CMSC 313 Preview Slides
|
|
- Alison Bryant
- 6 years ago
- Views:
Transcription
1 CMSC 33 Preview Slies These are raft slies. The actual slies presente in lecture may be ifferent ue to last minute changes, scheule slippage,... UMBC, CMSC33, Richar Chang
2 CMSC 33 Lecture 2 Introuction to Circuit Simplification Karnaugh Map eamples Quine-McCluskey (Tabular Reuction) UMBC, CMSC33, Richar Chang <chang@umbc.eu>
3 B-3 Appeni B: Reuction of Digital Logic Reuction (Simplification) of Boolean Epressions It is usually possible to simplify the canonical SOP (or POS) forms. A smaller Boolean equation generally translates to a lower gate count in the target circuit. We cover three methos: algebraic reuction, Karnaugh map reuction, an tabular (Quine-McCluskey) reuction. Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
4 B-7 Appeni B: Reuction of Digital Logic Karnaugh Maps: Venn Diagram Representation of Majority Function Each istinct region in the Universe represents a minterm. This iagram can be transforme into a Karnaugh Map. ABC A ABC AB C AB C A BC A B C A BC A B C B C Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
5 B-8 Appeni B: Reuction of Digital Logic K-Map for Majority Function Place a in each cell that correspons to that minterm. Cells on the outer ege of the map wrap aroun Minterm Ine A B C F C AB -sie -sie A balance tips to the left or right epening on whether there are more s or s. Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
6 B-9 Appeni B: Reuction of Digital Logic Ajacency Groupings for Majority Function C AB F = BC + AC + AB Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
7 B- Appeni B: Reuction of Digital Logic Minimize AND-OR Majority Circuit A B C F F = BC + AC + AB The K-map approach yiels the same minimal two-level form as the algebraic approach. Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
8 B- Appeni B: Reuction of Digital Logic K-Map Groupings Minimal grouping is on the left, non-minimal (but logically equivalent) grouping is on the right. To obtain minimal grouping, create smallest groups first. AB CD AB CD F = A B C + A C D + A B C + A C D F = B D + A B C + A C D + A B C + A C D Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
9 Eample Requiring More Rules A AB CD D C B UMBC, CMSC33, Richar Chang <chang@umbc.eu>
10 B-2 Appeni B: Reuction of Digital Logic K-Map Corners are Logically Ajacent CD AB F = B C D + B D + A B Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
11 B-3 Appeni B: Reuction of Digital Logic K-Maps an Don t Cares There can be more than one minimal grouping, as a result of on t cares. AB CD AB CD F = B C D + B D F = A B D + B D Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
12 Two bits:,,, Gray Coe Three bits:,,,,,,, Successive bit patterns only iffer at position For Karnaugh maps, ajacent s represent minterms that can be simplifie using the rule: ABC + A BC = (A + A )BC = BC = BC A AB C UMBC, CMSC33, Richar Chang <chang@umbc.eu> B
13 Karnaugh Maps Implicant: rectangle with, 2, 4, 8, 6... s Prime Implicant: an implicant that cannot be etene into a larger implicant Essential Prime Implicant: the only prime implicant that covers some K-map Algorithm (not from M&H):. Fin ALL the prime implicants. Be sure to check every an to use on t cares. 2. Inclue all essential prime implicants. 3. Try all possibilities to fin the minimum cover for the remaining s. UMBC, CMSC33, Richar Chang <chang@umbc.eu>
14 K-map Eample A AB CD A AB CD D D C C B B A B + AC D + AB D UMBC, CMSC33, Richar Chang <chang@umbc.eu>
