SPECIAL TOPICS ECE 599
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Date Created: 10/19/15
EE 679 Lecture 4 Patrick Chiang Review Review Definition of BER Sending one bit onto the line Summation of two bits pchianghspicerlcspicedeck Overlay the bit times over each other Assume ideal sampling clock at receiver Creates Eye Diagram Intersymbol Interference Ideally a transmission system is memorye35 no history of previous bits History A c In reaiity the state of the system is affected by previous bits re ections on transmission History B C iines magnitude and phase of excited resonances signais that don t reach the rails by the end of the cycie This history affects the transmission of the current bit 0 993 cmngmmymnam new Aiimgnls esenea 7 Impedance Mismatches Echos of previous bits reflect up and down transmission Z w t iines A mismatch of X gives to RT first order a reflection of kR x2 Worstcase superposition of entire echo train is r k k k 1quot g R 17k EE273 LB omz i993 CuViiyiilic w miwi mi AiiRiyi Managing Noise We manage noise using noise budgets Proportional Independent Categorize noise along two 8 axes cs Parallel Xtalk Rcvr Offset C 3 ISI Rcvr Sens proportional vs independent 3 bounded vs statistical E Some noise sources could 3 PS N go in either category eg Perp Xtalk else crosstalk to perpendicular 5g Thermal NO39SE lines I Allocate noise to various sources Constrain design to meet the budget E273 Ls omz iesa M cuwngm m 2y wu mi 1 Dali All m4 Reserved The Noise Budget 1 n Signal Swing 400mV Threshold n quot Net margin 70mV Gross Margin 200mV Bounded Sources 50mV VSNR P t39 S 200 RMS ro or Iona ources O quotV V p D NO Se quot71 W WW W cnpmgmmy mHzmw saw AHR gmsResened quot WorstCase Analysis Noise sources are an unknown With worstcase analysis e all noise sources have the greatest possible magnitude they all sum up in the same direction Unliker in any single unit at any instant in time but if you make enough units and run them long enough this case will occur Signal SWing Vni Rcvr offsense X 3 Uncancelled PS noise Vni Crosstalk Reflections TOTAL Kn KnVs Gross Margin Vni KnVs Vn Net Margin Margin Ratio EE273 LE call was Cnpyrigh in by Wil mm 1 Duly All mm Reserved VSNR signalnoise vsnr VGNM IVNOISE Gross Noise Margin signalnoise before unbounded noise sources unbounded noise Statistical Analysis Gaussian Noise and BitError Rate For some noise sources we consider the probability distribution of values rather than the worstcase value truly random noise sources thermal noise shot noise uncorrelated bounded noise sources crosstalk power supply noise 52273 LS OcHZ i923 Camrlghi lit by Wll lam v om All aim mam Typically we model these sources with a Gaussian distribution Most sources are zero mean The relevant parameter is their standard deviation or rootmeansquared RMS value Adding Gaussian Noise Sources To sum up zeromean m Gaussran norse sources Vm LEIV3 sum the variance not the N standard deviation A 10mV RMS source added to a 20mV RMS source gives a 224mV RMS source Crosstaik 100 mVRMS PS Noise 200 mV RMS TOTAL 224 mV RMS EH 0 995 Cawrigh 5ib l v iiiam Univ AiiRigNsReserved 15 Statistical Analysis Gaussian Noise and BitError Rate For some noise sources we consider the probability distribution of values rather than the worstcase value truly random noise sources thermal noise shot noise uncorrelated bounded noise sources crosstalk power supply noise 527 LB omz i993 caWrigm to by Wil mm 1 Jaw All mm Reserved Typically we model these sources with a Gaussian distribution Most sources are zero mean The relevant parameter is their standard deviation or rootmeansquared RMS value An Example Noise Calculation attenuation 5 