LatticeReductionAided Equalization and Generalized Partial Response Signaling for PointtoPoint Transmission over Flat Fading MIMO Channels


 Allison Henderson
 3 years ago
 Views:
Transcription
1 LtticeReductioAided Equliztio d Geerlized rtil Respose Siglig for oittooit Trsmissio over Flt Fdig MIMO Chels Robert F.. Fischer Lehrstuhl für Iformtiosübertrgug, Friedrich Aleder Uiversität Erlge Nürberg, Cuerstrße 7/LIT, Erlge, Germ, Emil: Abstrct oittopoit commuictio over fltfdig MIMO chels is coside. Kow equliztio d preequliztio schemes re reviewed d comp. I prticulr, ltticeuctioided techiques re coside, d strtig from them ew pproch is derived. This proposed techique c be iterpreted s geerliztio of prtilrespose siglig, kow from itersmboliterferece chels. The mi dvtge of this lowcompleit scheme is tht ol ver little chelstte iformtio is requi t the trsmitter d performce close to mimumlikelihood detectio is possible. Moreover, chel codig c directl be pplied. 1 Itroductio Commuictio over MIMO chels, e.g., b usig te rrs, is ver iterestig becuse of the ver high spectrl efficiecies d hece dt rtes which c be chieved. The multite iterferece preset i such scerio hs to be combtted with some mes of equliztio. Over the lst ers, equliztio techiques kow from itersmboliterferece (ISI) chels hve bee trsfe to the MIMO settig, cf., e.g., [3]. Besides this, ew techiques bsed o lttice bsis uctio hve bee recetl developed [1], [15]. I this pper, equliztio schemes for poittopoit trsmissio over fltfdig MIMO chels, i prticulr ltticeuctioided techiques, re reviewed d comp. Equliztio/preequliztio structures beig the respective couterprt to ech other re idetified. Besides this, ew pproch which c be iterpreted s geerliztio of prtilrespose siglig (well kow o ISI chels) is derived. These schemes require ol ver limited chel kowledge t the trsmitter side but perform rther close to much more comple mimumlikelihood detectio. I Sectio the chel model is give d Sectio 3 briefl reviews covetiol (pre)equliztio schemes. Sectio 4 discusses ltticeuctioided detectio d presets geerlized prtilrespose siglig for MIMO chels. Numericl results re give d discussed i Sectio 5; Sectio 6 drws some coclusios. Chel Model Cosider multiplete poittopoit trsmissio over fltfdig chels. Over ech of the N T trsmit tes dt smbols c,µ, µ =1,,...,N T, tke from M r squre QAM sigl costelltio A c (vrice σ def =E{ c,µ }, µ), re trsmitted. Vritio of the costelltio (rte/power lodig, dptive modultio) is ot coside here. The dt smbols re combied ito the vector 1 c =[ c,1,..., c,nt ] T. The chel is chrcterized i the equivlet comple bsebd [1] b the fdig coefficiets h c,kl, collected i the N R N T chel mtri c =[h c,kl ], betwee ech pir of trsmit/receive te. Sice, due to the fltfdig ssumptio, ech smbol itervl c be processed o its ow, we do ot itroduce time ide. The receive smbols c,k t the N R receive tes (combied ito the vector c = [ c,1,..., c,nr ] T ) re give b the usul iput/output reltio c = c c + c. c =[ c,1,..., c,nr ] T is the vector cotiig the complevlued oise smbols d is epected to be white with vrice σ per compoet, i.e., E{ c c } = σi. Throughout the pper we ssume N T = N R ; the geerliztio is esil possible. The complevlued MIMO chel model is equivletl writte s Kdimesiol (K =N def T ) relvlued MIMO chel model ccordig to [ ] [ ][ ] [ ] R c R c I = c R c R c + I c I c R c I c I c (R d I deote rel d imgir prt, respectivel), or i short usig obvious defiitios = +. (1) Due to the ssumptio of squre QAM costelltios A c, ech compoet µ of is idepedetl drw 1 Nottio: A T : trspose of mtri A; A : ermiti (i.e., cojugte) trspose. I: Idetit mtri. E{ }: epecttio.
