Hybrid Digital-Analog Joint Source-Channel Coding for Broadcasting Correlated Gaussian Sources
|
|
- Rosalind Butler
- 5 years ago
- Views:
Transcription
1 Hybrid Digitl-Alog Joit ource-chel Codig for Brodcstig Correlted Gussi ources Hmid Behroozi, Fdy Aljji d Tmás Lider Deprtmet of Mthemtics d ttistics, Quee s iversity, Kigsto, Otrio, Cd, K7L 3N6 Emil: {behroozi, fdy, lider}@mstqueesuc Abstrct We cosider the trsmissio of bivrite Gussi source =, cross power-limited two-user Gussi brodcst chel ser i i =, observes the trsmitted sigl corrupted by Gussi oise with power i d wts to estimte i We study hybrid digitl-log HDA joit source-chel codig schemes d lyze these schemes to obti chievble squred-error distortio regios Two cses re cosidered: source d chel bdwidths re equl, brodcstig with bdwidth compressio We dpt HDA schemes of Wilso et l [] d rbhkr et l [] to provide vrious chievble distortio regios for both cses sig umericl exmples, we demostrte tht for bdwidth compressio, three-lyered codig scheme cosistig of log, superpositio, d codig performs well compred to the other provided HDA schemes I the cse of mtched bdwidth, three-lyered codig scheme with log lyer d two lyers, ech cosistig of Wyer-Ziv coder followed by coder, performs best I INTRODCTION This pper cosiders brodcstig correlted Gussi sources d ims to chrcterize me squred-error ME distortio pirs tht re simulteously chievble t two receivers usig hybrid digitl-log HDA codig schemes It is kow tht the seprte desig of source d chel codig due to ho does ot i geerl led to the optiml performce theoreticlly ttible OTA i etworks O the other hd, for the poit-to-poit trsmissio of sigle Gussi source through dditive white Gussi oise AWGN chel it is well kow tht if the chel d source bdwidths re equl, simple ucoded trsmissio chieves OTA coded or log trsmissio i this cse d i the rest of this pper mes sclig the ecoder iput subject to the chel power costrit d trsmittig it without explicit chel codig I order to exploit the dvtges of both log trsmissio d digitl techiques, vrious HDA schemes hve bee itroduced i the literture, see eg, [], [3] [9] Brodcstig sigle memoryless Gussi source uder bdwidth mismtch usig HDA schemes is cosidered i [5], [8] Bross et l [] show tht there exists cotiuum of HDA schemes with optiml performce for the trsmissio of Gussi source over verge-power-limited Gussi chel with mtched bdwidth Ti d hmi [] geerlize this result to the mismtched bdwidth cse Brodcstig Gussi source with memory is lyzed i [9] This work ws supported i prt by ostdoctorl Fellowship from the Otrio Miistry of Reserch d Iovtio MRI d by the Nturl cieces d Egieerig Reserch Coucil NERC of Cd Bi vrite ource Ecoder V V Receiver Receiver Fig Brodcstig bivrite Gussi source over two-user powerlimited Gussi brodcst chel Our system model is illustrted i Fig We im to determie chievble distortio regios usig HDA schemes for two cses; the source bdwidth equls the chel bdwidth, brodcstig with bdwidth compressio To our kowledge, prt from [] i which Bross et l lyzed ucoded trsmissio for brodcstig correlted Gussi sources, o explicit distortio-regios hve bee estblished i the literture for brodcstig correlted Gussi sources We re lso ot wre of y prior work o HDA schemes for brodcstig correlted Gussis either whe the source d chel bdwidths re equl or whe there is bdwidth mismtch Note tht the source-chel seprtio theorem does ot hold i brodcstig correlted sources II ROBLEM TATEMENT We cosider brodcstig bivrite Gussi source cross two-user power-limited Gussi brodcst chel ser i i =, receives the trsmitted sigl corrupted by Gussi oise with power i d wts to estimte the ith compoet of the source We ssume > d cll user the wek user d user the strog user Let d be correlted Gussi rdom vribles d let { t, t} t= be sttiory Gussi memoryless vector source with mrgil distributio tht of, We ssume tht t d t hve zero me d vrice d, respectively, d correltio coefficiet ρ, We represet the first source smples by the dt sequeces = {,,, } d = {,,, }, respectively The system is show i Fig The source sequeces d re joitly ecoded to = ϕ,, where the ecoder fuctio is of the form ϕ : R R R The trsmitted sequece is verge-power limited to >, ie, E t= [ t ] ser i observes the trsmitted sigl t corrupted by Gussi oise V i t with power i so tht ech observtio time t =,,3, receiver i observes i t = t + V i t, i =,
