High-Speed Decoding of the Binary Golay Code
|
|
- Wendy Sullivan
- 6 years ago
- Views:
Transcription
1 Hgh-Speed Decodng of the Bnary Golay Code H. P. Lee *1, C. H. Chang 1, S. I. Chu 2 1 Department of Computer Scence and Informaton Engneerng, Fortune Insttute of Technology, Kaohsung 83160, Tawan *hpl@fotech.edu.tw 2 Department of Electronc Engneerng, Natonal Kaohsung Unversty of Appled Scences, Kaohsung 807, Tawan ABSTRACT Recently, some table-lookup decodng algorthms (TLDAs) have been used to correct the bnary Golay code. Ths paper proposes an effcent hgh-speed TLDA called message-syndrome decodng algorthm (MSDA) by usng the message syndrome to correct the bnary systematc Golay code. The proposed MSDA s based on the novel message-syndrome lookup table (MSLT). The MSLT merely conssts of 12 canddate syndromes and ts memory sze s sgnfcantly smaller than other exstng lookup tables. Computer software smulaton results show that the decodng speed of the proposed MSDA s superor to other proposed TLDAs. Keywords: Golay code, weght, syndrome, error pattern 1. Introducton The well-known bnary Golay code, also called the bnary (23, 12, 7) quadratc resdue (QR) code [2], was frst dscovered by Golay [1] n It s a very useful perfect lnear errorcorrectng code; partcularly, t has been used n the past decades for a varety of applcatons nvolvng a party bt beng added to each word to yeld a half-rate code called the bnary (24, 12, 8) extended Golay code. One of ts most nterestng applcatons s the provson of error control n the Voyager mssons [2]. The Golay code can allow the correcton of up to t ( d 1)/ 2 = ( 7 1)/ 2 3 errors, where x denotes the greatest nteger less than or equal to x, t s the error-correctng capablty, and d s the mnmum Hammng dstance of the code. In the past few decades, several decodng technques have been developed to decode the bnary Golay codes, for example, the famous algebrac decodng algorthm (ADA) developed by Ela [3], the notable shft-search algorthm proposed by Reed et al. [4] after that, and the nverse-free Berlekamp Massey algorthm used to obtan the error-locator polynomal by Chen et al. [5]. The above s some representatve lterature on ADAs used to correct Golay codes. Recently, the TLDAs gven n [6-9] have acqured a promnent role n error-correctng decodng. A TLDA wth a refned lookup table (RLT) has been proposed by Ln et al. [8]; the RLT only needs 168 bytes memory sze. The memory sze of the RLT s the smallest memory sze requred to correct the Golay code untl now. In general, the TLDA makes consderably faster decodng speeds possble for the cyclc codes, n comparson wth the ADA. However, the TLDA requres an extra memory space n the decoder chp or software and t ncreases the decodng cost rapdly when the code length s large. Although the ADA does not need any lookup table, t requres a large number of complcated computatons n the fnte feld. These complcated computatons wll degrade the decodng performance and lead to a serous decodng delay as the code length ncreases []. Chen et al. [7] proposed the fast lookup table decodng algorthm (FLTDA) wth a lookup table of 1.35K bytes by usng the shft-search algorthm [4] to decode the Golay code. The famous syndrome decodng algorthm (SDA) wth a reduced-sze lookup table (RSLT) gven n [2, p119] requres 89 syndromes and ther correspondng error patterns, whereas the RSLT requres 445 bytes of memory. JournalofAppledResearchandTechnology 331
2 HghSpeedDecodngoftheBnaryGolayCode,H.P.Leeetal./ Ln et al. [8] proposed the syndrome-weght decodng algorthm (SWDA) wth RLT whch needs 168 bytes of memory, and the smulaton result shows that the decodng speed s approxmately 9.7 tmes faster than the mproved Ela s decodng algorthm (EDA). In ths paper, an effcent hgh-speed MSDA wth a MSLT s proposed to decode the bnary systematc Golay code. The novel MSLT conssts only of 12 canddate syndromes whch have one error n the message part of the receved vector, and t merely requres 24 bytes of memory. Software smulaton results show that the average decodng speed s the fastest among the TLDAs as mentoned above. In the four-error soft-decson decoder n [11], the mproved EDA s the role of the hard-decson decoder. We substtuted the proposed MSDA for the mproved EDA n the four-error soft-decson decoder. Apparently, the decodng speed of the soft-decson decoder was faster than before. The remander of ths paper s organzed as follows: The background of the bnary systematc Golay code s brefly revewed n Secton 2. Some related TLDAs and related theorems are brefly ntroduced n Secton 3. The proposed MSDA s presented n Secton 4. Computer smulaton results are gven n Secton 5. Fnally, ths paper concludes wth a bref summary n Secton Background of the bnary systematc Golay code The bnary (23, 12, 7) Golay code can be defned as a lnear cyclc code or QR code [2]. For the bnary Golay code over fnte feld GF(2 11 ), ts quadratc resdue set s gven by Q 23 = { j 2 mod 