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- EECS150 Digital Design Lecture 17 Memory 2 October 22, 2002 John Wawrzynek Fall 2002 EECS150 Lec17-mem2 Page 1 SDRAM Recap General Characteristics Optimized for high density and therefore low cost/bit Special fabrication process usually on a separate chip from processor Needs periodic refresh (in most applications) Relatively slow because: High capacity leads to large cell arrays with high word- and bit-line capacitance Complex read/write cycle. Read needs precharge and write-back word line DRAM bit cell bit line Multiple clock cycles per read or write access Multiple reads and writes are often grouped together to amortize overhead. Referred to as bursting . Fall 2002 EECS150 Lec17-mem2 Page 2 Functional Block Diagram 8 Meg x 16 SDRAM Fall 2002 EECS150 Lec17-mem2 Page 3 SDRAM Details Multiple banks of cell arrays are used to reduce access time: Each bank is 4K rows by 512 columns by 16 bits (for our part) ACTIVE command opens a row for operation transfers the contents of the entire to a row buffer Read and Write operations as split into RAS (row access) followed by CAS (column access) These operations are controlled by sending commands Commands are sent using the RAS, CAS, CS, & WE pins. Subsequent READ or WRITE commands modify the contents of the row buffer. For burst reads and writes during READ or WRITE the starting address of the block is supplied. Burst length is programmable as 1, 2, 4, 8 or a full page (entire row) with a burst terminate option. Address pins are time multiplexed During RAS operation, address lines select the bank and row During CAS operation, address lines select the column. Special commands are used for initialization (burst options etc.) A burst operation takes 4 + n cycles (for n words) Page 4 Fall 2002 EECS150 Lec17-mem2 READ burst (with auto precharge) Fall 2002 EECS150 Lec17-mem2 Page 5 WRITE burst (with auto precharge) See datasheet for more details. Fall 2002 Verilog simulation models available. Page 6 EECS150 Lec17-mem2 First-in-first-out (FIFO) Memory Used to implement queues. These find common use in computers and communication circuits. Generally, used for rate matching data producer and consumer: stating state after write after read Producer can perform many writes without consumer performing any reads (or vis versa). However, of because finite buffer size, on average, need equal number of reads and writes. Typical uses: interfacing I/O devices. Example network interface. Data bursts from network, then processor bursts to memory buffer (or reads one word at a time from interface). Operations not synchronized. Example: Audio output. Processor produces output samples in bursts (during process swap-in time). Audio DAC clocks it out at constant sample rate. Page 7 Fall 2002 EECS150 Lec17-mem2 FIFO Interfaces DIN RST CLK WE FULL HALF FULL EMPTY RE FIFO Address pointers pointers are used internally to keep next write position and next read position into a dualport memory. write ptr read ptr DOUT After write or read operation, FULL and EMPTY indicate status of buffer. Used by external logic to control own reading from or writing to the buffer. FIFO resets to EMPTY state. HALF FULL (or other indicator of partial fullness) is optional. Fall 2002 If pointers equal after write FULL: write ptr read ptr If pointers equal after read EMPTY: write ptr read ptr EECS150 Lec17-mem2 Page 8 FIFO Implementation Details Assume, dual-port memory with asynchronous read, synchronous write. Binary counter for each of read and write address. CEs controlled by WE and RE. Equal comparator to see when pointers match. Flip-flop each for FULL and EMPTY flags: WE RD equal EMPTYi FULLi 0 0 0 0 0 0 0 1 EMPTYi-1 FULLi-1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 1 1 1 EMPTYi-1 FULLi-1 With this memory, read happens before write. Therefore if WE and RD are asserted: When FULL, correct operation will happen. When EMPTY, would like to write before read. Correct this case with a bypass mux: DIN DOUT 1 0 Da memory block Db EMPTY RD WR Fall 2002 EECS150 Lec17-mem2 Page 9 Xilinx BlockRam Versions Our simple version has latency problems. FULL and EMPTY signals are asserted near the end of the clock period. Xilinx version solves this by predicting when full or empty will be asserted on next write/read operation. Also uses BlockRam with synchronous reads. Available on Xilinx website along with app note application note. These are linked to our page. Two versions available. Easy to modify to change the width if necessary. Will use the cc (common clock) version for checkpoint 2. Can use the ic (independent clock) version later for bridging network to video. Fall 2002 EECS150 Lec17-mem2 Page 10