2 where the V i t N,i re idepedetly distributed over i d t, d re idepedet of the t Bsed o its chel output i, user i provides estimte Ŝi = ψ i i, where ψ i : R R is decodig fuctio The qulity of the estimte is mesured by the verge ME distortio i = E[ i t Ŝit ] Let F t= deote ll ecoder d decoder fuctios ϕ,ψ,ψ defied s bove For prticulr codig scheme ϕ,ψ,ψ, the performce is determied by the chel power costrit d the icurred distortios d t the receivers For y give power costrit >, the distortio regio D is defied s the covex closure of the set of ll distortio pirs D,D for which,d,d is chievble, where power-distortio pir,d,d is chievble if for y δ >, there exists δ such tht for y δ there exists ϕ,ψ,ψ F with distortios i D i + δ i =, III DITORTION REGION WITH MATCHED BANDWIDTH A coded Trsmissio I [] for the bove problem chievble distortio regio is obtied bsed o lyzig the ucoded trsmissio i brodcstig bivrite Gussi source I this pproch, lier combitio of both compoets of bivrite Gussi source is trsmitted cross powerlimited Gussi brodcst chel The trsmitted sigl c be expressed s t = α i i t, 3 where α = Vr i it i= i=, i d Vr i i t = i= + + ρ The scle fctor α is chose such tht the chel power costrit is stisfied with equlity The received sigl t receiver i is the give by i t = t + V i t = α i i t + V i t 4 i= By evlutig the resultig ME distortio, the set of simulteously chievble distortio pirs t two users re s follows: D i = i α i i + j ρ i j + i, i,j =,, j i 5 It is show i [] tht the ucoded scheme is optiml below certi NR-threshold B Joit ource-chel Codig chemes I our schemes, we will closely follow the ottio d code costructios i [] Here we oly give high-level descriptio d lyses of the schemes without detiled proofs I prticulr, i my steps of the lysis we tret fiite-blocklegth codig schemes s idelized systems with symptoticlly lrge blocklegths Lyerig with Alog d Codig: This codig scheme hs three lyers d is similr to the scheme i [] for brodcstig sigle memoryless Gussi source The oly differece betwee the two schemes is tht we use Wyer-Ziv ecoder followed by ecoder i the secod lyer, while the secod lyer of the scheme i [] employs HDA coder which will be explied i ectio IV-A Block digrms of the ecoder d the decoder re show i Fig The first lyer is the log trsmissio lyer Here t = α i i t, where α = i= This lyer is met for both strog d Vr i it i= wek users Now fix d to stisfy = + + I the secod lyer, the first compoet of the source is first Wyer-Ziv coded t rte R = log + + usig estimte of t the receiver s side iformtio The Wyer-Ziv idex, m {,,, R }, is the ecoded usig s dirty pper codig tretig the log trsmissio lyer,, s iterferece Let be uxiliry rdom vrible give by = +α, where N, is idepedet of N, d the sclig fctor α is set to be + + We geerte legth iid Gussi codebook with I; codewords, where ech compoet of the codeword is Gussi with zero me d vrice + α, d ech codeword is the rdomly plced ito oe of R bis Let i be the idex of the bi cotiig For give m, we look for such tht i = m d d re joitly typicl The, we trsmit = α, where is met to be decoded by the wek user I the third lyer, which is met for the strog user, the secod compoet of the source,, is lso Wyer Ziv coded t rte R = log + usig the estimte of t the receiver s side iformtio The Wyer-Ziv idex, m {,,, R }, is the ecoded usig digitl codig tht trets both d s iterferece d uses power Let be uxiliry rdom vrible give by = + α +, where N,, d re idepedet from ech other d α = + Here we lso crete legth iid Gussi codebook with I; codewords, where ech compoet of the codeword is Gussi with zero me d vrice + α + d rdomly evely distribute them over R bis Let i be the idex of the bi cotiig For give m, we look for such tht i = m d,, re joitly typicl The, we trsmit = α + As show i Fig, we merge ll three lyers d trsmit = + + A chievble distortio-regio c be obtied by vryig, d subject to = + + For give, d, the chievble distortio pirs c be computed s follows At the receiver Fig b, estimte of the first compoet of the source,, is first obtied from the log lyer This estimte cts s side iformtio tht c be used i refiig the estimte of for the wek user usig the R decoded Wyer-Ziv bits obtied by the decoder of the secod lyer ice R equls the cpcity of the chel with kow iterferece t the ecoder oly, I ; I ; = log + the distortio i estimtig t the wek user is give by the Wyer-Ziv distortio-rte fuctio, D R, where D = E[ E[ ] ] is the idelized MME from +,
