Digital Signal Processing, Fall 2006

Size: px
Start display at page:

Download "Digital Signal Processing, Fall 2006"

Transcription

1 Digital Sigal Procssig, Fall 6 Lctur 9: Th Discrt Fourir Trasfor Zhg-Hua Ta Dpartt of Elctroic Systs Aalborg Uivrsity, Dar zt@o.aau.d Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Cours at a glac MM Discrt-ti sigals ad systs Syst MM Fourir-doai rprstatio Saplig ad rcostructio Syst aalysis MM5 Syst structur MM6 MM4 Filtr dsig z-trasfor DFTFFT MM7, MM8 MM MM9, MM Digital Sigal Procssig, I, Zhg-Hua Ta, 6

2 Th discrt-ti Fourir trasfor DTFT Th DTFT is usful for th thortical aalysis of sigals ad systs. But, accordig to its dfiitio coputatio of DTFT by coputr has svral probls: Th suatio ovr is ifiit Th idpdt variabl w is cotiuous Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th discrt Fourir trasfor DFT I ay cass, oly fiit duratio is of cocr Th sigal itslf is fiit duratio Oly a sgt is of itrst at a ti Sigal is priodic ad thus oly fiit uiqu valus For fiit duratio squcs, a altrativ Fourir rprstatio is DFT Th suatio ovr is fiit DFT itslf is a squc, rathr tha a fuctio of a cotiuous variabl Thrfor, DFT is coputabl ad iportat for th ipltatio of DSP systs DFT corrspods to sapls of th Fourir trasfor 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6

3 Part I: Th discrt Fourir sris Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT 5 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th discrt Fourir sris A priodic squc with priod + r Priodic squc ca b rprstd by a Fourir sris, i.. a su of copl potial squcs with frqucis big itgr ultipls of th fudatal frqucy associatd with th Th frqucy of th priodic squc. Oly uiqu haroically rlatd copl potials sic 6 so + Digital Sigal Procssig, I, Zhg-Hua Ta, 6

4 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 7 Th Fourir sris cofficits Th cofficits Th squc is priodic with priod For covic, dfi + + W Aalysisquatio Sythsis quatio W W Vry siilar quatios duality Digital Sigal Procssig, I, Zhg-Hua Ta, 6 8 DFS of a priodic ipuls trai Priodic ipuls trai Th discrt Fourir sris cofficits By usig sythsis quatio, a altrativ rprstatio of is r r δ W W W δ

5 5 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 9 Part II: Th Fourir trasfor of priodic sigals Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th Fourir trasfor of priodic sigals Fourir trasfor of copl potials Fourir trasfor of has th rquird priodicity with priod δ + < < r r a a, δ

6 Fourir trasfor of a priodic ipuls trai Priodic ipuls trai Th discrt Fourir sris cofficits P δ W Fourir trasfor P δ Fiit duratio sigal outsid of, Costruct * p * δ r r Its Fourir trasfor p δ r r r P δ Digital Sigal Procssig, I, Zhg-Hua Ta, 6 r Th Fourir trasfor of priodic sigals Copar P δ δ Coclud that First rprst it as Fourir sris ad th calculat Fourir trasfor i.. th DFS cofficits of ar sapls of th Fourir trasfor of th o priod of,, othrwis Digital Sigal Procssig, I, Zhg-Hua Ta, 6 6

7 Part III: Saplig th Fourir trasfor Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Saplig th Fourir trasfor A apriodic squc ad its Fourir trasfor Saplig th Fourir trasfor grats a priodic squc i with priod sic th Fourir trasfor is priodic i with priod d 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 7

8 Saplig th Fourir trasfor ow w wat to s if th saplig squc is th squc of DFS cofficits of a squc this ca b do by usig th sythsis quatio W 5 r r W W Digital Sigal Procssig, I, Zhg-Hua Ta, 6 p A priodic squc rsultig fro apriodic covolutio Eapls Cas Fig 8.8 I this cas, th Fourir sris cofficits for a priodic squc ar sapls of th Fourir trasfor of o priod 6 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 8

9 Eapls Cas Fig 8.9 I this cas, still th Fourir sris cofficits for ar sapls of th Fourir trasfor of. But, o priod of is o logr idtical to This is ust saplig i th frqucy doai as copard i th ti doai discussd bfor. 7 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Saplig i th frqucy doai Th rlatioship btw ad o priod of i th udrsapld cas is cosidrd a for of ti doai aliasig. Ti doai aliasig ca b avoidd oly if has fiit lgth, ust as frqucy doai aliasig ca b avoidd oly for sigals big badliitd. If has fiit lgth ad w ta a sufficit ubr of qually spacd sapls of its Fourir trasfor spcifically, a ubr gratr tha or qual to th lgth of, th th Fourir trasfor is rcovrabl fro ths sapls, quivaltly is rcovrabl fro. 8 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 9

10 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 9 Saplig i th frqucy doai Rcovrig i.. rcovrig dos ot rquir to ow its Fourir trasfor at all frqucis Applicatio: rprst fiit lgth squc by usig Fourir sris cofficits DFT othrwis,,, DFS Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Saplig th Fourir trasfor Fourir trasfor Discrt-ti Fourir trasfor Discrt Fourir trasfor d Ω Ω Ω Ω Ω d t dt t t t

