Hybrid Digital-Analog Coding for Interference Broadcast Channels
|
|
- Emmeline Sanders
- 5 years ago
- Views:
Transcription
1 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 Abstrct We consider the trnsmission of bivrite Gussin sources (V, V over the two-user Gussin brodcst chnnel in the presence of interference tht is correlted to the source nd known to the trnsmitter. Ech user i is interested in estimting V i. We study hybrid digitl-nlog (HDA schemes nd nlyze the chievble (squre-error distortion region under mtched nd expnsion bndwidth regimes. These schemes require proper combintions of power splitting, bndwidth splitting, rte splitting, nd HDA Cost coding. An outer bound on the distortion region is lso derived by ssuming knowledge of V t the second user nd full/prtil knowledge of the interference t both users. umericl results show tht the HDA schemes outperform tndem nd liner schemes nd perform close to the derived bound for certin system settings. I. ITRODUCTIO The trditionl pproch for nlog source trnsmission over noisy chnnels is to use seprte source nd chnnel coders. This pproch is well known to be optiml for point-to-point communictions. For multi-terminl systems, tndem coding is no longer optiml; joint source-chnnel coding (JSCC scheme my be required to chieve optimlity. One simple scenrio where tndem scheme is suboptiml concerns the brodcst of Gussin sources over Gussin chnnels []. For single Gussin source sent over Gussin brodcst chnnel with mtched source-chnnel bndwidth, the distortion region is known, nd cn be relized by liner scheme []. For mismtched source-chnnel bndwidth, the best known coding schemes re bsed on JSCC with hybrid signlling [] [5]. One extension to this problem is the brodcsting of two correlted sources to two users, ech of which is interested in recovering one of the two sources; in [6], it ws proven tht the liner scheme is optiml when the system s signl-to-noise rtio is below certin threshold under mtched bndwidth. In [7], hybrid digitl-nlog (HDA scheme is proposed for the sme mtched bndwidth system nd is shown to be optiml whenever the liner scheme of [6] is not; hence providing complete chrcteriztion of the distortion region. Under mismtched bndwidth, vrious HDA schemes re proposed in [8], consisting of different combintions of severl known schemes using either superposition or dirty pper coding. Recently, in [9], tndem scheme bsed on successive coding is studied nd shown to outperform the HDA schemes of [8]. In [0], the uthors investigte the trnsmission of Gussin source over correlted interference chnnel nd propose hybrid lyered scheme for point-to-point communictions. This work ws supported in prt by SERC of Cnd. In this work, we consider the trnsmission of two correlted sources over brodcst interference chnnels, where the interference is ssumed to be correlted with the sources. This brodcst system with correlted source-interference cn model situtions where two nerby nodes re trnsmitting correlted informtion simultneously. One node sends directly its signl; the other, however, hs knowledge bout its neighbour signl nd trets it s correlted interference. We propose nd nlyze HDA schemes for this system bsed on Wyner- Ziv [], Cost [] nd HDA Cost coding [3]. The rest of the pper is orgnized s follows. Section II presents the problem formultion. Section III introduces n outer bound on the system s distortion region nd some reference schemes. In Section IV, inner bounds on the distortion region under mtched nd expnsion bndwidth re studied by proposing HDA schemes. umericl results re included in Section V. Finlly, conclusions re drwn in Section VI. II. PROBLEM FORMULATIO We consider the trnsmission (Fig. of pir of correlted Gussin sources (V,V over two-user Gussin brodcst chnnel in the presence of Gussin interference S known to the trnsmitter. User i (i =, receives the trnsmitted signl corrupted by dditive white Gussin noise W i nd interference S with vrinces Wi nd S, respectively. Ech user i ims to estimte Vi =(V i (,V i (,...,V i (, where ech smple V i (j,j =,...,,is drwn from n independent nd identiclly distributed (i.i.d. Gussin. In this work, we ssume tht (V (i,v (i,s(i, i =,...,, re correlted vi the following covrince mtrix 3 V V V V S VV S = 4 5 ( V V V V S V S V S where V nd V re the vrinces of V nd V, respectively,, nd re the correltion coefficients between V nd V, S nd V nd S nd V, respectively. The covrince mtrix in ( being positive definite restricts the possible vlues of, nd. As shown in Fig., the source pir vector (, S is trnsformed into n dimensionl chnnel input X R vi (, mpping from (R R R! R. The received vector t user i is Yi = X S Wi, where ddition is component-wise, X = (,,S, S is the i.i.d. Gussin interference (S (0, S known to the trnsmitter, nd ech smple in the dditive noise Wi is drwn from n i.i.d. Gussin distribution (W i (0, Wi