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Berkeley >> CS >> 150 (Fall, 1996)
EECS150 Components and Design Techniques for Digital Systems John Wawrzynek (Warznek) Professor of EECS johnw@eecs.berkeley.edu 643-9434 631 Soda Hall Office hours: Tu, Th 1-2 Teaching Assistants: Vinay Krishnan Liang Teck Pang Yury Markovskiy Yatish...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 - Digital Design Lecture 13 - Combinational Logic & Arithmetic Circuits Part 3 October 8, 2002 John Wawrzynek Fall 2002 EECS150 - Lec13-cla3 Page 1 Multiplication a3 b3 X a3b1 a2b2 a1b3 a3b0 a2b1 a1b2 a0b3 . a2 b2 a2b0 a1b1 a0b2 a1 b1 a1b...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 - Digital Design Lecture 20 - Finite State Machines 2 October 31, 2002 John Wawrzynek Fall 2002 EECS150 Lec20-fsm2 Page 1 Outline Moore versus Mealy style state machines. FSM optimization State Reduction State Assignment Fall 2002 ...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 - Digital Design Lecture 23 - High-level Design and Optimization 3, Parallelism and Pipelining Nov 12, 2002 John Wawrzynek Fall 2002 EECS150 - Lec23-HL3 Page 1 Parallelism Parallelism is the act of doing more than one thing at a time. Opt...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 - Digital Design Lecture 27 - Asynchronous Sequential Circuits Nov 26, 2002 John Wawrzynek Fall 2002 EECS150 - Lec27-asynch Page 1 Outline SR Latches and other storage elements Synchronizers Figures from Digital Design, John F. Wakerly ...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 - Digital Design Lecture 11 - Combinational Logic Circuits Part 1 - Adders October 1, 2002 John Wawrzynek Fall 2002 EECS150 - Lec11-cl1 Page 1 Adders Full-adder cell (FA) revisited: a bcin a b cin cout s 000 0 0 001 0 1 010 0 1 011 1 0 10...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 - Digital Design Lecture 3 - Timing September 3, 2002 John Wawrzynek Fall 2002 EECS150 - Lec03-Timing Page 1 Outline Finish up from lecture 2 General Model of Synchronous Systems Performance Limits Announcements Delay in logic...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 - Digital Design Lecture 8 - Hardware Description Languages September 19, 2002 John Wawrzynek Fall 2002 EECS150 - Lec08-HDL Page 1 Outline Netlists Design flow What is a HDL? Verilog history examples Fall 2002 EECS150 - Lec08-HDL ...
Berkeley >> CS >> 150 (Fall, 1996)
Parallelism Parallelism is the act of doing more than one thing at a time. Optimization in hardware design often involves using parallelism to trade between cost and performance. EECS150 - Digital Design Lecture 23 - High-level Design and Optimizati...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 Spring 2008 Checkpoint 2.5 UNIVERSITY OF CALIFORNIA AT BERKELEY COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE CHECKPOINT 2.5 FOUR PORT ARBITER AND USER INTERFACE 1.0 MOTIVATION Please note that this check...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150: Lab 1, FPGA CAD Tools UC Berkeley College of Engineering Department of Electrical Engineering and Computer Science 1 Time Table Friday, August 29th Week 3: September 7th 13th , 10 minutes after your lab section starts ASSIGNED DUE 2 Mo...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 - Digital Design Lecture 4 - Verilog Introduction Feb 3, 2009 John Wawrzynek Spring 2009 EECS150 - Lec05-Verilog Page 1 Outline Background and History of Hardware Description Brief Introduction to Verilog Basics Lots of examples struc...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 Homework4Solutions Fall2008 Problem1:Youhavea100MHzclock,andneedtogenerate3separateclocksat differentfrequencies:20MHz,1kHz,and1Hz.Howmanyflipflopsdoyouneedto implementeachclockifyouuse: a)aringcounter? b)abinarycounter? Showanexampleofeac...
Berkeley >> CS >> 150 (Fall, 1996)
University of California at Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences EECS 150 Fall 2000 R. Katz N. Zhou Checkpoint 4 Packet Transmission Reception 1 Objective For this checkpoint you will make additi...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 Midterm2Solutions Fall2008 1)Shortanswerquestionsontheproject: a) GiventhefollowingsynthesismessagefromSynplifyPro: @W:CG133:\"C:\\Test.v\":26:15:26:19|NoassignmenttoIfOut Isthissynthesismessageanote,awarning,oranerror?WhatlineinTest.v trig...