15 Notes on K-maps Also works for POS Takes 2 n time for formulas with n variables Only optimizes two-level logic Reuces number of terms, then number of literals in each term Assumes inverters are free Does not consier minimizations across functions Circuit minimization is generally a har problem Quine-McCluskey can be use with more variables CAD tools are available if you are serious UMBC, CMSC33, Richar Chang <chang@umbc.eu>
16 Karnaugh Maps Implicant: rectangle with, 2, 4, 8, 6... s Prime Implicant: an implicant that cannot be etene into a larger implicant Essential Prime Implicant: the only prime implicant that covers some K-map Algorithm (not from M&H):. Fin ALL the prime implicants. Be sure to check every an to use on t cares. 2. Inclue all essential prime implicants. 3. Try all possibilities to fin the minimum cover for the remaining s. UMBC, CMSC33, Richar Chang <chang@umbc.eu>
17 Notes on K-maps Also works for POS Takes 2 n time for formulas with n variables Only optimizes two-level logic Reuces number of terms, then number of literals in each term Assumes inverters are free Does not consier minimizations across functions Circuit minimization is generally a har problem Quine-McCluskey can be use with more variables CAD tools are available if you are serious UMBC, CMSC33, Richar Chang <chang@umbc.eu>
18 Circuit Minimization is Har Uni systems store passwors in encrypte form. User types in, system computes f() an looks for f() in a file. Suppose we us 64-bit passwors an I want to fin the passwor, such that f() = y. Let gi() = if f() = y an the ith bit of is otherwise. If the ith bit of is, then gi() outputs for every an has a very, very simple circuit. If you can simplify every circuit quickly, then you can crack passwors quickly. UMBC, CMSC33, Richar Chang <chang@umbc.eu>
19 B-6 Appeni B: Reuction of Digital Logic 3-Level Majority Circuit K-Map Reuction results in a reuce two-level circuit (that is, AND followe by OR. Inverters are not inclue in the two-level count). Algebraic reuction can result in multi-level circuits with even fewer logic gates an fewer inputs to the logic gates. A B C M Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
20 CD AB A CD AB A CD AB A D D D C C C B B B CD AB A CD AB A CD AB A D D D C C C B B B
21 B-4 Appeni B: Reuction of Digital Logic Five-Variable K-Map Visualize two 4-variable K-maps stacke one on top of the other; groupings are mae in three imensional cubes. CDE AB AB CDE F = A C D E + A B D E + B E Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
22 B-5 Si-Variable K-Map Appeni B: Reuction of Digital Logic Visualize four 4-variable K-maps stacke one on top of the other; groupings are mae in three imensional cubes. ABC DEF ABC DEF ABC ABC DEF DEF Principles of Computer Architecture by M. Murocca an V. Heuring G = B C E F + A B D E 999 M. Murocca an V. Heuring
23 Appeni B: Reuction of Digital Logic B-9 Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring Truth Table with Don t Cares A truth table representation of a single function with on t cares. C D B F A
24 Appeni B: Reuction of Digital Logic B-2 Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring Tabular (Quine-McCluskey) Reuction Tabular reuction begins by grouping minterms for which F is nonzero accoring to the number of s in each minterm. Don t cares are consiere to be nonzero. The net step forms a consensus (the logical form of a cross prouct) between each pair of ajacent groups for all terms that iffer in only one variable. C D B A C D B A * * * C D B A * * * Initial setup After first reuction After secon reuction (a) (b) (c)