crosstalk from adjacent lines 5 ISI from reflections 20mV receiver offsetsensitivity 10mV RMS perpendicular crosstalk 10mV RMS other39 noise What is the Bit Error Rate L 250mV differential signal 15 highfrequency gtltgtlt EOmV f ram L5 omz l aai Noise Budgets and Design Rules Do we do a noise budget for every signal Of course not Noise budgets are done for classes of signals that obey a set of design rules or ground rules board to board signals intraboard signals chipto chip global onchip signals local onchip signals special signals eg domino or RAM bit lines For each class of signals a set of rules constrains the signal to meet the noise budget line geometry and spacmg maximum parallel length ground shields impedance and termination tolerance driver and receiver specifications If each signal meets these specs you know it meets the budget and hence will have a BER at least as good as calculated 2227 LE on 2 less Camnght c 2 Wu lam v Duly All Righls Reserved 21 Vsignalswing 500mV 15 high frequency attenuation 5 crosstalk from adjacent line 5 ISI from reflections 20mV receiver offsetsensitivity 10mV RMS perpendicular crosstalk 10mV RMS noise Lecture Note of Information Theory Class Date Oct 8 2008 Written by Majid Adeli Till now we have discussed about Joint Entropy HX Y Conditional Entropy HXY Chain Rule HX Y Z HX HYX HZX Y Generally the chain rule is presented as HX1 Xn 2 EL HXiX1 Xi1 We defined HX panama XEX So we have 0 S HX S logIXI HX Y HX HYX HYX HX Y H00 Z pxHYX XEX HYX s HY 0 Mutual Information Mutual information is the average amount of information that we get about X when observe the value of Y It is calculated by the equation as below IX39 Y HX HXY HX HY HXY HX Y HX Note Mutual information is symmetrical in the other words X Y Y X reverse the position ofX and Y I Proof X Y HX HY HX Y gt same equation forX and Y 39aAJno am 0 g JOJ monaq pue V JOJ aAoqe s alum am 0 slugod OM1813QUUOD 4ng augaq1u01ugod papaps Lpeg alum XQAUOQ g alum QAEDUOQ V My I 41 gt 111 I M g q e uo XQAUOI Aneonsnels s x suogpun SAEDUOD pue XSAUOD 0 0 A IX lt AH XH A XH XAH XH A XH 3 AHXAH1uapuadapuz A puv X jg A XH AH XH AXH XH A fX q 0 E 15X1e samN o I1 A HX m 39IXZXH HX m 39IXZXH Z u A HX m 39IXZXHIZ HX 39IXZXH HZ A Xquotquot ZX39IXH Xquotquot 39ZX39IXNHI A Xquotquot ZX XI oo1d HX m 39IXA 11133 A Xquotquot ZX UOI Heme UI ZX39IXA 55X IxA 52X A 51X A ISX ZX IXN uoneuuow en1nu1 Jo an4 ugqu o AXH XH A IX 1eq1JaqLuaLuaa Z AXH ZXH ZA IX uoneuuowI emnw euongpu03 o Lecture Note of Information Theory Class Date Oct 10 2008 Written by Majid Adeli o Jensen s Inequality a fx is convex and X is DRV Efx 2 fEX b fx strictly convex Efx gt fEX unless X is constant Proof By induction Suppose n 1 2 IXI 1 Efx fx fEX Assume for IXI n 1 we have Efx 2 fEX So for IXI n we have mm ismmi pn xn 1 pn2111i nfxi 2 pumaquot 1 pnf2111f nxt2 fpnxn 1 wt quot xi 11500 11 1pn Example for Jenson s Inequality Givenfx x2 and X 11 and P 12 12 so we have EX 0 and fEX 0 2 Efx 1 gt 0 2 Efx gt fEX 2 strictly convex Relative Entropy KL Divergence M Px i706 0p u q gm log N 5 1096105 Ep logqx Hon Example for KL divergence IfX 12 6 and P and Q then Hp 1096 2585 bits and Hq 2161 bits gt Dp u q Ep lagqx Hp 2935 2585 035 bit Dq II p Ep logpx Hq 2585 2161 0424 bit o Gibb s Inequality Information Inequality For all distributions we have Dp H q 2 0 Proof Define A xpx gt 0 Q X then we have m m m Dp u q zxexmxnogm zxexmnagm s log2xexpx pm 1me c106 S lUQZXEXqOC 0 gt 00 H q S 0 gt DP H q 2 0 Now let discuss when the equality is held If wrist equality is heldsa px cqx 2 C 1 Another proof for HX S longl i i L Let Q W mm somp u q Ep logqx Hp lag m mm 2 0 gt HP S 10ng o More corollaries 0 S 1X Y HX HYIX 2 HYlX