3 α 5 Alog Trsmissio Alog, uperpositio d Codig Alog d Codig Bi vrite ource Wyer Ziv m Ecoder Wyer Ziv m Ecoder Ecoder Ecoder Ecoder log D 5 5 V V MME Estimtor Decoder Decoder Decoder b Decoder Wyer Ziv Decoder Wyer Ziv Decoder MME Estimtor Fig Brodcstig bivrite source, by doptig the lyerig scheme with log d codig lyers i [] the received o the overll distortio see t the wek, user c be expressed s D = D + + where D = α + ρ The, estimte of c be determied from the first d the secod lyers This estimte cts s side iformtio for estimtig for the strog user from the R decoded Wyer-Ziv bits Here, gi, R equls the cpcity of the chel with kow iterferece, d, t the ecoder oly, ie, R = I ; I ;, = log+ Thus, the distortio i estimtig t the strog user is give by the Wyer-Ziv distortio-rte fuctio, D R, where D is the MME from the received d the decoded o the overll distortio for the strog user, is give by D = D + where D = Γ T Υ Γ, d Υ = Γ = α + ρ α α, + ρ α + α + α 7 Lyerig with Alog, uperpositio d Codig: This scheme lso hs three codig lyers: log, superpositio, d codig I the secod lyer, we hve two merged strems, similr to the cse of brodcstig sigle memoryless source over brodcst chel [4], [3] The first compoet of the source is brodcsted to two users The first source ecoder is optiml Wyer-Ziv ecoder with rte R = log + λ λ + + +, d the secod source ecoder is optiml Wyer-Ziv ecoder for the residul error of the first ecoder with rte R R = log + λ + + The, we ecode the Wyer-Ziv bits with cpcity-chievig chel codes d trsmit with log D Fig 3 Distortio regios i brodcstig bivrite source with the correltio coefficiet ρ = powers λ d λ, respectively ice we require rte of oe chel use per source symbol, d the Gussi source is successively refible, by combiig the Wyer-Ziv rte-distortio fuctio with the pir of chievble rtes for brodcst chel R,R, the correspodig chievble distortio pirs re [4]: D R d D R, where D is give i 6 The codig scheme i the third lyer is similr to tht i the previous scheme The fil distortio i estimtig t the wek user is D = D D R = + λ λ At the strog user, first estimte of the first compoet of the source c be obtied withi distortio D = D R D = R + λ + + = + D λ + + The we obti estimte of from the bove estimte of with the followig distortio: D = ρ D 9 This estimte of cts s side iformtio i refiig the estimte of for the strog user usig the decoded Wyer-Ziv bits The overll distortio for the strog user i estimtig is thus give by D = D + 3 Numericl Exmple: We trsmit smples of bivrite [ Gussi ] source with the covrice mtrix Λ = i uses of power-limited brodcst chel to two users with observtio oise vrices = 5dB d = db, respectively The two-user brodcst chel hs the power costrit = db The boudries of the distortio regios for the schemes preseted i this sectio re show i Fig 3 We observe tht the lyerig with log trsmissio d codig outperforms ll other JCC schemes, icludig log trsmissio IV DITORTION REGION WITH BANDWIDTH COMREION We ext cosider the problem of brodcstig bivrite Gussi source with : bdwidth compressio We wt to trsmit k = smples of bivrite Gussi source k, k i uses of power-limited brodcst chel to
4 Bi vrite ource k= k k,, k, k, Hybrid Digitl Alog HDA Ecoder Wyer Ziv Ecoder α Ecoder Fig 4 Brodcstig bivrite source k, k with bdwidth compressio usig three-lyered codig provided i [] two users The two-user brodcst chel hs the power costrit We split both compoets of the bivrite Gussi source ito two equl legth prts, ie, we split smples of ech source vector i ito two vectors of legth : i, d i, A Lyerig with Alog, HDA d Codig This scheme is itroduced i [] for brodcstig memoryless Gussi source with bdwidth compressio; see Fig 4 I the first log trsmissio lyer, lier combitio of the first smples of the bivrite Gussi source compoets re scled such tht the power of the trsmitted sigl i this lyer becomes Here t = α i i, t where α = i= + + ρ I the secod d the third lyers, we work o the remiig smples of the source compoets, ie,, d,, respectively I the secod lyer, we pply the HDA codig, preseted i [], to, i order to produce with power Here, the source is ot explicitly qutized d it ppers i log form i the trsmitted sigl [] Let be uxiliry rdom vrible give by = + α + K,, where N,, N,, d, re idepedet of ech other, α = + +, d K = + + As i [], we geerte rdom iid codebook with R codewords, where ech compoet of ech codeword is Gussi with zero me d vrice + α + K d R = log +α +K For give, d, we fid such tht,,, is joitly typicl d trsmit = α K, I the third lyer, smples of the secod compoet of the source,, re Wyer Ziv coded t rte R = log + usig the estimte of, t the receiver s side iformtio The Wyer-Ziv idex is the ecoded usig codig tht trets both d s iterferece d uses power = The code costructio s well s the ecodig d decodig procedures re logous to the oes described i ectio III-B Therefore, we trsmit = α + We merge ll three lyers d trsmit = + + At the