Digital Communications I: Modulation and Coding Course. Term Catharina Logothetis Lecture 9

Transcription

1 Diital Commicatios I: Modlatio ad Codi Corse Term 3-8 Catharia Loothetis Lectre 9

2 Last time we taled abot: Ealati the aerae probability of symbol error for differet badpass modlatio schemes Compari differet modlatio schemes based o their error performaces. Lectre 9

3 Today, we are oi to tal abot: Chael codi Liear bloc codes The error detectio ad correctio capability Ecodi ad decodi Hammi codes Cyclic codes Lectre 9 3

4 Bloc diaram of a DCS Format Sorce ecode Chael ecode Plse modlate Badpass modlate Diital modlatio Diital demodlatio Chael Format Sorce decode Chael decode Detect Demod. Sample Lectre 9 4

5 What is chael codi? Chael codi: Trasformi sials to improe commicatios performace by icreasi the robstess aaist chael impairmets oise, iterferece, fadi,... Waeform codi: Trasformi waeforms to better waeforms Strctred seqeces: Trasformi data seqeces ito better seqeces, hai strctred reddacy. - Better i the sese of mai the decisio process less sbject to errors. Lectre 9 5

6 Error cotrol techiqes Atomatic Repeat reqest ARQ Fll-dplex coectio, error detectio codes The receier seds feedbac to the trasmitter, sayi that if ay error is detected i the receied pacet or ot Not-Acowledemet NACK ad Acowledemet ACK, respectiely. The trasmitter retrasmits the preiosly set pacet if it receies NACK. Forward Error Correctio FEC Simplex coectio, error correctio codes The receier tries to correct some errors Hybrid ARQ ARQFEC Fll-dplex, error detectio ad correctio codes Lectre 9 6

7 Why si error correctio codi? Error performace s. badwidth Power s. badwidth Data rate s. badwidth Capacity s. badwidth P B Coded Codi ai: For a ie bit-error probability, the redctio i the Eb/N that ca be realized throh the se of code: G [db] E N b [db] E N b c [db] C A E F B D E b coded / N db Lectre 9 7

8 Chael models Discrete memory-less chaels Discrete ipt, discrete otpt Biary Symmetric chaels Biary ipt, biary otpt Gassia chaels Discrete ipt, cotios otpt Lectre 9 8

9 Liear bloc codes Let s reiew some basic defiitios first that are sefl i derstadi Liear bloc codes. Lectre 9 9

10 Lectre 9 Some defiitios Biary field : The set {,}, der modlo biary additio ad mltiplicatio forms a field. Biary field is also called Galois field, GF. Additio Mltiplicatio

11 Fields : Some defiitios Let F be a set of objects o which two operatios ad. are defied. F is said to be a field if ad oly if. F forms a commtatie rop der operatio. The additie idetity elemet is labeled. a, b F a b b a F 3. F-{} forms a commtatie rop der. Operatio. The mltiplicatie idetity elemet is labeled. a, b F a b b a F 5. The operatios ad. are distribtie: a b c a b a c Lectre 9

12 Lectre 9 Some defiitios Vector space: Let V be a set of ectors ad F a fields of elemets called scalars. V forms a ector space oer F if:. Commtatie:. 3. Distribtie: 4. Associatie: 5. V V a F a, a a a b a b a ad F V,,, b a b a V F b a V,

13 Some defiitios Examples of ector spaces The set of biary -tples, deoted by V 4 {,,,,,,,,,,,,,} Vector sbspace: A sbset S of the ector space sbspace if: is called a The all-zero ector is i S. The sm of ay two ectors i S is also i S. Example: {,,,} is a sbspace of V4. V V Lectre 9 3

14 Spai set: Some defiitios A collectio of ectors, is said to be a spai set for V or to spa V if liear combiatios of the ectors i G iclde all ectors i the ector space V, Example:,,,, spas V Bases: {,,, } G { }. 4 The spai set of V that has miimal cardiality is called the basis for V. Cardiality of a set is the mber of objects i the set. Example: {,,, } is a basis for V. 4 Lectre 9 4

15 Liear bloc codes Liear bloc code, A set C V with cardiality is called a liear bloc code if, ad oly if, it is a sbspace of the ector space. V C V V Members of C are called code-words. The all-zero codeword is a codeword. Ay liear combiatio of code-words is a codeword. Lectre 9 5

16 Liear bloc codes cot d V mappi V C Bases of C Lectre 9 6

17 Liear bloc codes cot d The iformatio bit stream is chopped ito blocs of bits. Each bloc is ecoded to a larer bloc of bits. The coded bits are modlated ad set oer the chael. The reerse procedre is doe at the receier. Data bloc Chael ecoder Codeword bits bits - R c Reddat Code rate bits Lectre 9 7

18 Liear bloc codes cot d The Hammi weiht of the ector, deoted by w, is the mber of o-zero elemets i. The Hammi distace betwee two ectors ad V, is the mber of elemets i which they differ. d, V w V The miimm distace of a bloc code is d mi mi d, i j i j mi w i i Lectre 9 8

19 Liear bloc codes cot d Error detectio capability is ie by e d mi Error correcti-capability t of a code is defied as the maximm mber of arateed correctable errors per codeword, that is t d mi Lectre 9 9

20 Lectre 9 Liear bloc codes cot d For memory less chaels, the probability that the decoder commits a erroeos decodi is is the trasitio probability or bit error probability oer chael. The decoded bit error probability is j j t j M p p j P j j t j B p p j j P p

21 Liear bloc codes cot d Discrete, memoryless, symmetric chael model p -p p Tx. bits Note that for coded systems, the coded bits are modlated ad trasmitted oer the chael. For example, for M-PSK modlatio o AWGN chaels M>: lo M Q where p -p Rx. bits lo M E π lo M Ec N c si M lo Q is eery per coded bit, ie by M N E b R c E c si R π M c E b Lectre 9