25 B-2 Appeni B: Reuction of Digital Logic Table of Choice The prime implicants form a set that completely covers the function, although not necessarily minimally. A table of choice is use to obtain a minimal cover set. Prime Implicants Minterms * * * Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
26 B-22 Appeni B: Reuction of Digital Logic Reuce Table of Choice In a reuce table of choice, the essential prime implicants an the minterms they cover are remove, proucing the eligible set. F = ABC + ABC + BD + AD Eligible Set Minterms Set Set 2 X Y Z Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
27 B-23 Appeni B: Reuction of Digital Logic Multiple Output Truth Table The power of tabular reuction comes into play for multiple functions, in which minterms can be share among the functions. Minterm A B C F F F 2 m m m 2 m 3 m 4 m 5 m 6 m 7 Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
28 B-24 Appeni B: Reuction of Digital Logic Multiple Output Table of Choice F (A,B,C) = ABC + BC F (A,B,C) = AC + AC + BC F 2 (A,B,C) = B Minterms Prime Implicants F F * * F (A,B,C) F (A,B,C) F 2 (A,B,C) m m 3 m 7 m m 3 m 4 m 6 m 7 m 2 m 3 m 6 m 7 F * F 2 * F,2 F,,2 * Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
29 Simplifying Finite State Machines State Reuction: equivalent FSM with fewer states State Assignment: choose an assignment of bit patterns to states (e.g., B is ) that results in a smaller circuit Choice of flip-flops: use D flip-flops, J-K flip-flops or a T flip-flops? a goo choice coul lea to simpler circuits. UMBC, CMSC33, Richar Chang <chang@umbc.eu>
30 B-32 Appeni B: Reuction of Digital Logic The State Assignment Problem Two state assignments for machine M 2. Input X P.S. A B/ A/ B C/ D/ C C/ D/ D B/ A/ Input X S S A: / / B: / / C: / / D: / / Input X S S A: / / B: / / C: / / D: / / Machine M 2 State assignment SA State assignment SA Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
31 B-33 Appeni B: Reuction of Digital Logic State Assignment SA Boolean equations for machine M 2 using state assignment SA. S S X X S S X S S S = S S + S S S = S S X + S S X + S S X + S S X Z = S S + S X + S S X Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
32 B-34 Appeni B: Reuction of Digital Logic State Assignment SA Boolean equations for machine M 2 using state assignment SA. S S X X S S X S S S = S S = X Z = S X + S X Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
33 State Assignment Heuristics No known efficient alg. for best state assignment Some heuristics (rules of thumb): The initial state shoul be simple to reset all zeroes or all ones. Minimize the number of state variables that change on each transition. Maimize the number of state variables that on't change on each transition. Eploit symmetries in the state iagram. If there are unuse states (when the number of states s is not a power of 2), choose the unuse state variable combinations carefully. (Don't just use the first s combination of state variables.) Decompose the set of state variables into bits or fiels that have well-efine meaning with respect to the input or output behavior. Consier using more than the minimum number of states to achieve the objectives above. UMBC, CMSC33, Richar Chang <chang@umbc.eu>
34 B-35 Appeni B: Reuction of Digital Logic Sequence Detector State Transition Diagram / / D B / / / E / A / / / F / / C / Input: Output: / G Time: / Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
35 B-36 Appeni B: Reuction of Digital Logic Sequence Detector State Table Present state Input X A B/ C/ B D/ E/ C F/ G/ D D/ E/ E F/ G/ F D/ E/ G F/ G/ Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
36 Sequence Detector State Reuction Table A B C D E F G A B C D E F G UMBC, CMSC33, Richar Chang <chang@umbc.eu>