decoder, we look for tht is joitly typicl with The wek user estimtes k =,, by MME estimtio from the received sigl d the decoded Thus, the overll distortio see t the wek user is []: D = k D + k D = D + D, where D j j =,, the MME distortio i estimtig,j from d, is give by where d Γ = Υ HDA = D j = Γ T jυ HDA Γ j, α + ρ α α, Γ + ρ = K, α + α + α + K The, estimte of k is obtied from the first d the secod lyers This estimte cts s side iformtio for estimtig for the strog user usig the decoded Wyer- Ziv bits The strog user estimtes the secod compoet of the source k =,, from, the decoded d Hece the overll distortio for the strog user is give by D = D + D, where D j j =,, the distortio i estimtig,j, is determied vi the Wyer- Ziv distortio-rte fuctio: where Γ = d D j = Γ T jυ HDA Γ j + j, Υ HDA = α + ρ α α,γ + ρ =, K ρ α + α + α + K B Lyerig with Alog d Codig Here, we lso use three codig lyers d they re the sme s the oes i ectio IV-A, except for the secod lyer I the secod lyer, the smples of the secod hlf of the first compoet of the source,,, re qutized t rte R = log + + The qutiztio idex is the ecoded usig codig tht trets s iterferece d uses power Therefore, we trsmit = α, where α = + + We merge ll three lyers d trsmit = + + At the receiver, the wek user estimtes =,, by MME estimtio from the received sigl d the decoded Thus the overll distortio see t the wek user is give by D = Γ T Υ Γ +, where Γ is give i d + Υ = α + α + α The strog user estimtes the secod compoet of the
5 source =,, withi the overll distortio Γ T Υ Γ D = + ρ D + 4 where Γ is give i, Υ is provided i 7 d D = C Lyerig with Alog, uperpositio d Codig Alogously to the previous codig schemes, this scheme is three-lyered with its lyers ideticl to the oes preseted i ectio IV-A, except for the secod lyer I the secod lyer, s i ectio III-B, we use two merged strems The secod prt of the first compoet of the source,,, is brodcsted to two users The first source ecoder is optiml source ecoder with rte R log+ λ = λ + + +, d the secod source ecoder is optiml ecoder for the residul error of the first ecoder with rte R R = λ log+ + + The, we ecode the qutiztio bits with cpcity-chievig chel codes d trsmit the resultig strems uder powers λ d λ, respectively The wek user forms MME estimte of with the followig distortio: D = α + ρ λ λ λ At the strog user, first estimte of, c be obtied withi distortio D = + + λ + + λ λ This estimte cts s side iformtio for obtiig the estimte of, usig the decoded Wyer-Ziv bits The resultig distortio for the strog user is thus give by D = Γ T Υ Γ + ρ D + 6 Filly, ote tht if we set ρ = d =, the the results of [], [9], which curretly pper to be the best kow results for brodcstig Gussi source with bdwidth compressio, re obtied D Numericl Results We trsmit k = smples of bivrite Gussi ρ source, k k with the covrice mtrix Λ = ρ i uses of power-limited brodcst chel to two users with observtio oise vrices = 5dB d = db, respectively The distortio regios for the schemes preseted i this sectio re show i Fig 5 for two differet correltio coefficiets, ρ = d ρ = 8 We observe tht the lyerig with log, superpositio log D 5 5 Alog d Codig Alog, HDA d Codig Alog, uperpositio d Codig ρ=8 ρ= log D Fig 5 Distortio regios of differet HDA codig schemes ystem prmeters re = db, = 5dB d = db d codig of ectio IV-C outperforms ll other schemes i both cses Whe the source compoets re highly correlted, lyerig with log, HDA, d codig scheme performs better th the lyerig with log d codig scheme; however, the two two schemes perform similrly for smll vlues of the correltio coefficiet REFERENCE [] M Wilso, K Nry, d G Cire, Joit source chel codig with side iformtio usig hybrid digitl log codes, i roc IEEE If Theory d Applictios ITA Workshop, L Joll, CA, J 7, pp [] V M rbhkr, R uri, d K Rmchdr, Colored Gussi source-chel brodcst for heterogeeous log/digitl receivers, IEEE Trs If Theory, vol 54, o 4, pp 87 84, Apr 8 [3] hmi, Verdu, d R Zmir, ystemtic lossy source/chel codig, IEEE Trs If Theory, vol 44, o, pp , Mr 998 [4] B Che d G W Worell, Alog error-correctig codes bsed o chotic dymicl systems, IEEE Trs Commu, vol 46, o 7, pp 88 89, Jul 998 [5] Mittl d N hmdo, Hybrid digitl-log HDA joit sourcechel codes for brodcstig d robust commuictios, IEEE Trs If Theory, vol 48, o 5, pp 8, My [6] esi, G Cire, d G Vivier, Lossy trsmissio over slowfdig AWGN chels: compriso of progressive, superpositio d hybrid pproches, i roc IEEE IIT, Adelide, Austrli, ep 5 [7] M koglud, N hmdo, d F Aljji, Hybrid digitl-log source-chel codig for bdwidth compressio/expsio, IEEE Trs If