11 Part IV: Th DFT Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th discrt Fourir trasfor Cosidr a fiit lgth squc of lgth sapls if sallr tha, appdig zros Costruct a priodic squc r r Assuig o ovrlap btw r odulo Rcovr th fiit lgth squc,, othrwis To aitai a duality btw th ti ad frqucy doais, choos o priod of as th DFT,, othrwis Digital Sigal Procssig, I, Zhg-Hua Ta, 6

12 Th DFT Priodic squc ad DFS cofficits W W Sic suatios ar calculatd btw ad - W, Grally, othrwis W, W, othrwis W Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th DFT A fiit or priodic squc has oly uiqu valus, for << Spctru is copltly dfid by distict frqucy sapls DFT: uifor saplig of DTFT spctru 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6

13 Th DFT of a rctagular puls Eapl 8.7 pp.56 5 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Th DFT of a rctagular puls 6 Digital Sigal Procssig, I, Zhg-Hua Ta, 6

14 Part V: Proprtis of th DFT Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT 7 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Proprtis of th DFT liarity Liarity DFT a + b a + b Th lgths of squcs ad thir DFTs ar all qual to th aiu of th lgths of ad 8 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 4

15 Circular shift of a squc Giv DFT DFT Th,, othrwis 9 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Circular shift of a squc a apl Digital Sigal Procssig, I, Zhg-Hua Ta, 6 5

16 6 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Duality, DFT DFT Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Circular covolutio I liar covolutio, o squc is ultiplid by a ti rvrsd ad liarly shiftd vrsio of th othr. For covolutio hr, th scod squc is circularly ti rvrsd ad circularly shiftd. So it is calld a -poit circular covolutio,,,

17 Circular covolutio with a dlayd ipuls Th dlayd ipuls squc δ W W Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Suary of proprtis of th DFT 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 7

18 Part VI: Liar covolutio of th DFT Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT 5 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Liar covolutio usig th DFT Procdur Coput th -poit DFTs ad of two squcs ad, rspctivly Coput th product of for Coput th squc as th ivrs DFT of As w ow, th ultiplicatio of DFTs corrspods to a circular covolutio of th squcs. To obtai a liar covolutio, w ust sur that circular covolutio has th ffct of liar covolutio. 6 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 8

19 9 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 7 Liar covolutio of two fiit-lgth squcs Digital Sigal Procssig, I, Zhg-Hua Ta, 6 8 Th circular covolutio corrspodig to is idtical to th liar covolutio corrspodig to if th lgth of DFTs satisfis Circular covolutio as liar covolutio with alaisig othrwis,, : th ivrs DFT of So,, Also, DFT : Dfi a : Fourir trasfor of r r p p + P L

20 Circular covolutio as liar covolutio with alaisig 9 Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Suary Th discrt Fourir sris Th Fourir trasfor of priodic sigals Saplig th Fourir trasfor Th discrt Fourir trasfor Proprtis of th DFT Liar covolutio usig th DFT 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6

21 Cours at a glac MM Discrt-ti sigals ad systs Syst MM Fourir-doai rprstatio Saplig ad rcostructio Syst aalysis MM5 Syst structur MM6 MM4 Filtr dsig z-trasfor DFTFFT MM7, MM8 MM MM9, MM 4 Digital Sigal Procssig, I, Zhg-Hua Ta, 6

DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1

DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1 DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT

More information

Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia

Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia Part B: Trasform Mthods Chaptr 3: Discrt-Tim Fourir Trasform (DTFT) 3. Discrt Tim Fourir Trasform (DTFT) 3. Proprtis of DTFT 3.3 Discrt Fourir Trasform (DFT) 3.4 Paddig with Zros ad frqucy Rsolutio 3.5

More information

Digital Signal Processing

Digital Signal Processing Digital Sigal Procssig Brli C 4 Rfrcs:.. V. Oppi ad R. W. Scafr, Discrt-ti Sigal Procssig, 999.. uag t. al., Spo Laguag Procssig, Captrs 5, 6. J. R. Dllr t. al., Discrt-Ti Procssig of Spc Sigals, Captrs

More information

Discrete Fourier Transform. Nuno Vasconcelos UCSD

Discrete Fourier Transform. Nuno Vasconcelos UCSD Discrt Fourir Trasform uo Vascoclos UCSD Liar Shift Ivariat (LSI) systms o of th most importat cocpts i liar systms thory is that of a LSI systm Dfiitio: a systm T that maps [ ito y[ is LSI if ad oly if

More information

15/03/1439. Lectures on Signals & systems Engineering

15/03/1439. Lectures on Signals & systems Engineering Lcturs o Sigals & syms Egirig Dsigd ad Prd by Dr. Ayma Elshawy Elsfy Dpt. of Syms & Computr Eg. Al-Azhar Uivrsity Email : aymalshawy@yahoo.com A sigal ca b rprd as a liar combiatio of basic sigals. Th

More information

Chapter 6: DFT/FFT Transforms and Applications 6.1 DFT and its Inverse

Chapter 6: DFT/FFT Transforms and Applications 6.1 DFT and its Inverse 6. Chaptr 6: DFT/FFT Trasforms ad Applicatios 6. DFT ad its Ivrs DFT: It is a trasformatio that maps a -poit Discrt-tim DT) sigal ] ito a fuctio of th compl discrt harmoics. That is, giv,,,, ]; L, a -poit