S HY and equality holds if Xahd Y are independent HX1Xn Z1HXiX1 Xi1 S Z1HXi and equality holds if the RV s are independent Conditional entropy independent bound Um 52 TI TI HX1X2 XnY1Y2 my 1HltXiIX1 Xi1Y1 WYquot 5 1mm 1 1 d Mutual Information independent bound Ifall Xi are independent and if all Yi are independent so 1X1 Xn Y1 Yn HX1Xn HX1JX2 erlllY1 Y2 Yn 2 Zi1HX EmmiIn Emma Yi 1X Y HX HY HX Y E 109 gt Dpxypxpy 2 0 L Bayeux An Architecture for Scalable and Fault Tolerant Widearea Data Dissemination By Shelley ZhuangBen ZhaoAnthony Joseph Randy Kal2John Kublatowicz Introduction 0 Multimedia Streaming typically involves a single source and multiple receivers 0 Unicast and IP multicast not feasible Solution 0 Application Level Protocols 0 Build network of unicast connections and construct distribution trees over it Introduction 0 Bayeux protocol lncurs minimum delay and bandwidth penalties and handles fault at both links and routlng nodes 0 Utilizes pre x based routing of Tapestry whld1 ls an applicatlon level routing protocol 0 Organizes recelvers into a tree mated at the source 0 Provides load balancing across replicated root nodes TaPESW Routing Layer 0 Incremental Routing of overlay messages 0 Eadwnodehasmapofmul plelevelthl39leadilevel havinganumberofentries 0 Any destination will be found in logN hops 0 EadwentyhasBmatdiesforagivensuf x Data Location 0 Each object is associated with a location root 039 Server sends publish message to the root 0 At each hopobjectld and serverld is stored 0 For multiple oopiesmapping sorted by distance from node Tapestry Bene ts 0 Powerful Fault Handling 0 Scalable 0 Proportional Route Distance Bayeux Base Architecture Bayeux session identi ed by ltseesionnameUIDgt Session Advertisement 0 Hash the above tupie into a 160 bit unique identi er 0 Rootorsouroeservera39eatesa ie usingtheidenti er 0 Advertise it 0 Receive messages from interested client Tree Maintenance JOIN and TREE messages 0 When a router receives TREE message it adds new member to its list of receivers 0 LEAVE and PRUNE messages Evaluation of Base Design We compare Bayeux algorithm against IP multicast and naive unicast Performance Metrics 0 Relative Delay Penalty The increase in delay that applications incur while using overlay routing oPhyslcal Unk Stress Measure of how effective Bayeux is in distributing network load across multiple links For a majority of pair wise oonnedjonsRDP is low Stress Value is number of duplicate packets going through a link In Bayeuxoverall distribution of link stress is lower and naive unicast has a much larger tail Scalability Enhancements Source speci c model has scalability drawbach Tree Partitioning 0 Idea is to create multiple roots and partition receivers 0 AddBayemrootnodestotapestrynetwork 0 PutobjectOineadwoftherootnodes 0 Leteadi rootnodeadvertiseOtothetapestchhosen looationnode 0 On JOIN client gets 0 from its nearest root node 0 No need of periodic advertisements by roots 0 See Graph for number of join request handled per root as number of roots Increase Scalability Enhancement Receiver Identi er Clustering 0 Aim is to reduce packet duplication 0 Delivery of packets approadies destination digit by digit 0 Local nodes should share longest possible suf x 0 Padtet duplication is thus delayed till LAN is reached thus bandwidth consumption at intermediate nodes is reduced Fault Resilient Packet Delivery 0 At each routerevery outgoing hop has 2 backup polnters 0 See figure for reachablllty comparison with IP 0 Another aspect of tapestry is hierarchical routing 0 Eadwhopdeaeasesexpectednumberofnatthopsbya factorequaltothebaseoftapestryiden er 0 Paths converge to the destination in logN hops 0 