2 α(. X S W Y Y γ (. γ (. ˆ ˆ W Fig.. System model structure. independently from both sources nd interference. The system opertes under n verge power constrint P given by E[ (,,S ]/ pple P ( where E[( ] denotes the expecttion opertor. The reconstructed signl is given by ˆV i = i (Yi, where the decoder functions i(. re mppings from R! R. The system s rte is given by = chnnel use/source symbol nd the reconstruction qulity t ech user is the men squre error (MSE D i = E[ Vi ˆV i ]/ for i =,. We ssume tht W > W, nd hence user is the wek user nd user is the strong one. For given power constrint P nd rte the distortion region is defined s the closure of ll distortion pirs ( D, D for which (P, D, D is chievble, where powerdistortion triple is chievble if for ny > 0, there exist sufficiently lrge integers nd with / =, encoding nd decoding functions (,, stisfying (, such tht D i < D i, i =,. In this work, we re interested in nlyzing the distortion region of this system under mtched ( = nd expnsion bndwidth modes ( >. ote tht for >, Vi nd the first interference smples S in S =[S,S ] re correlted vi the covrince mtrix in (, while Vi nd S re independent. III. OUTER BOUD AD REFERECE SCHEMES A. Outer Bound In [4] nd [8], n outer bound on the distortion region for sending correlted sources over the brodcst chnnel without interference ws obtined for =nd 6=, respectively, by ssuming knowledge of the source t the strong user. In [0], [5], severl bounds re derived for point-to-point communictions under correlted interference. In this section, we derive n outer bound on the distortion region for the interference brodcst chnnel for. Since S(i nd V (i re correlted for i =,...,, we hve S(i =S I (is D (i, with S D (i = S V V (i nd S I (0, ( S.To derive n outer bound, we ssume knowledge of t the strong user (this is resonble ssumption for smll correltion coefficients; this bound, however, might not be tight for high correltion vlues nd ( S,S t both users, where S = SI SD. The knowledge of the liner combintion S is motivted by [5]. The outer bound on the distortion region cn be expressed s follows Vr(V S( P W D (MSE(Y ; S(P W,D Vr(V V,S P W where [0, ], Vr(V V,S= V Vr(V S = V ( (3, nd MSE(Y ; S is the distortion from estimting Y bsed on S using liner minimum MSE estimtor (LMMSE. This distortion is function of,, E[XS I ] nd E[XS D ]. By Cuchy-Schwrtz, we hve E[XS I ] pple p p E[X ]E[SI ] nd E[XS D] pple E[X ]E[SD ]. For given nd, the mximum vlue of MSE(Y ; S hs to be used in (3. ote tht we need to mximize D over the prmeters nd. Proof: For : system with, we hve log V pple I( ; D ˆV pple I( ; Y,, S,S = I( ;, S,S I( ; Y, S,S where the first nd the second terms re nd h( h(,s = log V Vr(V V,S, h(y,s h(y,,s = log e( P W log e( W = log ( P W, (4 respectively. ote tht we used log e( W pple h(y,s pple log e(p W ; hence there exists n [0 ] such tht h(y,s = log e( P W. To get bound on estimting, we cn write the following log V pple I( ; D ˆV pple I( ; Y, S,S = I( ; S,S I( ; Y S,S (5 with the first nd the second terms stisfy nd h( h( S,S = log V Vr(V S, h(y S,S h(y,s pple h(y S h(y S h(y pple log e(mse(y ; S log e(p W,,S log e( P W respectively, where h(y,s log e( P W due to the entropy power inequlity nd since Y = Y Z with Z (0, W W. Moreover, we used h(y S pple h(y lmse( S, where lmse ( S is the LMMSE estimtor of Y bsed on S. ote tht most inequlities follow from rte-distortion theory, the dt processing inequlity, the nonnegtivity of mutul informtion, conditioning reduces differentil entropy nd the fct tht the Gussin distribution mximizes differentil entropy. Remrk: The bound in (3 reduces to the one in [4] when there is no interference. This cn be seen by setting = = 0 nd = =0. eglecting the strong user (i.e., reducing the brodcst problem to point-to-point communictions, the bound on V in (3 reduces to the bounds derived in [0], [5] for point-to-point communictions over the interference chnnel under equl bndwidth. This cn be seen by setting =0nd =for different vlues of nd.
