron(p,p)p2 =

0.8100 0.0900 0.0900 0.0100>> p3 = kron(p2,p)p3 =

Columns 1 through 70.7290 0.0810 0.0810 0.0090 0.0810 0.0090 0.0090

Column 80.0010

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]

Final Remarks about Huffman Code

Huffman coding compresses an i.i.d. source with a known pmf pX(x) to its entropy limit H(X).

Sensitive to the assumed distribution. If the code is designed for some incorrect distribution, a penalty

is incurred.

What compression can be achieved if the true pmf pX(x) is unknown? One may assume uniform pmf Inefficient if the actual pmf is not uniform.

Lempel-Ziv algorithm

Often used in practice to compress data that cannot be modeled simply

Could also be described as adaptive dictionary compression algorithms.

Ziv and Lempel wrote their papers in 1977 and 1978.

The two papers describe two distinct versions of the algorithm. LZ77: sliding window Lempel–Ziv LZ78: tree-structured Lempel–Ziv

Lempel-Ziv algorithm

Arithmetic Coding

The Huffman coding procedure is optimal for encoding a random variable with a known pmf that has to be encoded symbol by symbol.

Coding Inefficiency of Huffman Code: The codeword lengths for a Huffman code were restricted to be

integer-valued There could be a loss of up to 1 bit per symbol in coding efficiency.

We could alleviate this loss by encoding blocks of source symbols The complexity of this approach increases exponentially with block

length n. In arithmetic coding, instead of using a sequence of bits to

represent a symbol, we represent it by a subinterval of the unit interval.

3 Discrete Memoryless Channel (DMC)

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(compression)

Add systematic redundancy to combat errors introduced by the channel

System considered previously

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

System considered in this section

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

EquivalentChannel

X: channel input

Y: channel output

3.1 DMC Models

MATLAB

%% Generating the channel input xx = randsrc(1,n,[S_X;p_X]); % channel input

%% Applying the effect of the channel to create the channel output yy = DMC_Channel_sim(x,S_X,S_Y,Q); % channel output

function y = DMC_Channel_sim(x,S_X,S_Y,Q)%% Applying the effect of the channel to create the channel output yy = zeros(size(x)); % preallocationfor k = 1:length(x)

% Look at the channel input one by one. Choose the corresponding row% from the Q matrix to generate the channel output.y(k) = randsrc(1,1,[S_Y;Q(find(S_X == x(k)),:)]);

[DMC_Analysis_demo.m]

[DMC_Channel_sim.m]

Ex: BSC

>> BSC_demoans =1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1ans =1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 1 1 1

Elapsed time is 0.134992 seconds.

%% Simulation parameters% The number of symbols to be transmittedn = 20; % Channel Input S_X = [0 1]; S_Y = [0 1];p_X = [0.3 0.7];% Channel Characteristicsp = 0.1; Q = [1-p p; p 1-p];

[BSC_demo.m]

[Example 3.2]

p_X =0.3000 0.7000

p_X_sim =0.1500 0.8500

q =0.3400 0.6600

q_sim =0.1500 0.8500

Q =0.9000 0.10000.1000 0.9000

Q_sim =0.6667 0.33330.0588 0.9412

PE_sim =0.1000

PE_theretical =0.1000

Rel. freq. from the simulation

%% Statistical Analysis% The probability values for the channel inputsp_X % Theoretical probabilityp_X_sim = hist(x,S_X)/n % Relative frequencies from the simulation% The probability values for the channel outputsq = p_X*Q % Theoretical probabilityq_sim = hist(y,S_Y)/n % Relative frequencies from the simulation% The channel transition probabilities from the simulationQ_sim = [];for k = 1:length(S_X)

I = find(x==S_X(k)); LI = length(I);rel_freq_Xk = LI/n; yc = y(I);cond_rel_freq = hist(yc,S_Y)/LI; Q_sim = [Q_sim; cond_rel_freq];

endQ % Theoretical probabilityQ_sim % Relative frequencies from the simulation

[DMC_Analysis_demo.m]

Ex: BSC

>> BSC_demoans =1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1ans =1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 1 1 1

%% Simulation parameters% The number of symbols to be transmittedn = 20; % Channel Input S_X = [0 1]; S_Y = [0 1];p_X = [0.3 0.7];% Channel Characteristicsp = 0.1; Q = [1-p p; p 1-p];

[BSC_demo.m]

p_X =0.3000 0.7000

p_X_sim =0.1500 0.8500

q =0.3400 0.6600

q_sim =0.1500 0.8500

Q =0.9000 0.10000.1000 0.9000

Q_sim =0.6667 0.33330.0588 0.9412

PE_sim =0.1000

[Example 3.2]

Because there are only 20 samples, we can’t expect the relative freq. from the simulation to match the specified or calculated probabilities.

