Optimum Selection Combining for M-QAM on Fading Channels

Size: px
Start display at page:

Download "Optimum Selection Combining for M-QAM on Fading Channels"

Transcription

1 Optmum Seecton Combnng for M-QAM on Fadng Channes M. Surendra Raju, Ramesh Annavajjaa and A. Chockangam Insca Semconductors Inda Pvt. Ltd, Bangaore-56000, Inda Department of ECE, Unversty of Caforna, San Dego, La Joa, CA 92093, U.S.A Department of ECE, Indan Insttute of Scence, Bangaore 56002, Inda Abstract In ths paper, we present the optmum seecton combnng SC) scheme for M-QAM whch mnmzes the average bt error rate on fadng channes. We show that the seecton combnng scheme where each bt n a QAM symbo seects the dversty branch wth the argest magntude of the og-kehood rato LLR) of that bt s optmum n the sense that t mnmzes the average bt error rate BER). In ths optmum SC scheme, dfferent bts n a gven QAM symbo may seect dfferent dversty branches snce the argest LLRs for dfferent bts may occur on dfferent dversty branches), and hence ts compexty s hgh. However, ths scheme provdes the best possbe BER performance for M-QAM wth seecton combnng, and can serve as a benchmark to compare the performance of other SC schemes e.g., seecton based on maxmum SNR). We compare the BER performance of ths optmum SC scheme wth other SC schemes where the dversty seecton s done based on maxmum SNR and maxmum symbo LLR. Keywords M-QAM, bt og-kehood rato, seecton combnng. I. INTRODUCTION Muteve quadrature amptude moduaton M-QAM)san attractve moduaton scheme for wreess communcatons due to the hgh spectra effcency t provdes []. Dversty recepton s a we known technque for mtgatng the effects of fadng on wreess channes [3],[2]. Typca dverstycombnng schemes ncude maxma rato combnng MRC), equa gan combnng EGC), seecton combnng SC), and generazed seecton combnng GSC). Seecton combnng s the smpest of a, as t processes ony one of the dversty branches. In ths paper, we are concerned wth seecton combnng for M-QAM. The dversty branch seecton n SC schemes can be done n severa ways. One way s to choose the dversty branch wth the argest nstantaneous SNR. It s known that choosng the dversty branch wth the maxmum SNR s not the optmum. An aternate way s to choose the branch wth the argest magntude of the og-kehood rato LLR) of the transmtted symbo we ca ths as the symbo LLR - SLLR), as proposed n [4], where the authors show that choosng the branch wth the argest SLLR mnmzes the symbo error rate SER) for M-ary sgnas. We, n ths paper, obtan the optmum seecton combnng scheme for M-QAM whch mnmzes the average bt error rate BER), rather than mnmzng the SER. In our scheme, we compute the LLR for each bt n a gven QAM symbo we Ths work was supported n part by the Indo-French Centre for Promoton of Advanced Research, New Deh, under Project 2900-IT. ca ths as the bt LLR - BLLR) on each dversty branch. For a gven bt n a QAM symbo, the dversty branch havng the argest magntude of the BLLR s chosen. We show that the above BLLR based dversty branch seecton mnmzes the average BER for M-QAM, and hence s optmum. In ths optmum SC scheme, t can be noted that dfferent bts n a gven QAM symbo may seect dfferent dversty branches snce the argest LLRs for dfferent bts may occur on dfferent dversty branches), and hence ts compexty s hgh,.e., the scheme needs a the L receve RF chans to be present for the bts to choose ther respectve best antennas. We however note that ths scheme provdes the best possbe BER performance for M-QAM wth seecton combnng, and can serve as a benchmark to compare the performance of other SC schemes e.g., seecton based on maxmum SNR). We present a BER performance comparson of the BLLR based optmum SC scheme wth other SC schemes where the dversty branch seecton s done based on maxmum SNR and maxmum symbo LLR. We show that, for 6-QAM wth one transmt antenna and L receve antennas, at a BER of 0 2, maxmum SLLR based SC performance s away from the BLLR based optmum SC performance by 0.9 db for L 2,by.4dBforL 3, and by.6 db for L 4. Lkewse, the maxmum SNR based SC performance s away from the optmum SC performance by.4 db for L 2,by 2. db for L 3, and by 2.6 db for L 4. We aso provde smar comparsons for 6-QAM wth two transmt antennas usng Aamout code [5] and L receve antennas. For 6- QAM wth two transmt antennas and L receve antennas, the SLLR based SC performance s away from the optmum SC performance by. db for L 2,by.6dB for L 3, and by.9 db for L 4at a BER of 0 2. We present smar performance comparson for 32-QAM as we. Athough the resuts are shown ony for 6- and 32-QAM n ths paper, the method for BLLR dervaton can be extended for any M-ary QAM. The rest of the paper s organzed as foows. In Secton 2, we derve BLLR expressons for 6-QAM on a gven receve antenna n a system wth one-tx/two-tx antennas. In Secton 3, we show that the SC scheme that chooses the branch wth the argest BLLR mnmzes the BER, and hence s optmum. Smuaton resuts of the BER performance of the optmum SC scheme n comparson wth the performance of other SC schemes are presented n Secton 4. Concusons are gven n Secton 5.