3 B. Liner Scheme In this section, we ssume tht the encoder trnsforms the dimensionl sources (, into n dimensionl chnnel input X using liner trnsformtion ccording to X = A B CS (6 where A, B re mtrices nd C is mtrix. At the receiver side, we use liner decoder tht minimizes the MSE distortion. The estimted source is ˆV i = F i G i Yi, where F i is the correltion mtrix between Vi nd Yi, nd G i is the covrince mtrix of Yi, for i =,. C. Tndem Digitl Scheme This strtegy is bsed on successive coding where the sources re encoded jointly t both the common nd the refinement lyers. Using [9], the chievble source coding rte (R,R for ny distortion (D,D is given by R ( = log D ( ( pple R ( = log (7 D h where, min (,, p i ( D /, [x] = mx (x, 0, nd = D. For Gussin interference brodcst chnnel, the rte (R,R cn be chieved if nd only if there exists 0 pple pple such tht R pple log nd R pple log ( P W P ( P W. By plugging these rtes into (7, we get the chievble distortions for D nd D. The bove rtes cn be chieved vi Cost coding. ote tht this is the best tndem scheme for uncorrelted interference in terms of chievble distortion region. IV. HDA CODIG SCHEMES A. HDA Scheme for Mtched Bndwidth As shown from the encoder structure in Fig., this scheme hs four lyers tht re merged to output X. The first lyer, which uses n verge power of P u, outputs Xu = p u( V V 3 S, liner combintion of the sources nd the interference, where,, 3 [, ], nd u = P u /( V V 3 S V V 3 V S 3 V S is gin fctor relted to power constrint P u. The second lyer, which outputs X with power P, uses HDA Cost coding on the liner combintion X 0 = p Xu, where = P /P u is gin fctor relted to power constrint P. This lyer is ment for both users nd trets Xu nd S s known interference. The uxiliry rndom vrible of the HDA Cost encoder is U = X (S Xu pple X 0, where X (0,P, P = P P u, pple P W = (P P u W D, nd D is defined in (8 below. The HDA Cost encoder forms codebook U with codeword length nd R codewords (R is defined lter. Every codeword is generted following the rndom vrible U. The codebook is reveled to both the encoder nd decoder. The encoder serches for U U such tht (X 0, (S Xu,U re jointly typicl. The third lyer encodes the source using coding t rte R = log ( P P P u P P. The index W m is then encoded using Cost coding tht trets S, Xu, nd X s interference nd uses n verge power of P ; the output of this lyer is denoted by X. Similrly, in the fourth lyer the source is first encoded using t rte R 0 = P log (, where P = P P u P P, W followed by Cost coder tht trets S s well s the outputs of the first three lyers s interference nd outputs X. S β β β 3 u Encoder Encoder X HDA Cost Encoder Cost Encoder Cost Encoder X u X X X Fig.. HDA Scheme encoder structure for rte = =. X At the wek user, from the noisy received signl Y, LMMSE estimtor is used to get n estimte of X 0 denoted 0 by ˆX. The distortion D = E[(X 0 ˆX 0 ], used in the prmeter pple, is given by E[X D = P Y 0 ] p ( Pu E[X E[Y ] = P S] 0 P S W E[SX u ]. (8 With our choice of prmeters nd pple, the HDA Cost decoder cn estimte U with low probbility of error (for sufficiently lrge by serching for U such tht (U,Y 0, ˆX is jointly typicl. ote tht the HDA codeword U cn be decoded t both users. This cn be proved by noting tht the HDA Cost rte R stisfies I(U ;(S X u,x 0 pple R pple I(U ; Y i, ˆX, 0 for i =,. The HDA Cost decoder then forms LMMSE estimte of, denoted by ˆV, bsed on Y nd the decoded codeword U. The resulting distortion is given by D = V T (9 where is the covrince mtrix of [U Y ], nd is the correltion vector between V nd [U Y ]. ote tht fter some mnipultions, the distortion in (9 cn be written s D = V D x 0 v x (0 0 P where vx 0 is the correltion coefficient between X0 nd V, D nd D x 0 = P /(P P u P. A better estimte of V W is obtined from the third lyer by using the decoded Wyner- Ziv codeword T nd the previous estimte r ˆV. ote tht T = wz H, where wz = P W P P W, nd H (0,D/( P P. The overll distortion in W reproducing is then given by D D =. ( P P W