37 B-37 Appeni B: Reuction of Digital Logic Sequence Detector Reuce State Table Present state Input X A: A' BD: B' C: C' E: D' F: E' G: F' B'/ B'/ E'/ E'/ B'/ E'/ C'/ D'/ F'/ F'/ D'/ F'/ Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
38 6-State Sequence Detector / B/D / E / / / / A F / / / / C / G UMBC, CMSC 33, Richar Chang <chang@umbc.eu>
39 B-38 Appeni B: Reuction of Digital Logic Sequence Detector State Assignment Present state Input X S 2 S S S 2 S S Z S 2 S S Z A': / / B': / / C': / / D': / / E': / / F': / / Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
40 B-4 Appeni B: Reuction of Digital Logic Sequence Detector K-Maps K-map reuction of net state an output functions for sequence etector. S 2 S S X S = S 2 S X + S X + S 2 S + S X S 2 S S X S = S 2 S X + S 2 S X S 2 S S X S 2 S S X S 2 = S 2 S + S Principles of Computer Architecture by M. Murocca an V. Heuring Z = S 2 S X + S S X + S 2 S X 999 M. Murocca an V. Heuring
41 Improve Sequence Detector? Formulas from the 7-state FSM: s2 = (s + )(s2 + s + s) s = s + s = s or s = z = s2 s + s2 s Formulas from the 6-state FSM: s2 = s2 s + s s = s2 s + s2 s s = s2 s + s + s2 s + s z = s2 s + s s + s2 s UMBC, CMSC33, Richar Chang <chang@umbc.eu>
42 Sequence Detector State Assignment 7-state s2 s s s2' s' s' z A = E = B = F = C = G = D = new 6-state s2 s s s2' s' s' z A = E = B/D = F = C = G = D = UMBC, CMSC33, Richar Chang <chang@umbc.eu>
43 6-State Sequence Detector 7-state s2 s s2 s new 6-state s2 s s2 s s s s s s2 = (s + )(s2 + s + s) s2 = (s + )(s2 + s + s) UMBC, CMSC33, Richar Chang <chang@umbc.eu>
44 6-State Sequence Detector 7-state s2 s s2 s new 6-state s2 s s2 s s s s s s = s + s s = s UMBC, CMSC33, Richar Chang <chang@umbc.eu>
45 6-State Sequence Detector 7-state s2 s s2 s new 6-state s2 s s2 s s s s s s = s = UMBC, CMSC33, Richar Chang <chang@umbc.eu>
46 6-State Sequence Detector 7-state s2 s s2 s new 6-state s2 s s2 s s s s s z = s2 s + s2 s z = s2 s + s2 s UMBC, CMSC33, Richar Chang <chang@umbc.eu>
47 Improve Sequence Detector Tetbook formulas for the 6-state FSM: s2 = s2 s + s s = s2 s + s2 s s = s2 s + s + s2 s + s z = s2 s + s s + s2 s New formulas for the 6-state FSM: s2 = (s + )(s2 + s + s) s = s s = z = s2 s + s2 s UMBC, CMSC33, Richar Chang <chang@umbc.eu>
48 B-43 Appeni B: Reuction of Digital Logic Ecitation Tables Each table shows the settings that must be applie at the inputs at time t in orer to change the outputs at time t+. S-R flip-flop Q t Q t+ Q t Q t+ S J R K D flip-flop Q t Q t+ Q t Q t+ D T J-K flip-flop T flip-flop Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
49 B-44 Appeni B: Reuction of Digital Logic Serial Aer Time (t) Time (t) X Y C in Serial Aer Z C out i y i No carry state / / A / Carry state B / / z i / / / State transition iagram, state table, an state assignment for a serial aer. Input XY Present state A A/ A/ A/ B/ B A/ B/ B/ B/ Net state Output Present state (S t ) Input XY A: / / / / B: / / / / Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
50 Appeni B: Reuction of Digital Logic B-45 Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring Serial Aer Net-State Functions Truth table showing net-state functions for a serial aer for D, S-R, T, an J-K flip-flops. Shae functions are use in the eample. Y S t X D S R J K T Z Present State (Set) (Reset)
51 B-46 Appeni B: Reuction of Digital Logic J-K Flip-Flop Serial Aer Circuit X Y X Y CLK J Q S K Q X Y Y Z X Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