2 from oedimesiol Mr ASK costelltio A = {±1/, ±3/,...,±(M 1)/}. All subsequet discussios re purel bsed o the relvlued MIMO chel model (1). Lier equliztio techiques (icludig sigulrvlue decompositio) d mimumlikelihood pproches perform ectl the sme whe derivig them for the comple or rel chel descriptio, respectivel. owever, due to lrger umber of degrees of freedom, successive decodig or ecodig schemes show better performce whe strtig from the relvlued model, cf. [5]. 3 Equliztio Strtegies We ow briefl review trsmissio schemes for poittopoit trsmissio over MIMO chels. Cotrr to multipoittopoit (multiple ccess; uplik) or poittomultipoit scerios (brodcst chel; dowlik) here processig c be either doe t receiver or trsmitter d splittig betwee both sides is possible. For tht we ssume throughout tht perfect chel stte iformtio (CSI) (the chel mtri ) is vilble t the receiver side. If (prtil) CSI is lso requi t the trsmitter side (TX CSI), this fct is eplicitel stted. lese ote tht ll equliztio techiques re optimized ccordig to the zeroforcig (ZF) criterio. I ech cse optimiztio ccordig to the miimum mesqu error (MMSE) criterio is possible, too. This strteg i fct offers dditiol gi but o ew isight ito the fudmetl properties of the equliztio schemes d their compriso is provided. 3.1 Lier Equliztio d Lier re Equliztio I lier equliztio t the receiver side the decisio vector is geerted ccordig to r = 1. It is wellkow tht lier equliztio suffers from oise ehcemet d hece hs poor power efficiec. If the chel mtri is kow to the trsmitter (e.g., b commuictig it to the trsmitter vi bckwrd chel) the dt strems c be seprted vi lier preequliztio [4], [8]. ere, the vector of chel smbols µ, which re fed ito the chel isted of the dt smbols µ itself, is geerted s = ς 1. The rel costt ς is chose such tht verge trsmit power equls tht of direct trsmissio of the dt vector (short term power costrit). ς is compested t the receiver side b proper sclig (utomtic gi cotrol, AGC). Eforcig the short term power costrit, lier preequliztio shows (ecept mior differeces) the sme performce s lier receiver side equliztio. This strteg is lso kow uder the me chel iversio (e.g., [8]) d the mbiguous deomitio zeroforcig, which, however, should ol be used s ddedum to emphsize the optimiztio criterio. Figure 1 (top) shows the trsmissio schemes emploig lier equliztio d lier preequliztio. 3. DecisioFeedbck Equliztio d recodig erformce c be improved if decisiofeedbck equliztio (DFE) is pplied. ere, decisios l vilble re used for iterferece ccelltio. DFE is lso kow s successive ccelltio i multiuser detectio [] d is the equliztio priciple emploed i the (V)BLAST (Bell Lbortories Le Spce Time) scheme [19], [7]. The process of successive ccelltio (subtrctio) of iterferece of l detected dt smbols, lier filterig for suppressio of iterferece cotributed b ot et detected dt smbols, d detectio (threshold decisio) c be depicted s i the middle of Figure 1. The receive vector is processed b feedforwrd mtri F, which hs the tsk to trsform the chel ito lower trigulr shpe with uit mi digol. Due to this structure, the smbols c be detected i sequece. The iterferece of l detected smbols is subtrcted b the strictl lower trigulr mtri, where B deotes the feedbck mtri. I geerl, for optimum performce the dt smbols should be detected i optimized order [7], described b permuttio mtri. I the lst step, vi, the origil order is reestblished. Give, the mtrices re clculted such tht [4], [16] = F 1 B, () i.e., sorted QRtpe fctoriztio 3 hs to be performed. Defiig criterio of optimlit (usull mimum sigltooise rtio i ech detectio step), the fctoriztio is uique d c be performed s, e.g., give i [19], [7]. DFE suffers from error propgtio i the feedbck loop d requires immedite decisios, which complictes the pplictio of chel codig. Both problems c be voided if DFE is replced b trsmitter side precodig. Bsicll, Tomlisorshimtpe precodig [4] is derived from DFE b flippig the etire structure, see Figure 1. ere, the dt smbols re first permuted b d the processed b the olier feedbck loop with feedbck mtri B. The olier modulo opertio, which hs to mtch up with the ctul sigl costelltio A, restricts the output smbols b ddig iteger multiples of M to the support regio [ M/, M/) 3 Give the permuttio, the requested fctoriztio c be performed usig stdrd QR decompositio, i.e., M = QR, where Q is uitr (QQ = I) d R is upper (right) trigulr. Usig the tidigol idetit mtri J (idetit mtri I with reversed colum order; J = I), we perform: J = QR. ece: = QJ JRJ = def Q L, d usig D = dig(l 11,l,...), with L =[L ij ], we rrive t = Q D D 1 L = def F 1 B, where F 1 hs orthogol colums d B is lower trigulr with uit mi digol.