23 for 1 j 22} = {1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18}. (1) The generator polynomal g(x) of the bnary Golay code s defned by g( x) x Q x ( x x 7 ) x 6 x 5 x 1, (2) where s a prmtve 23rd root of unty n GF(2 11 ). n Let the codeword polynomal 1 C ( x) 0C x, where C GF(2) for 0 22, and the message k polynomal 1 m ( x) 0 m x, where k = 12 s the message length and m GF(2) for C(x) s a multple of g(x), namely, C(x) = m(x)g(x). Also, let the product m(x)x 11 dvde by g(x), then one has m(x)x 11 = q(x)g(x) + d(x). Multplyng both sdes of last expresson by x 12 and usng x 23 = 1, one yelds d(x)x 12 + m(x) = (q(x)x 12 )g(x). The term (d(x)x 12 + m(x)), whch s a multple of g(x), s a systematc codeword gven by c(x) = d(x)x 12 + m(x) = p(x) + m(x), (3) 22 where p ( x ) 12 p x denotes the party-check polynomal [12] of a codeword and p GF(2) for The bnary systematc Golay code s practcally appled n the standard of the communcaton system for automatc lnk establshment (ALE) by the Unted States Department of Defense [13]. Now, let a 12-bt message be encoded nto a 23- bt systematc codeword and be transmtted through a nosy channel to obtan a receved word of the form r(x) = c(x) + e(x). 22 e ( x ) 0 e x s the error pattern polynomal, where e GF(2) for The syndromes of the code are defned by s r( ) c( ) e( ), (4) e( ) where Q j j 0 e j ( ) To smplfy the polynomal expressons above, the message, codeword, error pattern, receved word, and syndrome polynomals can be expressed as the bnary vector forms m = (m m 1 m 0 ), c = (c c 1 c 0 ), e = (e e 1 e 0 ), r = c + e = (r r 1 r 0 ), and s = (s... s 1 s 0 ), respectvely. For the bnary systematc Golay code, t follows from [2, p85] that the systematc generator matrx G can be expressed as follows: 332 Vol.11,June2013
3 HghSpeedDecodngoftheBnaryGolayCode,H.P.Leeetal./ I12 G PI , (5) where P and I 12 denote the 1211 matrx and the 1212 dentty matrx, respectvely. The systematc vector form of the codeword can be obtaned by c mg ( p p p m m 0 ). (6) m The party-check polynomal h(x) = x 12 + x + x 7 + x 4 + x 3 + x 2 + x + 1 s a factor of x The 1123 party-check matrx H can be expressed as follows: T I P H, (7) 11 where P T s a 1112 transpose matrx of P and I 11 s a 1111 dentty matrx. The H T denotes the transpose matrx of H; I11 that s, H T. The vector form of the P 2311 syndrome can be defned by s = rh T. (8) The followng lemma gven n [14] shows that the mappng between the syndromes and error patterns under the error-correctng capablty s one-to-one. Lemma 1. The mappng between the syndromes of a code and the correspondng error patterns of weght t s one-to-one. 3. Related work and theorems In ths secton, some related TLDAs and theorems are brefly ntroduced n order to develop the proposed MSDA. For the TLDA, the followng defnton, theorems and corollary gven n [9] are needed. Defnton 1. Let the Hammng norm or weght of a bnary vector a = (a n-1... a 1 a 0 ) be desgnated by w(a). The Hammng dstance between two bnary vectors a and b s defned by d(a, b) = w(a + b). Theorem 1. Let a = (a n-1... a 1 a 0 ) and b = (b n-1... b 1 b 0 ) be two bnary vectors, then w(a + b) = w(a) + w(b) 2 1 a b. (9) Corollary 1. Let a and b, defned n Defnton 1, be two bnary vectors. If a b = 0 for 1 n, then w(a + b) = w(a) + w(b). () Theorem 2. For the bnary systematc (n, k, d) QR codes, let us assume that there are v errors n the receved word, where 1 v t. All v errors occur n the n k party-check bts f and only f the weght of syndrome w(s) = v. As mentoned above, the error-correctng capablty of the bnary Golay code s t =3. The full lookup table (FLT) n the conventonal table lookup decodng 3 23 algorthm (CTLDA) requres 1 = 2,047 syndromes and ther correspondng error patterns by Lemma 1. Each syndrome and error pattern needs 2 bytes and 3 bytes, respectvely, to store n the FLT. Thus, the total memory sze needed for the FLT s (2047 (2 + 3)) =,235 bytes. However, such a large memory s less effcent n practce. Recently, some useful TLDAs have been presented n the lterature to reduce the memory sze of the FLT. The LTDA [7] used the shftsearch method [4] to reduce the memory sze of the FLT and the bnary-search method to reduce the search tme n the FLT. The shft-search method deletes all v = t error patterns and keeps 1 v t 1 error patterns. n JournalofAppledResearchandTechnology 333