52 B-47 Appeni B: Reuction of Digital Logic D Flip-Flop Serial Aer Circuit X Y X Y CLK D Q S Q X Y Y Z X Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
53 B-35 Appeni B: Reuction of Digital Logic Sequence Detector State Transition Diagram / / D B / / / E / A / / / F / / C / Input: Output: / G Time: / Principles of Computer Architecture by M. Murocca an V. Heuring 999 M. Murocca an V. Heuring
54 6-State Sequence Detector / B/D / E / / / / A F / / / / C / G UMBC, CMSC 33, Richar Chang <chang@umbc.eu>
55 Improve Sequence Detector? Formulas from the 7-state FSM: s2 = (s + )(s2 + s + s) s = s + s = s or s = z = s2 s + s2 s Formulas from the 6-state FSM: s2 = s2 s + s s = s2 s + s2 s s = s2 s + s + s2 s + s z = s2 s + s s + s2 s UMBC, CMSC33, Richar Chang <chang@umbc.eu>
56 Sequence Detector State Assignment 7-state s2 s s s2' s' s' z A = E = B = F = C = G = D = new 6-state s2 s s s2' s' s' z A = E = B/D = F = C = G = D = UMBC, CMSC33, Richar Chang <chang@umbc.eu>
57 Improve Sequence Detector Tetbook formulas for the 6-state FSM: s2 = s2 s + s s = s2 s + s2 s s = s2 s + s + s2 s + s z = s2 s + s s + s2 s New formulas for the 6-state FSM: s2 = (s + )(s2 + s + s) s = s s = z = s2 s + s2 s UMBC, CMSC33, Richar Chang <chang@umbc.eu>
58 6-State Sequence Detector s2 s s s2' s' s' z j2 k2 j k j k Q Q' J K UMBC, CMSC33, Richar Chang <chang@umbc.eu>
59 6-State Sequence Detector J2 s2 s s2 s K2 s2 s s2 s s s s s J2 = s + s K2 = s UMBC, CMSC33, Richar Chang <chang@umbc.eu>
60 6-State Sequence Detector J s2 s s2 s K s2 s s2 s s s s s J = s K = UMBC, CMSC33, Richar Chang <chang@umbc.eu>
61 6-State Sequence Detector J s2 s s2 s K s2 s s2 s s s s s J = K = UMBC, CMSC33, Richar Chang <chang@umbc.eu>
62 Improve Sequence Detector Formulas for the 6-state FSM with D Flip-flops: s2 = (s + )(s2 + s + s) s = s s = Formulas for the 6-state FSM with J-K Flip-flops: J2 = s + s J = s J = K2 = s K = K = UMBC, CMSC33, Richar Chang <chang@umbc.eu>
63 CMSC 33 Computer Organization & Assembly Language Programming Section Fall 24 Homework 5 Due: Tuesay, December 7, 23. ( points) Question A.3, page 494, Murocca & Heuring 2. ( points) Question A.29, page 497, Murocca & Heuring 3. ( points) Question B., page 542, Murocca & Heuring 4. ( points) Question B., page 542, Murocca & Heuring 5. (6 points) This problem asks you to take the steps involve in the esign process of a finite state machine. You will esign a finite state machine that has a one bit input an a one bit output z. Themachinemustoutputforeveryinputsequenceening in the string or. The output shoul be in all other cases. [Aapte from Contemporary Logic Design, Rany H.Katz,Benjamin/Cummings Publishing, 994.] (a) ( points) In the space provie on the net page, raw the minimumstatetransition iagram for the finite state machine escribe above. You must use the state-minimization algorithm escribe in class to show that the finite state machine has the minimum number of states. (Hint: You shoul have fewer than 8statesinyourmachine.) (b) (5 points) Use the state assignment heuristics escribe in class an pick two ifferent state assignments for your finite state machine. Note: the bit pattern for the initial state must be. (c) (4 points) For each of the two state assignments: i. Fill in the truth tables with values for D flip-flops, for the output bit an for J-K flip-flops. ii. Use the Karnaugh maps provie to minimize the formulas for each column of the truth table. iii. Count the number of gates neee for each implementation. () (5 points) Shoul you use your first or secon state assignment? D flip-flops or J-K flip-flops? Note: Keep a copy of your work for the last question. You will nee it for DigSim Assignment 3.