3 r 1 ς 1 F ςf Algorithm Algorithm d ς 1 Fig. 1. Trsmissio sstems for poittopoit trsmissio over fltfdig MIMO chels. Left colums: receiver side equliztio; right colum: trsmitter side techiques. Top to bottom: lier equliztio/preequliztio; decisiofeedbck equliztio / precodig; mimumlikelihood detectio / vector precodig. of A. Due to the modulo uctio, the iitil sigl set A is eteded periodicll; ll sigl poits which differ b multiples of M represet the sme dt. The chel iput smbols re obtied b feedforwrd filterig with ςf. At the receiver side, ol sclig (AGC) b ς 1 remis, followed b threshold decisio which tkes the modulo cogruece ito ccout. Detils o precodig c be foud, e.g., i [4]. Give, the mtrices re ow obtied ccordig to 4 [4], [16] = BF 1, (3) where the mtrices hve the sme properties s bove d ς is djusted such tht the shortterm power costrit is met. Ecept smll icresed umber of erest eighbor sigl poits d (usull) egligible precodig loss (smbols t the output of the modulo device hve slightl lrger vrice comp to the iitil dt smbols), precodig performs s good s DFE would without error propgtio. 3.3 MimumLikelihood Detectio d Vector recodig All bove metioed schemes perform smbolbsmbol decisios d hve moderte compleit. Best performce, t the price of highest cost, is obtied b mimumlikelihood detectio (MLD). ere, o Gussi chels lgorithm serches through the dt vectors for which the (oiseless) chel output hs lest (squ) Euclide distce to the observed receive vector, mthemticll [1] ( deotes Euclide orm) =rgmi A K. (4) 4 This fctoriztio c be obtied b QR decompositio of T = def QR. With D = dig(l 11,l,...), L =[L ij ],we hve: ( T ) = = R Q = def LQ = LD 1 DQ = def BF 1. MLD c be efficietl implemeted usig soclled sphere decoder [1]. The trsmissio scheme usig MLD is visulized i Figure 1, too. vig CSI t the trsmitter, the lgorithmic serch c be trsfe to the trsmitter, resultig i combied precodig/shpig techique sometimes clled vector precodig (V) [11], cf. lso [13] d [4, Sectio 5.3]. ere, give the dt vector, lgorithm (lttice decoder) serches through vectors d M K, for which the (oormlized) vector = 1 ( + d) of trsmit smbols hs lest orm (power), see Figure 1. ς is the gi chose to gurtee fied (shortterm) power. Cotrr to lier schemes d DFE/precodig (diversit order 1), MLD d V chieve the full diversit order (K/) of the chel. owever, this improvemet i performce is ped tpicll with (much) higher compleit. 4 LtticeReductioAided Detectio d recodig Recetl, lowcompleit equliztio schemes, bridgig the gp betwee DFE/precodig d MLD/V, hve bee proposed [1], [15], [18], [0]. The ide is to combie priciples kow from lttice theor i prticulr lttice (bsis) uctio (the choice of more suited represettio of lttice) which hs to be doe ol oce i preprocessig step with covetiol equliztio schemes such s lier equliztio or DFE discussed bove. Astoishigl, these cocepts chieve the full diversit orders [14], i.e., the error rte curves whe usig such detectors ru prllel to those for MLD/V. 4.1 Bsic Opertio Usull (ecept SK), the sigl poits i ech qudrture compoet re drw from ( trslte of) the i
4 1 Z 1 F Z 1 1 F Z 1 ς 1 ςf Fig.. Trsmissio sstems usig ltticebsisuctioided equliztio. Top to bottom: receiver side LRA techique (lier equliztio/dfe), o TX CSI requi; prtil respose LRA techique (lier/dfe), prtil TX CSI requi; trsmitter side LRA techique (lier/precodig), full TX CSI requi. (The requi shift of the sigls to the iteger grid is ot show.) teger lttice. ece, t the receiver side (disregrdig the oise) the lttice K is preset. Applig lttice bsis uctio, e.g., b usig the LLL lgorithm [10], [6], the chel mtri m be fcto s = Z, (5) where Z is mtri with iteger etries tht hs uit determit, i.e., Z 1 lso cotis ol iteger etries. is more suited chel descriptio s it specifies the sme lttice of chel output sigl poits, K K, but its colums re closer to orthogol. Isted of performig equliztio of the etire chel, ol the fctor is equlized which cuses less oise ehcemet. The, sice Z K = K, idividul threshold decisio i ech compoet c be performed. To recover dt, vi Z 1 estimtes µ of the iitil dt smbols re geerted. Thereb, error multiplictio will occur. Lier equliztio of c be replced b DFE s give bove. For tht, is decomposed ccordig to (). The trsmissio scheme usig (upper brch) ltticebsisuctioided (LRA) lier equliztio d LRA DFE (lower brch) is depicted o top of Figure. 4. rtilrespose Siglig I poittopoit trsmissio, ssumig pproprite CSI, we hve the freedom to choose t which side equliztio is performed. A iterestig pproch is to move the iteger mtri Z 1 to the trsmitter. Thereb, error multiplictio i dt recover is voided. owever, direct pplictio of Z 1 would icrese verge trsmit power. The solutio is to use modulo uctio similr to Tomlisorshim precodig. The proposed trsmissio scheme either usig lier prtil equliztio or DFE is depicted i the middle row of Figure. Sice K (igorig the shift b 1/) d Z is uimodulr mtri, Z 1 K. The modulo opertio (fter gi itroducig the trsltio) results smbols µ drw from the sme sigl set A s the dt smbols µ, formll =mod M ( Z 1 ( + M 1 ) 1) M 1 1, (6) where 1 is the lloe vector d mod M ( ) is the covetiol modulo uctio of ech compoet to the itervl [0, M). At the slicer (lier equliztio) the vector M 1 (Z I)1 + M K + 1 is preset. After elimitig the offset (secod term) threshold device tkig the periodic etesio (cused b +M K ) ito ccout c recover the dt. Similr cosidertios hold for DFE. This scheme, up to ow ot preset i literture, c be iterpreted s geerliztio of prtilrespose (R) siglig [9], populr o ISI chels. At the trsmitter, ol the kowledge of Z is requi (prtil TX CSI), which c be commuicted ver efficietl due to its iteger coefficiets. Additioll, the chel smbols µ re tke from the sme costelltio s the dt smbols µ ; hece o icrese i verge trsmit power is cused. Fill, o error multiplictio occurs t the receiver d, sice ol periodic etesio is preset, chel codig c be pplied immeditel whe lier equliztio is performed or the code words re rrged i time directio (cf. BLAST).