Ex: BSC

%% Simulation parameters% The number of symbols to be transmittedn = 1e4; % Channel Input S_X = [0 1]; S_Y = [0 1];p_X = [0.3 0.7];% Channel Characteristicsp = 0.1; Q = [1-p p; p 1-p];

[BSC_demo.m]

>> BSC_demo

p_X =0.3000 0.7000

p_X_sim =0.3037 0.6963

q =0.3400 0.6600

q_sim =0.3407 0.6593

Q =0.9000 0.10000.1000 0.9000

Q_sim =0.9078 0.09220.0934 0.9066

PE_sim =0.0930

Ex: DMC

p_X =0.2000 0.8000

p_X_sim =0.2000 0.8000

q =0.3400 0.3600 0.3000

q_sim =0.4000 0.3500 0.2500

Q =0.5000 0.2000 0.30000.3000 0.4000 0.3000

Q_sim =0.7500 0 0.25000.3125 0.4375 0.2500

>> sym(Q_sim)ans =[ 3/4, 0, 1/4][ 5/16, 7/16, 1/4]PE_sim =

0.7500

>> DMC_demoans =1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1ans =1 3 2 2 1 2 1 2 2 3 1 1 1 3 1 3 2 3 1 2

%% Simulation parameters% The number of symbols to be transmittedn = 20; % General DMC% Ex. 3.6 amd 3.12 in lecture note% Channel Input S_X = [0 1]; S_Y = [1 2 3];p_X = [0.2 0.8];% Channel CharacteristicsQ = [0.5 0.2 0.3; 0.3 0.4 0.3];

[DMC_demo.m]

Ex: DMC

>> p = [0.2 0.8]p =

0.2000 0.8000>> p = [0.2 0.8];>> Q = [0.75 0 0.25; 0.3125 0.4375 0.25];>> p*Qans =

0.4000 0.3500 0.2500

Block Matrix Multiplications

10 6 6 4 39 7 3 5 9

2 2 5 10 2 10 2 53 3 4 5 10 5 3 63 3 4 1 1 5 5 67 2 5 3 10 6 10 38 3 6 9 8 3 6 5

108 73 136 175 150 193 126 149155 85 164 224 213 197 158 165

A BC D

AC+BE AD+BF

10 6 6 4 39 7 3 5 9

2 2 5 10 2 10 2 53 3 4 5 10 5 3 63 3 4 1 1 5 5 67 2 5 3 10 6 10 38 3 6 9 8 3 6 5

108 73 136 175 150 193 126 149155 85 164 224 213 197 158 165

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

Ex: DMC

>> p = [0.2 0.8];>> Q = [0.5 0.2 0.3; 0.3 0.4 0.3];>> p*Qans =

0.3400 0.3600 0.3000>> P = (diag(p))*QP =

0.1000 0.0400 0.06000.2400 0.3200 0.2400

>> sum(P)ans =

0.3400 0.3600 0.3000

3.2 Decoder and

for Naïve Decoder

20 [DMC_Analysis_demo.m]

%% Naive Decoderx_hat = y;

%% Error ProbabilityPE_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)

t = S_X(k);i = find(S_Y == t);if length(i) == 1

PC = PC+ p_X(k)*Q(k,i);end

endPE_theretical = 1-PC

Formula derived in 3.19 of lecture notes

Ex: Naïve Decoder and BAC

p_X =0.5000 0.5000

p_X_sim =0.7000 0.3000

q =0.5500 0.4500

q_sim =0.6500 0.3500

Q =0.7000 0.30000.4000 0.6000

Q_sim =0.7143 0.28570.5000 0.5000

PE_sim =0.3500

>> BAC_demoans =0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0ans =0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 0 0 1 0

%% Simulation parameters% The number of symbols to be transmittedn = 20; % Binary Assymmetric Channel (BAC)% Ex 3.8 in lecture note (11.3 in [Z&T, 2010])% Channel Input S_X = [0 1]; S_Y = [0 1];p_X = [0.5 0.5];% Channel CharacteristicsQ = [0.7 0.3; 0.4 0.6];

[BAC_demo.m]

[Ex. 3.18]

p_X =0.5000 0.5000

p_X_sim =0.5043 0.4957

q =0.5500 0.4500

q_sim =0.5532 0.4468

Q =0.7000 0.30000.4000 0.6000

Q_sim =0.7109 0.28910.3928 0.6072

PE_sim =0.3405

%% Simulation parameters% The number of symbols to be transmittedn = 1e4; % Binary Assymmetric Channel (BAC)% Ex 3.8 in lecture note (11.3 in [Z&T, 2010])% Channel Input S_X = [0 1]; S_Y = [0 1];p_X = [0.5 0.5];% Channel CharacteristicsQ = [0.7 0.3; 0.4 0.6];