64 CMSC 33 Homework 5 Name: Minimize State Transition Diagram (show work) State Assignment: A B C D E F unuse unuse Assignment # Assignment#2
65 ASSIGNMENT # Ecitation Table for J-K Flip-Flops Q Q' J K Truth Table: s2 s s s2' s' s' z j2 k2 j k j k
66 Assignment #: Karnaugh Maps for D Flip-Flops an the output s2 s2 s s s2 s2 s s s s s s s2 = s = # of gates = # of gates = s2 s2 s s s2 s2 s s s s s s s = z = # of gates = # of gates = Total # of gates for D flip-flops (on t count z) =
67 Assignment #: Karnaugh Maps for J-K Flip-Flops s2 s2 s s s2 s2 s s s s s j2 = k2 = # of gates = # of gates = s s2 s2 s s s2 s2 s s s s s j = k = # of gates = # of gates = s s2 s2 s s s2 s2 s s s s s j = k = # of gates = # of gates = s Total # of gates for J-K flip-flops (on t count z) =
68 ASSIGNMENT #2 Ecitation Table for J-K Flip-Flops Q Q' J K Truth Table: s2 s s s2' s' s' z j2 k2 j k j k
69 Assignment #2: Karnaugh Maps for D Flip-Flops an the output s2 s2 s s s2 s2 s s s s s s s2 = s = # of gates = # of gates = s2 s2 s s s2 s2 s s s s s s s = z = # of gates = # of gates = Total # of gates for D flip-flops (on t count z) =
70 Assignment #2: Karnaugh Maps for J-K Flip-Flops s2 s2 s s s2 s2 s s s s s j2 = k2 = # of gates = # of gates = s s2 s2 s s s2 s2 s s s s s j = k = # of gates = # of gates = s s2 s2 s s s2 s2 s s s s s j = k = # of gates = # of gates = s Total # of gates for J-K flip-flops (on t count z) =
Principles of Computer Architecture. Appendix B: Reduction of Digital Logic. Chapter Contents
B-1 Principles of Computer Architecture Miles Murdocca and Vincent Heuring Appendix B: Reduction of Digital Logic B-2 Chapter Contents B.1 Reduction of Combinational Logic and Sequential Logic B.2 Reduction
More informationCMSC 313 Lecture 19 Combinational Logic Components Programmable Logic Arrays Karnaugh Maps
CMSC 33 Lecture 9 Combinational Logic Components Programmable Logic rrays Karnaugh Maps UMC, CMSC33, Richard Chang Last Time & efore Returned midterm exam Half adders & full adders Ripple
More informationCMSC 313 Lecture 19 Homework 4 Questions Combinational Logic Components Programmable Logic Arrays Introduction to Circuit Simplification
CMSC 33 Lecture 9 Homework 4 Questions Combinational Logic Components Programmable Logic rrays Introduction to Circuit Simplification UMC, CMSC33, Richard Chang CMSC 33, Computer Organization
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More informationLecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University
Lecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Aby K George, ECE Department, Wayne State University Contents The Map method Two variable
More informationThe Karnaugh Map COE 202. Digital Logic Design. Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals
The Karnaugh Map COE 202 Digital Logic Design Dr. Muhamed Mudawar King Fahd University of Petroleum and Minerals Presentation Outline Boolean Function Minimization The Karnaugh Map (K-Map) Two, Three,
More informationMotivation. CS/EE 3700 : Fundamentals of Digital System Design
Motivation CS/EE 37 : Funamentals o Digital System Design Chris J. Myers Lecture 4: Logic Optimization Chapter 4 Algebraic manipulation is not systematic. This chapter presents methos that can be automate
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal
More informationUNIT 5 KARNAUGH MAPS Spring 2011
UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two- and three-variable Four-variable Determination of minimum expressions using essential prime implicants Five-variable
More informationKarnaugh Maps Objectives
Karnaugh Maps Objectives For Karnaugh Maps of up to 5 variables Plot a function from algebraic, minterm or maxterm form Obtain minimum Sum of Products and Product of Sums Understand the relationship between
More informationOutcomes. Unit 14. Review of State Machines STATE MACHINES OVERVIEW. State Machine Design
4. Outcomes 4.2 Unit 4 tate Machine Design I can create a state iagram to solve a sequential problem I can implement a working state machine given a state iagram 4.3 Review of tate Machines 4.4 TATE MACHINE
More informationUnit 2 Session - 6 Combinational Logic Circuits
Objectives Unit 2 Session - 6 Combinational Logic Circuits Draw 3- variable and 4- variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the Product-of-Sums
