CHAPTER XI DATAPATH ELEMENTS
|
|
- Grant Underwood
- 6 years ago
- Views:
Transcription
1 CHAPTER XI- CHAPTER XI CHAPTER XI READ REE-DOC ON COURSE WEBPAGE
2 CHAPTER XI-2 INTRODUCTION -INTRODUCTION So far we have discussed may small compoets ad buildig blocks. Oe fial step i our buildig blocks before we ca start to piece together a microprocessor is various datapath elemets. We have already discussed portios of these datapath elemets i terms of other compoets ad buildig blocks. We will ow cosider some of these compoets ad buildig blocks i ways that will make the desig of a microprocessor a little easier i the ext chapter.
3 CHAPTER XI-3 REGISTER ILES REGISTER LAYOUT -INTRODUCTION A geeral m register file with m registers that are each -bits wide is illustrated below. Data I Register 0 Data Out w 0 Register r 0 w r Register m - w m r m r k w j The ad sigals idicate which register to read/write, respectively.
4 CHAPTER XI-4 REGISTER ILES WRITE DECODER REGISTER ILES -REGISTER LAYOUT or writig to a register, we iclude a write address with decoder. Write Address Write Eable Data I Decoder 0 m- w 0 w Register 0 Register Register m - r 0 r Data Out w m r m A give Write Address (with Write Eable = ) selects which register, 0 through m -, to store the iput from Data I.
5 CHAPTER XI-5 REGISTER ILES READ DECODER REGISTER ILES -REGISTER LAYOUT -WRITE DECODER or readig from a register, we iclude a read address with decoder. Data Data Register 0 I Out Read Address Read Eable Decoder 0 m- r 0 r r m w 0 w w m Register Register m - A give Read Address (with Read Eable = ) selects which register, 0 through m -, to read from ad output to Data Out. Could have multiple data outputs with multiple read address decoders.
6 CHAPTER XI-6 REGISTER ILES 32-BIT WORD, 32 REGISTERS REGISTER ILES -REGISTER LAYOUT -WRITE DECODER -READ DECODER or the upcomig datapath desigs i the ext chapter, we wat to have a 32x32 register file with oe write iput ad two read outputs. X ra - X read address Y ra - Y read address Z wa - Z write address X do - X data out Y do - Y data out Z di - Z data i rwe - register write eable Clk Z di Z wa X ra Y ra X do Y do Note: Two data outputs implemeted with two read address decoders. 32 rwe x32 register file 32 32
7 CHAPTER XI-7 ADDER/SUBTRACTOR GENERAL UNIT DIAGRAM REGISTER ILES -WRITE DECODER -READ DECODER -32X32 REGISTER ILE A -bit adder/subtractor uit is ofte illustrated as follows. Eable uit () or disable uit (0) A B adder/subtrator uit eable a s Select either additio (0) or subtractio () This uit would have full-adders iterally.
8 CHAPTER XI-8 ADDER/SUBTRACTOR OTHER UNIT SIGNALS REGISTER ILES ADDER/SUBTRACTOR -GENERAL UNIT DIAGRAM Other sigals ofte icluded with a adder/subtractor are show below. Carry-i or Borrow-i A C i eable C out Carry-out or Borrow-out a s B lags - Overflow - Negative (<0?) - Zero (=0?)
9 CHAPTER XI-9 LOGICAL UNIT INTRODUCTION REGISTER ILES ADDER/SUBTRACTOR -GENERAL UNIT DIAGRAM -OTHER UNIT SIGNALS A useful uit would be oe that ca take two -bit iputs ad perform some logical operatio betwee each of the bits to get a -bit output. or example, give the 8-bit values ad , we might wat to fid the bit-wise logical OR. bit-wise logical OR Or similarly, the bit-wise logical AND of the two 8-bit values. bit-wise logical AND These types of operatios are ofte used for maskig ad settig bits.
