Digital Signal Processing, Fall 2006
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1 Digitl Sigl Processig, Fll 6 Lecture 6: Sstem structures for implemettio Zeg-u T Deprtmet of Electroic Sstems Alorg Uiversit, Demr t@om.u.d Digitl Sigl Processig, VI, Zeg-u T, 6 Course t glce Discrete-time sigls d sstems Sstem Fourier-domi represettio Smplig d recostructio Sstem lsis 5 Sstem structure 6 Filter desig -trsform DFTFFT 7, 8 3 9, Digitl Sigl Processig, VI, Zeg-u T, 6
2 Sstem implemettio LTI sstems it rtiol sstem fuctio e.g., > Impulse respose u u Lier costt-coefficiet differece equtio Tree equivlet represettios! o to implemet, i.e. covert to lgoritm or structure? 3 Digitl Sigl Processig, VI, Zeg-u T, 6 Sstem implemettio Te iput-output trsformtio c e computed i differet s ec is clled implemettio A implemettio is specific descriptio of its iterl computtiol structure Te coice of implemettio cocers it computtiol requiremets memor requiremets, effects of fiite-precisio, d so o Digitl Sigl Processig, VI, Zeg-u T, 6
3 Prt I: Bloc digrm represettio Bloc digrm represettio of computtiol structures Sigl flo grp descriptio Bsic structures for IIR sstems Trsposed forms Bsic structures for FIR sstems 5 Digitl Sigl Processig, VI, Zeg-u T, 6 Sstem implemettio Impulse respose u * u is ifiite-durtio, impossile to implemet i tis. oever, lier costt-coefficiet differece equtio provides mes for recursive computtio of te output 6 Digitl Sigl Processig, VI, Zeg-u T, 6 3
4 Bsic elemets Implemettio sed o te recurrece formul derived from differece equtio requires dders multipliers memor for storig deled sequece vlues Fig Digitl Sigl Processig, VI, Zeg-u T, 6 Emple of loc digrm represettio A secod-order differece equtio Demostrtes te compleit, te steps, te mout of resources required. 8 Digitl Sigl Processig, VI, Zeg-u T, 6
5 5 Digitl Sigl Processig, VI, Zeg-u T, 6 9 Geerl t-order differece equtio v A cscde of to sstems! X v, v Digitl Sigl Processig, VI, Zeg-u T, 6 Rerrgemet of loc digrm A loc digrm c e rerrged i m s itout cgig overl fuctio, e.g. reversig te order of te to cscded sstems.
6 6 Digitl Sigl Processig, VI, Zeg-u T, 6 Sstem fuctio decompositio v Digitl Sigl Processig, VI, Zeg-u T, 6 I te time domi v v
7 iimum del implemettio Oe ig differece t te to implemettios cocers te umer of del elemets m, 3 Digitl Sigl Processig, VI, Zeg-u T, 6 Direct form I d II Direct form I s so i Fig. 6.3 A direct relitio of te differece equtio Direct form II or coic direct form s so i Fig. 6.5 Tere is direct li etee te sstem fuctio differece equtio d te loc digrm Digitl Sigl Processig, VI, Zeg-u T, 6 7
8 A emple Direct form I d direct form II implemettio Digitl Sigl Processig, VI, Zeg-u T, 6 Prt II: Sigl flo grp descriptio Bloc digrm represettio of computtiol structures Sigl flo grp descriptio Bsic structures for IIR sstems Trsposed forms Bsic structures for FIR sstems 6 Digitl Sigl Processig, VI, Zeg-u T, 6 8
9 Sigl flo grp SFG As ltertive to loc digrms it fe ottiol differeces. A etor of directed rces coectig odes. vrile 7 Digitl Sigl Processig, VI, Zeg-u T, 6 Sigl flo grp odes i SFG represet ot rcig poits d dders depedig o te umer of icomig rces, ile i te digrm specil smol is used for dders d ode s ol oe icomig rc. SGF is simpler to dr. 8 Digitl Sigl Processig, VI, Zeg-u T, 6 9
10 Digitl Sigl Processig, VI, Zeg-u T, 6 9 From flo grp to sstem fuctio Fig. 6. ot direct form, cot oti ispectio. But c rite equtio for ec ode ivolve feedc, difficult to solve B -trsform lier equtios 3 3 α 3 Digitl Sigl Processig, VI, Zeg-u T, 6 From flo grp to sstem fuctio 3 3 Y X X α Y X X α u u X Y α α α α α α te sstem is? rel, is If α All-pss Cusl!