5 BER lier preequl. lier equliztio SVD DFE precodig MLD vector precodig log 10 (Ēb/N 0)[dB] BER LRA lier equl. LRA DFE LRA lier preeq. LRA precodig referece log 10 (Ēb/N 0)[dB] Fig. 3. Bit error rte over Ēb/N 0 (i db). Top to bottom: lier preequliztio, lier equliztio, SVD, DFE (dotted: geieided), precodig, vector precodig, MLD. K =8; 4r ASK trsmissio per rel compoet. Fig. 4. Bit error rte over Ēb/N 0 (i db). Top to bottom: LRA lier equliztio, LRA DFE (dotted: geieided), LRA lier preequliztio, LRA precodig. Gr: referece lier equl./mld. K =8; 4r ASK trsmissio per rel compoet. 4.3 LRA reequliztio d recodig The ide of ltticebsisuctioided equliztio c lso be pplied to preequliztio techiques [15], [18] ere, (5) hs to be replced b = Z, (7) i.e., lttice uctio is performed o T. Usig precodig, is dditioll fcto ccordig to (3). The respective LRA schemes which re bsicll obtied b flippig the structures for receiver side equliztio re depicted i the bottom row of Figure. ere, complete chel kowledge is requi t the trsmitter side (full TX CSI). 5 Numericl Results d Discussio The performce of the vrious schemes is ow ssessed b umericl simultios. We ssume N T = N R =4(K =8), d the coefficiets of the chel mtri c re chose i.i.d. comple Gussi with uit vrice. Ech of the 8 prllel rel dt strems uses ucoded (M =4)r ASK siglig. The results re verged over lrge umber of chel reliztios. I ech cse (chel reliztio/trsmissio scheme) the trsmissio burst (10000 smbols) is scled for fied verge power (sme s for direct trsmissio of the dt smbols). The bit error rte (BER) results re displed over the rtio of the verge trsmitted eerg per bit Ē b d the (oesided) oise power spectrl desit N 0. I the preset cse we hve Ē b /N 0 = σ /(log (M )σ )=σ /(4σ ). Clssicl Schemes: I Figure 3 the performce of clssicl trsmissio schemes is comp. As o rte or power lodig is ctive, lier receiver side equliztio, lier preequliztio d sigulr vlue decompositio (SVD) show lmost the sme performce. I prticulr, the diversit order (egtive slope of the error rte curves i doublelogrithmic scle) is ol 1. Usig DFE (with optiml orderig) some gi i SNR c be chieved but the diversit order does ot chge. Moreover, much gi is lost due to error propgtio i the feedbck loop s the compriso with the geieided (perfect feedbck) DFE revels. recodig performs close to geieided DFE; the error rte is somewht icresed due the lrger umber of erest eighbor sigl poits. Iterestigl, MLD d V lso perform ver similr. Agi, the slightl higher error rte c be eplied b the periodic etesio of the sigl set d hece lrger umber of erest eighbors. Both schemes chieve full diversit order; 4 i the preset cse. LRA Schemes: Net, LRA equliztio schemes re ssessed i Figure 4. The curves of lier equliztio (worst cse) d MLD (best cse) re repeted for referece. The performce dvtge of LRA schemes (lier d DFE) is clerl visible. I prticulr, the full diversit order is chieved, cf. [14], however, sigifict gp to the MLD curve b up to 4 db remis. LRA DFE offers ol gi of 1 db over LRA lier equliztio; sice is close to orthogol, error propgtio i the DFE feedbck loop is of mior iterest comp to error multiplictio t Z 1. Trsferrig (prts of) the equliztio i prticulr the iteger mtri Z 1 to the trsmitter is clerl dvtgeous. Now, error multiplictio whe recoverig dt vi Z 1 is voided. LRA Tomlisorshimtpe precodig performs ver close to MLD d V (cf. lso [18]) followed b LRA lier preequliztio. owever, i these cses full TX CSI is requi. Implemetig ol Z 1 t the trsmitter (prtil respose siglig) offers ver good performce but requires ol little TX CSI, see Figure 5. R LRA lier equliztio performs slightl worse th LRA lier preequliztio but eve better th LRA DFE. The slight loss comp to pure trsmitter side techiques is due to the fct tht whe implemetig