10 CHAPTER XI-0 LOGICAL UNIT GENERAL UNIT DIAGRAM REGISTER ILES ADDER/SUBTRACTOR LOGICAL UNIT -INTRODUCTION Below is a geeral uit diagram for a -bit logical uit. Eable uit () or disable uit (0) A logical uit eable L B 4 Logical uctio (L) o 2 bits Logical operatios, such as AND/OR/NOT/NAND/NOR/etc., are doe for each bit of A ad B to form.
11 CHAPTER XI- LOGICAL UNIT 4-BIT LOGICAL UNCTIONS (L) ADDER/SUBTRACTOR LOGICAL UNIT -INTRODUCTION -GENERAL UNIT DIAGRAM Recall the possible logic fuctios for two bits, A ad B. We ca use the colum as the 4-bit L iput for the logical uit. A B AB A B A + B B A AB Null A B A B Idetity Ihibitio A + B Implicatio
12 CHAPTER XI-2 LOGICAL UNIT BIT SLICE IMPLEMENTATION LOGICAL UNIT -INTRODUCTION -GENERAL UNIT DIAGRAM -4-BIT LOGICAL UNCTIONS A umber of iteral implemetatios exist for the logical uit. The easiest is to use a 4-to- multiplexer for each bit as follows Take colum from previous slide as L iput L 0 L L 2 L 3 Module Eable E 4X MULTIPLEXER S S 0 A B Require of these to form our -bit logical uit. Note: Whe you look at a desig for each bit, it is kow as a bit slice
13 CHAPTER XI-3 LOGICAL UNIT BIT SLICE IMPLEMENTATION LOGICAL UNIT -GENERAL UNIT DIAGRAM -4-BIT LOGICAL UNCTIONS -BIT SLICE IMPLEMENTAT. The followig are example L iputs for a logical uit bit slice. OR fuctio Module Eable E NAND fuctio Module Eable E A+ B X MULTIPLEXER AB X MULTIPLEXER S S 0 S S 0 A B A B
14 CHAPTER XI-4 SHIT UNIT INTRODUCTION LOGICAL UNIT -GENERAL UNIT DIAGRAM -4-BIT LOGICAL UNCTIONS -BIT SLICE IMPLEMENTAT. We have already discussed the bulk about shift uits i previous chapters. As give i the ree-doc, there are differet types of shift uits. Logical shift Arithmetic shift Circular shift (this is just a rotate uit) We wat to discuss a implemetatio, the barrel shifter, that will be useful i our sigle cycle datapath computer we will desig ext chapter.
15 CHAPTER XI-5 SHIT UNIT GENERAL UNIT DIAGRAM LOGICAL UNIT SHIT UNIT -INTRODUCTION Below is a geeral uit diagram for a -bit shift uit. -bit value to shift Eable uit () or disable uit (0) A shift uit eable +log 2 Notice that the -bit value A will be shifted accordig to the distace idicated with siged umber B. ST B 2 Distace of shift (siged #) + ive = right - ive = left Shift type 0 = logical = arithmetic 2 = rotate
16 CHAPTER XI-6 SHIT UNIT P-SHITER BIT SLICE LOGICAL UNIT SHIT UNIT -INTRODUCTION -GENERAL UNIT DIAGRAM Previously, we discussed the p-shifter but ot its implemetatio. A p-shifter shifts the value to the left or right by p-bits. A bit slice view of a p-shifter for th bit could be as follows. A +p s d S S 0 3 A -p A s 0 = o shift = shift 2 0 4X MULTIPLEXER E Notice that this ca ONLY shift by p-bits. It is hardwired to shift p-bits. Module Eable d 0 = left = right
17 CHAPTER XI-7 SHIT UNIT 2 K -SHITER BIT SLICE SHIT UNIT -INTRODUCTION -GENERAL UNIT DIAGRAM -P-SHITER BIT SLICE A useful type of p-shifter is whe p = 2 k for some positive iteger k. s d A +2 k S S 0 3 A -2 k 2 0 4X MULTIPLEXER A E Module Eable s 0 = o shift = shift d 0 = left = right A 2 k -shifter allows use to build a barrel shifter.