2 symbos wth r and S 0) comprses symbos wth r 0 n the consteaton. Then, from 2), we have ) Pra α y,h LLR r ) og. 3) Pra β y,h Fg.. 6-QAM Consteaton II. BIT LOG-LIKELIHOOD RATIOS In ths secton, we derve expressons for the BLLRs for 6- QAM.e., M 6) scheme shown n Fg., where 4 bts r,r 2,r 3,r 4 ) are mapped on to a compex symbo a a I + ja Q. The horzonta/vertca ne peces n Fg. denote that a bts under these nes take the vaue, and the rest take the vaue 0. For exampe, the symbo wth coordnates 3d, 3d) maps the 4-bt combnaton r, r 2 0, r 3 r 4. A. -Tx and L-Rx Antennas Frst, consder the case of one transmt antenna and L receve antennas. Assumng that the transmtted symbo a undergoes mutpcatve and ndependent fadng on each dversty path the fadng s assumed to be sow, frequency non-seectve and reman constant over one symbo nterva on each dversty path), the receved sgna y at the th receve antenna correspondng to the transmtted symbo a can be wrtten as y h a + n, 0,,L ) where h,0,,l, s the compex channe coeffcent on the th receve antenna wth E h 2 and the r.v s h s are assumed to be..d, and n n I + jn Q s a compex Gaussan nose r.v of zero mean and varance σ 2 /2 per dmenson. We defne the og-kehood rato of bt r,, 2, 3, 4, of the receved symbo on the th antenna, LLR r ),as[6] ) Prr y,h LLR r ) og. 2) Prr 0 y,h Ceary, the optmum decson rue for the th branch s to decde ˆr f LLR r ) 0, and 0 otherwse. Defne two set parttons, S ) and S 0), such that S ) comprses BLLR expressons for other vaues of M can be derved kewse. Assume that a the symbos are equay key and that fadng s ndependent of the transmtted symbos. Usng Bayes rue, we then have ) f y h,ay h,a α) LLR r ) og. 4) f y h,ay h,a β) Snce f y h,ay h,a α πσ exp 2 σ y 2 h α ),4) 2 can be wrtten as exp σ y LLR r ) og 2 h α 2) ) exp σ y 2 h β 2). 5) Usng og j exp X j) ) mn j X j ), whch s a good approxmaton [7], we can approxmate 5) as LLR r ) σ 2 [ Defne z as mn z y h y h β 2 mn y h α 2 ]. 6) a + n h a + n, 7) where n s compex Gaussan wth varance σ 2 / h 2.Usng 7) n 6), and normazng LLR r ) by 4/σ 2, we get [ LLR r h 2 ) mn z β 2 mn z α ]. 2 8) 4 Further smpfcaton of 8) gves [ LLR r ) h 2 β 2 2z I β I 2z Q β Q 4 mn ] mn α 2 2z I α I 2z Q α Q, 9) where z z I + jz Q, α α I + jα Q and β β I + jβ Q. Note that the set parttons S ) and S 0) are demted by horzonta or vertca boundares. As a consequence, two symbos n dfferent sets cosest to the receved symbo aways e ether on the same row f the demtng boundares are vertca) or on the same coumn f the demtng boundares are horzonta). Usng the above fact, the LLRs for bt r,r 2,r 3 and r 4 are gven by h 2 z I d z I 2d LLR r ) 2 h 2 dd z I ) z I > 2d 2 h 2 dd + z I ) z I < 2d,

3 h 2 z Q d z Q 2d LLR r 2 ) 2 h 2 dd z Q ) z Q > 2d 2 h 2 dd + z Q ) z Q < 2d, LLR r 3 ) h 2 d z I 2d), LLR r 4 ) h 2 d z Q 2d) 0) where 2d s the mnmum dstance between pars of sgna ponts. B. 2-Tx and L-Rx Antennas Next, we consder the case of two transmt antennas and L receve antennas. Durng a gven symbo nterva, two symbos are transmtted smutaneousy on the two antennas usng Aamout code [5]. Let a, a 2 be the symbos transmtted on the frst and the second transmt antennas, respectvey, durng a symbo nterva. Durng the next symbo nterva, a 2, a are transmtted on the frst and the second transmt antennas, respectvey [5]. We denote the fadng coeffcents as foows: h 2 represents the fadng coeffcent from transmt antenna to receve antenna,,,l, and h 2 represents the fadng coeffcent from transmt antenna 2 to receve antenna,,,l.lety 2 and y 2,,,Lbe the receved sgnas at the th antenna durng two successve symbo ntervas, respectvey. Assumng that the channe reman constant over two consecutve symbo ntervas, the receved sgnas durng the two consecutve symbo ntervas can be wrtten as y 2 a h 2 a 2h 2 + n 2 y 2 a 2 h 2 + a h 2 + n 2, ) where h 2 L and h 2 L are the compex fadng coeffcents and n 2 and n 2 are compex Gaussan random varabes of zero mean and varance σ 2. Assumng perfect knowedge of the fadng coeffcents at the recever, we form â and â 2,forthe th receve branch as â h 2 y 2 + h 2 y2 ) â 2 h 2 y 2 h 2 y2 ). 2) After further smpfcaton, â and â 2 can be rewrtten as â h h 2 2) a + ζ â 2 h h 2 2) a 2 + ζ 2, 3) where ζ and ζ 2 are compex Gaussan random varabes wth of mean and varance h h 2 2 )σ 2. Foowng smar steps as n the case of one transmt and L receve antennas above, the LLR of bts r,, 2, 3, 4 of symbo a j, j, 2 on the th antenna, LLR aj r ),forthe two transmt and L receve antennas, can be derved as r ) h2 2 + h 2 2 ẑj I d ẑi j 2d 2 h h 2 2 dd ẑj I ) ẑi j > 2d 2 h h 2 2 dd +ẑj I ) ẑi j < 2d, h h 2 2) ẑ Q d ẑq j 2d j r 2 ) 2 h h 2 2) dd ẑ Q j ) ẑq j > 2d 2 h h 2 2) dd +ẑ Q j ) ẑq j < 2d, r 3) h h 2 2) d ẑ I j 2d ), r 4) h h 2 2) d ẑ Q j 2d). 4) In the above equatons, ẑj I and ẑq j are the rea and magnary parts of ẑ j, respectvey, where ẑ j s gven by â j ẑ j h h ) It s noted that the LLRs of the varous bts n any M-QAM consteaton of order M and for any arbtrary mappng of bts to the M-QAM symbos can be derved foowng smar steps gven above for 6-QAM. III. BLLR BASED OPTIMUM SC In ths secton, we derve the rue for optma seecton combnng so as to mnmze the BER of each of the bts formng the QAM symbo. We prove that n order to mnmze the BER of bt r, we must seect the dversty branch whch has the argest LLR r ). The proof s as foows. The BER for bt r, P b, s gven as P b Pr ˆr r y, h f y,h dy dh, 6) y,h where y y 0,y 2,,y L ), h h 0,h 2,,h L ), and f y,h s the jont probabty densty functon of y, h. It foows from the above equaton that P b n mnmzed by maxmzng Pr ˆr r y, h for a y, h. Now, Pr ˆr r y, h L 0 Pr ˆr r th branch seected, y, h Pr th branch seected y, h L Pr ˆr r y,h 0 Pr th branch seected y, h max Pr ˆr r y,h. 7) Note that P b s mnmzed by seectng the branch that provdes the maxmum Pr ˆr r y,h, or, equvaenty, seectng the branch that provdes the mnmum Pr ˆr r y,h, whch can be wrtten as Pr ˆr r y,h Pr ˆr,r 0 y,h + Pr ˆr 0,r y,h Pr LLR r ) 0,r 0 y,h + Pr LLR r ) < 0,r y,h. 8)