Berkeley >> CS >> 150 (Fall, 1996)
EECS 150 - Components and Design Techniques for Digital Systems FSMs 9/11/2007 Sarah Bird Electrical Engineering and Computer Sciences University of California, Berkeley Slides borrowed from David Culler Fa04 and Randy Katz Sp07 Sequential Logic Im...
Berkeley >> CS >> 150 (Fall, 1996)
Endgame Today: Miscellaneous 1 clock skew multi-port memory FIFOs, implementation version 1 EECS150 - Digital Design Lecture 26 - Miscellaneous 1 April 30, 2002 John Wawrzynek Thursday 5/2: Miscellaneous 2 FIFO version 2 LFSR Division ...
Berkeley >> CS >> 150 (Fall, 1996)
EECS150 Spring 2002 Lab 2 Logic Gates UNIVERSITY OF CALIFORNIA AT BERKELEY COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE Lab 2 Logic Gates 1 Motivation In this lab you will get to build and test a simple combinat...
Berkeley >> CS >> 150 (Fall, 1996)
University of California at Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences CS 150 Fall 1998 Lab 7 Wire-Wrap and SRAMS J. Wawrzynek and N. Weaver Later revisions by R. Fearing, X. Zhang, and B. Choi 1 Obj...
Berkeley >> CS >> 150 (Fall, 1996)
8QLYHUVLW\\ RI ROOHJH RI (QJLQHHULQJ \'HSDUWPHQW RI (OHFWULFDO (QJLQHHULQJ DQG 6 6SULQJ - :DZU]\\QHN DQG 1 :HDYHU 5HYLVHG E\\ 5 )HDULQJ DQG & (QJ /DE 1DVW\\ 5HDOLWLHV 2EMHFWLYHV )RU WKLV ODE \\RX ZLOO OHDUQ...
Berkeley >> CS >> 150 (Fall, 1996)
University of California at Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences EECS150 Fall 1998 R. Fearing and Kevin Cho Checkpoint 2 Video Interface 1. Objective In this checkpoint, you will: 1. Wire-up the ...
Berkeley >> CS >> 150 (Fall, 1996)
University of California at Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences EECS 150 Fall 1998 Checkpoint 1 Serial Transmitter J. Wawrzynek and N. Weaver Later revisions by R. Fearing, J. Shih and D. Chinn...
Berkeley >> CS >> 150 (Fall, 1996)
Problem 7.12 The sequence for the Johnson counter gives a state transition table which looks like this: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q3 Q2 Q1 Q0 Q3+ Q2+ Q1+ Q0+ 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 x x x x 0 0 1 1 0 0 0 1 0 1 0 0 x x x ...
Berkeley >> CS >> 4 (Fall, 2008)
Problem 7.12 The sequence for the Johnson counter gives a state transition table which looks like this: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q3 Q2 Q1 Q0 Q3+ Q2+ Q1+ Q0+ 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 x x x x 0 0 1 1 0 0 0 1 0 1 0 0 x x x ...
Berkeley >> CS >> 150 (Fall, 1996)
Outline m Last time: Introduction to number systems: sign/magnitude, ones complement, twos complement Review of latches , flip flops, counters Clocked Synchronous Finite-State Machines m Example: Consider the student association coffee vending mac...
Berkeley >> CS >> 150 (Fall, 1996)
Problem set 4 solutions 7.10 Were designing a counter that counts from 0 to 9 and returns to 0, so well implement with 4 bits, and encode the states in binary. Because we only need 10 states, well have six rows of dont care, which will make the imple...
Berkeley >> CS >> 4 (Fall, 2008)
Problem set 4 solutions 7.10 Were designing a counter that counts from 0 to 9 and returns to 0, so well implement with 4 bits, and encode the states in binary. Because we only need 10 states, well have six rows of dont care, which will make the imple...
Berkeley >> CS >> 150 (Fall, 1996)
3.2a) F= a\\b\\c+\\ac+\\ab A \\B \\C \\A C \\A B 3.2b) F=(\\a+\\b+\\c)(\\a+\\b) (\\a+\\c) \\A \\B \\C \\A \\B A B C A B \\A \\C A C 3.2c) F=\\ab+a+\\c+\\d \\A B A \\C \\D \\A B \\A C D ...