18 CHAPTER XI-8 SHIT UNIT BARREL SHITER SHIT UNIT -GENERAL UNIT DIAGRAM -P-SHITER BIT SLICE -2 K -SHITER BIT SLICE We wat to be able to shift a vector by a arbitrary distace istead of hardwired like the p-shifter ad 2 k -shifter. The top level ca shift A by bits, depedig o s. Subsequet levels ca shift result by /2 bits, depedig o their iput s q. s d s s 0 A 2 -shifter 2 -shifter 2 0 -shifter
19 CHAPTER XI-9 SHIT UNIT SAMPLE BARREL SHITER SHIT UNIT -P-SHITER BIT SLICE -2 K -SHITER BIT SLICE -BARREL SHITER We will do some examples with the followig arbitrary -shifter o a 6-bit iput. Note that this barrel shifter ca shift the iput by 5 bits i either directio. s 3 d A 6 s 2 6 s s shifter 2 2 -shifter 6 2 -shifter shifter 6
20 CHAPTER XI-20 SHIT UNIT BARREL SHITER: EXAMPLE # SHIT UNIT -2 K -SHITER BIT SLICE -BARREL SHITER -SAMPLE BARREL SHITER or example, cosider the iput of If we wat to shift this value to the left by 3, we eed the iput d = 0 s = (s 3 s 2 s s 0 ) = 0 Note: This example is for a logical shift. s 3 = d=0 s 2 = s =0 s 0 = shifter shifter shifter shifter
21 CHAPTER XI-2 SHIT UNIT BARREL SHITER: EXAMPLE #2 SHIT UNIT -BARREL SHITER -SAMPLE BARREL SHITER -BARREL SHITER EX. # As aother example, cosider the iput of If we wat to shift this value to the right by 6, we eed the iput d = s = (s 3 s 2 s s 0 ) = 00 Note: This example is for a logical shift. s 3 =0 d= s 2 = s = s 0 = shifter shifter shifter shifter
Chapter 9 Computer Design Basics
Logic ad Computer Desig Fudametals Chapter 9 Computer Desig asics Part Datapaths Charles Kime & Thomas Kamiski 008 Pearso Educatio, Ic. (Hyperliks are active i View Show mode) Overview Part Datapaths Itroductio
More informationEE260: Digital Design, Spring n Binary Addition. n Complement forms. n Subtraction. n Multiplication. n Inputs: A 0, B 0. n Boolean equations:
EE260: Digital Desig, Sprig 2018 EE 260: Itroductio to Digital Desig Arithmetic Biary Additio Complemet forms Subtractio Multiplicatio Overview Yao Zheg Departmet of Electrical Egieerig Uiversity of Hawaiʻi
More informationChapter 9 Computer Design Basics
Logic ad Computer Desig Fudametals Chapter 9 Computer Desig Basics Part 1 Datapaths Overview Part 1 Datapaths Itroductio Datapath Example Arithmetic Logic Uit (ALU) Shifter Datapath Represetatio Cotrol
More informationSignals & Systems Chapter3
Sigals & Systems Chapter3 1.2 Discrete-Time (D-T) Sigals Electroic systems do most of the processig of a sigal usig a computer. A computer ca t directly process a C-T sigal but istead eeds a stream of
More informationEE260: Digital Design, Spring n MUX Gate n Rudimentary functions n Binary Decoders. n Binary Encoders n Priority Encoders
EE260: Digital Desig, Sprig 2018 EE 260: Itroductio to Digital Desig MUXs, Ecoders, Decoders Yao Zheg Departmet of Electrical Egieerig Uiversity of Hawaiʻi at Māoa Overview of Ecoder ad Decoder MUX Gate
More informationFig. 7-6 Single Bus versus Dedicated Multiplexers
7- Select SSS2 LLL2 LLL2 S 2 to MUX R R Select S 2 to MUX R S S 3 to MUX 2 us R S 2 to MUX R2 R2 (a) Dedicated multiplexers (b) Sigle us Fig. 7-6 Sigle us versus Dedicated Multiplexers 2 Pretice Hall,
More informationChapter Vectors
Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it
More informationEECE 301 Signals & Systems
EECE 301 Sigals & Systems Prof. Mark Fowler Note Set #8 D-T Covolutio: The Tool for Fidig the Zero-State Respose Readig Assigmet: Sectio 2.1-2.2 of Kame ad Heck 1/14 Course Flow Diagram The arrows here
More informationProblem Set # 5 Solutions
MIT./8.4/6.898/8.435 Quatum Iformatio Sciece I Fall, 00 Sam Ocko October 5, 00 Problem Set # 5 Solutios. Most uitar trasforms are hard to approimate. (a) We are dealig with boolea fuctios that take bits
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 40 Digital Sigal Processig Prof. Mark Fowler Note Set #3 Covolutio & Impulse Respose Review Readig Assigmet: Sect. 2.3 of Proakis & Maolakis / Covolutio for LTI D-T systems We are tryig to fid y(t)