More information

DFT: Discrete Fourier Transform

DFT: Discrete Fourier Transform : Discrt Fourir Trasform Cogruc (Itgr modulo m) I this sctio, all lttrs stad for itgrs. gcd m, = th gratst commo divisor of ad m Lt d = gcd(,m) All th liar combiatios r s m of ad m ar multils of d. a b

More information

Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform

Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform Discrt Fourir Trasform Dfiitio - T simplst rlatio btw a lt- squc x dfid for ω ad its DTFT X ( ) is ω obtaid by uiformly sampli X ( ) o t ω-axis btw ω < at ω From t dfiitio of t DTFT w tus av X X( ω ) ω

More information

Discrete Fourier Series and Transforms

Discrete Fourier Series and Transforms Lctur 4 Outi: Discrt Fourir Sris ad Trasforms Aoucmts: H 4 postd, du Tus May 8 at 4:3pm. o at Hs as soutios wi b avaiab immdiaty. Midtrm dtais o t pag H 5 wi b postd Fri May, du foowig Fri (as usua) Rviw

More information

Discrete Fourier Transform (DFT)

Discrete Fourier Transform (DFT) Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial

More information

PURE MATHEMATICS A-LEVEL PAPER 1

PURE MATHEMATICS A-LEVEL PAPER 1 -AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio

More information

Chapter 4 - The Fourier Series

Chapter 4 - The Fourier Series M. J. Robrts - 8/8/4 Chaptr 4 - Th Fourir Sris Slctd Solutios (I this solutio maual, th symbol,, is usd for priodic covolutio bcaus th prfrrd symbol which appars i th txt is ot i th fot slctio of th word

More information

Chapter 3 Fourier Series Representation of Periodic Signals

Chapter 3 Fourier Series Representation of Periodic Signals Chptr Fourir Sris Rprsttio of Priodic Sigls If ritrry sigl x(t or x[] is xprssd s lir comitio of som sic sigls th rspos of LI systm coms th sum of th idividul rsposs of thos sic sigls Such sic sigl must:

More information

Derivation of a Predictor of Combination #1 and the MSE for a Predictor of a Position in Two Stage Sampling with Response Error.

Derivation of a Predictor of Combination #1 and the MSE for a Predictor of a Position in Two Stage Sampling with Response Error. Drivatio of a Prdictor of Cobiatio # ad th SE for a Prdictor of a Positio i Two Stag Saplig with Rspos Error troductio Ed Stak W driv th prdictor ad its SE of a prdictor for a rado fuctio corrspodig to

More information

Problem Value Score Earned No/Wrong Rec -3 Total

Problem Value Score Earned No/Wrong Rec -3 Total GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING ECE6 Fall Quiz # Writt Eam Novmr, NAME: Solutio Kys GT Usram: LAST FIRST.g., gtiit Rcitatio Sctio: Circl t dat & tim w your Rcitatio

More information

VI. FIR digital filters

VI. FIR digital filters www.jtuworld.com www.jtuworld.com Digital Sigal Procssig 6 Dcmbr 24, 29 VI. FIR digital filtrs (No chag i 27 syllabus). 27 Syllabus: Charactristics of FIR digital filtrs, Frqucy rspos, Dsig of FIR digital

More information

8(4 m0) ( θ ) ( ) Solutions for HW 8. Chapter 25. Conceptual Questions

8(4 m0) ( θ ) ( ) Solutions for HW 8. Chapter 25. Conceptual Questions Solutios for HW 8 Captr 5 Cocptual Qustios 5.. θ dcrass. As t crystal is coprssd, t spacig d btw t plas of atos dcrass. For t first ordr diffractio =. T Bragg coditio is = d so as d dcrass, ust icras for

More information

[ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is

[ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is Discrt-tim ourir Trsform Rviw or discrt-tim priodic sigl x with priod, th ourir sris rprsttio is x + < > < > x, Rviw or discrt-tim LTI systm with priodic iput sigl, y H ( ) < > < > x H r rfrrd to s th

More information

SIGNALS AND LINEAR SYSTEMS UNIT-1 SIGNALS

SIGNALS AND LINEAR SYSTEMS UNIT-1 SIGNALS SIGNALS AND LINEAR SYSTEMS UNIT- SIGNALS. Dfi a sigal. A sigal is a fuctio of o or mor idpdt variabls which cotais som iformatio. Eg: Radio sigal, TV sigal, Tlpho sigal, tc.. Dfi systm. A systm is a st

More information

Frequency Measurement in Noise

Frequency Measurement in Noise Frqucy Masurmt i ois Porat Sctio 6.5 /4 Frqucy Mas. i ois Problm Wat to o look at th ct o ois o usig th DFT to masur th rqucy o a siusoid. Cosidr sigl complx siusoid cas: j y +, ssum Complx Whit ois Gaussia,

More information

Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series

Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris

More information

EC1305 SIGNALS & SYSTEMS

EC1305 SIGNALS & SYSTEMS EC35 SIGNALS & SYSTES DEPT/ YEAR/ SE: IT/ III/ V PREPARED BY: s. S. TENOZI/ Lcturr/ECE SYLLABUS UNIT I CLASSIFICATION OF SIGNALS AND SYSTES Cotiuous Tim Sigals (CT Sigals Discrt Tim Sigals (DT Sigals Stp