Theory, vol 5, o 8, pp , Aug 6 [8] Z Rezic, M Feder, d R Zmir, Distortio bouds for brodcstig with bdwidth expsio, IEEE Trs If Theory, vol 5, o 8, pp , Aug 6 [9] V M rbhkr, R uri, d K Rmchdr, Hybrid logdigitl strtegies for source-chel brodcst, i roc 43rd Allerto Cof Commu, Cotr, Comput, Allerto, IL, ep 5 [] Bross, A Lpidoth, d Tiguely, uperimposed coded d ucoded trsmissios of Gussi source over the Gussi chel, i roc IEEE IIT, ettle, WA, Jul 6, pp [] C Ti d hmi, A uified codig scheme for hybrid trsmissio of Gussi source over Gussi chel, i roc IEEE IIT, Toroto, ON, Jul 8 [] Bross, A Lpidoth, d Tiguely, Brodcstig correlted Gussis, i roc IEEE IIT, Toroto, ON, Jul 8 [3] M C Gstpr, eprtio theorems d prtil orderigs for sesor etwork problems, I ligrm, Vektesh Ed, Networked esig Iformtio d Cotrol, priger, 8
Hybrid Digital-Analog Coding for Interference Broadcast Channels
Hybrid Digitl-Anlog Coding for Interference Brodcst Chnnels Ahmd Abou Sleh, Fdy Aljji, nd Wi-Yip Chn Queen s University, ingston, O, 7L 36 Emil: hmd.bou.sleh@queensu.c, fdy@mst.queensu.c, chn@queensu.c
More informationError-free compression
Error-free compressio Useful i pplictio where o loss of iformtio is tolerble. This mybe due to ccurcy requiremets, legl requiremets, or less th perfect qulity of origil imge. Compressio c be chieved by
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationBackground 1. Cramer-Rao inequality
Bro-06, Lecture, 60506 D/Stt/Bro-06/tex wwwmstqueesuc/ blevit/ Observtio vector, dt: Bckgroud Crmer-Ro iequlity X (X,, X X R Here dt c be rel- or vector vlued, compoets of the dt c be idepedet or depedet,
More informationAvd. Matematisk statistik
Avd. Mtemtisk sttistik TENTAMEN I SF94 SANNOLIKHETSTEORI/EAM IN SF94 PROBABILITY THE- ORY WEDNESDAY 8th OCTOBER 5, 8-3 hrs Exmitor : Timo Koski, tel. 7 3747, emil: tjtkoski@kth.se Tillåt hjälpmedel Mes
More informationINFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
More informationReview of Sections
Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationFrequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
More information=> PARALLEL INTERCONNECTION. Basic Properties LTI Systems. The Commutative Property. Convolution. The Commutative Property. The Distributive Property
Lier Time-Ivrit Bsic Properties LTI The Commuttive Property The Distributive Property The Associtive Property Ti -6.4 / Chpter Covolutio y ] x ] ] x ]* ] x ] ] y] y ( t ) + x( τ ) h( t τ ) dτ x( t) * h(
More informationFig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg
ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced
More informationSchrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
More informationSimilar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication
Next. Covered bsics of simple desig techique (Divided-coquer) Ch. of the text.. Next, Strsse s lgorithm. Lter: more desig d coquer lgorithms: MergeSort. Solvig recurreces d the Mster Theorem. Similr ide
More informationSection 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
More informationlecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
More information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationPrior distributions. July 29, 2002
Prior distributios Aledre Tchourbov PKI 357, UNOmh, South 67th St. Omh, NE 688-694, USA Phoe: 4554-64 E-mil: tchourb@cse.ul.edu July 9, Abstrct This documet itroduces prior distributios for the purposes
More informationConvergence rates of approximate sums of Riemann integrals
Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationThe Definite Integral
The Defiite Itegrl A Riem sum R S (f) is pproximtio to the re uder fuctio f. The true re uder the fuctio is obtied by tkig the it of better d better pproximtios to the re uder f. Here is the forml defiitio,
More informationKing Fahd University of Petroleum & Minerals
Kig Fhd Uiversity of Petroleum & Mierls DEPARTMENT OF MATHEMATICAL CIENCE Techicl Report eries TR 434 April 04 A Direct Proof of the Joit Momet Geertig Fuctio of mple Me d Vrice Awr H. Jorder d A. Lrdji
More informationNumbers (Part I) -- Solutions
Ley College -- For AMATYC SML Mth Competitio Cochig Sessios v.., [/7/00] sme s /6/009 versio, with presettio improvemets Numbers Prt I) -- Solutios. The equtio b c 008 hs solutio i which, b, c re distict
More informationMATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
More informationDefinite Integral. The Left and Right Sums
Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give
More informationStudents must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...
Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More informationNumerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials
Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationDouble Sums of Binomial Coefficients
Itertiol Mthemticl Forum, 3, 008, o. 3, 50-5 Double Sums of Biomil Coefficiets Athoy Sofo School of Computer Sciece d Mthemtics Victori Uiversity, PO Box 448 Melboure, VIC 800, Austrli thoy.sofo@vu.edu.u
More informationChapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationChapter 11 Design of State Variable Feedback Systems
Chpter Desig of Stte Vrible Feedbck Systems This chpter dels with the desig of cotrollers utilizig stte feedbck We will cosider three mjor subjects: Cotrollbility d observbility d the the procedure for
More information2017/2018 SEMESTER 1 COMMON TEST
07/08 SEMESTER COMMON TEST Course : Diplom i Electroics, Computer & Commuictios Egieerig Diplom i Electroic Systems Diplom i Telemtics & Medi Techology Diplom i Electricl Egieerig with Eco-Desig Diplom
More informationChapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
More informationB. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i
Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio
More informationWhy study large deviations? The problem of estimating buer overow frequency The performance of many systems is limited by events which have a small pr
Why study lrge devitios? The problem of estimtig buer overow frequecy The performce of my systems is ited by evets which hve smll probbility of occurrig, but which hve severe cosequeces whe they occur.
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationELG4156 Design of State Variable Feedback Systems
ELG456 Desig of Stte Vrible Feedbck Systems This chpter dels with the desig of cotrollers utilizig stte feedbck We will cosider three mjor subjects: Cotrollbility d observbility d the the procedure for
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationVectors. Vectors in Plane ( 2
Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationAngle of incidence estimation for converted-waves
Agle of icidece estimtio for coverted-wves Crlos E. Nieto d Robert R. tewrt Agle of icidece estimtio ABTRACT Amplitude-versus-gle AA lysis represets li betwee te geologicl properties of roc iterfces d
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More information( ) k ( ) 1 T n 1 x = xk. Geometric series obtained directly from the definition. = 1 1 x. See also Scalars 9.1 ADV-1: lim n.
Sclrs-9.0-ADV- Algebric Tricks d Where Tylor Polyomils Come From 207.04.07 A.docx Pge of Algebric tricks ivolvig x. You c use lgebric tricks to simplify workig with the Tylor polyomils of certi fuctios..
More informationThe Weierstrass Approximation Theorem
The Weierstrss Approximtio Theorem Jmes K. Peterso Deprtmet of Biologicl Scieces d Deprtmet of Mthemticl Scieces Clemso Uiversity Februry 26, 2018 Outlie The Wierstrss Approximtio Theorem MtLb Implemettio
More informationINTEGRATION IN THEORY
CHATER 5 INTEGRATION IN THEORY 5.1 AREA AROXIMATION 5.1.1 SUMMATION NOTATION Fibocci Sequece First, exmple of fmous sequece of umbers. This is commoly ttributed to the mthemtici Fibocci of is, lthough
More information: : 8.2. Test About a Population Mean. STT 351 Hypotheses Testing Case I: A Normal Population with Known. - null hypothesis states 0
8.2. Test About Popultio Me. Cse I: A Norml Popultio with Kow. H - ull hypothesis sttes. X1, X 2,..., X - rdom smple of size from the orml popultio. The the smple me X N, / X X Whe H is true. X 8.2.1.