More informationLecture 2.5: Sequences
Lecture.5: Sequeces CS 50, Discrete Structures, Fall 015 Nitesh Saxea Adopted from previous lectures by Zeph Gruschlag Course Admi HW posted Covers Chapter Due Oct 0 (Tue) Mid Term 1: Oct 15 (Thursday)
More informationArithmetic Circuits. (Part I) Randy H. Katz University of California, Berkeley. Spring 2007
rithmetic Circuits (Part I) Rady H. Katz Uiversity of Califoria, erkeley prig 27 Lecture #23: rithmetic Circuits- Motivatio rithmetic circuits are excellet examples of comb. logic desig Time vs. pace Trade-offs
More informationLoad. Load. Load 1 0 MUX B. MB select. Bus A. A B n H select S 2:0 C S. G select 4 V C N Z. unit (ALU) G. Zero Detect.
9- Write D data Load eable A address A select B address B select Load R 2 2 Load Load R R2 UX 2 3 UX 2 3 2 3 Decoder D address 2 Costat i Destiatio select 28 Pearso Educatio, Ic.. orris ao & Charles R.
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationArithmetic Circuits. (Part I) Randy H. Katz University of California, Berkeley. Spring Time vs. Space Trade-offs. Arithmetic Logic Units
rithmetic rcuits (art I) Rady H. Katz Uiversity of Califoria, erkeley otivatio rithmetic circuits are excellet examples of comb. logic desig Time vs. pace Trade-offs Doig thigs fast requires more logic
More informationAdvanced Course of Algorithm Design and Analysis
Differet complexity measures Advaced Course of Algorithm Desig ad Aalysis Asymptotic complexity Big-Oh otatio Properties of O otatio Aalysis of simple algorithms A algorithm may may have differet executio
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationsubcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016
subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationRADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify
Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationEnd-of-Year Contest. ERHS Math Club. May 5, 2009
Ed-of-Year Cotest ERHS Math Club May 5, 009 Problem 1: There are 9 cois. Oe is fake ad weighs a little less tha the others. Fid the fake coi by weighigs. Solutio: Separate the 9 cois ito 3 groups (A, B,
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationHashing and Amortization
Lecture Hashig ad Amortizatio Supplemetal readig i CLRS: Chapter ; Chapter 7 itro; Sectio 7.. Arrays ad Hashig Arrays are very useful. The items i a array are statically addressed, so that isertig, deletig,
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationSummary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function
Summary: Cogrueces j whe divided by, ad determiig the additive order of a iteger mod. As described i the Prelab sectio, cogrueces ca be thought of i terms of coutig with rows, ad for some questios this
More information1. n! = n. tion. For example, (n+1)! working with factorials. = (n+1) n (n 1) 2 1
Biomial Coefficiets ad Permutatios Mii-lecture The followig pages discuss a few special iteger coutig fuctios You may have see some of these before i a basic probability class or elsewhere, but perhaps
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationThe Discrete Fourier Transform
The Discrete Fourier Trasform Complex Fourier Series Represetatio Recall that a Fourier series has the form a 0 + a k cos(kt) + k=1 b k si(kt) This represetatio seems a bit awkward, sice it ivolves two
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationCHAPTER 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
More information7. Modern Techniques. Data Encryption Standard (DES)
7. Moder Techiques. Data Ecryptio Stadard (DES) The objective of this chapter is to illustrate the priciples of moder covetioal ecryptio. For this purpose, we focus o the most widely used covetioal ecryptio
More informationComputing the output response of LTI Systems.