4 HghSpeedDecodngoftheBnaryGolayCode,H.P.Leeetal./ t Therefore, the LTDA needs syndromes and ther correspondng error patterns; that s, the FLT needs 276 (2 + 3) = 1,380 bytes memory sze. Nevertheless, the memory sze of the FLT s stll large and the smulaton result shows that the average decodng tme of the proposed MSDA s about.6 tmes faster that the LTDA. To develop the followng TLDAs, Theorem 3 s useful to reduce the memory sze of the FLT. The well-known SDA wth RSLT [2, p119] has the followng useful theorem of the cyclc codes to dramatcally reduce the memory sze of the FLT. For more detaled proof of ths theorem, see [2, p118]. Theorem 3. Let s(x) be the syndrome polynomal correspondng to a receved polynomal r(x). Also, let r (1) (x) be the polynomal obtaned by cyclcally shftng the coeffcents of r(x) one bt to the left. Then the remander obtaned when dvdng xs(x) by g(x) s the syndrome s (1) (x) correspondng to r (1) (x). Decodng the bnary systematc Golay code by Theorem 3, the RSLT of the SDA only needs 1/23 memory sze of the FLT, namely,235/23 = 445 bytes memory sze [8]. Nonetheless, the memory sze of the RSLT s stll large and can be further reduced. Smlarly, by Theorem 3, r (j) (x) s the polynomal obtaned by cyclcally shftng the coeffcents of r(x) j bts to the left. Next, the syndrome dfference s d s defned by s d = s s for 1 N, where s s the syndrome of r, s s the syndrome n lookup table, and N s the table sze. ( j ) Smlarly, the syndrome dfference s s defned by s ( j ) d s ( j ) s for 1 j n, where s (j) d s the syndrome of r (j). By usng the property of the syndrome dfference, the RLT of the SWDA only requres 168 bytes. However, the decodng speed s not very outstandng accordng to the smulaton result. To overcome ths problem, the MSDA s presented by usng the syndrome that only one error occurred n the message part of the receved word. 4. MSDA for the bnary Golay code In ths secton, an effcent hgh-speed MSDA wth a MSLT for decodng the bnary systematc Golay code s developed to further reduce the decodng tme and the memory sze of the RLT. By Theorem 2 and (8), t s obvous that at least one error s n the message bts f the weght of syndrome w(s) > 3. Defnton 2. The message-syndrome lookup table (MSLT) s a table of all the error patterns occurred n the message part wth weght 1 v t / 2 together wth ther correspondng syndromes, where x denotes the smallest nteger less than or equal to x. Therefore, the novel MSLT merely conssts of 12 canddate syndromes whch are the syndromes of havng one error n the message part; that s, s s the syndrome of e whch s one error located at bt for The MSLT s lsted n Table 1. Syndrome Syndrome s 1 = (0011) s 2 = (0) s 3 = (001111) s 4 = (011) s 5 = (1) s 6 = (00011) s 7 = (00) s 8 = (00) s 9 = (00111) s = (01) s 11 = (1101) s 12 = (01) Table 1. MSLT for the Bnary Systematc Golay Code. To develop the proposed MSDA by usng the MSLT, the followng lemma and propertes are necessary. Lemma 2. For the bnary Golay code, let e be an error pattern, and let e p and e m be ts party-check part and message part, respectvely. Let s p and s m be the syndromes correspondng to e p and e m, respectvely. Assume that w(e) 3, then we have () w(s p ) = w(e p ); () w(s m ) d w(e m ), where d = 7; () ((s m << 12) + e m ) s always a codeword, where << denotes the logcal left shft operator n programmng or the extenson by zeros on the rght n mathematcs. Proof: By Equaton (8), () s obvous. Agan, by Equaton (8), gven that ((s m << 12) + e m )H T = ((s m I << 12) + e m ) 11 I = (sm << 12) 11 I P + e 11 m P = s m P + s m = 0, ((s m << 12)+ e m ) s thus a codeword, whch proves (). Hence, w((s m << 12) + e m ) = w(s m ) + w(e m ) d, that s, w(s m ) d w(e m ). The proof s thus completed. 334 Vol.11,June2013
5 HghSpeedDecodngoftheBnaryGolayCode,H.P.Leeetal./ By Lemma 2, f v t, then one can obtan the followng two propertes. Property 1. For the bnary Golay code, the weght of the syndrome dfference w(s d ) has the followng cases: Case 1: If w(s) 3, then all three errors are n the party-check part and no error s n the message part. Case 2: If w(s) > 3 and w(s d ) < 3, then only 1 error s n the message part. Case 3: If w(s) > 3 and w(s d ) > 3, than at least 2 errors are n the message part. Next, for Case 3 of Property 1, the r and s are cyclcally shfted left by 11 bts, then one obtans r and s, respectvely. Thus, one yelds the followng property. Property 2 For the bnary Golay code, f w(s) > 3 and w(s d ) > 3, then the weght of s and s d have the followng cases: Case 1: If w(s ) = 2 or 3, then r has at least 2 errors n the message part and no error s n the party-check part. d Case 2: If w(s ) > 3 and w( s ) 2, then r has at least 2 errors n the message part. d Case 3: If w(s ) > 3 or w( s ) > 3, then 2 errors are n the message part and one of them s at the poston of bt 0. By usng the MSLT, the decodng process of the MSDA s as follows: (1) Gvng a receved vector r. (2) Compute the syndrome s = rh T and ts weght w(s). (3) If w(s) 3, then return c = r (s << 12), and go to step 13. (4) For 1 12, compute the syndrome s d = s s and ts weght w(s d ). (5) If w(s d ) 2, then return c = r (s d << 12) (1 << ( 1)), and go to step 13. (6) Cyclcally shft r and s left by 11 bts obtanng r and s, and then compute w(s ). (7) If w(s ) = 2 or 3, then return c = ((r (s << 12)) << 12) ((r (s << 12)) >> 11), and go to step 13. (8) For 1 12, compute the syndrome d = s s and ts weght w( s ). d s d (9) If w( s ) = 1 or 2, then return c = ((r ( s d << 12) (1 << ( 1))) << 12) ((r d ( s << 12) (1 << ( 1))) >> 11), and go to step 13. () Compute r = r 1 and s = s s 1. For 2 12, compute the syndrome s d = s s and ts weght w(s d ). (12) If w(s d ) = 1, then return c = r (s d << 12) (1 << ( 1)). (13) Stop. An example s gven below. By (6), a message vector m = ( ) s encoded nto a bnary systematc Golay code c = ( ). Assume that a nose vector s e = ( ), then the receved word becomes r = c + e = ( ). The decodng steps proceed as follows: (1) The receved vector s r = ( ). (2) Compute the syndrome s = ( ) and ts weght w(s) = 5. (3) Gven that w(s) > 3, go to step 4. (4) For = 1, compute the syndrome s d1 = s s 1 = () and ts weght w(s d1 ) = 7. JournalofAppledResearchandTechnology 335