22 Lectre 9 Liear bloc codes cot d A matrix G is costrcted by tai as its rows the ectors of the basis,. V V C Bases of C mappi },,, { V V V V V G

23 Lectre 9 3 Liear bloc codes cot d Ecodi i, bloc code The rows of G are liearly idepedet. mg m m m m m m V V V V V V,,,,,,,,,

24 Lectre 9 4 Liear bloc codes cot d Example: Bloc code 6,3 3 V V V G Messae ector Codeword

25 Liear bloc codes cot d Systematic bloc code, For a systematic code, the first or last elemets i the codeword are iformatio bits. G I P [ P I ] idetity matrix matrix,,..., p, p,..., p, m, m,..., m parity bits messae bits Lectre 9 5

26 Liear bloc codes cot d For ay liear code we ca fid a matrix H, sch that its rows are orthooal to the rows of G : GH H is called the parity chec matrix ad its rows are liearly idepedet. For systematic liear bloc codes: H T [ I P T ] Lectre 9 6

27 Liear bloc codes cot d Data sorce Data si Format Format m mˆ Chael ecodi Chael decodi r Modlatio Demodlatio Detectio chael r e r, r e, e,..., r,..., e Sydrome testi: r receied codeword or ector error patter or ector S is the sydrome of r, correspodi to the error patter e. T T S rh eh e Lectre 9 7

28 Liear bloc codes cot d Stadard array For row fid a ector i of miimm weiht that is ot already listed i the array. Call this patter ad form the row as the zero codeword coset leaders correspodi coset e i e,3,..., e e e i i : th e e V coset Lectre 9 8

29 Liear bloc codes cot d Stadard array ad sydrome table decodi T. Calclate S rh. Fid the coset leader, e ˆ e i, correspodi to S. 3. Calclate ˆ r eˆ ad the correspodi mˆ. Note that ˆ r eˆ e eˆ e If e ˆ e, the error is corrected. If eˆ e, detectable decodi error occrs. eˆ Lectre 9 9

30 Liear bloc codes cot d Example: Stadard array for the 6,3 code codewords coset Coset leaders Lectre 9 3

31 Liear bloc codes cot d Error patter Sydrome trasmitted. r is receied. The sydrome of r is compted : S Error patter correspodi to this sydrome is eˆ The corrected ector is estimated ˆ rh T H T r eˆ Lectre 9 3

32 Hammi codes Hammi codes Hammi codes are a sbclass of liear bloc codes ad belo to the cateory of perfect codes. Hammi codes are expressed as a fctio of a sile iteer m. Code leth : Nmber of Nmber of iformatio bits : parity bits : Error correctio capability : The colms of the parity-chec matrix, H, cosist of all o-zero biary m-tples. t - m m m m Lectre 9 3

33 Lectre 9 33 Hammi codes Example: Systematic Hammi code 7,4 ] [ 3 3 T P I H ] [ 4 4 I P G

34 Cyclic bloc codes Cyclic codes are a sbclass of liear bloc codes. Ecodi ad sydrome calclatio are easily performed si feedbac shiftreisters. Hece, relatiely lo bloc codes ca be implemeted with a reasoable complexity. BCH ad Reed-Solomo codes are cyclic codes. Lectre 9 34

35 Cyclic bloc codes A liear, code is called a Cyclic code if all cyclic shifts of a codeword are also codewords. i Example:, i,,,..., i,...,, 3, i cyclic shifts of,,..., 4 i Lectre 9 35

36 Lectre 9 36 Cyclic bloc codes Alebraic strctre of Cyclic codes, implies expressi codewords i polyomial form Relatioship betwee a codeword ad its cyclic shifts: Hece: deree , modlo i i By extesio modlo

37 Cyclic bloc codes Basic properties of Cyclic codes: Let C be a biary, liear cyclic code. Withi the set of code polyomials i C, there is a iqe moic polyomial with miimal deree r <. is called the eerator polyomial. r... r 3. Eery code polyomial i C ca be expressed iqely as m 4. The eerator polyomial is a factor of Lectre 9 37

38 Lectre 9 38 Cyclic bloc codes The orthooality of G ad H i polyomial form is expressed as. This meas is also a factor of. The row, of the eerator matrix is formed by the coefficiets of the cyclic shift of the eerator polyomial. r r r r G h h i i,...,, " " i

39 Cyclic bloc codes Systematic ecodi alorithm for a, Cyclic code: m. Mltiply the messae polyomial by 3. Diide the reslt of Step by the eerator polyomial. Let p be the remider. p m 5. Add to to form the codeword Lectre 9 39

40 Lectre 9 4 Cyclic bloc codes Example: For the systematic 7,4 Cyclic code with eerator polyomial. Fid the codeword for the messae Form the codeword polyomial: : by Diide 3 4, 7, messae bits parity bits remaider eerator 3 qotiet m p m m m m m p q m 3

41 Lectre 9 4 Cyclic bloc codes Fid the eerator ad parity chec matrices, G ad H, respectiely.,,, 3 3 G Not i systematic form. We do the followi: row4 row4 row row row3 row3 row G H 4 I I T P P

42 Cyclic bloc codes Sydrome decodi for Cyclic codes: Receied codeword i polyomial form is ie by Receied codeword r e Error patter The sydrome is the remaider obtaied by diidi the receied polyomial by the eerator polyomial. r q S Sydrome With sydrome ad Stadard array, the error is estimated. I Cyclic codes, the size of stadard array is cosiderably redced. Lectre 9 4

43 Example of the bloc codes P B 8PSK QPSK E b / N [db] Lectre 9 43