11 From flo grp to sstem fuctio Fig. 6.3 Fig. 6. Differet implemettios, differet mouts of computtiol resources Digitl Sigl Processig, VI, Zeg-u T, 6 Prt III: Bsic structures for IIR sstems Bloc digrm represettio of computtiol structures Sigl flo grp descriptio Bsic structures for IIR sstems Trsposed forms Bsic structures for FIR sstems Digitl Sigl Processig, VI, Zeg-u T, 6
12 Digitl Sigl Processig, VI, Zeg-u T, 6 3 Direct form I Digitl Sigl Processig, VI, Zeg-u T, 6 Direct form II
13 3 Digitl Sigl Processig, VI, Zeg-u T, 6 5 Emple.5.75 Digitl Sigl Processig, VI, Zeg-u T, 6 6 Cscde form Fctor te umertor d deomitor polomils * * d d c g g f A s
14 Digitl Sigl Processig, VI, Zeg-u T, 6 7 A emple: from d -order to st-order cscde Digitl Sigl Processig, VI, Zeg-u T, 6 8 Prllel form prtil frctio epsio * d d e B c A C p
15 Feedc i IIR sstems u Feedc loop: closed pt ecessr ut ot sufficiet coditio for IIR sstem Feedc itroduced poles couldeccelled eros All loops must coti t lest oe uit del elemet 9 Digitl Sigl Processig, VI, Zeg-u T, 6 Prt IV: Trsposed forms Bloc digrm represettio of computtiol structures Sigl flo grp descriptio Bsic structures for IIR sstems Trsposed forms Bsic structures for FIR sstems 3 Digitl Sigl Processig, VI, Zeg-u T, 6 5
16 Trsposed form for first-order sstem Flo grp reversl or trspositio lso provides ltertives: reversig te directios of ll rces d reversig te iput d output Resultig i sme 3 Digitl Sigl Processig, VI, Zeg-u T, 6 Trsposed direct form II d direct form II Te trsposed direct form II implemets te eros first d te te poles, eig importt effect for fiite-precisio eistig 3 Digitl Sigl Processig, VI, Zeg-u T, 6 6
17 Prt V: Bsic structures for FIR sstems Bloc digrm represettio of computtiol structures Sigl flo grp descriptio Bsic structures for IIR sstems Trsposed forms Bsic structures for FIR sstems 33 Digitl Sigl Processig, VI, Zeg-u T, 6 Direct form So fr, sstem fuctio s ot poles d eros. FIR sstems s specil cse. Cusl FIR sstem fuctio s ol eros ecept for poles s,,,...,, oterise Form I d form II re te sme. 3 Digitl Sigl Processig, VI, Zeg-u T, 6 7
18 8 Digitl Sigl Processig, VI, Zeg-u T, 6 35 Cscde form Fctorig te polomil sstem fuctio s Digitl Sigl Processig, VI, Zeg-u T, 6 36 Lier-pse FIR sstems Geerlied lier-pse sstem Cusl FIR sstems ve geerlied lier-pse if stisfies te smmetr coditio,,...,, or,,...,, costts re rel d, fuctio of rel is β α ω ω β ωα ω ω j j j j j e A e e A e
19 9 Digitl Sigl Processig, VI, Zeg-u T, 6 37 Lier-pse FIR sstems if is eve iteger if Digitl Sigl Processig, VI, Zeg-u T, 6 38 Lier pse FIR sstems is eve iteger d -
20 Discussios Implemettio of FIR d IIR sstems Use sigl loc digrm flo grp represettio to so te computtiol structures Altoug to structures m ve equivlet iputoutput crteristics for ifiite-precisio represetios of coefficiets d vriles, te m ve drmticll differet eviour e te umericl precisio is limited. 39 Digitl Sigl Processig, VI, Zeg-u T, 6 Summr Bloc digrm represettio of computtiol structures Sigl flo grp descriptio Bsic structures for IIR sstems Trsposed forms Bsic structures for FIR sstems Digitl Sigl Processig, VI, Zeg-u T, 6
21 Course t glce Discrete-time sigls d sstems Sstem Fourier-domi represettio Smplig d recostructio Sstem lsis 5 Sstem structure 6 Filter desig -trsform DFTFFT 7, 8 3 9, Digitl Sigl Processig, VI, Zeg-u T, 6
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Exam. Notes: A single A4 sheet of paper (double sided; hand-written or computer typed)
Exam February 8th, 8 Sigals & Systems (5-575-) Prof. R. D Adrea Exam Exam Duratio: 5 Mi Number of Problems: 5 Number of Poits: 5 Permitted aids: Importat: Notes: A sigle A sheet of paper (double sided;
Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
CHAPTER 1 INTRODUCTION NUMBER SYSTEMS AND CONVERSION
Fudmetls of Logic Desig Chp. CHAPTE /9 INTODUCTION NUMBE SYSTEMS AND CONVESION This chpter i the book icludes: Objectives Study Guide. Digitl Systems d Switchig Circuits. Number Systems d Coversio. Biry
(1) Functions A relationship between two variables that assigns to each element in the domain exactly one element in the range.
-. ALGEBRA () Fuctios A reltioship etwee two vriles tht ssigs to ech elemet i the domi ectly oe elemet i the rge. () Fctorig Aother ottio for fuctio of is f e.g. Domi: The domi of fuctio Rge: The rge of
denominator, think trig! Memorize the following two formulas; you will use them often!