4 If LLR r ) 0, then Pr ˆr r y,h Pr r 0 y,h +e LLR r ). 9) If LLR r ) < 0, then Pr ˆr r y,h Pr r y,h Hence, we have +e LLR r ). 20) Pr ˆr r y,h +e LLR r ). 2) Therefore, to mnmze Pr ˆr r y,h, we need to maxmze the denomnator n 2), or, equvaenty, maxmze the term, LLR r ). Hence, by seectng the branch that provdes the argest magntude of LLR r ), we mnmze the BER, P b, and hence mnmze the average BER. It s noted that dfferent bts n a gven symbo may choose dfferent antennas, snce the argest BLLRs for dfferent bts may occur on dfferent antennas, and hence w requre that a the L receve RF chans are present for the bts to choose ther respectve best antennas. Ths scheme however provdes the best possbe BER performance of M-QAM wth seecton combnng, and can serve as a benchmark to compare the performance of other SC schemes as ustrated n the next secton). IV. SIMULATION RESULTS In ths secton, we present the smuated BER performance of the BLLR optmum SC scheme derved n the prevous secton n comparson wth the performance of other SC schemes where the dversty branch seecton s done based on maxmum SNR.e., choose the branch wth argest nstantaneous SNR) and maxmum SLLR.e., choose the branch wth the argest magntude of the symbo LLR). The channe gan coeffcents h s are taken to be..d compex Gaussan.e., fade amptudes are Rayegh dstrbuted) wth zero mean and E h 2. Fgure 2 shows the smuated average BER performance as functon of average SNR per branch for the foowng a) BLLR based optmum SC scheme, b) SNR based SC scheme, and c) SLLR based optmum SC scheme, for 6- QAM wth one transmt and L, 2, 3, 4 receve antennas. From Fg. 2, t s observed that, at a BER of 0 2, the SLLR based SC performance s away from the BLLR based optmum SC performance by 0.9 db for L 2,by.4dBfor L 3, and by.6 db for L 4. Lkewse, the SNR based SC performance s away from the optmum SC performance by.4 db for L 2,by2.dBforL 3, and by 2.6 db for L 4. Snce both the SNR based SC as we as the SLLR based SC have the same compexty.e., ony one of the dversty branches needs to be processed n both cases), SLLR Average Bt Error Rate L, No SC L2, BLLR based Opt. SC L2, SLLR based SC L2, SNR based SC L3, BLLR based Opt. SC L3, SLLR based SC L3, SNR based SC L4, BLLR based Opt. SC L4, SLLR based SC L4, SNR based SC Average Receved SNR per Branch db) Fg. 2. Comparson of varous seecton combnng schemes for 6-QAM for Tx antenna and L, 2, 3, 4 Rx antennas BLLR based optmum SC, SLLR based SC, and SNR based SC. based SC s preferred over SNR based SC snce t acheves BER performance coser to that of the BLLR based optmum SC Fgure 3 shows smar comparson for 6-QAM wth two transmt antennas usng Aamout code and L receve antennas. It s ponted out that the pots correspondng to the SLLR based seecton n ths fgure has been obtaned by dervng the expressons for the symbo LLRs for the two transmt and L receve antennas case.e., by extendng dervaton n [4] to the 2 Tx antennas case usng Aamout code). From Fg. 3, t s observed that, for 2-Tx and L 4Rx antennas, the SLLR based SC performance s away from the BLLR based optmum SC performance by. db for L 2,by.6dB for L 3, and by.9 db for L 4, at a BER of 0 2. Smary, the SNR based SC performance s away from the optmum SC performance by.5 db for L 2,by2.6for L 3, and by 3. for L 4,ataBERof0 2. We aso derved the BLLR expressons for 32-QAM dervaton not gven n ths paper), and evauated the BER performance of the three SC schemes for 32-QAM. Fgure 4 shows the BER performance of the three SC schemes for 32-QAM for -Tx and L 2, 4 Rx antennas. It can be observed that for 32-QAM, L 4, at a BER of 0 2, the SLLR based SC s worse by.6 db and the SNR based SC by 2.5 db compared to the BLLR based optmum SC. V. CONCLUSIONS We presented the optmum seecton combnng SC) scheme for M-QAM whch mnmze the average bt error rate BER) on fadng channes. We showed that the SC scheme whch chooses the dversty branch wth the argest magntude of the og-kehood ratos LLRs) of the ndvdua bts n the QAM symbo mnmzes the BER, and hence s optmum. It was ponted out that the compexty of ths optmum SC scheme s hgher snce dfferent bts n a gven QAM symbo