4 The bove distortion is found by equting log D to R. The strong user, tht is ble to decode ll codewords used by the wek user, estimtes the source by first finding liner MMSE estimte of, denoted by ˆV, bsed on the HDA Cost codeword U, the codeword T, nd Y. The distortion in reproducing is D = V T ( where is the covrince mtrix of [U Y T ], nd is the correltion vector between V nd [U Y T ].A better estimte ˆV is then found using the decoded Wyner- Ziv codeword T nd ˆV. The resulting overll distortion in estimting is given by equting log D D to R 0 s follows D = D. (3 P W The inner bound for HDA Scheme is given by ( nd (3. B. HDA Scheme for Bndwidth Expnsion D This scheme comprises two lyers tht re conctented to output the trnsmitted signl s shown in Fig. 3. The first lyer, which outputs X, consists of the HDA Scheme encoder for =(composed of four sublyers s described in the previous section. The second lyer is composed of two sublyers. The first sublyer encodes using t rte R = log ( P 0 P P 0 followed by Cost coder W with n verge power P. 0 ote tht the Cost coder trets S s interference nd outputs X tht is ment for both users. The second sublyer encodes using Wyner- Ziv t rte R 0 = log ( P 0 followed by Cost coder W with n verge power P 0 = P P 0 tht trets X nd S s interference nd outputs X. The output of the second lyer is then given by X = X X. ote tht X is the conctention of X nd X Encoder3 Encoder4 HDA Scheme Encoder Cost Encoder3 Cost Encoder4 X X Fig. 3. HDA Scheme encoder structure for rte = >.. X X X At the wek user, LMMSE decoder bsed on the decoded HDA Cost codeword U nd the first received smples Y is used to get n estimte of denoted by. The distortion in estimting cn be expressed in similr wy s given in (0. A better estimte, V 0, is then chieved using the decoder (of the HDA scheme. The resulting distortion is then D 0 D =. Using the lst P P W smples of the received signl Y, better refinement of cn be obtined using the decoder 3. The overll distortion in reconstructing is then D = D 0 P 0 P 0 W. (4 At the strong user, using the received signl Y, the HDA Cost codeword U, the decoded codeword T nd T3 of the encoder (of HDA scheme nd 3, we cn obtin n estimte of using LMMSE estimtor. The resulting distortion is D = V T, where is the covrince mtrix of [U Y T T 3 ], nd is the correltion vector between V nd [U Y T T 3 ]. ote tht T 3 = wz3 V H 3, where wz3 = s P 0 W P 0P 0 W nd H 3 (0,D. A refinement of this estimte cn be obtined using the decoder (of HDA scheme nd 4. The resulting distortion in estimting is then D = P W D P 0 W. (5 The inner bound for HDA Scheme is given by (4 nd (5. Remrk: The presented lyered schemes use purely nlog lyer tht consists of liner combintion of the sources nd the interference. The use of this lyer is to benefit from the interference when the source-interference correltion is high. V. UMERICAL RESULTS In this section, we ssume tht the source pirs, with vrince V = V =, re brodcsted to two users with interference vrince S =0dB, nd observtion noise vrince W =0dB nd W = 5 db, respectively. The system s verge power is set to P =. To evlute the performnce, we plot the inner nd outer bounds derived in the previous sections for =nd. Fig. 4 focuses on the 0 log 0 (D 0 ρ =0.8, ρ = ρ =0 λ = ρ =0., ρ = ρ =0 λ = HDA Scheme Liner scheme Tndem scheme log 0 (D Fig. 4. Distortion regions for HDA Scheme for =. uncorrelted source-interference cse ( = = 0 under mtched bndwidth ( =. For low correltion between the source pirs ( =0., the proposed scheme gives some improvement over the tndem scheme nd considerbly performs very close to the outer bound (derived in Section III; for high source correltion levels ( =0.8, however, the HDA scheme outperforms the tndem system but hs lrger gp with respect to the outer bound. ote tht for uncorrelted sourceinterference, the liner scheme gives poor performnce. Fig. 5 shows the performnce of the HDA scheme for =0.5,
5 = =0.nd =. We cn notice tht the purely nlog scheme outperforms slightly the tndem scheme without being ble to pproch the HDA scheme. This cn be explined from the fct tht we operte t high noise levels, nd since the liner scheme cn benefit from the source-interference correltion. For the tndem scheme, which uses Cost coding, the trnsmitted signl is designed to be orthogonl to the interference; hence it cnnot exploit the source-interference correltion nd no performnce improvement cn be detected. ote tht for moderte to low noise levels, the liner scheme does not outperform the tndem scheme for low sourceinterference correltions. Moreover, from other simultions, we noticed tht for =, =0.8, nd = =0.5, the