Berkeley >> CS >> 4 (Fall, 2008)
3.2a) F= a\\b\\c+\\ac+\\ab A \\B \\C \\A C \\A B 3.2b) F=(\\a+\\b+\\c)(\\a+\\b) (\\a+\\c) \\A \\B \\C \\A \\B A B C A B \\A \\C A C 3.2c) F=\\ab+a+\\c+\\d \\A B A \\C \\D \\A B \\A C D ...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Part A: Denitions Midterm Solutions Fall 2008 (a) A matrix P Cnn is called positive denite if (i) P = P (Hermitian) and (ii) for all v = 0, we have v P v > 0. (b) The Cauchy-Schwartz inequality asserts that is an inner-product space ...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A 1 (a) Let f (x, t) = x1 + et cos(x1 x2 ) x2 + 15 sin(x1 x2 ) Solutions # 5 (1) then, given a xed time t, the Jacobian of f is: J(x) = 1 et sin(x1 x2 ) et sin(x1 x2 ) 15 cos(x1 x2 ) 1 15 cos(x1 x2 ) (2) and its innity induced no...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Issued: September 5, 2008 Assignment # 1 Due: September 12, 2008 1 Vector Spaces Determine conclusively which of the following are elds: (a) GLn = the set of all n n nonsingular matrices. (b) GF2 = the set {0, 1} with addition being bin...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A 1 Consider the system x = Ax + Bu, where: 3 6 13 7 B= 1 A = 2 4 2 1 2 4 3 (a) Lets compute the controllabitlity matrix: 3 2 5 C = B AB A2 B = 1 0 2 1 1 1 thus rank(C) = 3 and the pair (A, B) is controllable. Solutions # 9 ...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A A State Feedback B Observers C Output Feedback D Integral Error Feedback E Gain Scheduling LTI Systems: Feedback Aspects 1 A. State Feedback 1 Introduction We begin with the transfer function H(s) of some m input, p output linear time...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Issued: September 17, 2008 Assignment # 2 Due: September 25, 2008 1 Linear Matrix Equations. Let A Cmn , B Cnq , C Cmn , and D Cnq . (a) When is the matrix equation AX = C solvable for X Cnn . When is the solution unique? (b) When i...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Solutions # 8 1 Consider a linear time varying system R, completely controlable on [t0 , t1 ]. Let s0 t0 and s1 t1 , we want to show that R is completely controllable on [s0 , s1 ]. Fix x0 , x1 Rn , and assume is the state transition...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Issued: October 28, 2008 Assignment # 5 Due: November 6, 2008 1 Existence and uniqueness of solutions to dierential equations. Consider the following two systems of dierential equations: x1 = x1 + et cos(x1 x2 ) x2 = x2 + 15 sin(x1 x2...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Solutions # 4 1 (a) By the Spectral Mapping Theorem, the eigenvalues of f (A) are at f () including multiplicity. Thus, the eigenvalues of (I + A) are at 1 + A = 0, proving the claim. (b) Again, using the Spectral Mapping Theorem, the ei...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A 1 Recall that the pair (A, C) is completely observable if and only if: rank A I C =n Solutions # 7 (1) for each (A), i.e. if v is an eigenvector of A then Cv = 0. Suppose there exists a positive denite matrix P = P T such that AT P +...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Issued: December 04, 2008 Assignment # 9 Due: December 11, 2008 1 Controllable canonical form Consider the system x(t) = Ax(t) + Bu(t) where 3 6 13 7 A = 2 4 2 , B = 1 1 2 4 3 (a) Verify that (A, B) is controllable. (b) Tran...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A 1 Consider x(t) = u(t), then the system can be formulated as: y(t) = 0 0 1 u(t) y(0) = y(t) + 1 0 0 x0 v0 Solutions # 6 (1) where x(t) is the position, v(t) the velocity, and y(t) = [x(t) v(t)]T the state of the system. We want to sta...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Issued: October 07, 2008 Assignment # 4 Due: October 16, 2008 1 Spectral Mapping Theorem Suppose A Cnn has all its eigenvalues in the right half complex plane. In other words, Spec (A) {s C : real (s) 0}. (a) Show that (I + A) is inv...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Issued: November 6, 2008 o 1 Linear Quadratic Optimization. Assignment # 6 Due: November 13, 2008 Consider an object of mass m = 1 moving along the x-axis in response to a force input u(t). The objects dynamics can be described simply as...