Intentionally fork of duplicates onto secondary and primary paths expecting them to merge qulddy Fault Resilient Packet Delivery 0 Proactive Duplication 0 Application Speci c Duplication 0 Prediction Based Selective Duplication 0 Explicit Knowledge Path selection 0 First Reachable Link selection NOTE Each of the rst three create duplicate packets But the duplicates converge qulddy Duplicate suppression is done using sequence numbers First Reachable Link Selection C Delivers padtets with high reliability in face of link failures 0 No packet duplication e Overhead in the form of Bandwidth used for transmitting membership information s Size of membership state transmitted decreases for routers further away from the root node 0 Delay for mul cast data directly proportional to size of member state transmltted Conclusion 0 Bayeux is an architecture for Internet Content distribution that leverages Tapestry an existing fault tolerant routing lnfrashuclure 0 Bayeuxshowsthatef dentnetworkprotoool n be designed with simplicity while inheriting desirable properties form underlying application infrastructure Introduction to Wavelet Based on A Mukherjee s lecture notes Contents El History of Wavelet Problems of Fourier Transform El Uncertainty Principle I The Short time Fourier Transform El Continuous Wavelet Transform El Fast Discrete Wavelet Transform El Multiresolution Analysis El Haar Wavelet Wavelet De nition The wavelet transform is a tool that cuts up data functions or operators into different frequency components and then studies each component with a resolution matched to its scale Dr Ingrid Daubechies Lucent Princeton U Sine Wave Wavelet db10 Wavelet Coding Methods El EZW Shapiro 1993 I Embedded Zerotree coding El SPIHT Said and Pearlman 1996 I Set Partitioning in Hierarchical Trees coding Also uses zerotrees El ECECOW Wu 1997 I Uses arithmetic coding with context El EBCOT Taubman 2000 I Uses arithmetic coding with different context El JPEG 2000 new standard based largely on EBCOT Comparison of Wavelet Based JPEG 2000 and DCT Based JPEG n JPEGZOOO image shows almost no quality loss from current JPEG even at 1581 compression Introduction to Wavelets n quot the new computational paradigm wavelets eventually may swallow Fourier transform methods quot n quot a new approach to data crunching that if successful could spawn new computer architectures and bring about commercial realization of such difficult datacompression tasks as sending images over telephone lines quot from quotNew wave number crunchingquot C Brown Electronic Engineering Times 11590 Timeline Huuuleus have had an uuusuaI suuurmcumum marker by many Memudumdiscuuerus and u isuuuuuea mu um mum umgrusu has same am the EM 1mg Humu a muequ mathemama luau u uuuuuu nally emude 13quot 1930 93 W W 5W Jam Luumau mu mu 1934 1 Inns mun any mum Iumuuu Pam M WWW quotminim 1W5 39 mm mm w mm 11 a WWW 5 a sum Hm luau inlunnaiiuu uuuul m WW5 mm and BMW MW Inuian sum u smu au uuumn a mm mm as W mm M a Salami and amui mm mm New in mm u vuriuuu luauuuncius mm mlmmm 1n H mm Esiauuu disuuuur hluuuul m Whammy lam Bmusu I minus Mia mm H mm my Wm M 33 MM 3 W M muiiuu mm mm mm m mammal l Ms uranium mm mm m Mam H mm lrammisuim luv Ms mm is nul publishali um 39 mu luaeuhuuu 15 mm mm 1931 1909 1945 Pa lru uum anuummuuu Mum ul Mimi Hun a Hungarian DEW quot1995 53M EllAquitaine mm a warm damm ma uamuldmu Mama I H mr m m WW5quot um seismfu siua u iullu what u basis at uuuuliuus um auu WM WWW WWW uul s uranium Ill camsham shape IIW39H uauugmmd as M um WWW 59W m H9 tum lu quamln pmia Max wmluis Thug uusjsl u a mi WWW BMW Buussmalm lnu nun I39m usuuiug um shun mum puts Iullmd m quotGm WIFEquot uhu mu w mks Jhy