More information

Lectures 9 IIR Systems: First Order System

Lectures 9 IIR Systems: First Order System EE3054 Sigals ad Systms Lcturs 9 IIR Systms: First Ordr Systm Yao Wag Polytchic Uivrsity Som slids icludd ar xtractd from lctur prstatios prpard by McCllla ad Schafr Lics Ifo for SPFirst Slids This work

More information

Australian Journal of Basic and Applied Sciences, 4(9): , 2010 ISSN

Australian Journal of Basic and Applied Sciences, 4(9): , 2010 ISSN Australia Joural of Basic ad Applid Scics, 4(9): 4-43, ISSN 99-878 Th Caoical Product of th Diffrtial Equatio with O Turig Poit ad Sigular Poit A Dabbaghia, R Darzi, 3 ANaty ad 4 A Jodayr Akbarfa, Islaic

More information

Motivation. We talk today for a more flexible approach for modeling the conditional probabilities.

Motivation. We talk today for a more flexible approach for modeling the conditional probabilities. Baysia Ntworks Motivatio Th coditioal idpdc assuptio ad by aïv Bays classifirs ay s too rigid spcially for classificatio probls i which th attributs ar sowhat corrlatd. W talk today for a or flibl approach

More information

H2 Mathematics Arithmetic & Geometric Series ( )

H2 Mathematics Arithmetic & Geometric Series ( ) H Mathmatics Arithmtic & Gomtric Sris (08 09) Basic Mastry Qustios Arithmtic Progrssio ad Sris. Th rth trm of a squc is 4r 7. (i) Stat th first four trms ad th 0th trm. (ii) Show that th squc is a arithmtic

More information

Fractional Sampling using the Asynchronous Shah with application to LINEAR PHASE FIR FILTER DESIGN

Fractional Sampling using the Asynchronous Shah with application to LINEAR PHASE FIR FILTER DESIGN Jo Dattorro, Summr 998 Fractioal Samplig usig th sychroous Shah with applicatio to LIER PSE FIR FILER DESIG bstract W ivstigat th fudamtal procss of samplig usig a impuls trai, calld th shah fuctio [Bracwll],

More information

Chapter Taylor Theorem Revisited

Chapter Taylor Theorem Revisited Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o

More information

1985 AP Calculus BC: Section I

1985 AP Calculus BC: Section I 985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b

More information

SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C

SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C Joural of Mathatical Aalysis ISSN: 2217-3412, URL: www.ilirias.co/ja Volu 8 Issu 1 2017, Pags 156-163 SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C BURAK

More information

(looks like a time sequence) i function of ˆ ω (looks like a transform) 2. Interpretations of X ( e ) DFT View OLA implementation

(looks like a time sequence) i function of ˆ ω (looks like a transform) 2. Interpretations of X ( e ) DFT View OLA implementation viw of STFT Digital Spch Procssig Lctur Short-Tim Fourir Aalysis Mthods - Filtr Ba Dsig j j ˆ m ˆ. X x[ m] w[ ˆ m] ˆ i fuctio of ˆ loos li a tim squc i fuctio of ˆ loos li a trasform j ˆ X dfid for ˆ 3,,,...;

More information

A Simple Proof that e is Irrational

A Simple Proof that e is Irrational Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural

More information

Chapter 9 Computation of the Discrete. Fourier Transform

Chapter 9 Computation of the Discrete. Fourier Transform Chapter 9 Coputatio of the Discrete Fourier Trasfor Itroductio Efficiet Coputatio of the Discrete Fourier Trasfor Goertzel Algorith Deciatio-I-Tie FFT Algoriths Deciatio-I-Frequecy FFT Algoriths Ipleetatio

More information

The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA

The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA Applid ad Coputatioal Mathatics 0; 7(3: 6-66 http://www.scicpublishiggroup.co/j/ac doi: 0.6/j.ac.00703. ISS: 3-5605 (Prit; ISS: 3-563 (Oli Th Applicatio of Eigvctors for th Costructio of Miiu-Ergy Wavlt

More information

ELEC9721: Digital Signal Processing Theory and Applications

ELEC9721: Digital Signal Processing Theory and Applications ELEC97: Digital Sigal Pocssig Thoy ad Applicatios Tutoial ad solutios Not: som of th solutios may hav som typos. Q a Show that oth digital filts giv low hav th sam magitud spos: i [] [ ] m m i i i x c

More information

Hadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms

Hadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of

More information

z 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z

z 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist

More information

Systems in Transform Domain Frequency Response Transfer Function Introduction to Filters

Systems in Transform Domain Frequency Response Transfer Function Introduction to Filters LTI Discrt-Tim Systms i Trasform Domai Frqucy Rspos Trasfr Fuctio Itroductio to Filtrs Taia Stathai 811b t.stathai@imprial.ac.u Frqucy Rspos of a LTI Discrt-Tim Systm Th wll ow covolutio sum dscriptio

More information

Chapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1

Chapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1 Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida

More information

Probability & Statistics,

Probability & Statistics, Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus Poisso Distributio Poisso Distributio: A radom variabl X is said

More information

Wavelet Transform Theory. Prof. Mark Fowler Department of Electrical Engineering State University of New York at Binghamton