6 BER Equliztio schemes for poittopoit commuictio over fltfdig MIMO chels hve bee reviewed d comp. Strtig from ltticeuctioided techiques, ew pproch is derived, which c be iterpreted s geerlized prtilrespose siglig. This lowcompleit scheme requires ol ver limited TX CSI d performce is close to MLD. The combitio with chel codig is strightforwrd. Sice the trsmitter opertes ol o iteger coefficiets w, the R LRA schemes re robust gist certi degree of chel vritios. I decisio directed mer, the receive filters c esil b dpted to the ctul situtio. Ol if the chel chges sigifictl, ew iteger mtri Z hs to be commuicted to the trsmitter. ece, such schemes re well suited for mobile pplictios with slowl to modertel vrig chel coditios R LRA lier eq. R LRA DFE referece log 10 (Ēb/N 0)[dB] Fig. 5. Bit error rte over Ēb/N 0 (i db). Top to bottom: prtil respose LRA lier equliztio, prtil respose LRA DFE (dotted: geieided). Gr: referece (LRA curves from Figure 4). K =8; 4r ASK trsmissio per rel compoet. (or F ) t the trsmitter isted of the receiver, some form of power lodig over the prllel chel is ctive (lso vlid for the covetiol trsmitter side techiques). All prllel chels ehibit the sme error rte, wheres i pure receiver side equliztio the prllel dt strems usull hve differet error rtes d the worst cse error rte domites. owever, sice iteger ture of the sigls i R LRA schemes is requi, o trsmitter side sclig for power djustmet is possible. Nevertheless, R LRA DFE is ttrctive trsmissio scheme: it chieves ver good performce with ol little TX CSI. Fill it should be oted tht, give the chel reliztio, it would be possible to decide whether equliztio is preferbl doe t the trsmitter or receiver side. E.g., i DFE or precodig, the feedforwrd mtri F c be implemeted either t the trsmitter or receiver, cf. [17]. Eve though o verge both pproches perform ectl the sme, gi is possible if for give the istteous better versio is chose. This however, requires full TX CSI. Due to the iteger structure of the sigls, such procedure is ot possible i LRA schemes. ere, feedforwrd mtri d feedbck structure lws hve to be implemeted t the sme side; o splittig is llowed. 1 6 Coclusios Refereces [1] E. Agrell, T. Eriksso, A. Vrd, K. Zeger. Closest oit Serch i Lttices. IEEE Tr. If. Theor, pp , Aug. 00. [] A. Duellle. A Fmil of Multiuser DecisioFeedbck Detectors for Aschroous CodeDivisio MultipleAccess Chels. IEEE Tr. Comm., pp , Feb./Mr./Apr [3] R.F.. Fischer, C. Widpssiger, A. Lmpe, J.B. uber. SpceTime Trsmissio usig Tomlisorshim recodig. ITG Cof. Source d Chel Codig, Berli, Germ, J. 00. [4] R.F.. Fischer. recodig d Sigl Shpig for Digitl Trsmissio, Joh Wile & Sos, New York, 00. [5] R.F.. Fischer, C. Widpssiger. Rel vs. complevlued equlistio i VBLAST sstems. Electroics Letters, pp , Mr [6] J. v.z. Gthe, J. Gerhrd. Moder Computer Algebr. Cmbridge Uiversit ress, Cmbridge, UK, d editio, 00. [7] G.D. Golde, G.J. Foschii, R.A. Vlezuel,.W. Wolisk. Detectio lgorithm d iitil lbortor results usig V BLAST spcetime commuictio rchitecture. Electroics Letters, pp , J [8] T. ustei, C. v. elmolt, E. Jorswieck, V. Jugickel, V. ohl. erformce of MIMO Sstems with Chel Iversio. VTC Sprig 00, Birmighm, Albm, M 00. [9] Kbl, S. supth. rtilrespose Siglig. IEEE Tr. Comm., pp , Sep [10] A.K. Lestr,.W. Lestr, L. Lovász. Fctorig polomils with rtiol coefficiets, Mth. A., pp , 198. [11] C.B. eel, B.M. ochwld, B.M., A.L. Swidlehurst. A Vectorerturbtio Techique for NerCpcit Multite Multiuser Commuictio rts I d II. IEEE Tr. Comm., pp , J. 005, d pp , Mr [1] J.G. rokis. Digitl Commuictios. McGrwill, New York, 4. editio, 001. [13] D. Schmidt, M. Johm, W. Utschick. Miimum Me Squre Error Vector