a mu nauauuu plum 1995 1935 199 mar swam Mama mm mm Sl qlhzm HaHIHMMMM 39 MWW39 WWWHrsnmwwnmler Mammy a pmsmm 51mm WWW EH Slim IMMM cannon lm ma seqmelL that Ilaa Hm Msisi me Linlmnd quotWWW 9 mm W WW3 WW WW 3 P3 quotlave Gamma mth w mum Imaws imam W WMWMW 5513395 and me mm mm m Gama mm quotW 5quot WW I WWW 5 malh ma39ii l v and Ex iluan are 3 Mai 0 m H mmquot mm W WWW5v mamham algerimms 1992 1999 1937 m F l chums a mum me WWW 3mm 1935 Ingrid Bauhau i amps mmslmcls minnow swamped mum gmzjngnf gi wmmw m3 Mm M m line ms swam n hogma Hummer n ma ml Eatmim gm g g fmr em quotmm M ram WWW Willi WWW SHBWN Jam ice Imma nn Swims my I m d n a Humanmama lira H WM er Wth Divisionaml Jamaal MM fl s wmm mm H mm WWW mm a mum on mil m in and cm Er slaavm nmm L m gt than Mg m mama MW marman 311M301 Mamas Rational la limlm i H I f w h m W W 59mm WM 3 mm to campus ii mm m m W 3 gm mm a mamamalicali aiming damnm u l39mamrinnsi m mm m mm B 5 513mm Mr m1 Model and Prediction El Compression is m i There are many decomposition approaches to modeling the signal I Every signal is a function I Modeling is function representationapproximation Probability Distribution Probability P b bilit m a y Distribution Estimates Transmission System Probability Estimates Original Source Source Messages L Compressed Bit Stream Methods of Function Approximation El Sequence of samples I Time domain El Pyramid hierarchical El Polynomial El Piecewise polynomials I Finite element method El Fourier Transform I Frequency domain I Sinusoids of various frequencies El Wavelet Transform I Timefrequency domain The Fourier Transform Analysis forward transform Fu ne mdr El Synthesis inverse transform ft j Fuej2 tdu El Forward transform decomposes ft into sinusoids I Fu represents how much of the sinusoid with frequency u is in ft Inverse transform synthesizes ft from sinusoids weighted by Fu The Fourier Transform Properties Linear Transform El Analysis decomposition of signals into sines ancl cosines has physical significance l tones vibrations El Fast algorithms exist I The fast Fourier transform requires Onlogn computations Problems With the Fourier Transform El Fourier transform well suited for stationary 2 l l signals statistics of the E 17 A ll ll ll ill W signals do not vary with i f ll j l iii lwl ill ii time This model does not 2 l i l 391 ll fit real signals well lt5 5 in n 0 240 210 El For time varying signals or signals with abrupt transitions the Fourier 2 J I transform does not provide E information on when 5 transitions occur ituda nl Fnuriartransiun39n Problems With the Fourier Transform ii Fourier transform is a global analysis A small perturbation of the function at any one point on the time axis influences all points on the frequency axis and vise versa n A qualitative explanation of why Fourier transform fails to capture time information is the fact that the set of basis functions sines and cosines are infinitely long and the transform picks up the frequencies regardless of where it appears in the signal El Need a better way to represent functions that are localized in both time and frequency Uncertainty Principle Preliminaries for the STFT n The time and frequency domains are complimentary I If one is local the other is global I For an impulse signal which assumes a constant value for a very brief period of time the frequency spectrum is infinite I If a sinusoidal signal extends over infinite time its frequency spectrum is a single vertical line at the given frequency n We can always localize a signal or a frequency but we cannot do that simultaneously I If the signal has a short duration its band of frequency is wide and vice versa Uncertainty Principle El