Wavelet Transform Theory. Prof. Mark Fowler Department of Electrical Engineering State University of New York at Binghamton Wavelet Trasfor Theory Prof. Mark Fowler Departet of Electrical Egieerig State Uiversity of New York at Bighato What is a Wavelet Trasfor? Decopositio of a sigal ito costituet parts Note that there are

More information

Analysis of a Finite Quantum Well

Analysis of a Finite Quantum Well alysis of a Fiit Quatu Wll Ira Ka Dpt. of lctrical ad lctroic girig Jssor Scic & Tcology Uivrsity (JSTU) Jssor-748, Baglads ika94@uottawa.ca Or ikr_c@yaoo.co Joural of lctrical girig T Istitutio of girs,

More information

10. Joint Moments and Joint Characteristic Functions

10. Joint Moments and Joint Characteristic Functions 0 Joit Momts ad Joit Charactristic Fctios Followig sctio 6 i this sctio w shall itrodc varios paramtrs to compactly rprst th iformatio cotaid i th joit pdf of two rvs Giv two rvs ad ad a fctio g x y dfi

More information

2D DSP Basics: Systems Stability, 2D Sampling

2D DSP Basics: Systems Stability, 2D Sampling - Digital Iage Processig ad Copressio D DSP Basics: Systes Stability D Saplig Stability ty Syste is stable if a bouded iput always results i a bouded output BIBO For LSI syste a sufficiet coditio for stability:

More information

Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia

Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia Par B: rasform Mhods Profssor E. Ambikairaah UNSW, Ausralia Chapr : Fourir Rprsaio of Sigal. Fourir Sris. Fourir rasform.3 Ivrs Fourir rasform.4 Propris.4. Frqucy Shif.4. im Shif.4.3 Scalig.4.4 Diffriaio

More information

ln x = n e = 20 (nearest integer)

ln x = n e = 20 (nearest integer) H JC Prlim Solutios 6 a + b y a + b / / dy a b 3/ d dy a b at, d Giv quatio of ormal at is y dy ad y wh. d a b () (,) is o th curv a+ b () y.9958 Qustio Solvig () ad (), w hav a, b. Qustio d.77 d d d.77

More information

Frequency Response & Digital Filters

Frequency Response & Digital Filters Frquy Rspos & Digital Filtrs S Wogsa Dpt. of Cotrol Systms ad Istrumtatio Egirig, KUTT Today s goals Frquy rspos aalysis of digital filtrs LTI Digital Filtrs Digital filtr rprstatios ad struturs Idal filtrs

More information

UNIT 2: MATHEMATICAL ENVIRONMENT

UNIT 2: MATHEMATICAL ENVIRONMENT UNIT : MATHEMATICAL ENVIRONMENT. Itroductio This uit itroducs som basic mathmatical cocpts ad rlats thm to th otatio usd i th cours. Wh ou hav workd through this uit ou should: apprciat that a mathmatical

More information

Topic 5:Discrete-Time Fourier Transform (DTFT)

Topic 5:Discrete-Time Fourier Transform (DTFT) ELEC64: Sigals Ad Systms Tpic 5:Discrt-Tim Furir Trasfrm DTFT Aishy Amr Ccrdia Uivrsity Elctrical ad Cmputr Egirig Itrducti DT Furir Trasfrm Sufficit cditi fr th DTFT DT Furir Trasfrm f Pridic Sigals DTFT

More information

ECE 6560 Chapter 2: The Resampling Process

ECE 6560 Chapter 2: The Resampling Process Capter 2: e Resaplig Process Dr. Bradley J. Bazui Wester iciga Uiversity College of Egieerig ad Applied Scieces Departet of Electrical ad Coputer Egieerig 1903 W. iciga Ave. Kalaazoo I, 49008-5329 Capter

More information

Chapter 11.00C Physical Problem for Fast Fourier Transform Civil Engineering

Chapter 11.00C Physical Problem for Fast Fourier Transform Civil Engineering haptr. Physical Problm for Fast Fourir Trasform ivil Egirig Itroductio I this chaptr, applicatios of FFT algorithms [-5] for solvig ral-lif problms such as computig th dyamical (displacmt rspos [6-7] of

More information

Электронный архив УГЛТУ ЭКО-ПОТЕНЦИАЛ 2 (14),

Электронный архив УГЛТУ ЭКО-ПОТЕНЦИАЛ 2 (14), УДК 004.93'; 004.93 Электронный архив УГЛТУ ЭКО-ПОТЕНЦИАЛ (4), 06 V. Labuts, I. Artmov, S. Martyugi & E. Osthimr Ural dral Uivrsity, Ykatriburg, Russia Capricat LLC, USA AST RACTIOAL OURIER TRASORMS BASED

More information

10. The Discrete-Time Fourier Transform (DTFT)

10. The Discrete-Time Fourier Transform (DTFT) Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w

More information

Vtusolution.in FOURIER SERIES. Dr.A.T.Eswara Professor and Head Department of Mathematics P.E.S.College of Engineering Mandya

Vtusolution.in FOURIER SERIES. Dr.A.T.Eswara Professor and Head Department of Mathematics P.E.S.College of Engineering Mandya LECTURE NOTES OF ENGINEERING MATHEMATICS III Su Cod: MAT) Vtusoutio.i COURSE CONTENT ) Numric Aysis ) Fourir Sris ) Fourir Trsforms & Z-trsforms ) Prti Diffrti Equtios 5) Lir Agr 6) Ccuus of Vritios Tt

More information

07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n

07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n 07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l

More information

MATH 681 Notes Combinatorics and Graph Theory I. ( 4) n. This will actually turn out to be marvelously simplifiable: C n = 2 ( 4) n n + 1. ) (n + 1)!