6 HghSpeedDecodngoftheBnaryGolayCode,H.P.Leeetal./ (5) Gven that w(s d1 ) > 2, go to step 4. repeatng steps 4 and 5 (4) For = 12, compute the syndrome s d12 = s s 12 = (001111) and ts weght w(s d12 ) = 7. (5) Gven that w(s d12 ) > 2, go to step 6. (6) Cyclcally shft r and s left by 11 bts obtanng r = ( ) and s = (011), and then compute w(s ) = 7. (7) Gven that w(s ) 2 or 3, go to step 8. (8) For = 1, compute the syndrome s d1 = s s 1 = (00111) and ts weght d1 w( s ) = 7. d1 (9) Gven that w( s ) > 2, go to step 8. repeatng steps 8 and 9 (8) For = 12, compute the syndrome s d12= s s 12 = (000) and ts weght d12 w( s ) = 5. d12 (9) Gven that w( s ) > 2, go to step. () Compute r = r 1 = ( ) and s = s s 1 = (). For = 2, compute the syndrome s d2 = s s 2 = (00) and ts weght w(s d2 ) = 5. (12) Gven that w(s d2 ) > 1, go to step 11. repeatng steps 11 and 12 (13) For = 8, compute the syndrome s d8 = s s 8 = ( ) and ts weght w(s d8 ) = 1. (12) (14) If w(s d8 ) = 1, then return the correct codeword c = r1 (s d8 << 12) (1 << ( 1)) = ( ) ( ) (000000) = ( ). Go to step 13. (13) (15) Stop. Therefore, the correct message can be retreved from the receved vector. 5. Smulaton results The proposed MSDA has been programmed n C++ language. On an Intel Q6600 PC, all codewords wth up to three errors, namely C 8, receved vectors, were created to check every possble error pattern. The detaled decodng tmes for decodng the bnary systematc Golay code wth dfferent decodng algorthms are gven n Table 2. The proposed MSDA has the fastest average decodng tme and works perfectly wth an average decodng tme of s per error pattern. Algorthms Proposed MSDA SDA wth RSLT FLTDA wth table SWDA wth RLT Improved EDA Number of errors Average Table 2. Computer Decodng Tme for Decodng the Bnary Golay Code (n s). From Table 2, the proposed MSDA has the fastest average decodng tme and s about 44.5 tmes faster than the mproved EDA. Furthermore, we substtuted the proposed MSDA for the mproved EDA n the four-error soft-decson decoder [11]. Apparently, the decodng speed of the softdecson decoder was about 44.5 tmes faster than before. The decodng tme of the two hard-decson decoders wth respect to the SNR values s shown n Fgure Conclusons Ths paper presented a smple and more effcent decodng algorthm for decodng the bnary systematc Golay code. The key deas of the MSDA are based on the propertes of cyclc codes, the one-error syndrome n the message part, the weght of syndrome, the weght of syndrome 336 Vol.11,June2013
7 HghSpeedDecodngoftheBnaryGolayCode,H.P.Leeetal./ dfference, Property 1, and Property 2. The MSLT of the MSDA merely requres a very small memory sze of 24 bytes. These facts lead to a substantal mprovement n decodng the bnary systematc Golay code. Smulaton results show that, followng the new decodng strateges, the proposed MSDA acheves the best effcency. Hence, t s naturally sutable and proper for DSP or embedded system software mplementaton. [7] Y. H. Chen et al., Effcent decodng of systematc (23, 12, 7) and (41, 21, 9) quadratc resdue codes, J Inf Sc Eng, vol. 26, no. 5, pp , 20. [8] T. C. Ln et al., On the decodng of the (24, 12, 8) Golay code, Inf Sc, vol. 180, no. 23, pp , 20. [9] T. C. Ln et al., Hgh speed decodng of the bnary (47, 24, 11) quadratc resdue code, Inf Sc, vol. 180, no. 20, pp , 20. [] S. Ln and E. J. Jr Weldon, Long BCH codes are bad, Inf Contr, vol. 11, no. 4, pp , [11] S. I. Chu et al., Fast decodng of the (23, 12, 7) Golay code wth four-error-correctng capablty, Eur Trans Telecomm, vol, 22, no. 7, pp , [12] C. A. Vázquez-Fernández and G. Vega-Hernández, On the Weght Dstrbuton of the Dual of some Cyclc Codes wth Two Non Conjugated Zeros, J Appl Res Technol, vol.9, no.1, pp , [13] MIL-STD-188/141B. Department of Defense Interface Standard: Interoperablty and Performance Standards for Medum and Hgh Frequency Rado Systems. STD+( )/MIL_STD_188_141B_1703. Fgure 1. Decodng Tme of Two Hard-Decson Decoders at SNR of 0~6 db. References [14] I. S Reed et al., The algebrac decodng of the (41, 21, 9) quadratc resdue code, IEEE Trans Inf Theory, vol. 38, no. 3, pp , [1] M. Golay, Notes on dgtal codng, Proc. IRE, vol. 37, p. 657, [2] S. Wcker, "Error Control Systems for Dgtal Communcaton and Storage, Prentce Hall: Upper Saddle Rver, NJ, [3] M. Ela, Algebrac decodng of the (23, 12, 7) Golay code, IEEE Trans Inf Theory, vol. 33, no. 1, pp , [4] I. S. Reed et al., Decodng the (24, 12, 8) Golay code, IEE P-COMPUT DIG T, vol. 137, no. 3, pp , [5] Y. H. Chen et al., Algebrac decodng of quadratc resdue codes usng Berlekamp-Massey algorthm, J Inf Sc Eng, vol. 23, no. 1, 2007, pp [6] H. Chang et al., A weght method of decodng the (23, 12, 7) Golay code Usng Reduced Lookup Table, 2008 Internatonal Conference On Communcaton, Crcuts and Systems (ICCCAS 2008), Xamen, Chna, 2008, pp JournalofAppledResearchandTechnology 337