Heisenberg s uncertainty principle was enunciated in the context of quantum physics which stated that the position and the momentum of a particle cannot be precisely determined simultaneously El This principle is also applicable to signal processing Uncertainty Principle In Signal Processing Let gt be a function with the property Then gt2 dt 1 00 2 2 00 2 2 1 lt r rm igltr dr 1 f 1 Gf 010216112 where tmfm denote average values of t and fand Gf is the Fourier transform of gt rm jzgr2 dt fm inGor dr Gabor s Proposal Short time Fourier Transform n The STl I39 is an attempt to alleviate the problems with FT Am litude El It takes a non stationary signal and breaks it down Time39 into windows of signals for a specified short period of time and does Fourier transform on the window by considering the signal to consist of repeated Time 39 windows over time STFT L r Frequency The Short time Fourier Transform Time frequency Resolution Small Window Large Window 6 gt lt 5t A 39 A E f v time The Short time Fourier Transform El Analysis STFTT u lfrw r 239ej2 dt n Synthesis ft j STFm uwt rej2 tdrdu El where wt is a window function localized in time and frequency a Analysis functions are sinusoids windowed by vvt a Common window functions I Gaussian Gabor Hamming Hanning Kaiser The Short time Fourier Transform Properties Linear transform El Time resolution At and frequency resolution Au are determined by wt and remain fixed regardless of Tor u El Biggest disadvantage l since At and Au are fixed by choice of wt need to know a priori what wt will work for the desired application Basic Idea of the Wavelet Transform Time frequency Resolution I Basic idea I At Au vary as a function of scale scale 1frequency 6t lt gt 6f I O O I O O O I O C O O i frequency time Wavelet Transformation El Ana lysis er wsfrgt f i ftwtTTdt 3350 El S Y S WT coefficient xt CW1 J LO Wm TwtTTdsdT cltlgtl where lt is the mother wavelet l admissibility condition Cw Iowwdfltoo EWsTZO Scaling n Scaling frequency band a Small scale I Rapidly changing details I Like high frequency a Large scale I Slowly changing details I Like low frequency Scale V l 1A 3A Wavelet Basis functions at 3 different scales More on Scale El It lets you either narrow down the frequency band of interest or determine the frequency content in a narrower time interval El Good for non stationary data El Low scale a Compressed wavelet Rapidly changing details High frequency El High scale 9a Stretched wavelet 9 Slowly changing coarse features 9 Low frequency Shifting o Shifting a wavelet simply means delaying or hastening its onset o Mathematically shifting a function ft by k is represented by ft k Wavelet function Shifted wavelet fu nation will wit kl The Wavelet Transform Properties El Linear transform a All analysis functions 3 wt T are shifts and dilations of the mother wavelets zt a Time resolution and frequency resolution vary as a function of scale El Continuous wavelet transform CWT I s and rare continuous El Discrete wavelet transform DWT I s and rare discrete Different Types of Mother Wavelets e we Meyer Daubechies W vew ve BattleLemarie ChuiWang Haar Calculate the CWT Coef cients El The result of the CWT are many wavelet coefficients WT 1 l T WT d m ltwsfrgt lftw S gt r l Function of scale and position El How to calculate the coefficient for each SCALE 3 for each POSITION t WT s t Signal x Wavelet s t end end Calculate the CWT Coef cients Take a wavelet and compare it to a section at the start of the original signal Calculate a correlation coefficient Shift the wavelet to the right and repeat steps 1 and 2 until you39ve covered the whole signal Scale stretch the wavelet and repeat steps 1 through 3 Repeat