More informationClosed Newton-Cotes Integration
Closed Newto-Cotes Itegrtio Jmes Keeslig This documet will discuss Newto-Cotes Itegrtio. Other methods of umericl itegrtio will be discussed i other posts. The other methods will iclude the Trpezoidl Rule,
More informationCHAPTER 6: USING MULTIPLE REGRESSION
CHAPTER 6: USING MULTIPLE REGRESSION There re my situtios i which oe wts to predict the vlue the depedet vrible from the vlue of oe or more idepedet vribles. Typiclly: idepedet vribles re esily mesurble
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More informationChapter System of Equations
hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probbility d Stochstic Processes: A Friedly Itroductio for Electricl d Computer Egieers Roy D. Ytes d Dvid J. Goodm Problem Solutios : Ytes d Goodm,4..4 4..4 4..7 4.4. 4.4. 4..6 4.6.8 4.6.9 4.7.4 4.7.
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationInference on One Population Mean Hypothesis Testing
Iferece o Oe Popultio Me ypothesis Testig Scerio 1. Whe the popultio is orml, d the popultio vrice is kow i. i. d. Dt : X 1, X,, X ~ N(, ypothesis test, for istce: Exmple: : : : : : 5'7" (ull hypothesis:
More informationOrthogonality, orthogonalization, least squares
ier Alger for Wireless Commuictios ecture: 3 Orthogolit, orthogoliztio, lest squres Ier products d Cosies he gle etee o-zero vectors d is cosθθ he l of Cosies: + cosθ If the gle etee to vectors is π/ (90º),
More informationGeneral properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
More informationUncertainty Analysis for Uncorrelated Input Quantities and a Generalization
WHITE PAPER Ucertity Alysis for Ucorrelted Iput Qutities d Geerliztio Welch-Stterthwite Formul Abstrct The Guide to the Expressio of Ucertity i Mesuremet (GUM) hs bee widely dopted i the differet fields
More informationParticle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise
Prticle i Box We must hve me where = 1,,3 Solvig for E, π h E = = where = 1,,3, m 8m d the stte fuctio is x A si for 0 x, d 0 otherwise x ˆ d KE V. m dx I this cse, the Hermiti opertor 0iside the box The
More informationPROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
More informationImproving XOR-Dominated Circuits by Exploiting Dependencies between Operands. Ajay K. Verma and Paolo Ienne. csda
Improvig XOR-Domited Circuits y Exploitig Depedecies etwee Operds Ajy K. Verm d Polo Iee csd Processor Architecture Lortory LAP & Cetre for Advced Digitl Systems CSDA Ecole Polytechique Fédérle de Luse
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationVariational Iteration Method for Solving Volterra and Fredholm Integral Equations of the Second Kind
Ge. Mth. Notes, Vol. 2, No. 1, Jury 211, pp. 143-148 ISSN 2219-7184; Copyright ICSRS Publictio, 211 www.i-csrs.org Avilble free olie t http://www.gem.i Vritiol Itertio Method for Solvig Volterr d Fredholm
More informationSome New Iterative Methods Based on Composite Trapezoidal Rule for Solving Nonlinear Equations
Itertiol Jourl of Mthemtics d Sttistics Ivetio (IJMSI) E-ISSN: 31 767 P-ISSN: 31-759 Volume Issue 8 August. 01 PP-01-06 Some New Itertive Methods Bsed o Composite Trpezoidl Rule for Solvig Nolier Equtios
More informationLEVEL I. ,... if it is known that a 1
LEVEL I Fid the sum of first terms of the AP, if it is kow tht + 5 + 0 + 5 + 0 + = 5 The iterior gles of polygo re i rithmetic progressio The smllest gle is 0 d the commo differece is 5 Fid the umber of
More informationCertain sufficient conditions on N, p n, q n k summability of orthogonal series
Avilble olie t www.tjs.com J. Nolier Sci. Appl. 7 014, 7 77 Reserch Article Certi sufficiet coditios o N, p, k summbility of orthogol series Xhevt Z. Krsiqi Deprtmet of Mthemtics d Iformtics, Fculty of
More informationSection IV.6: The Master Method and Applications
Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationSequence and Series of Functions
6 Sequece d Series of Fuctios 6. Sequece of Fuctios 6.. Poitwise Covergece d Uiform Covergece Let J be itervl i R. Defiitio 6. For ech N, suppose fuctio f : J R is give. The we sy tht sequece (f ) of fuctios
More informationSUCCESSIVE INTERFERENCE CANCELLATION DECODING FOR THE K -USER CYCLIC INTERFERENCE CHANNEL
Joural of Theoretical ad Applied Iformatio Techology 31 st December 212 Vol 46 No2 25-212 JATIT & LLS All rights reserved ISSN: 1992-8645 wwwatitorg E-ISSN: 1817-3195 SCCESSIVE INTERFERENCE CANCELLATION
More informationA general theory of minimal increments for Hirsch-type indices and applications to the mathematical characterization of Kosmulski-indices