Decoding of the Triple-Error-Correcting Binary Quadratic Residue Codes
Automatc Control and Informaton Scences, 04, Vol., No., 7- Avalable onlne at http://pubs.scepub.com/acs/// Scence and Educaton Publshng DOI:0.69/acs--- Decodng of the rple-error-correctng Bnary Quadratc
More informationChapter 6. BCH Codes
Wreless Informaton Transmsson System Lab Chapter 6 BCH Codes Insttute of Communcatons Engneerng Natonal Sun Yat-sen Unversty Outlne Bnary Prmtve BCH Codes Decodng of the BCH Codes Implementaton of Galos
More informationApplication of Nonbinary LDPC Codes for Communication over Fading Channels Using Higher Order Modulations
Applcaton of Nonbnary LDPC Codes for Communcaton over Fadng Channels Usng Hgher Order Modulatons Rong-Hu Peng and Rong-Rong Chen Department of Electrcal and Computer Engneerng Unversty of Utah Ths work
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More informationDC-Free Turbo Coding Scheme Using MAP/SOVA Algorithms
Proceedngs of the 5th WSEAS Internatonal Conference on Telecommuncatons and Informatcs, Istanbul, Turkey, May 27-29, 26 (pp192-197 DC-Free Turbo Codng Scheme Usng MAP/SOVA Algorthms Prof. Dr. M. Amr Mokhtar
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationLow Complexity Soft-Input Soft-Output Hamming Decoder
Low Complexty Soft-Input Soft-Output Hammng Der Benjamn Müller, Martn Holters, Udo Zölzer Helmut Schmdt Unversty Unversty of the Federal Armed Forces Department of Sgnal Processng and Communcatons Holstenhofweg
More informationA Robust Method for Calculating the Correlation Coefficient
A Robust Method for Calculatng the Correlaton Coeffcent E.B. Nven and C. V. Deutsch Relatonshps between prmary and secondary data are frequently quantfed usng the correlaton coeffcent; however, the tradtonal
More informationLecture 3: Shannon s Theorem
CSE 533: Error-Correctng Codes (Autumn 006 Lecture 3: Shannon s Theorem October 9, 006 Lecturer: Venkatesan Guruswam Scrbe: Wdad Machmouch 1 Communcaton Model The communcaton model we are usng conssts
More informationRefined Coding Bounds for Network Error Correction
Refned Codng Bounds for Network Error Correcton Shenghao Yang Department of Informaton Engneerng The Chnese Unversty of Hong Kong Shatn, N.T., Hong Kong shyang5@e.cuhk.edu.hk Raymond W. Yeung Department
More informationFrom the Euclidean Algorithm for Solving a Key Equation for Dual Reed Solomon Codes to the Berlekamp-Massey Algorithm
1 / 32 From the Eucldean Algorthm for Solvng a Key Equaton for Dual Reed Solomon Codes to the Berlekamp-Massey Algorthm Mara Bras-Amorós, Mchael E O Sullvan The Claude Shannon Insttute Workshop on Codng
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationSpeeding up Computation of Scalar Multiplication in Elliptic Curve Cryptosystem
H.K. Pathak et. al. / (IJCSE) Internatonal Journal on Computer Scence and Engneerng Speedng up Computaton of Scalar Multplcaton n Ellptc Curve Cryptosystem H. K. Pathak Manju Sangh S.o.S n Computer scence
More informationVQ widely used in coding speech, image, and video
at Scalar quantzers are specal cases of vector quantzers (VQ): they are constraned to look at one sample at a tme (memoryless) VQ does not have such constrant better RD perfomance expected Source codng
More informationCONTRAST ENHANCEMENT FOR MIMIMUM MEAN BRIGHTNESS ERROR FROM HISTOGRAM PARTITIONING INTRODUCTION
CONTRAST ENHANCEMENT FOR MIMIMUM MEAN BRIGHTNESS ERROR FROM HISTOGRAM PARTITIONING N. Phanthuna 1,2, F. Cheevasuvt 2 and S. Chtwong 2 1 Department of Electrcal Engneerng, Faculty of Engneerng Rajamangala
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationHongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationA New Scrambling Evaluation Scheme based on Spatial Distribution Entropy and Centroid Difference of Bit-plane
A New Scramblng Evaluaton Scheme based on Spatal Dstrbuton Entropy and Centrod Dfference of Bt-plane Lang Zhao *, Avshek Adhkar Kouch Sakura * * Graduate School of Informaton Scence and Electrcal Engneerng,
More informationFUZZY GOAL PROGRAMMING VS ORDINARY FUZZY PROGRAMMING APPROACH FOR MULTI OBJECTIVE PROGRAMMING PROBLEM
Internatonal Conference on Ceramcs, Bkaner, Inda Internatonal Journal of Modern Physcs: Conference Seres Vol. 22 (2013) 757 761 World Scentfc Publshng Company DOI: 10.1142/S2010194513010982 FUZZY GOAL