More informationThis form sometimes used in logic circuit, example:
Objectives: 1. Deriving of logical expression form truth tables. 2. Logical expression simplification methods: a. Algebraic manipulation. b. Karnaugh map (k-map). 1. Deriving of logical expression from
More informationLogic Simplification. Boolean Simplification Example. Applying Boolean Identities F = A B C + A B C + A BC + ABC. Karnaugh Maps 2/10/2009 COMP370 1
Digital Logic COMP370 Introduction to Computer Architecture Logic Simplification It is frequently possible to simplify a logical expression. This makes it easier to understand and requires fewer gates
More informationLogic Synthesis and Verification. Two-Level Logic Minimization (1/2) Quine-McCluskey Procedure. Complexity
Logic Synthesis an Verification Two-Level Logic Minimization (/2) Jie-Hong Rolan Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 24 Reaing: Logic Synthesis in a Nutshell
More informationReview. EECS Components and Design Techniques for Digital Systems. Lec 06 Minimizing Boolean Logic 9/ Review: Canonical Forms
Review EECS 150 - Components and Design Techniques for Digital Systems Lec 06 Minimizing Boolean Logic 9/16-04 David Culler Electrical Engineering and Computer Sciences University of California, Berkeley
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Charles Kime & Thomas Kaminski 2004 Pearson Education, Inc. Terms of Use (Hyperlinks are active
More informationLecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps
EE210: Switching Systems Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps Prof. YingLi Tian Feb. 21/26, 2019 Department of Electrical Engineering The City College of New York
More informationELC224C. Karnaugh Maps
KARNAUGH MAPS Function Simplification Algebraic Simplification Half Adder Introduction to K-maps How to use K-maps Converting to Minterms Form Prime Implicants and Essential Prime Implicants Example on
More informationKarnaugh Maps (K-Maps)
Karnaugh Maps (K-Maps) Boolean expressions can be minimized by combining terms P + P = P K-maps minimize equations graphically Put terms to combine close to one another B C C B B C BC BC BC BC BC BC BC
More informationEE40 Lec 15. Logic Synthesis and Sequential Logic Circuits
EE40 Lec 15 Logic Synthesis and Sequential Logic Circuits Prof. Nathan Cheung 10/20/2009 Reading: Hambley Chapters 7.4-7.6 Karnaugh Maps: Read following before reading textbook http://www.facstaff.bucknell.edu/mastascu/elessonshtml/logic/logic3.html
More information211: Computer Architecture Summer 2016
211: Computer Architecture Summer 2016 Liu Liu Topic: Storage Project3 Digital Logic - Storage: Recap - Review: cache hit rate - Project3 - Digital Logic: - truth table => SOP - simplification: Boolean
More informationL4: Karnaugh diagrams, two-, and multi-level minimization. Elena Dubrova KTH / ICT / ES
L4: Karnaugh diagrams, two-, and multi-level minimization Elena Dubrova KTH / ICT / ES dubrova@kth.se Combinatorial system a(t) not(a(t)) A combinatorial system has no memory - its output depends therefore
More informationLecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions
EE210: Switching Systems Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions Prof. YingLi Tian Feb. 15, 2018 Department of Electrical Engineering The City College of New York The
More informationCPE100: Digital Logic Design I
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu CPE100: Digital Logic Design I Midterm02 Review http://www.ee.unlv.edu/~b1morris/cpe100/ 2 Logistics Thursday Nov. 16 th In normal lecture (13:00-14:15)
More informationDigital Logic Design. Midterm #2
EECS: igital Logic esign r. nthony. Johnson f5m2s_il7.fm - igital Logic esign Miterm #2 Problems Points. 5 2. 4 3. 6 Total 5 Was the exam fair? yes no EECS: igital Logic esign r. nthony. Johnson f5m2s_il7.fm
More informationfor Digital Systems Simplification of logic functions Tajana Simunic Rosing Sources: TSR, Katz, Boriello & Vahid
SE140: omponents and Design Techniques for Digital Systems Simplification of logic functions Tajana Simunic Rosing 1 What we covered thus far: Number representations Where we are now inary, Octal, Hex,
More informationDIGITAL ELECTRONICS & it0203 Semester 3