Computig the output respose of LTI Systems. By breaig or decomposig ad represetig the iput sigal to the LTI system ito terms of a liear combiatio of a set of basic sigals. Usig the superpositio property
More informationDiscrete-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 [
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More information(VII.A) Review of Orthogonality
VII.A Review of Orthogoality At the begiig of our study of liear trasformatios i we briefly discussed projectios, rotatios ad projectios. I III.A, projectios were treated i the abstract ad without regard
More informationECE 308 Discrete-Time Signals and Systems
ECE 38-5 ECE 38 Discrete-Time Sigals ad Systems Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa ECE 38-5 1 Additio, Multiplicatio, ad Scalig of Sequeces Amplitude Scalig: (A Costat
More informationChapter 9 - CD companion 1. A Generic Implementation; The Common-Merge Amplifier. 1 τ is. ω ch. τ io
Chapter 9 - CD compaio CHAPTER NINE CD-9.2 CD-9.2. Stages With Voltage ad Curret Gai A Geeric Implemetatio; The Commo-Merge Amplifier The advaced method preseted i the text for approximatig cutoff frequecies
More informationVector Spaces and Vector Subspaces. Remarks. Euclidean Space
Vector Spaces ad Vector Subspaces Remarks Let be a iteger. A -dimesioal vector is a colum of umbers eclosed i brackets. The umbers are called the compoets of the vector. u u u u Euclidea Space I Euclidea
More informationELEC1200: A System View of Communications: from Signals to Packets Lecture 3
ELEC2: A System View of Commuicatios: from Sigals to Packets Lecture 3 Commuicatio chaels Discrete time Chael Modelig the chael Liear Time Ivariat Systems Step Respose Respose to sigle bit Respose to geeral
More informationProblem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient
Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case
More informationUC Berkeley CS 170: Efficient Algorithms and Intractable Problems Handout 17 Lecturer: David Wagner April 3, Notes 17 for CS 170
UC Berkeley CS 170: Efficiet Algorithms ad Itractable Problems Hadout 17 Lecturer: David Wager April 3, 2003 Notes 17 for CS 170 1 The Lempel-Ziv algorithm There is a sese i which the Huffma codig was
More informationFinite Automata. Reading: Chapter 2
Fiite Automata Readig: Chapter 2 Fiite Automato (FA) Iformally, a state diagram that comprehesively captures all possible states ad trasitios that a machie ca take while respodig to a stream or sequece
More informationLecture 10: Universal coding and prediction
0-704: Iformatio Processig ad Learig Sprig 0 Lecture 0: Uiversal codig ad predictio Lecturer: Aarti Sigh Scribes: Georg M. Goerg Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationSigma notation. 2.1 Introduction
Sigma otatio. Itroductio We use sigma otatio to idicate the summatio process whe we have several (or ifiitely may) terms to add up. You may have see sigma otatio i earlier courses. It is used to idicate
More informationLinear time invariant systems
Liear time ivariat systems Alejadro Ribeiro Dept. of Electrical ad Systems Egieerig Uiversity of Pesylvaia aribeiro@seas.upe.edu http://www.seas.upe.edu/users/~aribeiro/ February 25, 2016 Sigal ad Iformatio
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationGeneric datapath. Generic datapath architecture. Register file. Register file. Calcolatori Elettronici e Sistemi Operativi.