steps 1 through 4 for all scales Signal W av e let w WT 00102 Signal Wavelet Egt 6 Wavelet WT 02247 Discrete Wavelet Transform El Calculating wavelet coefficients at every possible scale is a fair amount of work and it generates an awful lot of data a What if we choose only a subset of scales and positions at which to make our calculations a It turns out rather remarkably that if we choose scales and positions based on powers of two so called dyadic scales and positions then our analysis will be much more efficient and just as accurate Discrete Wavelet Transform I If s T take discrete value in R2 we get DWT 1 A popular approach to select 5 T is 1 1 111 SZ m s02 gt sz mz 1 gt m integer so 2 2 4 8 mquot mO s02 101 121m 11 m integer s0 2 El So n t 1 t r m m m m WsrtlTZAW T2 WmnfZAl2 f n 2m Discrete Wavelet Transform n Wavelet Transform DWTm n lt foxyMr gt 24 few2m ndt n Inverse Wavelet Transform f0 22 DWTmwmltr echo in If ft is continuous while 5 T consists of discrete values the series is called the Discrete Time Wavelet Transform DTWT u If ft is sampled that is discrete in time as well as s TDWdiscrete we get Discrete Wavelet Transform 1 OStltl 2 114100 l1ot Jay2t i 1 1414 1414 wllt v2t 1 1 w2or2w4r 1 2 1414 W210 2W4t 1 W22I 2W4t 2 Haar Scaling function El The Haar transform uses a scaling function and wavelet functions El Scaling function I Calculate scaling function cltfltrlt1gtrgt gt lfltrlt1gtltrdr f 1 I Synthesis the original signal f0 Z Z Wall1m r echo Example of Fast DWT Haar Wavelet El Given input value l El Step 1 I Output Low Frequency 1 I Output High Frequency O5 O5 O5 O5 El Step 2 I Refine Low frequency output in Step 1 n L m El H El Step 3 I Refine Low frequency output in Step 2 El L 45 El H 2 Fast Discrete Wavelet Transform n Behaves like a filter bank signal in W39s coefficients out n RecILirsitye f l 1 a Ica Ion o a Jigsaw lter bank to the Iowpass band of the previous stage Filters Fast Discrete Wavelet Transform Step 1 Filter input with Step 2 Filter low band with f and Hf Lf and Hf d s a i m Ha L r E E lt lt a a 1m 12 0 18 1M 112 Normalized Frequency Normalized Frequency Step 3 Filter low band with Step 4 Filter low band with Lf and Hf Lf and Hf I 0 39U 39U 3 3 E lt 0 18 1I4 12 Normalized Frequency Normalized Frequency Fast Wavelet Transform Properties n Algorithm is very fast I On operations El Discrete wavelet transform is not shift invariant I A deficiency El Key to the algorithm is the design of hn and n I Can Mn and Kn be designed so that wavelets and scaling unctlon orm an orthonormal basns I Can filters of finite length be found I YES i Daubechies Family of Wavelets u Haar basis is a special case of Daubechies wavelet Daubechies Family of Wavelets Examples D4 Scaling function D4 Wavelet function 1 1 0 0 1 I I I I l 0 1 2 3 1 l 1 2 D10 scaling funwon D10 Wavelet function 10 1 05 0 00 1 I I I I I TWO Dimensional Transform low resolution subband horizontaf Vertical LL LH transform transform L H HL HH Transform Transform 3 detail each row each column subbands in L and H Two Dimensional Transform Continued LL LH HL HH horizontat transform gt Transform LLL HLL LH HL HH each row in LL 39 LLLL LHLL vertical H transform LLL HHLL gt HL HH Transform each cotumn in LLL and HLL 2 levels of transform gives 7 subbands k levels of transform gives 3k 1 subbands 1D Haar Wavelet El Mother Wavelet 1 1 O tlt 2 W030 1 1 tlt1 2 El Wavelet basis function 94 t 912139t i 1 1 0 tlt 2 W09 t Z 1 tlt1 A 1 W10 t W2t i 1 1 Wu t W2t 1 A W10 t 2 WV 05 1 W110 W4t 1 W22l ll4t 2 Wavelet Transformed Image Three levels of Wavelet transform l One low resolution subband I Nine detail subband 39
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