Berkeley >> EE >> 221 (Fall, 2008)
[ECS 2ZIA LFC-TtJR.E 5] tlWl.\" GOALS: - -lr1.}YO MeL ~a.l\'V1Ae~ rtf NJU Vi. t (}Jh om - inputs - (Jurvt~ - ST-lLtln - s1-0.t t jy 0. yv) \" th CYVl.IU n. e\'h (YYV - OLJiput - - vU VlUvu ~ ~.VV\\.L - . (rtVOvu\'<AYI ct il k ~-o,j m r {1J ...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Solutions # 3 1 Given a xed reference frame, let R3 be the rotation about x3 by 45o and R2 the rotation about x2 by 30o , then: cos(45o ) sin(45o ) 0 R3 = sin(45o ) cos(45o ) 0 0 0 1 cos(30o ) 0 sin(30o ) R2 = 0 1 0 o ) 0 cos...
Berkeley >> EE >> 221 (Fall, 2008)
L E\"E.CS2.21 A GOAL S ; LEcTveE NO-/eS 9] ~ ~t eJ avvtd.J.d. - t\'Vlfvt JOrJndJ.d -auYp v + (BIRO) Sfa.b, (). Yl J. c,h fA va. e.f uvt -a.~ fAt;} - R.effkte- irlfu t - Ovyt vt sf().bj libj -fo a. con cL\'!tine ()yJ % S<j s ti r \'s in1pu!sz....
Berkeley >> EE >> 221 (Fall, 2008)
1. _ _ _ .-L . .L-.-L-+- ) EGCS 2-2-1 A j Y1 ( Levl-vr\'~ L hr 0 a<vu e.,V-, m fa G0A L~ cr. : 0. n \'-I1v- b-v- () fA.- tJL ayv~ : ~cmU-fk OJ -. of fA\'lJ il\'\\J.Lr i1 - ()( Y1 oj mo~ rArtd ~/T1 w1J) c: .: ~ ~ OJ . . \\ \'-\' ~ -mo~J ~...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Website: Instructors: Linear System Theory http:/inst.eecs.berkeley.edu/ee221a Claire Tomlin 259M Cory 643-6610 tomlin@eecs.berkeley.edu Tu 3-4pm, W 1-2pm To be determined. Linear Algebra (Math 110 or equivalent). Kameshwar Poolla 5141 or...
Berkeley >> EE >> 221 (Fall, 2008)
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Berkeley >> EE >> 221 (Fall, 2008)
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Berkeley >> EE >> 221 (Fall, 2008)
IFEes Me.- 22-JA LECTl1RE NOT~5 6J sySftVVL r~Y\\L4.v\",.-to.~ (2() = LAC-)} tH-)/ ce- )/DC-)J .YV1 , It\\I ho.-c if it W\\tQ VlD so.~s.fUJJ ~ ur sc.t.;.f L, eftYl.L ~ fY\'YU . ~ sfa.tL \'kaNJ(~aYl. Wl.rA.W,\'X a.n.i its f\'0fW! Y-v.n \"th...
Berkeley >> EE >> 221 (Fall, 2008)
Linear Algebra for Scientists and Engineers Kameshwar Poolla University of California, Berkeley January 18, 2005 Contents Notation 1 Vectors and Matrices 1.1 Quantiers, Sets, Functions . . 1.2 Vectors and Matrices . . . . . . 1.3 Linear Simultaneous...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A LTI Systems: Open Loop Aspects A Transfer Functions and Realizations B The State and Output Response C Stability D Controllability E Observability F The Kalman Decomposition G Balanced Realizations and Model Reduction 1 A. Transfer Fu...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A A Notation B Algebraic Aspects C Normed Vector Spaces D Inner Product Spaces E The Projection Theorem Vector Spaces 1 A. ! R C Q Notation there exists there exists a unique for all eld of real numbers eld of complex numbers eld of ra...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A 1 (c) Let F be a eld. If , are additive identities, then + = = (because is an additive identity) (because is an additive identity) Solutions # 1 proving uniqueness of the additive identity in F. Proof of uniqueness of the multiplica...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Solutions # 2 1 (a) AX = C is solvable for X if and only if R(C) R(A). In the event a solution exists, the solution is unique if and only if N (A) = 0. These assertions follow from the examining the given matrix equation column-wise. In...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A A Notation B Linear Operators C Change of Basis, Range Spaces, and Null Spaces D Eigenvectors and Eigenvalues E Matrix polynomials and functions F Hermitian and Definite Matrices G The Singular-Value Decomposition H Operator Theory Linear...