alcolatori Elettroici e Sistei Operativi Geeric datapath architecture Set of registers Geeric datapath uber, size. addressig R[0], R[1],... Set of operatios arithetic, logic, shift Addressig (uber of operads,
More informationGenerating Functions. 1 Operations on generating functions
Geeratig Fuctios The geeratig fuctio for a sequece a 0, a,..., a,... is defied to be the power series fx a x. 0 We say that a 0, a,... is the sequece geerated by fx ad a is the coefficiet of x. Example
More informationDisjoint set (Union-Find)
CS124 Lecture 7 Fall 2018 Disjoit set (Uio-Fid) For Kruskal s algorithm for the miimum spaig tree problem, we foud that we eeded a data structure for maitaiig a collectio of disjoit sets. That is, we eed
More informationCOMPARISON OF FPGA IMPLEMENTATION OF THE MOD M REDUCTION
Lati America Applied Research 37:93-97 (2007) COMPARISON OF FPGA IMPLEMENTATION OF THE MOD M REDUCTION J-P. DESCHAMPS ad G. SUTTER Escola Tècica Superior d Egiyeria, Uiversitat Rovira i Virgili, Tarragoa,
More informationLecture 7: Fourier Series and Complex Power Series
Math 1d Istructor: Padraic Bartlett Lecture 7: Fourier Series ad Complex Power Series Week 7 Caltech 013 1 Fourier Series 1.1 Defiitios ad Motivatio Defiitio 1.1. A Fourier series is a series of fuctios
More informationMeasures of Spread: Standard Deviation
Measures of Spread: Stadard Deviatio So far i our study of umerical measures used to describe data sets, we have focused o the mea ad the media. These measures of ceter tell us the most typical value of
More informationWe start by describing a one bit memory circuit built of a couple of two inputs NAND gates.
Chapter 4: Sequetial Logic ( copyright by aiel Seider) Util ow we discussed circuits that are combiatioal. This meas that their outputs were fuctios of the iputs, ad oly the iputs, at all times. For each
More informationAnalysis of Experimental Measurements
Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example,
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationUnit 6: Sequences and Series
AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More information6.003: Signals and Systems. Feedback, Poles, and Fundamental Modes
6.003: Sigals ad Systems Feedback, Poles, ad Fudametal Modes February 9, 2010 Last Time: Multiple Represetatios of DT Systems Verbal descriptios: preserve the ratioale. To reduce the umber of bits eeded
More informationMath 299 Supplement: Real Analysis Nov 2013
Math 299 Supplemet: Real Aalysis Nov 203 Algebra Axioms. I Real Aalysis, we work withi the axiomatic system of real umbers: the set R alog with the additio ad multiplicatio operatios +,, ad the iequality
More informationLecture 2: April 3, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 2: April 3, 203 Scribe: Shubhedu Trivedi Coi tosses cotiued We retur to the coi tossig example from the last lecture agai: Example. Give,
More informationMath F215: Induction April 7, 2013
Math F25: Iductio April 7, 203 Iductio is used to prove that a collectio of statemets P(k) depedig o k N are all true. A statemet is simply a mathematical phrase that must be either true or false. Here
More informationUNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series
UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE
More informationSummer High School 2009 Aaron Bertram
Summer High School 009 Aaro Bertram 3 Iductio ad Related Stuff Let s thik for a bit about the followig two familiar equatios: Triagle Number Equatio Square Number Equatio: + + 3 + + = ( + + 3 + 5 + + (
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationFeedback in Iterative Algorithms
Feedback i Iterative Algorithms Charles Byre (Charles Byre@uml.edu), Departmet of Mathematical Scieces, Uiversity of Massachusetts Lowell, Lowell, MA 01854 October 17, 2005 Abstract Whe the oegative system
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationCS276A Practice Problem Set 1 Solutions
CS76A Practice Problem Set Solutios Problem. (i) (ii) 8 (iii) 6 Compute the gamma-codes for the followig itegers: (i) (ii) 8 (iii) 6 Problem. For this problem, we will be dealig with a collectio of millio
More informationLecture 2 Linear and Time Invariant Systems
EE3054 Sigals ad Systems Lecture 2 Liear ad Time Ivariat Systems Yao Wag Polytechic Uiversity Most of the slides icluded are extracted from lecture presetatios prepared by McClella ad Schafer Licese Ifo
More informationCALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
More informationIntroduction to Signals and Systems, Part V: Lecture Summary
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive
More informationVector Quantization: a Limiting Case of EM
. Itroductio & defiitios Assume that you are give a data set X = { x j }, j { 2,,, }, of d -dimesioal vectors. The vector quatizatio (VQ) problem requires that we fid a set of prototype vectors Z = { z
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationChapter Unary Matrix Operations
Chapter 04.04 Uary atrix Operatios After readig this chapter, you should be able to:. kow what uary operatios meas, 2. fid the traspose of a square matrix ad it s relatioship to symmetric matrices,. fid
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More information1. ARITHMETIC OPERATIONS IN OBSERVER'S MATHEMATICS
1. ARITHMETIC OPERATIONS IN OBSERVER'S MATHEMATICS We cosider a ite well-ordered system of observers, where each observer sees the real umbers as the set of all iite decimal fractios. The observers are
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationParallel Vector Algorithms David A. Padua
Parallel Vector Algorithms 1 of 32 Itroductio Next, we study several algorithms where parallelism ca be easily expressed i terms of array operatios. We will use Fortra 90 to represet these algorithms.