Mlysi Jourl of Librry & Iformtio Sciece, Vol. 9, o. 3, 04: 4-49 A geerl theory of miiml icremets for Hirsch-type idices d pplictios to the mthemticl chrcteriztio of Kosmulski-idices L. Egghe Uiversiteit
More informationDiscrete-Time Signals & Systems
Chpter 2 Discrete-Time Sigls & Systems 清大電機系林嘉文 cwli@ee.thu.edu.tw 03-57352 Discrete-Time Sigls Sigls re represeted s sequeces of umbers, clled smples Smple vlue of typicl sigl or sequece deoted s x =
More informationIn an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case
Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio
More informationDIGITAL SIGNAL PROCESSING LECTURE 5
DIGITAL SIGNAL PROCESSING LECTURE 5 Fll K8-5 th Semester Thir Muhmmd tmuhmmd_7@yhoo.com Cotet d Figures re from Discrete-Time Sigl Processig, e by Oppeheim, Shfer, d Buck, 999- Pretice Hll Ic. The -Trsform
More informationLecture 3: A brief background to multivariate statistics
Lecture 3: A brief bckgroud to multivrite sttistics Uivrite versus multivrite sttistics The mteril of multivrite lysis Displyig multivrite dt The uses of multivrite sttistics A refresher of mtrix lgebr
More informationConvergence rates of approximate sums of Riemann integrals
Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationSUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More informationOn the Performance of Hybrid Digital-Analog Coding for Broadcasting Correlated Gaussian Sources
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 59, NO, DECEMBER 0 3335 O the Performace of Hybrid Digital-Aalog Codig for Broadcastig Correlated Gaussia Sources Hamid Behroozi, Member, IEEE, Fady Alajaji, Seior
More informationLecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields
Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive
More informationELEG 3143 Probability & Stochastic Process Ch. 5 Elements of Statistics
Deprtet of Electricl Egieerig Uiversity of Arkss ELEG 3143 Probbility & Stochstic Process Ch. 5 Eleets of Sttistics Dr. Jigxi Wu wuj@urk.edu OUTLINE Itroductio: wht is sttistics? Sple e d sple vrice Cofidece
More informationLinear Programming. Preliminaries
Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio
More informationAPPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES
Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 3() 0, 5-0 APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Jolt Borows, Le Łcińs, Jowit Rychlews Istitute of Mthetics,
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More information2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r
Z-Trsforms. INTRODUCTION TO Z-TRANSFORM The Z-trsform is coveiet d vluble tool for represetig, lyig d desigig discrete-time sigls d systems. It plys similr role i discrete-time systems to tht which Lplce
More informationELEG 5173L Digital Signal Processing Ch. 2 The Z-Transform
Deprtmet of Electricl Egieerig Uiversity of Arkss ELEG 573L Digitl Sigl Processig Ch. The Z-Trsform Dr. Jigxi Wu wuj@urk.edu OUTLINE The Z-Trsform Properties Iverse Z-Trsform Z-Trsform of LTI system Z-TRANSFORM
More informationFourier Series and Applications
9/7/9 Fourier Series d Applictios Fuctios epsio is doe to uderstd the better i powers o etc. My iportt probles ivolvig prtil dieretil equtios c be solved provided give uctio c be epressed s iiite su o
More informationM3P14 EXAMPLE SHEET 1 SOLUTIONS
M3P14 EXAMPLE SHEET 1 SOLUTIONS 1. Show tht for, b, d itegers, we hve (d, db) = d(, b). Sice (, b) divides both d b, d(, b) divides both d d db, d hece divides (d, db). O the other hd, there exist m d
More informationNumerical Methods (CENG 2002) CHAPTER -III LINEAR ALGEBRAIC EQUATIONS. In this chapter, we will deal with the case of determining the values of x 1
Numericl Methods (CENG 00) CHAPTER -III LINEAR ALGEBRAIC EQUATIONS. Itroductio I this chpter, we will del with the cse of determiig the vlues of,,..., tht simulteously stisfy the set of equtios: f f...
More information( ) dx ; f ( x ) is height and Δx is
Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio
More informationECE 564/645 - Digital Communication Systems (Spring 2014) Final Exam Friday, May 2nd, 8:00-10:00am, Marston 220
ECE 564/645 - Digital Commuicatio Systems (Sprig 014) Fial Exam Friday, May d, 8:00-10:00am, Marsto 0 Overview The exam cosists of four (or five) problems for 100 (or 10) poits. The poits for each part
More informationn 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
More information