liner scheme gives the best performnce (our HDA scheme reduces to liner scheme in this cse. This cn be explined by noting tht in [6], the uthors proved tht under some conditions on the noise power nd source correltion (which re in ccordnce with the conditions for the lst simultion, the liner scheme is optiml for brodcsting bivrite Gussins under no interference. As result, under similr conditions, the liner scheme is expected to give good performnce for our problem when the source-interference correltion gets high s in [0] for point-to-point system. Fig. 6 0 log 0 (D Liner scheme Tndem digitl scheme HDA Scheme log 0 (D Fig. 5. Distortion regions for HDA Scheme for =. shows tht the HDA scheme outperforms the tndem scheme under bndwidth expnsion ( =. ote tht for =0. nd = =0, it is hrd to notice (from Fig. 6 the gin of the HDA scheme over the tndem system on the plotted scle; the outer bound for this cse is not shown, since both schemes perform very closely to it. Moreover, the tndem scheme cnnot benefit from the source-interference correltion nd its performnce depends solely on in Fig. 6. VI. SUMMARY AD COCLUSIOS In this pper, we consider the trnsmission of pir of correlted Gussin sources over the two-user Gussin brodcst chnnel in the presence of interference tht is correlted to the source. We propose lyered HDA schemes under mtched nd expnsion bndwidth scenrios bsed on nd HDA Cost coding nd nlyze their inner bounds. An outer bound 0 log 0 (D 9 0 ρ =0.8 ρ = ρ =0.5 λ = ρ =0.8 ρ = ρ =0 λ = HDA Scheme ρ =0. ρ = ρ =0 λ = Tndem scheme 0 0 log 0 (D Fig. 6. Distortion regions for HDA Scheme for =. on the system s distortion region is lso derived. umericl results indicte tht the HDA schemes outperform the best tndem scheme nd perform close to the derived outer bound under some system settings. REFERECES [] M. Gstpr, B. Rimoldi, nd M. Vetterli, To code, or not to code: lossy source-chnnel communiction revisited, IEEE Trns. Inform. Theory, vol. 49, no. 5, pp , My 003. [] U. Mittl nd. Phmdo, Hybrid digitl nlog (HDA joint source chnnel codes for brodcsting nd robust communictions, IEEE Trns. Inform. Theory, vol. 48, pp. 08 0, My 00. [3] M. Skoglund,. Phmdo, nd F. Aljji, Hybrid digitl-nlog sourcechnnel coding for bndwidth compression/expnsion, IEEE Trns. Inform. Theory, vol. 5, no. 8, pp , Aug 006. [4] Z. Reznic, M. Feder, nd R. Zmir, Distortion bounds for brodcsting with bndwidth expnsion, IEEE Trns. Inform. Theory, vol. 5, no. 8, pp , Aug 006. [5] V. M. Prbhkrn, R. Puri, nd. Rmchndrn, Hybrid nlogdigitl strtegies for source-chnnel brodcst, in Proc. 43rd Allerton Conf. Communiction, Control nd Computing, Allerton, IL, Sep 005. [6] A. Lpidoth nd S. Tinguely, Brodcsting correlted Gussins, IEEE Trns. Inform. Theory, vol. 56, no. 7, pp , July 00. [7] C. Tin, S.. Diggvi, nd S. Shmi, The chievble distortion region of sending bivrite Gussin source on the Gussin brodcst chnnel, IEEE Trns. Inform. Theory, vol. 57, no. 0, pp , Oct 0. [8] H. Behroozi, F. Aljji, nd T. Linder, On the performnce of hybrid digitl-nlog coding for brodcsting correlted Gussin sources, IEEE Trns. Communictions, vol. 59, no., pp , Dec 0. [9] Y. Go nd E. Tuncel, Seprte source-chnnel coding for brodcsting correlted Gussins, in Proc. IEEE Int. Symp. on Inform. Theory, Sint Petersburg, Russi, July 0. [0] Y.-C. Hung nd. R. rynn, Joint source-chnnel coding with correlted interference, IEEE Trns. Commun., vol. 60, no. 5, pp , My 0. [] A. D. Wyner nd J. Ziv, The rte-distortion function for source coding with side informtion t the decoder, IEEE Trns. Inform. Theory, vol., pp. 0, Jn 976. [] M. Cost, Writing on dirty pper, IEEE Trns. Inform. Theory, vol. 9, no. 3, pp , My 983. [3] M. P. Wilson,. R. rynn, nd G. Cire, Joint source-chnnel coding with side informtion using hybrid digitl nlog codes, IEEE Trns. Inform. Theory, vol. 56, no. 0, pp , Oct 00. [4] R. Soundrrjn nd S. Vishwnth, Hybrid coding for Gussin brodcst chnnels with Gussin sources, in Proc. IEEE Int. Symp. on Inform. Theory, Seoul, ore, July 009. [5] Y.-. Chi, R. Soundrrjn, nd T. Weissmn, Estimtion with helper who knows the interference, in Proc. IEEE Int. Symp. on Inform. Theory, Cmbridge, MA, July 0.
Tutorial 4. b a. h(f) = a b a ln 1. b a dx = ln(b a) nats = log(b a) bits. = ln λ + 1 nats. = log e λ bits. = ln 1 2 ln λ + 1. nats. = ln 2e. bits.