recodig. IMRC 05, Berli, Germ, Sep [14] M. Therzdeh, A. Mobsher, A. Khdi. LLL LtticeBsis Reductio Achieves Mimum Diversit i MIMO Sstems. IEEE ISIT 05, pp , Adelide, Austrli, Sep [15] C. Widpssiger, R.F.. Fischer. LowCompleit NerMimumLikelihood Detectio d recodig for MIMO Sstems usig Lttice Reductio. IEEE If. Theor Workshop 003, pp , ris, Frce, Mr./Apr [16] C. Widpssiger. Detectio d recodig for Multiple Iput Multiple Output Chels. Disserttio, Erlge, Jue 004. [17] C. Widpssiger, R.F.. Fischer, T. Vecel, J.B. uber. recodig i MultiAte d MultiUser Commuictios. IEEE Tr. Wireless Comm., pp , Jul 004. [18] C. Widpssiger, R.F.. Fischer, J.B. uber. LtticeReductioAided Brodcst recodig. IEEE Tr. Comm., pp , Dec [19]. Wolisk, G. Foschii, G. Golde, R. Vlezuel. V BLAST: A Architecture for Relizig Ver igh Dt Rtes Over the RichSctterig Wireless Chel. ISSSE 98, ise, Itl, Sep [0] D. Wübbe, R. Böhke, V. Küh, K.D. Kmmeer. NerMimumLikelihood Detectio of MIMO Sstems usig MMSE Bsed Lttice Reductio. IEEE ICC 004, pp , ris, Frce, Jue 004. [1]. Yo, G.W. Worell. LtticeReductioAided Detectors for MIMO Commuictio Sstems. IEEE Globecom 00, Tipei, Tiw, Nov. 00.
Gray level image enhancement using the Bernstein polynomials
Buletiul Ştiiţiic l Uiersităţii "Politehic" di Timişor Seri ELECTRONICĂ şi TELECOMUNICAŢII TRANSACTIONS o ELECTRONICS d COMMUNICATIONS Tom 47(6), Fscicol , 00 Gry leel imge ehcemet usig the Berstei polyomils
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL  INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.
More informationChapter 04.05 System of Equations
hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vicevers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationAuthorized licensed use limited to: University of Illinois. Downloaded on July 27,2010 at 06:52:39 UTC from IEEE Xplore. Restrictions apply.
Uiversl Dt Compressio d Lier Predictio Meir Feder d Adrew C. Siger y Jury, 998 The reltioship betwee predictio d dt compressio c be exteded to uiversl predictio schemes d uiversl dt compressio. Recet work
More informationn Using the formula we get a confidence interval of 80±1.64
9.52 The professor of sttistics oticed tht the rks i his course re orlly distributed. He hs lso oticed tht his orig clss verge is 73% with stdrd devitio of 12% o their fil exs. His fteroo clsses verge
More informationRepeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re shorthd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
More informationA. Description: A simple queueing system is shown in Fig. 161. Customers arrive randomly at an average rate of
Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig
More informationPREMIUMS CALCULATION FOR LIFE INSURANCE
ls of the Uiversity of etroşi, Ecoomics, 2(3), 202, 97204 97 REIUS CLCULTIO FOR LIFE ISURCE RE, RI GÎRBCI * BSTRCT: The pper presets the techiques d the formuls used o itertiol prctice for estblishig
More informationApplication: Volume. 6.1 Overture. Cylinders
Applictio: Volume 61 Overture I this chpter we preset other pplictio of the defiite itegrl, this time to fid volumes of certi solids As importt s this prticulr pplictio is, more importt is to recogize
More informationMATHEMATICS SYLLABUS SECONDARY 7th YEAR
Europe Schools Office of the SecretryGeerl Pedgogicl developmet Uit Ref.: 201101D41e2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig
More informationSummation Notation The sum of the first n terms of a sequence is represented by the summation notation i the index of summation
Lesso 0.: Sequeces d Summtio Nottio Def. of Sequece A ifiite sequece is fuctio whose domi is the set of positive rel itegers (turl umers). The fuctio vlues or terms of the sequece re represeted y, 2, 3,...,....