5 Average Bt Error Rate Tx, Rx, No SC 2Tx, 2Rx, BLLR based Opt. SC 2Tx, 2Rx, SLLR based SC 2Tx, 2Rx, SNR based SC 2Tx, 3Rx, BLLR based Opt. SC 2Tx, 3Rx, SLLR based SC 2Tx, 3Rx, SNR based SC 2Tx, 4Rx, BLLR based Opt. SC 2Tx, 4Rx, SLLR based SC 2Tx, 4Rx, SNR based SC [3] M. K Smon and M. S. Aoun, Dgta Communcatons Over Fadng Channes: A Unfed Approach to Performance Anayss, Wey Seres, Juy [4] Y. G. Km and S. W. Km, Optmum seecton combnng for M-ary sgnas n fadng channes, Proc. IEEE GLOBECOM 2002, November [5] S. M. Aamout, A smpe transmt dversty technque for wreess communcatons, IEEE J. Se. Areas n Commun., vo. 6, no. 8, pp , October 998. [6] R. Pyndah, A. Pcard and A. Gaveux, Performance of bock turbo coded 6-QAM and 64-QAM moduatons, Proc. IEEE GLOBE- COM 95, pp , November 995. [7] A. J. Vterb, An ntutve justfcaton and a smpfed mpementaton of the MAP decoder for convoutona codes, IEEE J. Se. Areas n Commun., vo. 6, no. 2, pp , Average Receved SNR per Branch db) Fg. 3. Comparson of varous seecton combnng schemes for 6-QAM for 2 Tx antennas usng Aamout code and L, 2, 3, 4 Rx antennas BLLR based optmum SC, SLLR based SC, and SNR based SC QAM 0 Average Bt Error Rate L2, BLLR based Opt. SC L2, SLLR based SC L2, SNR based SC L4, BLLR based Opt. SC L4, SLLR based SC L4, SNR based SC Average Receved SNR per Branch db) Fg. 4. Comparson of varous seecton combnng schemes for 32-QAM for Tx antenna and L 2, 4 Rx antennas BLLR based optmum SC, SLLR based SC, and SNR based SC. may seect dfferent dversty branches and snce the argest LLRs for dfferent bts may occur on dfferent dversty branches. However, ths scheme provdes the best possbe BER performance for M-QAM wth seecton combnng, and can serve as a benchmark to compare the performance of other SC schemes. We presented a BER performance comparson of ths optmum SC scheme wth other SC schemes where the dversty seecton s done based on maxmum SNR and maxmum symbo LLR. REFERENCES [] W. T. Webb and L. Hanzo, Modern Quadrature Amptude Moduaton: Prncpes and Appcatons for Fxed and Wreess Channes, IEEE Press, New York, 994. [2] W. C. Jakes, Mcrowave Mobe Communcatons, New York: IEEE Press, 974.

BER Analysis of QAM with Transmit Diversity in Rayleigh Fading Channels

BER Analysis of QAM with Transmit Diversity in Rayleigh Fading Channels BER Analyss of QAM wth Transmt Dversty n Raylegh Fadng Channels M. Surendra Raju,A. Ramesh and A. Chockalngam Inslca Semconductors Inda Pvt. Ltd.,Bangalore 56000,INDIA Department of ECE,Unversty of Calforna,San

More information

On the Power Function of the Likelihood Ratio Test for MANOVA

On the Power Function of the Likelihood Ratio Test for MANOVA Journa of Mutvarate Anayss 8, 416 41 (00) do:10.1006/jmva.001.036 On the Power Functon of the Lkehood Rato Test for MANOVA Dua Kumar Bhaumk Unversty of South Aabama and Unversty of Inos at Chcago and Sanat

More information

Consider the following passband digital communication system model. c t. modulator. t r a n s m i t t e r. signal decoder.

Consider 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 information

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 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 information

Error Probability for M Signals

Error Probability for M Signals Chapter 3 rror Probablty for M Sgnals In ths chapter we dscuss the error probablty n decdng whch of M sgnals was transmtted over an arbtrary channel. We assume the sgnals are represented by a set of orthonormal

More information

Research on Complex Networks Control Based on Fuzzy Integral Sliding Theory

Research on Complex Networks Control Based on Fuzzy Integral Sliding Theory Advanced Scence and Technoogy Letters Vo.83 (ISA 205), pp.60-65 http://dx.do.org/0.4257/ast.205.83.2 Research on Compex etworks Contro Based on Fuzzy Integra Sdng Theory Dongsheng Yang, Bngqng L, 2, He

More information

Nested case-control and case-cohort studies

Nested case-control and case-cohort studies Outne: Nested case-contro and case-cohort studes Ørnuf Borgan Department of Mathematcs Unversty of Oso NORBIS course Unversty of Oso 4-8 December 217 1 Radaton and breast cancer data Nested case contro

More information

Downlink Power Allocation for CoMP-NOMA in Multi-Cell Networks

Downlink Power Allocation for CoMP-NOMA in Multi-Cell Networks Downn Power Aocaton for CoMP-NOMA n Mut-Ce Networs Md Shpon A, Eram Hossan, Arafat A-Dwe, and Dong In Km arxv:80.0498v [eess.sp] 6 Dec 207 Abstract Ths wor consders the probem of dynamc power aocaton n

More information

ECE559VV Project Report

ECE559VV Project Report ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUM-RATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sum-rate

More information

Neural network-based athletics performance prediction optimization model applied research

Neural network-based athletics performance prediction optimization model applied research Avaabe onne www.jocpr.com Journa of Chemca and Pharmaceutca Research, 04, 6(6):8-5 Research Artce ISSN : 0975-784 CODEN(USA) : JCPRC5 Neura networ-based athetcs performance predcton optmzaton mode apped

More information

Lecture Notes on Linear Regression

Lecture Notes on Linear Regression Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume

More information

COXREG. Estimation (1)

COXREG. Estimation (1) COXREG Cox (972) frst suggested the modes n whch factors reated to fetme have a mutpcatve effect on the hazard functon. These modes are caed proportona hazards (PH) modes. Under the proportona hazards

More information

One-sided finite-difference approximations suitable for use with Richardson extrapolation

One-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 information

Optimal and Suboptimal Linear Receivers for Time-Hopping Impulse Radio Systems

Optimal and Suboptimal Linear Receivers for Time-Hopping Impulse Radio Systems MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.mer.com Optma and Suboptma Lnear Recevers for Tme-Hoppng Impuse Rado Systems Snan Gezc Hsash Kbayash H. Vncent Poor Andreas F. Mosch TR2004-052 May

More information

Performance of SDMA Systems Using Transmitter Preprocessing Based on Noisy Feedback of Vector-Quantized Channel Impulse Responses