Berkeley >> EE >> 221 (Fall, 2008)
EECS 221 A Issued: September 25, 2008 Assignment # 3 Due: October 02, 2008 1 Operators Find the matrix representation of a linear operator T : R3 R3 (with respect to the standard basis) from the following description of T : T takes any vector, rst...
Berkeley >> EE >> 221 (Fall, 2008)
EE 221a Homework 4 Solutions Fall 2007 1 Poblem 1. (a). To have a constant state x(t) = x0 , we require x0 , such that 0 = Ax0 + b. That is, b R(A). Because u is a non-zero scalar, this is u u equivalent to b R(A). (b). Now, in addition, to mak...
Berkeley >> EE >> 221 (Fall, 2008)
EE 221a Homework 2 Solutions Fall 2007 1 Problem 1. Proof: Observe that I B I B I B A sI A sI I 0 sI B A I 0 I I B 0 sI BA A sI = = sI AB 0 Let Q = A . Take determinants, we get sI det(Q) = det(sI BA), det(Q)sm = sn det(sI AB) So, det(sI ...
Berkeley >> CS >> 61 (Fall, 2004)
CS61B Lecture #3: Containers Today: Chapter 2 from the Assorted Materials on Java reader, and (for lab) the section on \"Using Subversion\" from the CS61B Tools Documentation reader. Please read (again) Chapters 3 and 4 of Head First Java for Wednesd...
Berkeley >> CS >> 61 (Fall, 2004)
CS61B Summer 2006 Instructor: Erin Korber 1 Introduction CS61B is the second course in the CS61 Introduction to Computer Science series. You will using the Java programming language to study programming techniques, data structures, and algorithms,...
Berkeley >> CS >> 61 (Fall, 2004)
UNIVERSITY OF CALIFORNIA Department of Electrical Engineering and Computer Sciences Computer Science Division CS61B Fall 2007 P. N. Hilnger CS 61B: Introduction to Programming, Part II General Course Information Instructor: Paul N. Hilnger, 787 ...
Berkeley >> CS >> 61 (Fall, 2004)
Graphs 1. From Hilfinger Fall 2007: Suppose you have some weighted undirected graph. This graph has three nodes of interest. In one node, there is a cat. In another there is a rat. In a third, there is a mousehole. Each \"turn\", the cat and mouse can ...
Berkeley >> CS >> 61 (Fall, 2004)
CS 61B: Data Structures Summer 2005 Course Overview Instructor: George Porter Office: 473 Soda Hall Phone: (510) 642-8919 Email: gporter@cs.berkeley.edu Teaching Assistants: Sourav Chatterji, cs61b-ta@cory.eecs.berkeley.edu Nick Neely, cs61b-tb@cor...
Berkeley >> CS >> 61 (Fall, 2004)
UNIVERSITY OF CALIFORNIA Department of Electrical Engineering and Computer Sciences Computer Science Division CS61B Fall 2004 P. N. Hilnger CS 61B: Introduction to Programming, Part II General Course Information Instructor: Paul N. Hilnger, 787 ...
Berkeley >> CS >> 61 (Fall, 2004)
CS61B July9,2008 Exam1 Readandfillinthispagenow. DoNOTturnthepageuntilyouaretoldtodoso Yourname: Yourloginname: Yourdiscussionsectiondayandtime: Yourdiscussionsectiont.a.: Nameofthepersonsittingtoyourleft: Nameofthepersonsittingtoyourright: Prob...
Berkeley >> CS >> 61 (Fall, 2004)
Problem1 a.Suggestasequenceofinsertionoperationsthatwouldcreatethisbinarysearchtree. 4 /\\ 28 /\\ 12 b.Suggestasequenceofinsertionoperationsthatwouldcreatethis234tree. |3 5| /|\\ |1 2|4|6| Problem2 Ifa234treehasdepthh(thatis,the(empty)leavesareat...
Berkeley >> CS >> 61 (Fall, 2004)
Problem1 a.Suggestasequenceofinsertionoperationsthatwouldcreatethisbinarysearchtree. 4 /\\ 28 /\\ 12 Thereareanumberofsolutionstothis.Allofthefollowingwouldwork: 42812,48212,48221,42128,42218 Generally,theparentneedstobeinsertedbeforethetwochildrenbutt...
Berkeley >> CS >> 61 (Fall, 2004)
CS 61B Mock Exam July 30, 2008 Disclaimer: This is mock exam is designed to give you hints for your review. It by no means resembles the actual midterm and you should not use these questions as the only basis in your exam preparation. You have 45 mi...
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