Tutoril 4 Exercises on Differentil Entropy. Evlute the differentil entropy h(x) f ln f for the following: () The uniform distribution, f(x) b. (b) The exponentil density, f(x) λe λx, x 0. (c) The Lplce
More informationGeneralized Fano and non-fano networks
Generlized Fno nd non-fno networks Nildri Ds nd Brijesh Kumr Ri Deprtment of Electronics nd Electricl Engineering Indin Institute of Technology Guwhti, Guwhti, Assm, Indi Emil: {d.nildri, bkri}@iitg.ernet.in
More informationBinary Rate Distortion With Side Information: The Asymmetric Correlation Channel Case
Binry Rte Dtortion With Side Informtion: The Asymmetric Correltion Chnnel Cse Andrei Sechele, Smuel Cheng, Adrin Muntenu, nd Nikos Deliginn Deprtment of Electronics nd Informtics, Vrije Universiteit Brussel,
More informationX Z Y Table 1: Possibles values for Y = XZ. 1, p
ECE 534: Elements of Informtion Theory, Fll 00 Homework 7 Solutions ll by Kenneth Plcio Bus October 4, 00. Problem 7.3. Binry multiplier chnnel () Consider the chnnel Y = XZ, where X nd Z re independent
More informationTesting categorized bivariate normality with two-stage. polychoric correlation estimates
Testing ctegorized bivrite normlity with two-stge polychoric correltion estimtes Albert Mydeu-Olivres Dept. of Psychology University of Brcelon Address correspondence to: Albert Mydeu-Olivres. Fculty of
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationProblem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume
Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom r vrible. Define new rndom vrible Y = g. Find the pdf of Y. Method: Step : Step : Step 3: Plot Y = g( ). Find F ( y) by mpping
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationReinforcement learning II
CS 1675 Introduction to Mchine Lerning Lecture 26 Reinforcement lerning II Milos Huskrecht milos@cs.pitt.edu 5329 Sennott Squre Reinforcement lerning Bsics: Input x Lerner Output Reinforcement r Critic
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationSufficient condition on noise correlations for scalable quantum computing
Sufficient condition on noise correltions for sclble quntum computing John Presill, 2 Februry 202 Is quntum computing sclble? The ccurcy threshold theorem for quntum computtion estblishes tht sclbility
More informationHybrid Digital-Analog Joint Source-Channel Coding for Broadcasting Correlated Gaussian Sources
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,
More informationMethod: Step 1: Step 2: Find f. Step 3: = Y dy. Solution: 0, ( ) 0, y. Assume
Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom vrible. Define new rndom vrible Y g( ) ). Find the pdf of Y. Method: Step : Step : Step 3: Plot Y g( ). Find F ( ) b mpping
More information15. Quantisation Noise and Nonuniform Quantisation
5. Quntistion Noise nd Nonuniform Quntistion In PCM, n nlogue signl is smpled, quntised, nd coded into sequence of digits. Once we hve quntised the smpled signls, the exct vlues of the smpled signls cn
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationdecoder t the coecient level, s opposed to the bit level, so it is fundmentlly dierent from schemes tht use informtion bout the trnsmitted dt to produ
To pper in Proceedings of IEEE Dt Compression Conference 998 Lucent Technologies -- PROPRIETARY Optiml Multiple Description Trnsform Coding of Gussin Vectors Vivek K Goyl Dept. of Elec. Eng. & Comp. Sci.
More informationTests for the Ratio of Two Poisson Rates
Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson
More informationLecture Note 9: Orthogonal Reduction
MATH : Computtionl Methods of Liner Algebr 1 The Row Echelon Form Lecture Note 9: Orthogonl Reduction Our trget is to solve the norml eution: Xinyi Zeng Deprtment of Mthemticl Sciences, UTEP A t Ax = A
More informationSection 11.5 Estimation of difference of two proportions
ection.5 Estimtion of difference of two proportions As seen in estimtion of difference of two mens for nonnorml popultion bsed on lrge smple sizes, one cn use CLT in the pproximtion of the distribution
More informationNon-Linear & Logistic Regression
Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationData Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading
Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationAcceptance Sampling by Attributes
Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationReversals of Signal-Posterior Monotonicity for Any Bounded Prior
Reversls of Signl-Posterior Monotonicity for Any Bounded Prior Christopher P. Chmbers Pul J. Hely Abstrct Pul Milgrom (The Bell Journl of Economics, 12(2): 380 391) showed tht if the strict monotone likelihood
More informationStudent Activity 3: Single Factor ANOVA
MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether
More informationMonte Carlo method in solving numerical integration and differential equation
Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More informationREGULARITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2
EGULAITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2 OVIDIU SAVIN AND ENICO VALDINOCI Abstrct. We show tht the only nonlocl s-miniml cones in 2 re the trivil ones for ll s 0, 1). As consequence we obtin tht
More informationMIXED MODELS (Sections ) I) In the unrestricted model, interactions are treated as in the random effects model:
1 2 MIXED MODELS (Sections 17.7 17.8) Exmple: Suppose tht in the fiber breking strength exmple, the four mchines used were the only ones of interest, but the interest ws over wide rnge of opertors, nd
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationProbabilistic Investigation of Sensitivities of Advanced Test- Analysis Model Correlation Methods
Probbilistic Investigtion of Sensitivities of Advnced Test- Anlysis Model Correltion Methods Liz Bergmn, Mtthew S. Allen, nd Dniel C. Kmmer Dept. of Engineering Physics University of Wisconsin-Mdison Rndll
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationChapter 2 Fundamental Concepts
Chpter 2 Fundmentl Concepts This chpter describes the fundmentl concepts in the theory of time series models In prticulr we introduce the concepts of stochstic process, men nd covrince function, sttionry
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More information= a. P( X µ X a) σ2 X a 2. = σ2 X
Co902 problem sheet, solutions. () (Mrkov inequlity) LetX be continuous, non-negtive rndom vrible (RV) ndpositive constnt. Show tht: P(X ) E[X] Solution: Let p(x) represent the pdf of RVX. Then: E[X] =
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationMath Solutions to homework 1
Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationReinforcement Learning
Reinforcement Lerning Tom Mitchell, Mchine Lerning, chpter 13 Outline Introduction Comprison with inductive lerning Mrkov Decision Processes: the model Optiml policy: The tsk Q Lerning: Q function Algorithm
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationExtended nonlocal games from quantum-classical games
Extended nonlocl gmes from quntum-clssicl gmes Theory Seminr incent Russo niversity of Wterloo October 17, 2016 Outline Extended nonlocl gmes nd quntum-clssicl gmes Entngled vlues nd the dimension of entnglement
More informationA Matrix Algebra Primer
A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
More informationA NOTE ON ESTIMATION OF THE GLOBAL INTENSITY OF A CYCLIC POISSON PROCESS IN THE PRESENCE OF LINEAR TREND
A NOTE ON ESTIMATION OF THE GLOBAL INTENSITY OF A CYCLIC POISSON PROCESS IN THE PRESENCE OF LINEAR TREND I WAYAN MANGKU Deprtment of Mthemtics, Fculty of Mthemtics nd Nturl Sciences, Bogor Agriculturl
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More informationTESTING CATEGORIZED BIVARIATE NORMALITY WITH TWO-STAGE POLYCHORIC CORRELATION ESTIMATES
IE Working Pper WP 0 / 03 08/05/003 TESTING CATEGORIZED BIVARIATE NORMALITY WITH TWO-STAGE POLYCHORIC CORRELATION ESTIMATES Alberto Mydeu Olivres Instituto de Empres Mrketing Dept. C/ Mri de Molin, -5
More informationNew Integral Inequalities for n-time Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationTESTING CATEGORIZED BIVARIATE NORMALITY WITH TWO-STAGE POLYCHORIC CORRELATION ESTIMATES. IE Working Paper MK8-106-I 08 / 05 / 2003
TESTING CATEGORIZED BIVARIATE NORMALITY WITH TWO-STAGE POLYCHORIC CORRELATION ESTIMATES IE Working Pper MK8-06-I 08 / 05 / 2003 Alberto Mydeu Olivres Instituto de Empres Mrketing Dept. C / Mrí de Molin
More informationarxiv: v1 [cs.it] 19 Sep 2017
1 Secure Bemforming in Full-Duplex SWIPT Systems Ynjie Dong, Ahmed El Shfie, Md. Jhngir Hossin, Julin Cheng, Nofl Al-Dhhir, nd Victor C. M. Leung rxiv:1709.06623v1 [cs.it] 19 Sep 2017 Abstrct Physicl lyer
More informationVariational Techniques for Sturm-Liouville Eigenvalue Problems
Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationEstimation of Binomial Distribution in the Light of Future Data
British Journl of Mthemtics & Computer Science 102: 1-7, 2015, Article no.bjmcs.19191 ISSN: 2231-0851 SCIENCEDOMAIN interntionl www.sciencedomin.org Estimtion of Binomil Distribution in the Light of Future
More information1 Online Learning and Regret Minimization
2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationPredict Global Earth Temperature using Linier Regression
Predict Globl Erth Temperture using Linier Regression Edwin Swndi Sijbt (23516012) Progrm Studi Mgister Informtik Sekolh Teknik Elektro dn Informtik ITB Jl. Gnesh 10 Bndung 40132, Indonesi 23516012@std.stei.itb.c.id
More informationJoint distribution. Joint distribution. Marginal distributions. Joint distribution
Joint distribution To specify the joint distribution of n rndom vribles X 1,...,X n tht tke vlues in the smple spces E 1,...,E n we need probbility mesure, P, on E 1... E n = {(x 1,...,x n ) x i E i, i
More informationLYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN
Electronic Journl of Differentil Equtions, Vol. 203 (203), No. 28, pp. 0. ISSN: 072-669. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LYAPUNOV-TYPE INEQUALITIES FOR