7. Bsic Itegrtio Rules Some itegrls re esier to evlute th others. The three problems give i Emple, for istce, hve very similr itegrds. I fct, they oly differ by the power of i the umertor. Eve smll chges
Elementary Linear Algebra
Elemetry Lier Alger Ato & Rorres, th Editio Lecture Set Chpter : Systems of Lier Equtios & Mtrices Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules
COMM 602: Digital Signal Processing
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10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
Titus Beu University Babes-Bolyai Department of Theoretical and Computational Physics Cluj-Napoca, Romania
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[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
![[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.](/thumbs/78/77732161.jpg "[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.")
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
Repeated Root and Common Root
Repeted Root d Commo Root 1 (Method 1) Let α, β, γ e the roots of p(x) x + x + 0 (1) The α + β + γ 0, αβ + βγ + γα, αβγ - () (α - β) (α + β) - αβ (α + β) [ (βγ + γα)] + [(α + β) + γ (α + β)] +γ (α + β)
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. (2014 Admn. onwards) III Semester. B.Sc. Mathematics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION (0 Adm. owrds) III Semester B.Sc. Mthemtics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY Questio Bk & Aswer Key. l l () =... 0.00 b) 0 c). l d =... c
Filter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and
Filter bas Separately, the lowpass ad highpass filters are ot ivertible T removes the highest frequecy / ad removes the lowest frequecy Together these filters separate the sigal ito low-frequecy ad high-frequecy
Z-Transform of a discrete time signal x(n) is defined as the power series
Z-Trsform of discrete time sigl x is defied s the ower series x 3.. This reltio lso clled direct Z-Trsform. Z[ x ] 3.. x 3..3 Regio of covergece ROC of is the set of ll vlues of for which ttis fiite vlue.
The total number of permutations of S is n!. We denote the set of all permutations of S by
DETERMINNTS. DEFINITIONS Def: Let S {,,, } e the set of itegers from to, rrged i scedig order. rerrgemet jjj j of the elemets of S is clled permuttio of S. S. The totl umer of permuttios of S is!. We deote
DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1
NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit
Improving XOR-Dominated Circuits by Exploiting Dependencies between Operands. Ajay K. Verma and Paolo Ienne. csda
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Lincoln Land Community College Placement and Testing Office
Licol Ld Commuity College Plcemet d Testig Office Elemetry Algebr Study Guide for the ACCUPLACER (CPT) A totl of questios re dmiistered i this test. The first type ivolves opertios with itegers d rtiol
Chapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION
School Of Distce Eductio Questio Bk UNIVERSITY OF ALIUT SHOOL OF DISTANE EDUATION B.Sc MATHEMATIS (ORE OURSE SIXTH SEMESTER ( Admissio OMPLEX ANALYSIS Module- I ( A lytic fuctio with costt modulus is :
z-transform A generalization of the DTFT defined by
The DTFT provides frequecy-domi represettio of discrete-time sigs d LTI discrete-time systems Becuse of the covergece coditio, i my cses, the DTFT of sequece my ot exist As resut, it is ot possie to mke
Approximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
Discrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg
ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced
SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum
f(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
Chapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
Numerical Solution of Fuzzy Fredholm Integral Equations of the Second Kind using Bernstein Polynomials
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MULTIRESOLUTION TIME-DOMAIN ANALYSIS OF MICROWAVE STRUCTURES USING MATLAB. Ondřej Franek, Zbyněk Raida
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BITSAT MATHEMATICS PAPER. If log 0.0( ) log 0.( ) the elogs to the itervl (, ] () (, ] [,+ ). The poit of itersectio of the lie joiig the poits i j k d i+ j+ k with the ple through the poits i+ j k, i
CITY UNIVERSITY LONDON
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Advanced Algorithmic Problem Solving Le 6 Math and Search
Advced Algorithmic Prolem Solvig Le Mth d Serch Fredrik Heitz Dept of Computer d Iformtio Sciece Liköpig Uiversity Outlie Arithmetic (l. d.) Solvig lier equtio systems (l. d.) Chiese remider theorem (l.5
Math 3B Midterm Review
Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.1-7.8 i your ebook. The best idictio of wht
Section IV.6: The Master Method and Applications
Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio
Northwest High School s Algebra 2
Northwest High School s Algebr Summer Review Pcket 0 DUE August 8, 0 Studet Nme This pcket hs bee desiged to help ou review vrious mthemticl topics tht will be ecessr for our success i Algebr. Istructios:
FIR Filter Design: Part II
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we cosider how we might go about desigig FIR filters with arbitrary frequecy resposes, through compositio of multiple sigle-peak
Sect Simplifying Radical Expressions. We can use our properties of exponents to establish two properties of radicals: and
128 Sect 10.3 - Simplifyig Rdicl Expressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of expoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( Multiplictio d
THEORY OF EQUATIONS SYNOPSIS. Polyomil Fuctio: If,, re rel d is positive iteger, the f)x) = + x + x +.. + x is clled polyomil fuctio.. Degree of the Polyomil: The highest power of x for which the coefficiet
ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
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Introduction to Matrix Algebra
Itrodutio to Mtri Alger George H Olso, Ph D Dotorl Progrm i Edutiol Ledership Applhi Stte Uiversit Septemer Wht is mtri? Dimesios d order of mtri A p q dimesioed mtri is p (rows) q (olums) rr of umers,