Performance of SDMA Systems Using Transmitter Preprocessing Based on Noisy Feedback of Vector-Quantized Channel Impulse Responses Performance of SDMA Systems Usng Transmtter Preprocessng Based on Nosy Feedback of Vector-Quantzed Channe Impuse Responses Du Yang, Le-Lang Yang and Lajos Hanzo Schoo of ECS, Unversty of Southampton, SO7

More information

3. Stress-strain relationships of a composite layer

3. Stress-strain relationships of a composite layer OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman.. tress-stran reatonshps of a composte ayer Introducton

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Assuming that the transmission delay is negligible, we have

Assuming that the transmission delay is negligible, we have Baseband Transmsson of Bnary Sgnals Let g(t), =,, be a sgnal transmtted over an AWG channel. Consder the followng recever g (t) + + Σ x(t) LTI flter h(t) y(t) t = nt y(nt) threshold comparator Decson ˆ

More information

GENERATION OF GOLD-SEQUENCES WITH APPLICATIONS TO SPREAD SPECTRUM SYSTEMS

GENERATION OF GOLD-SEQUENCES WITH APPLICATIONS TO SPREAD SPECTRUM SYSTEMS GENERATION OF GOLD-SEQUENCES WITH APPLICATIONS TO SPREAD SPECTRUM SYSTEMS F. Rodríguez Henríquez (1), Member, IEEE, N. Cruz Cortés (1), Member, IEEE, J.M. Rocha-Pérez (2) Member, IEEE. F. Amaro Sánchez

More information

Diversity Combining of Signals with Different Modulation Levels in Cooperative Relay Networks 1

Diversity Combining of Signals with Different Modulation Levels in Cooperative Relay Networks 1 Dversty Combnng of gnas wth Dfferent oduaton Leves n Cooperatve Reay Networks Akram Bn edq and Ham Yankomerogu Broadband Communcatons and Wreess ystems BCW Centre Department of ystems and Computer Engneerng

More information

Supplementary Material: Learning Structured Weight Uncertainty in Bayesian Neural Networks

Supplementary Material: Learning Structured Weight Uncertainty in Bayesian Neural Networks Shengyang Sun, Changyou Chen, Lawrence Carn Suppementary Matera: Learnng Structured Weght Uncertanty n Bayesan Neura Networks Shengyang Sun Changyou Chen Lawrence Carn Tsnghua Unversty Duke Unversty Duke

More information

Joint Turbo Equalization for Relaying Schemes over Frequency-Selective Fading Channels

Joint Turbo Equalization for Relaying Schemes over Frequency-Selective Fading Channels Jont Turbo Equazaton for Reayng Schemes over Frequency-Seectve Fadng Channes Houda Chafnaj, Tark At-Idr, Ham Yankomerogu +, and Samr Saoud Communcatons Systems Department, INPT, Madnat A Irfane, Rabat,

More information

Chapter Newton s Method

Chapter Newton s Method Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve

More information

Communication with AWGN Interference

Communication with AWGN Interference Communcaton wth AWG Interference m {m } {p(m } Modulator s {s } r=s+n Recever ˆm AWG n m s a dscrete random varable(rv whch takes m wth probablty p(m. Modulator maps each m nto a waveform sgnal s m=m

More information

Signal space Review on vector space Linear independence Metric space and norm Inner product

Signal space Review on vector space Linear independence Metric space and norm Inner product Sgnal space.... Revew on vector space.... Lnear ndependence... 3.3 Metrc space and norm... 4.4 Inner product... 5.5 Orthonormal bass... 7.6 Waveform communcaton system... 9.7 Some examples... 6 Sgnal space

More information

Demodulation of PPM signal based on sequential Monte Carlo model

Demodulation of PPM signal based on sequential Monte Carlo model Internatona Journa of Computer Scence and Eectroncs Engneerng (IJCSEE) Voume 1, Issue 1 (213) ISSN 232 428 (Onne) Demoduaton of M sgna based on seuenta Monte Caro mode Lun Huang and G. E. Atkn Abstract

More information

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton

More information

MARKOV CHAIN AND HIDDEN MARKOV MODEL

MARKOV CHAIN AND HIDDEN MARKOV MODEL MARKOV CHAIN AND HIDDEN MARKOV MODEL JIAN ZHANG JIANZHAN@STAT.PURDUE.EDU Markov chan and hdden Markov mode are probaby the smpest modes whch can be used to mode sequenta data,.e. data sampes whch are not

More information

G : Statistical Mechanics

G : Statistical Mechanics G25.2651: Statstca Mechancs Notes for Lecture 11 I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS The probem of quantum statstca mechancs s the quantum mechanca treatment of an N-partce system. Suppose the

More information

8/25/17. Data Modeling. Data Modeling. Data Modeling. Patrice Koehl Department of Biological Sciences National University of Singapore

8/25/17. Data Modeling. Data Modeling. Data Modeling. Patrice Koehl Department of Biological Sciences National University of Singapore 8/5/17 Data Modelng Patrce Koehl Department of Bologcal Scences atonal Unversty of Sngapore http://www.cs.ucdavs.edu/~koehl/teachng/bl59 koehl@cs.ucdavs.edu Data Modelng Ø Data Modelng: least squares Ø

More information

Note 2. Ling fong Li. 1 Klein Gordon Equation Probablity interpretation Solutions to Klein-Gordon Equation... 2

Note 2. Ling fong Li. 1 Klein Gordon Equation Probablity interpretation Solutions to Klein-Gordon Equation... 2 Note 2 Lng fong L Contents Ken Gordon Equaton. Probabty nterpretaton......................................2 Soutons to Ken-Gordon Equaton............................... 2 2 Drac Equaton 3 2. Probabty nterpretaton.....................................