More information2D 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 informationIntensive Algorithms Lecture 11. DFT and DP. Lecturer: Daniel A. Spielman February 20, f(n) O(g(n) log c g(n)).
Itesive Algorithms Lecture 11 DFT ad DP Lecturer: Daiel A. Spielma February 20, 2018 11.1 Itroductio The purpose of this lecture is to lear how use the Discrete Fourier Trasform to save space i Dyamic
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationDesign of Digital Circuits Lecture 7: Sequential Logic Design. Prof. Onur Mutlu ETH Zurich Spring March 2018
Desig of Digital Circuits Lecture 7: Seuetial Logic Desig Prof. Our Mutlu ETH Zurich Sprig 2018 15 March 2018 Readigs Please study Slides 102-120 from Lecture 6 o your ow This week Seuetial Logic P&P Chapter
More informationComputability and computational complexity
Computability ad computatioal complexity Lecture 4: Uiversal Turig machies. Udecidability Io Petre Computer Sciece, Åbo Akademi Uiversity Fall 2015 http://users.abo.fi/ipetre/computability/ 21. toukokuu
More informationMatrices and vectors
Oe Matrices ad vectors This book takes for grated that readers have some previous kowledge of the calculus of real fuctios of oe real variable It would be helpful to also have some kowledge of liear algebra
More informationSIGNALS AND SYSTEMS I Computer Assignment 1
SIGNALS AND SYSTEMS I Computer Assigmet I MATLAB, sigals are represeted by colum vectors or as colums i matrices. Row vectors ca be used; however, MATLAB typically prefers colum vectors. Vector or matrices
More information, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)
Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso
More informationDiscrete-Time Signals and Systems. Discrete-Time Signals and Systems. Signal Symmetry. Elementary Discrete-Time Signals.
Discrete-ime Sigals ad Systems Discrete-ime Sigals ad Systems Dr. Deepa Kudur Uiversity of oroto Referece: Sectios. -.5 of Joh G. Proakis ad Dimitris G. Maolakis, Digital Sigal Processig: Priciples, Algorithms,
More informationLecture 11: Pseudorandom functions
COM S 6830 Cryptography Oct 1, 2009 Istructor: Rafael Pass 1 Recap Lecture 11: Pseudoradom fuctios Scribe: Stefao Ermo Defiitio 1 (Ge, Ec, Dec) is a sigle message secure ecryptio scheme if for all uppt
More informationQuantum Computing Lecture 7. Quantum Factoring
Quatum Computig Lecture 7 Quatum Factorig Maris Ozols Quatum factorig A polyomial time quatum algorithm for factorig umbers was published by Peter Shor i 1994. Polyomial time meas that the umber of gates
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationOverview EECS Components and Design Techniques for Digital Systems. Lec 15 Addition, Subtraction, and Negative Numbers. Positional Notation
Overview EEC 5 Compoets ad Desig Techiques for Digital ystems Lec 5 dditio, ubtractio, ad Negative Numbers David Culler Electrical Egieerig ad Computer cieces Uiversity of Califoria, erkeley Recall basic
More information