MATH 681 Notes Combinatorics and Graph Theory I. ( 4) n. This will actually turn out to be marvelously simplifiable: C n = 2 ( 4) n n + 1. ) (n + 1)! MATH 681 Nots Combiatorics ad Graph Thory I 1 Catala umbrs Prviously, w usd gratig fuctios to discovr th closd form C = ( 1/ +1) ( 4). This will actually tur out to b marvlously simplifiabl: ( ) 1/ C =

More information

Fourier Series: main points

Fourier Series: main points BIOEN 3 Lcur 6 Fourir rasforms Novmbr 9, Fourir Sris: mai pois Ifii sum of sis, cosis, or boh + a a cos( + b si( All frqucis ar igr mulipls of a fudamal frqucy, o F.S. ca rprs ay priodic fucio ha w ca

More information

Module 5: IIR and FIR Filter Design Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School of Electrical Engineering & Telecommunications

Module 5: IIR and FIR Filter Design Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School of Electrical Engineering & Telecommunications Modul 5: IIR ad FIR Filtr Dsig Prof. Eliathamby Ambiairaah Dr. Tharmaraah Thiruvara School of Elctrical Egirig & Tlcommuicatios Th Uivrsity of w South Wals Australia IIR filtrs Evry rcursiv digital filtr

More information

Statistics 3858 : Likelihood Ratio for Exponential Distribution

Statistics 3858 : Likelihood Ratio for Exponential Distribution Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai

More information

2D DSP Basics: 2D Systems

2D DSP Basics: 2D Systems - Digital Image Processig ad Compressio D DSP Basics: D Systems D Systems T[ ] y = T [ ] Liearity Additivity: If T y = T [ ] The + T y = y + y Homogeeity: If The T y = T [ ] a T y = ay = at [ ] Liearity

More information

CIVE322 BASIC HYDROLOGY Hydrologic Science and Engineering Civil and Environmental Engineering Department Fort Collins, CO (970)

CIVE322 BASIC HYDROLOGY Hydrologic Science and Engineering Civil and Environmental Engineering Department Fort Collins, CO (970) CVE322 BASC HYDROLOGY Hydrologic Scic ad Egirig Civil ad Evirotal Egirig Dpartt Fort Collis, CO 80523-1372 (970 491-7621 MDERM EXAM 1 NO. 1 Moday, Octobr 3, 2016 8:00-8:50 AM Haod Auditoriu You ay ot cosult

More information

Figure 2-18 Thevenin Equivalent Circuit of a Noisy Resistor

Figure 2-18 Thevenin Equivalent Circuit of a Noisy Resistor .8 NOISE.8. Th Nyquist Nois Thorm W ow wat to tur our atttio to ois. W will start with th basic dfiitio of ois as usd i radar thory ad th discuss ois figur. Th typ of ois of itrst i radar thory is trmd

More information

Slide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS

Slide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt

More information

Chapter 8. DFT : The Discrete Fourier Transform

Chapter 8. DFT : The Discrete Fourier Transform Chapter 8 DFT : The Discrete Fourier Trasform Roots of Uity Defiitio: A th root of uity is a complex umber x such that x The th roots of uity are: ω, ω,, ω - where ω e π /. Proof: (ω ) (e π / ) (e π )

More information

LECTURE 13 Filling the bands. Occupancy of Available Energy Levels

LECTURE 13 Filling the bands. Occupancy of Available Energy Levels LUR 3 illig th bads Occupacy o Availabl rgy Lvls W hav dtrmid ad a dsity o stats. W also d a way o dtrmiig i a stat is illd or ot at a giv tmpratur. h distributio o th rgis o a larg umbr o particls ad

More information

Page 1 BACI. Before-After-Control-Impact Power Analysis For Several Related Populations (Variance Known) October 10, Richard A.

Page 1 BACI. Before-After-Control-Impact Power Analysis For Several Related Populations (Variance Known) October 10, Richard A. Pag BACI Bfor-Aftr-Cotrol-Impact Powr Aalysis For Svral Rlatd Populatios (Variac Kow) Octobr, 3 Richard A. Hirichs Cavat: This study dsig tool is for a idalizd powr aalysis built upo svral simplifyig assumptios

More information

DEPARTMENT OF MATHEMATICS BIT, MESRA, RANCHI MA2201 Advanced Engg. Mathematics Session: SP/ 2017

DEPARTMENT OF MATHEMATICS BIT, MESRA, RANCHI MA2201 Advanced Engg. Mathematics Session: SP/ 2017 DEARMEN OF MAEMAICS BI, MESRA, RANCI MA Advad Egg. Mathatis Sssio: S/ 7 MODULE I. Cosidr th two futios f utorial Sht No. -- ad g o th itrval [,] a Show that thir Wroskia W f, g vaishs idtially. b Show

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1

More information

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH. C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH

More information

International Journal of Advanced and Applied Sciences

International Journal of Advanced and Applied Sciences Itratioal Joural of Advacd ad Applid Scics x(x) xxxx Pags: xx xx Cotts lists availabl at Scic Gat Itratioal Joural of Advacd ad Applid Scics Joural hompag: http://wwwscic gatcom/ijaashtml Symmtric Fuctios

More information

EE140 Introduction to Communication Systems Lecture 2

EE140 Introduction to Communication Systems Lecture 2 EE40 Introduction to Communication Systms Lctur 2 Instructor: Prof. Xiliang Luo ShanghaiTch Univrsity, Spring 208 Architctur of a Digital Communication Systm Transmittr Sourc A/D convrtr Sourc ncodr Channl

More information

ON THE RELATIONSHIP BETWEEN THE SPHERICAL WAVE EXPANSION AND THE PLANE WAVE EXPANSION FOR ANTENNA DIAGNOSTICS

ON THE RELATIONSHIP BETWEEN THE SPHERICAL WAVE EXPANSION AND THE PLANE WAVE EXPANSION FOR ANTENNA DIAGNOSTICS ON THE RELATIONSHIP BETWEEN THE SPHERICAL WAVE EXPANSION AND THE PLANE WAVE EXPANSION FOR ANTENNA DIAGNOSTICS Ccilia Capplli,, Aksl Frads, Olav Bribjrg TICRA, Lædrstræd 34, DK-0 Cophag K, Dark Ørstd DTU,

More information

PH4210 Statistical Mechanics

PH4210 Statistical Mechanics PH4 Statistical Mchaics Probl Sht Aswrs Dostrat that tropy, as giv by th Boltza xprssio S = l Ω, is a xtsiv proprty Th bst way to do this is to argu clarly that Ω is ultiplicativ W ust prov that if o syst

More information

A Review of Complex Arithmetic

A Review of Complex Arithmetic /0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd

More information

Power Spectrum Estimation of Stochastic Stationary Signals

Power Spectrum Estimation of Stochastic Stationary Signals ag of 6 or Spctru stato of Stochastc Statoary Sgas Lt s cosr a obsrvato of a stochastc procss (). Ay obsrvato s a ft rcor of th ra procss. Thrfor, ca say:

More information

COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II

COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II COLLECTION OF SUPPLEMENTARY PROBLEMS I. CHAPTER 6 --- Trscdtl Fuctios CALCULUS II A. FROM CALCULUS BY J. STEWART:. ( How is th umbr dfid? ( Wht is pproimt vlu for? (c ) Sktch th grph of th turl potil fuctios.

More information

The University of Manchester Analogue & Digital Filters 2003 Section D5: More on FIR digital filter design.

The University of Manchester Analogue & Digital Filters 2003 Section D5: More on FIR digital filter design. Th Uivrsity of achstr Aalogu & Digital Filtrs 3 Sctio D5: or o FIR digital filtr dsig. D5..Bacgroud: As w hav s bfor, a FIR digital filtr of ordr may b implmtd by programmig th sigal-flow graph show blow.

More information

Calculus & analytic geometry

Calculus & analytic geometry Calculus & aalytic gomtry B Sc MATHEMATICS Admissio owards IV SEMESTER CORE COURSE UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITYPO, MALAPPURAM, KERALA, INDIA 67 65 5 School of Distac

More information

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3 SOLVED EXAMPLES E. If f() E.,,, th f() f() h h LHL RHL, so / / Lim f() quls - (D) Dos ot ist [( h)+] [(+h) + ] f(). LHL E. RHL h h h / h / h / h / h / h / h As.[C] (D) Dos ot ist LHL RHL, so giv it dos

More information

Time : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120

Time : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120 Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing Aoucemets HW solutios posted -- self gradig due HW2 due Friday EE2 Digital Sigal Processig ham radio licesig lectures Tue 6:-8pm Cory 2 Lecture 6 based o slides by J.M. Kah SDR give after GSI Wedesday

More information

ONLINE SUPPLEMENT Optimal Markdown Pricing and Inventory Allocation for Retail Chains with Inventory Dependent Demand

ONLINE SUPPLEMENT Optimal Markdown Pricing and Inventory Allocation for Retail Chains with Inventory Dependent Demand Submittd to Maufacturig & Srvic Opratios Maagmt mauscript MSOM 5-4R2 ONLINE SUPPLEMENT Optimal Markdow Pricig ad Ivtory Allocatio for Rtail Chais with Ivtory Dpdt Dmad Stph A Smith Dpartmt of Opratios

More information

NET/JRF, GATE, IIT JAM, JEST, TIFR

NET/JRF, GATE, IIT JAM, JEST, TIFR Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Mathmatical Physics JEST-6 Q. Giv th coditio φ, th solutio of th quatio ψ φ φ is giv by k. kφ kφ lφ kφ lφ (a) ψ (b) ψ kφ (c) ψ

More information

Practical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement

Practical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement Practical Spectral Aaysis (cotiue) (from Boaz Porat s book) Frequecy Measuremet Oe of the most importat applicatios of the DFT is the measuremet of frequecies of periodic sigals (eg., siusoidal sigals),

More information

Contents Two Sample t Tests Two Sample t Tests

Contents Two Sample t Tests Two Sample t Tests Cotets 3.5.3 Two Saple t Tests................................... 3.5.3 Two Saple t Tests Setup: Two Saples We ow focus o a sceario where we have two idepedet saples fro possibly differet populatios. Our

More information

Linear Algebra Existence of the determinant. Expansion according to a row.