More information

An Upper Bound on SINR Threshold for Call Admission Control in Multiple-Class CDMA Systems with Imperfect Power-Control

An Upper Bound on SINR Threshold for Call Admission Control in Multiple-Class CDMA Systems with Imperfect Power-Control An Upper Bound on SINR Threshold for Call Admsson Control n Multple-Class CDMA Systems wth Imperfect ower-control Mahmoud El-Sayes MacDonald, Dettwler and Assocates td. (MDA) Toronto, Canada melsayes@hotmal.com

More information

Lecture 3: Shannon s Theorem

Lecture 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 information

[WAVES] 1. Waves and wave forces. Definition of waves

[WAVES] 1. Waves and wave forces. Definition of waves 1. Waves and forces Defnton of s In the smuatons on ong-crested s are consdered. The drecton of these s (μ) s defned as sketched beow n the goba co-ordnate sstem: North West East South The eevaton can

More information

Interference Alignment and Degrees of Freedom Region of Cellular Sigma Channel

Interference Alignment and Degrees of Freedom Region of Cellular Sigma Channel 2011 IEEE Internatona Symposum on Informaton Theory Proceedngs Interference Agnment and Degrees of Freedom Regon of Ceuar Sgma Channe Huaru Yn 1 Le Ke 2 Zhengdao Wang 2 1 WINLAB Dept of EEIS Unv. of Sc.

More information

The Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD

The 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 information

The Concept of Beamforming

The Concept of Beamforming ELG513 Smart Antennas S.Loyka he Concept of Beamformng Generc representaton of the array output sgnal, 1 where w y N 1 * = 1 = w x = w x (4.1) complex weghts, control the array pattern; y and x - narrowband

More information

Chapter 7 Channel Capacity and Coding

Chapter 7 Channel Capacity and Coding Chapter 7 Channel Capacty and Codng Contents 7. Channel models and channel capacty 7.. Channel models Bnary symmetrc channel Dscrete memoryless channels Dscrete-nput, contnuous-output channel Waveform

More information

Example: Suppose we want to build a classifier that recognizes WebPages of graduate students.

Example: Suppose we want to build a classifier that recognizes WebPages of graduate students. Exampe: Suppose we want to bud a cassfer that recognzes WebPages of graduate students. How can we fnd tranng data? We can browse the web and coect a sampe of WebPages of graduate students of varous unverstes.

More information

Pop-Click Noise Detection Using Inter-Frame Correlation for Improved Portable Auditory Sensing

Pop-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 information

OFDM is a good candidate for wireless multimedia communication

OFDM is a good candidate for wireless multimedia communication Detecton of OFDM-CPM Sgnals over Multpath Channels Imran A. Tasadduq and Raveendra K. Rao Department of Electrcal & Computer Engneerng Elborn College, 11 Western Road The Unversty of Western Ontaro London,

More information

Chapter 7 Channel Capacity and Coding

Chapter 7 Channel Capacity and Coding Wreless Informaton Transmsson System Lab. Chapter 7 Channel Capacty and Codng Insttute of Communcatons Engneerng atonal Sun Yat-sen Unversty Contents 7. Channel models and channel capacty 7.. Channel models

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 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 information

Comparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method

Comparison 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

Retrodirective Distributed Transmit Beamforming with Two-Way Source Synchronization

Retrodirective Distributed Transmit Beamforming with Two-Way Source Synchronization Retrodrectve Dstrbuted Transmt Beamformng wth Two-Way Source Synchronzaton Robert D. Preuss and D. Rchard Brown III Abstract Dstrbuted transmt beamformng has recenty been proposed as a technque n whch

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

Scientia Iranica, Vol. 13, No. 4, pp 337{347 c Sharif University of Technology, October 2006 Performance Evaluations and Comparisons of Several LDPC C

Scientia Iranica, Vol. 13, No. 4, pp 337{347 c Sharif University of Technology, October 2006 Performance Evaluations and Comparisons of Several LDPC C Scenta Iranca, Vo. 13, No. 4, pp 337{347 c Sharf Unversty of Technoogy, Octoer 006 Performance Evauatons and Comparsons of Severa LDPC Coded MC-FH-CDMA Systems H. Behrooz, J. Haghghat 1, M. Nasr-Kenar

More information

Lossy Compression. Compromise accuracy of reconstruction for increased compression.

Lossy Compression. Compromise accuracy of reconstruction for increased compression. Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

Pulse Coded Modulation

Pulse 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 information

On Uplink-Downlink Sum-MSE Duality of Multi-hop MIMO Relay Channel

On Uplink-Downlink Sum-MSE Duality of Multi-hop MIMO Relay Channel On Upn-Downn Sum-MSE Duat of Mut-hop MIMO Rea Channe A Cagata Cr, Muhammad R. A. handaer, Yue Rong and Yngbo ua Department of Eectrca Engneerng, Unverst of Caforna Rversde, Rversde, CA, 95 Centre for Wreess

More information

Cyclic Codes BCH Codes

Cyclic Codes BCH Codes Cycc Codes BCH Codes Gaos Feds GF m A Gaos fed of m eements can be obtaned usng the symbos 0,, á, and the eements beng 0,, á, á, á 3 m,... so that fed F* s cosed under mutpcaton wth m eements. The operator

More information

Xin Li Department of Information Systems, College of Business, City University of Hong Kong, Hong Kong, CHINA

Xin Li Department of Information Systems, College of Business, City University of Hong Kong, Hong Kong, CHINA RESEARCH ARTICLE MOELING FIXE OS BETTING FOR FUTURE EVENT PREICTION Weyun Chen eartment of Educatona Informaton Technoogy, Facuty of Educaton, East Chna Norma Unversty, Shangha, CHINA {weyun.chen@qq.com}

More information

Multispectral Remote Sensing Image Classification Algorithm Based on Rough Set Theory

Multispectral Remote Sensing Image Classification Algorithm Based on Rough Set Theory Proceedngs of the 2009 IEEE Internatona Conference on Systems Man and Cybernetcs San Antono TX USA - October 2009 Mutspectra Remote Sensng Image Cassfcaton Agorthm Based on Rough Set Theory Yng Wang Xaoyun

More information

Inductance Calculation for Conductors of Arbitrary Shape

Inductance Calculation for Conductors of Arbitrary Shape CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors

More information

Quantum Runge-Lenz Vector and the Hydrogen Atom, the hidden SO(4) symmetry

Quantum Runge-Lenz Vector and the Hydrogen Atom, the hidden SO(4) symmetry Quantum Runge-Lenz ector and the Hydrogen Atom, the hdden SO(4) symmetry Pasca Szrftgser and Edgardo S. Cheb-Terrab () Laboratore PhLAM, UMR CNRS 85, Unversté Le, F-59655, France () Mapesoft Let's consder

More information

Power Allocation for Distributed BLUE Estimation with Full and Limited Feedback of CSI

Power Allocation for Distributed BLUE Estimation with Full and Limited Feedback of CSI Power Allocaton for Dstrbuted BLUE Estmaton wth Full and Lmted Feedback of CSI Mohammad Fanae, Matthew C. Valent, and Natala A. Schmd Lane Department of Computer Scence and Electrcal Engneerng West Vrgna

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have

More information

A Novel Hierarchical Method for Digital Signal Type Classification

A Novel Hierarchical Method for Digital Signal Type Classification Proceedngs of the 6th WSEAS Internatona Conference on Apped Informatcs and Communcatons, Eounda, Greece, August 8-0, 006 (pp388-393) A Nove Herarchca Method for Dgta Sgna ype Cassfcaton AAOLLAH EBRAHIMZADEH,

More information

Image Classification Using EM And JE algorithms

Image Classification Using EM And JE algorithms Machne earnng project report Fa, 2 Xaojn Sh, jennfer@soe Image Cassfcaton Usng EM And JE agorthms Xaojn Sh Department of Computer Engneerng, Unversty of Caforna, Santa Cruz, CA, 9564 jennfer@soe.ucsc.edu

More information

Optimization of JK Flip Flop Layout with Minimal Average Power of Consumption based on ACOR, Fuzzy-ACOR, GA, and Fuzzy-GA

Optimization of JK Flip Flop Layout with Minimal Average Power of Consumption based on ACOR, Fuzzy-ACOR, GA, and Fuzzy-GA Journa of mathematcs and computer Scence 4 (05) - 5 Optmzaton of JK Fp Fop Layout wth Mnma Average Power of Consumpton based on ACOR, Fuzzy-ACOR, GA, and Fuzzy-GA Farshd Kevanan *,, A Yekta *,, Nasser

More information

Power Allocation/Beamforming for DF MIMO Two-Way Relaying: Relay and Network Optimization

Power Allocation/Beamforming for DF MIMO Two-Way Relaying: Relay and Network Optimization Power Allocaton/Beamformng for DF MIMO Two-Way Relayng: Relay and Network Optmzaton Je Gao, Janshu Zhang, Sergy A. Vorobyov, Ha Jang, and Martn Haardt Dept. of Electrcal & Computer Engneerng, Unversty

More information

VQ widely used in coding speech, image, and video

VQ 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 information

Research Article Green s Theorem for Sign Data

Research Article Green s Theorem for Sign Data Internatonal Scholarly Research Network ISRN Appled Mathematcs Volume 2012, Artcle ID 539359, 10 pages do:10.5402/2012/539359 Research Artcle Green s Theorem for Sgn Data Lous M. Houston The Unversty of

More information

Support Vector Machines. Vibhav Gogate The University of Texas at dallas

Support Vector Machines. Vibhav Gogate The University of Texas at dallas Support Vector Machnes Vbhav Gogate he Unversty of exas at dallas What We have Learned So Far? 1. Decson rees. Naïve Bayes 3. Lnear Regresson 4. Logstc Regresson 5. Perceptron 6. Neural networks 7. K-Nearest

More information

LOW-DENSITY Parity-Check (LDPC) codes have received

LOW-DENSITY Parity-Check (LDPC) codes have received IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 7, JULY 2011 1807 Successve Maxmzaton for Systematc Desgn of Unversay Capacty Approachng Rate-Compatbe Sequences of LDPC Code Ensembes over Bnary-Input

More information

Single-Source/Sink Network Error Correction Is as Hard as Multiple-Unicast

Single-Source/Sink Network Error Correction Is as Hard as Multiple-Unicast Snge-Source/Snk Network Error Correcton Is as Hard as Mutpe-Uncast Wentao Huang and Tracey Ho Department of Eectrca Engneerng Caforna Insttute of Technoogy Pasadena, CA {whuang,tho}@catech.edu Mchae Langberg

More information

A finite difference method for heat equation in the unbounded domain

A finite difference method for heat equation in the unbounded domain Internatona Conerence on Advanced ectronc Scence and Technoogy (AST 6) A nte derence method or heat equaton n the unbounded doman a Quan Zheng and Xn Zhao Coege o Scence North Chna nversty o Technoogy

More information

CHALMERS, GÖTEBORGS UNIVERSITET. SOLUTIONS to RE-EXAM for ARTIFICIAL NEURAL NETWORKS. COURSE CODES: FFR 135, FIM 720 GU, PhD

CHALMERS, GÖTEBORGS UNIVERSITET. SOLUTIONS to RE-EXAM for ARTIFICIAL NEURAL NETWORKS. COURSE CODES: FFR 135, FIM 720 GU, PhD CHALMERS, GÖTEBORGS UNIVERSITET SOLUTIONS to RE-EXAM for ARTIFICIAL NEURAL NETWORKS COURSE CODES: FFR 35, FIM 72 GU, PhD Tme: Place: Teachers: Allowed materal: Not allowed: January 2, 28, at 8 3 2 3 SB

More information

What would be a reasonable choice of the quantization step Δ?

What would be a reasonable choice of the quantization step Δ? CE 108 HOMEWORK 4 EXERCISE 1. Suppose you are samplng the output of a sensor at 10 KHz and quantze t wth a unform quantzer at 10 ts per sample. Assume that the margnal pdf of the sgnal s Gaussan wth mean

More information

Analysis of Bipartite Graph Codes on the Binary Erasure Channel

Analysis of Bipartite Graph Codes on the Binary Erasure Channel Anayss of Bpartte Graph Codes on the Bnary Erasure Channe Arya Mazumdar Department of ECE Unversty of Maryand, Coege Par ema: arya@umdedu Abstract We derve densty evouton equatons for codes on bpartte

More information

Chapter 6. Rotations and Tensors

Chapter 6. Rotations and Tensors Vector Spaces n Physcs 8/6/5 Chapter 6. Rotatons and ensors here s a speca knd of near transformaton whch s used to transforms coordnates from one set of axes to another set of axes (wth the same orgn).