10 20 30 40 50 60 70 80 90 100

10010 20 30 40 50 60 70 80 90 100

[BAC_demo.m]

Ex: Naïve Decoder and BAC [Ex. 3.18]

Ex: Naïve Decoder and DMC

p_X =0.2000 0.8000

p_X_sim =0.2000 0.8000

q =0.3400 0.3600 0.3000

q_sim =0.4000 0.3500 0.2500

Q =0.5000 0.2000 0.30000.3000 0.4000 0.3000

Q_sim =0.7500 0 0.25000.3125 0.4375 0.2500

PE_sim =0.7500

>> DMC_demoans =1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1ans =1 3 2 2 1 2 1 2 2 3 1 1 1 3 1 3 2 3 1 2

%% Simulation parameters% The number of symbols to be transmittedn = 20; % General DMC% Ex. 3.16 in lecture note% Channel Input S_X = [0 1]; S_Y = [1 2 3];p_X = [0.2 0.8];% Channel CharacteristicsQ = [0.5 0.2 0.3; 0.3 0.4 0.3];

20 420

[Same samples as in Ex. 3.6]

[Ex. 3.21]

[DMC_demo.m]

p_X =0.2000 0.8000

p_X_sim =0.2011 0.7989

q =0.3400 0.3600 0.3000

q_sim =0.3387 0.3607 0.3006

Q =0.5000 0.2000 0.30000.3000 0.4000 0.3000

Q_sim =0.4943 0.1914 0.31430.2995 0.4033 0.2972

PE_sim =0.7607

%% Simulation parameters% The number of symbols to be transmittedn = 1e4; % General DMC% Ex. 3.16 in lecture note% Channel Input S_X = [0 1]; S_Y = [1 2 3];p_X = [0.2 0.8];% Channel CharacteristicsQ = [0.5 0.2 0.3; 0.3 0.4 0.3];

10 20 30 40 50 60 70 80 90 100

10010 20 30 40 50 60 70 80 90 100

[DMC_demo.m]

Ex: Naïve Decoder and DMC [Ex. 3.21]

DIY Decoder

>> DMC_decoder_DIY_demoans =1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 1ans =2 1 1 3 3 1 2 2 1 2 1 2 3 1 1 3 1 3 1 1ans =1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0PE_sim =

0.5500PE_theretical =

0.5200Elapsed time is 0.081161 seconds.

%% Simulation parameters% The number of symbols to be transmittedn = 20; % General DMC% Ex. 3.16 in lecture note% Channel Input S_X = [0 1]; S_Y = [1 2 3];p_X = [0.2 0.8];% Channel CharacteristicsQ = [0.5 0.2 0.3; 0.3 0.4 0.3];

%% DIY DecoderDecoder_Table = [0 1 0]; % The decoded values corresponding to the received Y

[DMC_decoder_DIY_demo.m]

[Ex. 3.22]

DIY Decoder

% 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 = Q(k,:); PC = PC+ p_X(k)*sum(q(I));

[Ex. 3.22]

DIY Decoder

>> DMC_decoder_DIY_demoPE_sim =

0.5213PE_theretical =

0.5200Elapsed time is 2.154024 seconds.

%% Simulation parameters% The number of symbols to be transmittedn = 1e4; % General DMC% Ex. 3.16 in lecture note% Channel Input S_X = [0 1]; S_Y = [1 2 3];p_X = [0.2 0.8];% Channel CharacteristicsQ = [0.5 0.2 0.3; 0.3 0.4 0.3];

10 20 30 40 50 60 70 80 90 100

10010 20 30 40 50 60 70 80 90 100

[Ex. 3.22]

3.3 Optimal Decoder

Searching for the Optimal Detector

>> DMC_decoder_ALL_demoans =

0 0 0 0.80000 0 1.0000 0.62000 1.0000 0 0.52000 1.0000 1.0000 0.3400

1.0000 0 0 0.66001.0000 0 1.0000 0.48001.0000 1.0000 0 0.38001.0000 1.0000 1.0000 0.2000

Min_PE =0.2000

Optimal_Detector =1 1 1

Ex. 3.22

Ex. 3.23

Review: ECS315 (2016)

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

351 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

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)

%% 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));

[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)

%% 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));

[DMC_decoder_ML_demo.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

Operational Channel Capacity

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

Berger plaque

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)

Raymond Yeung

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

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