More informationPresent and future value formulae for uneven cash flow Based on performance of a Business
Advces i Mgemet & Applied Ecoomics, vol., o., 20, 9309 ISSN: 7927544 (prit versio), 7927552 (olie) Itertiol Scietific Press, 20 Preset d future vlue formule for ueve csh flow Bsed o performce of Busiess
More informationDiscontinuous Simulation Techniques for Worm Drive Mechanical Systems Dynamics
Discotiuous Simultio Techiques for Worm Drive Mechicl Systems Dymics Rostyslv Stolyrchuk Stte Scietific d Reserch Istitute of Iformtio Ifrstructure Ntiol Acdemy of Scieces of Ukrie PO Box 5446, Lviv3,
More informationCHAPTER10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS
Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER0 WAVEFUNCTIONS, OBSERVABLES d OPERATORS 0. Represettios i the sptil d mometum spces 0..A Represettio of the wvefuctio i
More informationDEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES
DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES The ultibioil odel d pplictios by Ti Kyg Reserch Pper No. 005/03 July 005 Divisio of Ecooic d Ficil Studies Mcqurie Uiversity Sydey NSW 09 Austrli
More informationGroundwater Management Tools: Analytical Procedure and Case Studies. MAF Technical Paper No: 2003/06. Prepared for MAF Policy by Vince Bidwell
Groudwter Mgemet Tools: Alyticl Procedure d Cse Studies MAF Techicl Pper No: 00/06 Prepred for MAF Policy by Vice Bidwell ISBN No: 07807778 ISSN No: 766 October 00 Disclimer While every effort hs bee
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationINVESTIGATION OF PARAMETERS OF ACCUMULATOR TRANSMISSION OF SELF MOVING MACHINE
ENGINEEING FO UL DEVELOENT Jelgv, 28.29.05.2009. INVESTIGTION OF ETES OF CCUULTO TNSISSION OF SELF OVING CHINE leksdrs Kirk Lithui Uiversity of griculture, Kus leksdrs.kirk@lzuu.lt.lt bstrct. Uder the
More informationMisspecification Effects in the Analysis of Longitudinal Survey Data
Misspecifictio Effects i te Alysis of Logitudil Survey Dt Mrcel de Toledo Vieir Deprtmeto de Esttístic, Uiversidde Federl de Juiz de For, Brsil mrcel.vieir@ufjf.edu.br M. Fátim Slgueiro ISCTE Busiess Scool
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 168040030 haupt@ieee.org Abstract:
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationSlowRate UtilityBased Resource Allocation in Wireless Networks
owrte UtiityBsed Resource Aoctio i Wireess Networks Peiju Liu, Rd Berry, Miche L. Hoig ECE Deprtmet, Northwester Uiversity herid Rod, Evsto, IL 68 UA peiju,rberry,mh @ece.wu.edu cott Jord ECE Deprtmet,
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationMANUFACTURERRETAILER CONTRACTING UNDER AN UNKNOWN DEMAND DISTRIBUTION
MANUFACTURERRETAILER CONTRACTING UNDER AN UNKNOWN DEMAND DISTRIBUTION Mrti A. Lriviere Fuqu School of Busiess Duke Uiversity Ev L. Porteus Grdute School of Busiess Stford Uiversity Drft December, 995
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationm n Use technology to discover the rules for forms such as a a, various integer values of m and n and a fixed integer value a.
TIth.co Alger Expoet Rules ID: 988 Tie required 25 iutes Activity Overview This ctivity llows studets to work idepedetly to discover rules for workig with expoets, such s Multiplictio d Divisio of Like
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationTransformer Maintenance Policies Selection Based on an Improved Fuzzy Analytic Hierarchy Process
JOURNAL OF COMPUTERS, VOL. 8, NO. 5, MAY 203 343 Trsformer Mitece Policies Selectio Bsed o Improved Fuzzy Alytic Hierrchy Process Hogxi Xie School of Computer sciece d Techology Chi Uiversity of Miig &
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationCompetitive Algorithms for an Online Rent or Buy Problem with Variable Demand
Competitive Algorithms for Olie Ret or Buy Prolem with Vrile Demd Roh Kodilm High Techology High School, Licroft, NJ rkodilm@ctemcorg Astrct We cosider geerliztio of the clssicl Ski Retl Prolem motivted
More informationResearch of PD online Monitoring System for DC Cable
Reserch Jourl of Applied Scieces, Eieeri d Techoloy 7(2): 263268, 2014 ISSN: 20407459; eissn: 20407467 Mxwell Scietific Oriztio, 2014 Submitted: Mrch 23, 2013 Accepted: My 10, 2013 Published: Jury
More informationMATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION. Itroductio Mthemtics distiguishes itself from the other scieces i tht it is built upo set of xioms d defiitios, o which ll subsequet theorems rely. All theorems c be derived, or
More informationInterference Alignment and the Generalized Degrees of Freedom of the X Channel
Iterferece Aligmet ad the Geeralized Degrees of Freedom of the X Chael Chiachi Huag, Viveck R. Cadambe, Syed A. Jafar Electrical Egieerig ad Computer Sciece Uiversity of Califoria Irvie Irvie, Califoria,
More informationStudy on the application of the software phaselocked loop in tracking and filtering of pulse signal
Advaced Sciece ad Techology Letters, pp.3135 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phaselocked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationSpace Vector Pulse Width Modulation Based Induction Motor with V/F Control
Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationReview: Classification Outline
Data Miig CS 341, Sprig 2007 Decisio Trees Neural etworks Review: Lecture 6: Classificatio issues, regressio, bayesia classificatio Pretice Hall 2 Data Miig Core Techiques Classificatio Clusterig Associatio
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationSection 1: Crystal Structure
Phsics 927 Section 1: Crstl Structure A solid is sid to be crstl if toms re rrnged in such w tht their positions re ectl periodic. This concept is illustrted in Fig.1 using twodimensionl (2D) structure.