More information

A Hybrid Variational Iteration Method for Blasius Equation

A 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 information

Maximizing the number of nonnegative subsets

Maximizing the number of nonnegative subsets Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum

More information

L-Edge Chromatic Number Of A Graph

L-Edge Chromatic Number Of A Graph IJISET - Internatona Journa of Innovatve Scence Engneerng & Technoogy Vo. 3 Issue 3 March 06. ISSN 348 7968 L-Edge Chromatc Number Of A Graph Dr.R.B.Gnana Joth Assocate Professor of Mathematcs V.V.Vannaperuma

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Composite Hypotheses testing

Composite Hypotheses testing Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter

More information

Associative Memories

Associative Memories Assocatve Memores We consder now modes for unsupervsed earnng probems, caed auto-assocaton probems. Assocaton s the task of mappng patterns to patterns. In an assocatve memory the stmuus of an ncompete

More information

Iterative Multiuser Receiver Utilizing Soft Decoding Information

Iterative Multiuser Receiver Utilizing Soft Decoding Information teratve Multuser Recever Utlzng Soft Decodng nformaton Kmmo Kettunen and Tmo Laaso Helsn Unversty of Technology Laboratory of Telecommuncatons Technology emal: Kmmo.Kettunen@hut.f, Tmo.Laaso@hut.f Abstract

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Analyss of Varance and Desgn of Experments- MODULE LECTURE - 6 EXPERMENTAL DESGN MODELS Dr. Shalabh Department of Mathematcs and Statstcs ndan nsttute of Technology Kanpur Two-way classfcaton wth nteractons

More information

Laboratory 1c: Method of Least Squares

Laboratory 1c: Method of Least Squares Lab 1c, Least Squares Laboratory 1c: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly

More information

Lecture 12: Classification

Lecture 12: Classification Lecture : Classfcaton g Dscrmnant functons g The optmal Bayes classfer g Quadratc classfers g Eucldean and Mahalanobs metrcs g K Nearest Neghbor Classfers Intellgent Sensor Systems Rcardo Guterrez-Osuna

More information

COMBINING SPATIAL COMPONENTS IN SEISMIC DESIGN

COMBINING SPATIAL COMPONENTS IN SEISMIC DESIGN Transactons, SMRT- COMBINING SPATIAL COMPONENTS IN SEISMIC DESIGN Mchae O Leary, PhD, PE and Kevn Huberty, PE, SE Nucear Power Technooges Dvson, Sargent & Lundy, Chcago, IL 6060 ABSTRACT Accordng to Reguatory

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models

More information

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011 Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected

More information

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1 P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the

More information

Finding Dense Subgraphs in G(n, 1/2)

Finding Dense Subgraphs in G(n, 1/2) Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng

More information

Chapter 6 Hidden Markov Models. Chaochun Wei Spring 2018

Chapter 6 Hidden Markov Models. Chaochun Wei Spring 2018 896 920 987 2006 Chapter 6 Hdden Markov Modes Chaochun We Sprng 208 Contents Readng materas Introducton to Hdden Markov Mode Markov chans Hdden Markov Modes Parameter estmaton for HMMs 2 Readng Rabner,

More information

COS 521: Advanced Algorithms Game Theory and Linear Programming

COS 521: Advanced Algorithms Game Theory and Linear Programming COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton

More information

Low Complexity Soft-Input Soft-Output Hamming Decoder

Low 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 information

Linear Approximation with Regularization and Moving Least Squares

Linear Approximation with Regularization and Moving Least Squares Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...

More information

Application of Nonbinary LDPC Codes for Communication over Fading Channels Using Higher Order Modulations

Application 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 information

A Lower Bound on SINR Threshold for Call Admission Control in Multiple-Class CDMA Systems with Imperfect Power-Control

A Lower Bound on SINR Threshold for Call Admission Control in Multiple-Class CDMA Systems with Imperfect Power-Control A ower Bound on SIR Threshold for Call Admsson Control n Multple-Class CDMA Systems w Imperfect ower-control Mohamed H. Ahmed Faculty of Engneerng and Appled Scence Memoral Unversty of ewfoundland St.

More information

8.592J: Solutions for Assignment 7 Spring 2005

8.592J: Solutions for Assignment 7 Spring 2005 8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1

More information

COMPUTATIONALLY EFFICIENT WAVELET AFFINE INVARIANT FUNCTIONS FOR SHAPE RECOGNITION. Erdem Bala, Dept. of Electrical and Computer Engineering,

COMPUTATIONALLY EFFICIENT WAVELET AFFINE INVARIANT FUNCTIONS FOR SHAPE RECOGNITION. Erdem Bala, Dept. of Electrical and Computer Engineering, COMPUTATIONALLY EFFICIENT WAVELET AFFINE INVARIANT FUNCTIONS FOR SHAPE RECOGNITION Erdem Bala, Dept. of Electrcal and Computer Engneerng, Unversty of Delaware, 40 Evans Hall, Newar, DE, 976 A. Ens Cetn,

More information

Assortment Optimization under MNL

Assortment Optimization under MNL Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

The optimal delay of the second test is therefore approximately 210 hours earlier than =2.

The optimal delay of the second test is therefore approximately 210 hours earlier than =2. THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple

More information

The internal structure of natural numbers and one method for the definition of large prime numbers

The internal structure of natural numbers and one method for the definition of large prime numbers The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract

More information

Goodness of fit and Wilks theorem

Goodness of fit and Wilks theorem DRAFT 0.0 Glen Cowan 3 June, 2013 Goodness of ft and Wlks theorem Suppose we model data y wth a lkelhood L(µ) that depends on a set of N parameters µ = (µ 1,...,µ N ). Defne the statstc t µ ln L(µ) L(ˆµ),

More information