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationTornado and Luby Transform Codes. Ashish Khisti Presentation October 22, 2003
Tornado and Luby Transform Codes Ashsh Khst 6.454 Presentaton October 22, 2003 Background: Erasure Channel Elas[956] studed the Erasure Channel β x x β β x 2 m x 2 k? Capacty of Noseless Erasure Channel
More informationDepartment of Electrical & Electronic Engineeing Imperial College London. E4.20 Digital IC Design. Median Filter Project Specification
Desgn Project Specfcaton Medan Flter Department of Electrcal & Electronc Engneeng Imperal College London E4.20 Dgtal IC Desgn Medan Flter Project Specfcaton A medan flter s used to remove nose from a sampled
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationThe Synchronous 8th-Order Differential Attack on 12 Rounds of the Block Cipher HyRAL
The Synchronous 8th-Order Dfferental Attack on 12 Rounds of the Block Cpher HyRAL Yasutaka Igarash, Sej Fukushma, and Tomohro Hachno Kagoshma Unversty, Kagoshma, Japan Emal: {garash, fukushma, hachno}@eee.kagoshma-u.ac.jp
More informationFinding Primitive Roots Pseudo-Deterministically
Electronc Colloquum on Computatonal Complexty, Report No 207 (205) Fndng Prmtve Roots Pseudo-Determnstcally Ofer Grossman December 22, 205 Abstract Pseudo-determnstc algorthms are randomzed search algorthms
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationDepartment of Mathematical Sciences, Clemson University. Some recent developments in permutation decoding
Department of Mathematcal Scences, Clemson Unversty http://www.ces.clemson.edu/ keyj/ Some recent developments n permutaton decodng J. D. Key keyj@ces.clemson.edu 1/32 P Abstract The method of permutaton
More informationSubset Topological Spaces and Kakutani s Theorem
MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered
More informationOn the Interval Zoro Symmetric Single-step Procedure for Simultaneous Finding of Polynomial Zeros
Appled Mathematcal Scences, Vol. 5, 2011, no. 75, 3693-3706 On the Interval Zoro Symmetrc Sngle-step Procedure for Smultaneous Fndng of Polynomal Zeros S. F. M. Rusl, M. Mons, M. A. Hassan and W. J. Leong
More informationPop-Click Noise Detection Using Inter-Frame Correlation for Improved Portable Auditory Sensing
Advanced Scence and Technology Letters, pp.164-168 http://dx.do.org/10.14257/astl.2013 Pop-Clc Nose Detecton Usng Inter-Frame Correlaton for Improved Portable Audtory Sensng Dong Yun Lee, Kwang Myung Jeon,
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationAn (almost) unbiased estimator for the S-Gini index
An (almost unbased estmator for the S-Gn ndex Thomas Demuynck February 25, 2009 Abstract Ths note provdes an unbased estmator for the absolute S-Gn and an almost unbased estmator for the relatve S-Gn for
More informationValuated Binary Tree: A New Approach in Study of Integers
Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Volume 4, Issue 3, March 6, PP 63-67 ISS 347-37X (Prnt) & ISS 347-34 (Onlne) wwwarcournalsorg Valuated Bnary Tree: A ew Approach
More informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More information} s ) was constructed [7]. C 0 with length n 0 and minimum distance d 0 over F q.
Ffty-Ffth Annual Allerton Conference Allerton House, UIUC, Illnos, USA October 3-6, 17 Mult-Erasure Locally Recoverable Codes Over Small Felds Pengfe Huang, Etan Yaakob, and Paul H Segel Electrcal and
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 1, July 2013
ISSN: 2277-375 Constructon of Trend Free Run Orders for Orthogonal rrays Usng Codes bstract: Sometmes when the expermental runs are carred out n a tme order sequence, the response can depend on the run
More informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
More informationAmusing Properties of Odd Numbers Derived From Valuated Binary Tree
IOSR Journal of Mathematcs (IOSR-JM) e-iss: 78-578, p-iss: 19-765X. Volume 1, Issue 6 Ver. V (ov. - Dec.016), PP 5-57 www.osrjournals.org Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree
More informationSimulated Power of the Discrete Cramér-von Mises Goodness-of-Fit Tests
Smulated of the Cramér-von Mses Goodness-of-Ft Tests Steele, M., Chaselng, J. and 3 Hurst, C. School of Mathematcal and Physcal Scences, James Cook Unversty, Australan School of Envronmental Studes, Grffth
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationMDS Array Codes for Correcting a Single Criss-Cross Error
1068 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000 MDS Array Codes for Correctng a Sngle Crss-Cross Error Maro Blaum, Fellow, IEEE, and Jehoshua Bruck, Senor Member, IEEE Abstract We
More informationOn the correction of the h-index for career length
1 On the correcton of the h-ndex for career length by L. Egghe Unverstet Hasselt (UHasselt), Campus Depenbeek, Agoralaan, B-3590 Depenbeek, Belgum 1 and Unverstet Antwerpen (UA), IBW, Stadscampus, Venusstraat
More informationFor now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.
Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson
More informationConsider the following passband digital communication system model. c t. modulator. t r a n s m i t t e r. signal decoder.
PASSBAND DIGITAL MODULATION TECHNIQUES Consder the followng passband dgtal communcaton system model. cos( ω + φ ) c t message source m sgnal encoder s modulator s () t communcaton xt () channel t r a n
More informationPulse Coded Modulation
Pulse Coded Modulaton PCM (Pulse Coded Modulaton) s a voce codng technque defned by the ITU-T G.711 standard and t s used n dgtal telephony to encode the voce sgnal. The frst step n the analog to dgtal
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationPerforming Modulation Scheme of Chaos Shift Keying with Hyperchaotic Chen System
6 th Internatonal Advanced echnologes Symposum (IAS 11), 16-18 May 011, Elazığ, urkey Performng Modulaton Scheme of Chaos Shft Keyng wth Hyperchaotc Chen System H. Oğraş 1, M. ürk 1 Unversty of Batman,
More informationHigh resolution entropy stable scheme for shallow water equations
Internatonal Symposum on Computers & Informatcs (ISCI 05) Hgh resoluton entropy stable scheme for shallow water equatons Xaohan Cheng,a, Yufeng Ne,b, Department of Appled Mathematcs, Northwestern Polytechncal
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationEGR 544 Communication Theory
EGR 544 Communcaton Theory. Informaton Sources Z. Alyazcoglu Electrcal and Computer Engneerng Department Cal Poly Pomona Introducton Informaton Source x n Informaton sources Analog sources Dscrete sources
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationA Comparison between Weight Spectrum of Different Convolutional Code Types
A Comparson between Weght Spectrum of fferent Convolutonal Code Types Baltă Hora, Kovac Mara Abstract: In ths paper we present the non-recursve systematc, recursve systematc and non-recursve non-systematc
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationLecture 10: May 6, 2013
TTIC/CMSC 31150 Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander,
More informationA Novel Feistel Cipher Involving a Bunch of Keys supplemented with Modular Arithmetic Addition
(IJACSA) Internatonal Journal of Advanced Computer Scence Applcatons, A Novel Festel Cpher Involvng a Bunch of Keys supplemented wth Modular Arthmetc Addton Dr. V.U.K Sastry Dean R&D, Department of Computer
More informationLecture 4: November 17, Part 1 Single Buffer Management
Lecturer: Ad Rosén Algorthms for the anagement of Networs Fall 2003-2004 Lecture 4: November 7, 2003 Scrbe: Guy Grebla Part Sngle Buffer anagement In the prevous lecture we taled about the Combned Input
More informationThe Quadratic Trigonometric Bézier Curve with Single Shape Parameter
J. Basc. Appl. Sc. Res., (3541-546, 01 01, TextRoad Publcaton ISSN 090-4304 Journal of Basc and Appled Scentfc Research www.textroad.com The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma
More informationCryptanalysis of pairing-free certificateless authenticated key agreement protocol
Cryptanalyss of parng-free certfcateless authentcated key agreement protocol Zhan Zhu Chna Shp Development Desgn Center CSDDC Wuhan Chna Emal: zhuzhan0@gmal.com bstract: Recently He et al. [D. He J. Chen
More informationA Fast Computer Aided Design Method for Filters
2017 Asa-Pacfc Engneerng and Technology Conference (APETC 2017) ISBN: 978-1-60595-443-1 A Fast Computer Aded Desgn Method for Flters Gang L ABSTRACT *Ths paper presents a fast computer aded desgn method
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationExercises of Chapter 2
Exercses of Chapter Chuang-Cheh Ln Department of Computer Scence and Informaton Engneerng, Natonal Chung Cheng Unversty, Mng-Hsung, Chay 61, Tawan. Exercse.6. Suppose that we ndependently roll two standard
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationDesign and Optimization of Fuzzy Controller for Inverse Pendulum System Using Genetic Algorithm
Desgn and Optmzaton of Fuzzy Controller for Inverse Pendulum System Usng Genetc Algorthm H. Mehraban A. Ashoor Unversty of Tehran Unversty of Tehran h.mehraban@ece.ut.ac.r a.ashoor@ece.ut.ac.r Abstract:
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationLecture 14 (03/27/18). Channels. Decoding. Preview of the Capacity Theorem.