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationHealth insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
More informationHealth insurance marketplace What to expect in 2014
Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum
More informationChapter 13 Volumetric analysis (acid base titrations)
Chpter 1 Volumetric lysis (cid se titrtios) Ope the tp d ru out some of the liquid util the tp coectio is full of cid d o ir remis (ir ules would led to iccurte result s they will proly dislodge durig
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationReleased Assessment Questions, 2015 QUESTIONS
Relesed Assessmet Questios, 15 QUESTIONS Grde 9 Assessmet of Mthemtis Ademi Red the istrutios elow. Alog with this ooklet, mke sure you hve the Aswer Booklet d the Formul Sheet. You my use y spe i this
More informationPHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS
PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationChair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics
Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationDENIAL OF SERVICE ATTACK IN DISTRIBUTED WIRELESS NETWORK BY DISTRIBUTED JAMMER NETWORK: A BIRTHDEATH RANDOM PROCESS ANALYSIS
Jourl of Coputer Sciece 0 (8): 39740, 04 ISSN: 5493636 04 Sciece Publictios doi:0.3844/jcssp.04.397.40 Published Olie 0 (8) 04 (http://www.thescipub.co/jcs.toc) DENIAL OF SERVICE ATTACK IN DISTRIBUTED
More informationFast Circuit Simulation Based on ParallelDistributed LIM using Cloud Computing System
JOURNAL OF SEMICONDUCTOR TECHNOLOGY AND SCIENCE, VOL.0, NO., MARCH, 00 49 Fst Circuit Simultio Bsed o PrllelDistriuted LIM usig Cloud Computig System Yut Ioue, Tdtoshi Sekie, Tkhiro Hsegw d Hideki Asi
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationMATHEMATICAL ANALYSIS
Mri Predoi Trdfir Băl MATHEMATICAL ANALYSIS VOL II INTEGRAL CALCULUS Criov, 5 CONTENTS VOL II INTEGRAL CALCULUS Chpter V EXTENING THE EFINITE INTEGRAL V efiite itegrls with prmeters Problems V 5 V Improper
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationThe Program and Evaluation of Internet of Things Used in Manufacturing Industry Hongyun Hu, Cong Yang. Intelligent procurement.
The Progrm d Evlutio of Iteret of Thigs Used i Mufcturig Idustry 1 Hogyu Hu, 2 Cog Yg 1 Xime Uiversity of Techology, xmldhy@163.com 2 Xime Uiversity of Techology, 474899564@qq.com Abstrct The mufcturig
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationTHE RISK ANALYSIS FOR INVESTMENTS PROJECTS DECISION
les Uiversittis pulesis Series Oecoomic, 11(1), 2009 THE RSK NLYSS FOR NVESTMENTS PROJECTS DECSON Cmeli Burj 1 Vsile Burj 2 BSTRCT: Te risk sigifies te possibility of existece of oe situtio i wic te obtied
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationSpam Detection. A Bayesian approach to filtering spam
Spam Detectio A Bayesia approach to filterig spam Kual Mehrotra Shailedra Watave Abstract The ever icreasig meace of spam is brigig dow productivity. More tha 70% of the email messages are spam, ad it
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70450) 18004186789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationModelDriven Hybrid and Embedded Software for Automotive Applications
1 ModelDrive Hybrid d Embedded Softwre for Automotive Applictios Aouck R. Girrd, Adm S. Howell d J. Krl Hedrick Abstrct Complex lrgescle embedded systems rise i my pplictios, i prticulr i the ig of utomotive
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationApplying Fuzzy Analytic Hierarchy Process to Evaluate and Select Product of Notebook Computers
Itertiol Jourl of Modelig d Optimiztio, Vol. No. April 202 Applyig Fuzzy Alytic Hierrchy Process to Evlute d Select Product of Noteook Computers Phrut Srichett d Wsiri Thurcho Astrct The ility, portility
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationOrdinal Classification Method for the Evaluation Of Thai Nonlife Insurance Companies
www.ijcsi.org 362 Ordil Method for the Evlutio Of Thi Nolife Isurce Compies Phiboo Jhopit, Sukree Sithupiyo 2 d Thitivdee Chiywt 3 Techopreeurship d Iovtio Mgemet Progrm Grdute School, Chullogkor Uiversity,
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More information