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationDriving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d
Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,
More informationChapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses
Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of
More informationProbability Distributions for Gradient Directions in Uncertain 3D Scalar Fields
Technicl Report 7.8. Technische Universität München Probbility Distributions for Grdient Directions in Uncertin 3D Sclr Fields Tobis Pfffelmoser, Mihel Mihi, nd Rüdiger Westermnn Computer Grphics nd Visuliztion
More informationA HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES. 1. Introduction
Ttr Mt. Mth. Publ. 44 (29), 159 168 DOI: 1.2478/v1127-9-56-z t m Mthemticl Publictions A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES Miloslv Duchoň Peter Mličký ABSTRACT. We present Helly
More informationThe steps of the hypothesis test
ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
More informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
More informationContents. Outline. Structured Rank Matrices Lecture 2: The theorem Proofs Examples related to structured ranks References. Structure Transport
Contents Structured Rnk Mtrices Lecture 2: Mrc Vn Brel nd Rf Vndebril Dept. of Computer Science, K.U.Leuven, Belgium Chemnitz, Germny, 26-30 September 2011 1 Exmples relted to structured rnks 2 2 / 26
More informationMath 135, Spring 2012: HW 7
Mth 3, Spring : HW 7 Problem (p. 34 #). SOLUTION. Let N the number of risins per cookie. If N is Poisson rndom vrible with prmeter λ, then nd for this to be t lest.99, we need P (N ) P (N ) ep( λ) λ ln(.)
More informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More informationZero-Sum Magic Graphs and Their Null Sets
Zero-Sum Mgic Grphs nd Their Null Sets Ebrhim Slehi Deprtment of Mthemticl Sciences University of Nevd Ls Vegs Ls Vegs, NV 89154-4020. ebrhim.slehi@unlv.edu Abstrct For ny h N, grph G = (V, E) is sid to
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationApplied Partial Differential Equations with Fourier Series and Boundary Value Problems 5th Edition Richard Haberman
Applied Prtil Differentil Equtions with Fourier Series nd Boundry Vlue Problems 5th Edition Richrd Hbermn Person Eduction Limited Edinburgh Gte Hrlow Essex CM20 2JE Englnd nd Associted Compnies throughout
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationA New Statistic Feature of the Short-Time Amplitude Spectrum Values for Human s Unvoiced Pronunciation
Xiodong Zhung A ew Sttistic Feture of the Short-Time Amplitude Spectrum Vlues for Humn s Unvoiced Pronuncition IAODOG ZHUAG 1 1. Qingdo University, Electronics & Informtion College, Qingdo, 6671 CHIA Abstrct:
More informationLocal orthogonality: a multipartite principle for (quantum) correlations
Locl orthogonlity: multiprtite principle for (quntum) correltions Antonio Acín ICREA Professor t ICFO-Institut de Ciencies Fotoniques, Brcelon Cusl Structure in Quntum Theory, Bensque, Spin, June 2013
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationInterference suppression in the presence of quantization errors
Forty-Seventh Annul Allerton Conference Allerton House, UIUC, Illinois, USA September - October, 009 Interference suppression in the presence of quntiztion errors Omr Bkr Mrk Johnson Rghurmn Mudumbi Upmnyu
More informationA recursive construction of efficiently decodable list-disjunct matrices
CSE 709: Compressed Sensing nd Group Testing. Prt I Lecturers: Hung Q. Ngo nd Atri Rudr SUNY t Bufflo, Fll 2011 Lst updte: October 13, 2011 A recursive construction of efficiently decodble list-disjunct
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationS. S. Dragomir. 2, we have the inequality. b a
Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 17
CS 70 Discrete Mthemtics nd Proility Theory Summer 2014 Jmes Cook Note 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion, y tking
More informationExtremal Problems of Information Combining
Extreml Problems of Informtion Combining Yibo Jing, Alexei Ashikhmin, Rlf Koetter, nd Andrew C Singer Coordinted Science Lbortory University of Illinois t Urbn-Chmpign, Urbn, IL 680, USA Emil: {yjing,
More informationAdministrivia CSE 190: Reinforcement Learning: An Introduction
Administrivi CSE 190: Reinforcement Lerning: An Introduction Any emil sent to me bout the course should hve CSE 190 in the subject line! Chpter 4: Dynmic Progrmming Acknowledgment: A good number of these
More informationMath Lecture 23
Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationSharp lower bounds on the least singular value of a random matrix without the fourth moment condition *
Electron. Commun. Probb. 2 (215), no. 44, 1 9. DOI: 1.1214/ECP.v2-489 ISSN: 183-589X ELECTRONIC COMMUNICATIONS in PROBABILITY Shrp lower bounds on the lest singulr vlue of rndom mtrix without the fourth
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More information