Lecture 14 (03/27/18). Channels. Decodng. Prevew of the Capacty Theorem. A. Barg The concept of a communcaton channel n nformaton theory s an abstracton for transmttng dgtal (and analog) nformaton from
More informationP R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering /
Theory and Applcatons of Pattern Recognton 003, Rob Polkar, Rowan Unversty, Glassboro, NJ Lecture 4 Bayes Classfcaton Rule Dept. of Electrcal and Computer Engneerng 0909.40.0 / 0909.504.04 Theory & Applcatons
More informationPower law and dimension of the maximum value for belief distribution with the max Deng entropy
Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationComments on a secure dynamic ID-based remote user authentication scheme for multiserver environment using smart cards
Comments on a secure dynamc ID-based remote user authentcaton scheme for multserver envronment usng smart cards Debao He chool of Mathematcs tatstcs Wuhan nversty Wuhan People s Republc of Chna Emal: hedebao@63com
More informationIntroduction to Algorithms
Introducton to Algorthms 6.046J/8.40J Lecture 7 Prof. Potr Indyk Data Structures Role of data structures: Encapsulate data Support certan operatons (e.g., INSERT, DELETE, SEARCH) Our focus: effcency of
More informationS Advanced Digital Communication (4 cr) Targets today
S.72-3320 Advanced Dtal Communcaton (4 cr) Convolutonal Codes Tarets today Why to apply convolutonal codn? Defnn convolutonal codes Practcal encodn crcuts Defnn qualty of convolutonal codes Decodn prncples
More informationNew Algebraic Decoding of (17,9,5) Quadratic Residue Code by using Inverse Free Berlekamp-Massey Algorithm (IFBM)
International Journal of Computational Intelligence Research (IJCIR). ISSN: 097-87 Volume, Number 8 (207), pp. 205 2027 Research India Publications http://www.ripublication.com/ijcir.htm New Algebraic
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationAn efficient algorithm for multivariate Maclaurin Newton transformation
Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v10065-008-0020-6 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,
More informationChapter 8 SCALAR QUANTIZATION
Outlne Chapter 8 SCALAR QUANTIZATION Yeuan-Kuen Lee [ CU, CSIE ] 8.1 Overvew 8. Introducton 8.4 Unform Quantzer 8.5 Adaptve Quantzaton 8.6 Nonunform Quantzaton 8.7 Entropy-Coded Quantzaton Ch 8 Scalar
More information2-Adic Complexity of a Sequence Obtained from a Periodic Binary Sequence by Either Inserting or Deleting k Symbols within One Period
-Adc Comlexty of a Seuence Obtaned from a Perodc Bnary Seuence by Ether Insertng or Deletng Symbols wthn One Perod ZHAO Lu, WEN Qao-yan (State Key Laboratory of Networng and Swtchng echnology, Bejng Unversty
More informationCocyclic Butson Hadamard matrices and Codes over Z n via the Trace Map
Contemporary Mathematcs Cocyclc Butson Hadamard matrces and Codes over Z n va the Trace Map N. Pnnawala and A. Rao Abstract. Over the past couple of years trace maps over Galos felds and Galos rngs have
More informationIntroduction to Information Theory, Data Compression,
Introducton to Informaton Theory, Data Compresson, Codng Mehd Ibm Brahm, Laura Mnkova Aprl 5, 208 Ths s the augmented transcrpt of a lecture gven by Luc Devroye on the 3th of March 208 for a Data Structures
More informationTELE4652 Mobile and Satellite Communication Systems
TELE4652 Moble and Satellte Communcaton Systems Lecture 7 Equalsaton, Dversty, and Channel Codng In ths lecture we ll loo at three complementary technologes to allow us to obtan hgh qualty transmsson over
More informationWavelet chaotic neural networks and their application to continuous function optimization
Vol., No.3, 04-09 (009) do:0.436/ns.009.307 Natural Scence Wavelet chaotc neural networks and ther applcaton to contnuous functon optmzaton Ja-Ha Zhang, Yao-Qun Xu College of Electrcal and Automatc Engneerng,
More informationCOEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN
Int. J. Chem. Sc.: (4), 04, 645654 ISSN 097768X www.sadgurupublcatons.com COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN R. GOVINDARASU a, R. PARTHIBAN a and P. K. BHABA b* a Department
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationSmarandache-Zero Divisors in Group Rings
Smarandache-Zero Dvsors n Group Rngs W.B. Vasantha and Moon K. Chetry Department of Mathematcs I.I.T Madras, Chenna The study of zero-dvsors n group rngs had become nterestng problem snce 1940 wth the
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationA new construction of 3-separable matrices via an improved decoding of Macula s construction
Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula
More informationDigital Signal Processing
Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationSection 3.6 Complex Zeros
04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationThe Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD
he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world X observatons g decson functon L[g,y] loss of predctng y wth g Bayes decson rule s
More informationOn quasiperfect numbers
Notes on Number Theory and Dscrete Mathematcs Prnt ISSN 1310 5132, Onlne ISSN 2367 8275 Vol. 23, 2017, No. 3, 73 78 On quasperfect numbers V. Sva Rama Prasad 1 and C. Suntha 2 1 Nalla Malla Reddy Engneerng
More informationHiding data in images by simple LSB substitution
Pattern Recognton 37 (004) 469 474 www.elsever.com/locate/patcog Hdng data n mages by smple LSB substtuton Ch-Kwong Chan, L.M. Cheng Department of Computer Engneerng and Informaton Technology, Cty Unversty
More information