Linear Algebra Existence of the determinant. Expansion according to a row. Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit

More information

2.29 Numerical Fluid Mechanics Spring 2015 Lecture 12

2.29 Numerical Fluid Mechanics Spring 2015 Lecture 12 REVIEW Lctur 11: Numrical Fluid Mchaics Sprig 2015 Lctur 12 Fiit Diffrcs basd Polyomial approximatios Obtai polyomial (i gral u-qually spacd), th diffrtiat as dd Nwto s itrpolatig polyomial formulas Triagular

More information

7. Differentiation of Trigonometric Function

7. Differentiation of Trigonometric Function 7. Diffrtiatio of Trigootric Fctio RADIAN MEASURE. Lt s ot th lgth of arc AB itrcpt y th ctral agl AOB o a circl of rais r a lt S ot th ara of th sctor AOB. (If s is /60 of th circfrc, AOB = 0 ; if s =

More information

The Interplay between l-max, l-min, p-max and p-min Stable Distributions

The Interplay between l-max, l-min, p-max and p-min Stable Distributions DOI: 0.545/mjis.05.4006 Th Itrplay btw lma lmi pma ad pmi Stabl Distributios S Ravi ad TS Mavitha Dpartmt of Studis i Statistics Uivrsity of Mysor Maasagagotri Mysuru 570006 Idia. Email:ravi@statistics.uimysor.ac.i

More information

ESS 265 Spring Quarter 2005 Time Series Analysis: Some Fundamentals of Spectral Analysis

ESS 265 Spring Quarter 2005 Time Series Analysis: Some Fundamentals of Spectral Analysis ESS 65 Srig Qurtr 5 Tim Sris ysis: Som Fudmts of Sctr ysis Lctur My, 5 Fourir Sris y riodic fuctio ttt whr ωt is th riod c xrssd s Fourir sris t c cos t s si t ω ω t must stisfy th coditio T t dt < y rso

More information

On the approximation of the constant of Napier

On the approximation of the constant of Napier Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of

More information

Mathematical Preliminaries for Transforms, Subbands, and Wavelets

Mathematical Preliminaries for Transforms, Subbands, and Wavelets Mahmaical Prlimiaris for rasforms, Subbads, ad Wavls C.M. Liu Prcpual Sigal Procssig Lab Collg of Compur Scic Naioal Chiao-ug Uivrsiy hp://www.csi.cu.du.w/~cmliu/courss/comprssio/ Offic: EC538 (03)5731877

More information

Law of large numbers

Law of large numbers Law of larg umbrs Saya Mukhrj W rvisit th law of larg umbrs ad study i som dtail two typs of law of larg umbrs ( 0 = lim S ) p ε ε > 0, Wak law of larrg umbrs [ ] S = ω : lim = p, Strog law of larg umbrs

More information

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig

More information

Partition Functions and Ideal Gases

Partition Functions and Ideal Gases Partitio Fuctios ad Idal Gass PFIG- You v lard about partitio fuctios ad som uss ow w ll xplor tm i mor dpt usig idal moatomic diatomic ad polyatomic gass! for w start rmmbr: Q( N ( N! N Wat ar N ad? W

More information

A Novel Approach to Recovering Depth from Defocus

A Novel Approach to Recovering Depth from Defocus Ssors & Trasducrs 03 by IFSA http://www.ssorsportal.com A Novl Approach to Rcovrig Dpth from Dfocus H Zhipa Liu Zhzhog Wu Qiufg ad Fu Lifag Collg of Egirig Northast Agricultural Uivrsity 50030 Harbi Chia

More information

Response of LTI Systems to Complex Exponentials

Response of LTI Systems to Complex Exponentials 3 Fourir sris coiuous-im Rspos of LI Sysms o Complx Expoials Ouli Cosidr a LI sysm wih h ui impuls rspos Suppos h ipu sigal is a complx xpoial s x s is a complx umbr, xz zis a complx umbr h or h h w will

More information

2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005

2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005 Mark Schm 67 Ju 5 GENERAL INSTRUCTIONS Marks i th mark schm ar plicitly dsigatd as M, A, B, E or G. M marks ("mthod" ar for a attmpt to us a corrct mthod (ot mrly for statig th mthod. A marks ("accuracy"

More information

Time Dependent Solutions: Propagators and Representations

Time Dependent Solutions: Propagators and Representations Tim Dpdt Solutios: Propagators ad Rprstatios Michal Fowlr, UVa 1/3/6 Itroductio W v spt most of th cours so far coctratig o th igstats of th amiltoia, stats whos tim dpdc is mrly a chagig phas W did mtio

More information

STIRLING'S 1 FORMULA AND ITS APPLICATION

STIRLING'S 1 FORMULA AND ITS APPLICATION MAT-KOL (Baja Luka) XXIV ()(08) 57-64 http://wwwimviblorg/dmbl/dmblhtm DOI: 075/МК80057A ISSN 0354-6969 (o) ISSN 986-588 (o) STIRLING'S FORMULA AND ITS APPLICATION Šfkt Arslaagić Sarajvo B&H Abstract:

More information