DIGITAL ELECTRONICS & it0203 Semester 3 P.Rajasekar & C.M.T.Karthigeyan Asst.Professor SRM University, Kattankulathur School of Computing, Department of IT 8/22/2011 1 Disclaimer The contents of the slides
More informationECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 9 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering
ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 9 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering TIMING ANALYSIS Overview Circuits do not respond instantaneously to input changes
More informationBoolean cubes EECS150. Mapping truth tables onto cubes. Simplification. The Uniting Theorem. Three variable example
EES5 Section 5 Simplification and State Minimization Fall 2 -cube X oolean cubes Visual technique for indentifying when the uniting theorem can be applied n input variables = n-dimensional "cube" Y 2-cube
More informationChapter 4 Optimized Implementation of Logic Functions
Chapter 4 Optimized Implementation of Logic Functions Logic Minimization Karnaugh Maps Systematic Approach for Logic Minimization Minimization of Incompletely Specified Functions Tabular Method for Minimization
More informationCMSC 313 Lecture 17. Focus Groups. Announcement: in-class lab Thu 10/30 Homework 3 Questions Circuits for Addition Midterm Exam returned
Focus Groups CMSC 33 Lecture 7 Need good sample of all types of CS students Mon /7 & Thu /2, 2:3p-2:p & 6:p-7:3p Announcement: in-class lab Thu /3 Homework 3 Questions Circuits for Addition Midterm Exam
More informationSimplifying Logic Circuits with Karnaugh Maps
Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression: f = abc + abc + abc Now, as we have seen, this expression can be simplified
More informationGate-Level Minimization
Gate-Level Minimization Dr. Bassem A. Abdullah Computer and Systems Department Lectures Prepared by Dr.Mona Safar, Edited and Lectured by Dr.Bassem A. Abdullah Outline 1. The Map Method 2. Four-variable
More informationUNIT 4 MINTERM AND MAXTERM EXPANSIONS
UNIT 4 MINTERM AND MAXTERM EXPANSIONS Spring 2 Minterm and Maxterm Expansions 2 Contents Conversion of English sentences to Boolean equations Combinational logic design using a truth table Minterm and
More informationENG2410 Digital Design Combinational Logic Circuits
ENG240 Digital Design Combinational Logic Circuits Fall 207 S. Areibi School of Engineering University of Guelph Binary variables Binary Logic Can be 0 or (T or F, low or high) Variables named with single
More informationOptimizations and Tradeoffs. Combinational Logic Optimization
Optimizations and Tradeoffs Combinational Logic Optimization Optimization & Tradeoffs Up to this point, we haven t really considered how to optimize our designs. Optimization is the process of transforming
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 2 Following the slides of Dr. Ahmed H. Madian ذو الحجة 438 ه Winter
More informationSpiral 1 / Unit 5. Karnaugh Maps
-. Spiral / Unit Karnaugh Maps -. Outcomes I know the difference between combinational and sequential logic and can name examples of each. I understand latency, throughput, and at least technique to improve
More informationSimplification of Boolean Functions. Dept. of CSE, IEM, Kolkata
Simplification of Boolean Functions Dept. of CSE, IEM, Kolkata 1 Simplification of Boolean Functions: An implementation of a Boolean Function requires the use of logic gates. A smaller number of gates,
More informationCS/EE 181a 2008/09 Lecture 4
CS/EE 181a 28/9 Lecture 4 General topic of today s lecture: Logic Optimization Karnaugh maps. Quine-McCluskey tabulation method (not in detail). Non series-parallel networks (some care is required). Reference
More informationChapter 7 Logic Circuits
Chapter 7 Logic Circuits Goal. Advantages of digital technology compared to analog technology. 2. Terminology of Digital Circuits. 3. Convert Numbers between Decimal, Binary and Other forms. 5. Binary
More informationOutcomes. Spiral 1 / Unit 5. Logic Function Synthesis KARNAUGH MAPS. Karnaugh Maps
-. -. Spiral / Unit Mark Redekopp Outcomes I know the difference between combinational and sequential logic and can name examples of each. I understand latency, throughput, and at least technique to improve
More informationCMSC 313 Lecture 15 Good-bye Assembly Language Programming Overview of second half on Digital Logic DigSim Demo
CMSC 33 Lecture 5 Good-bye ssembly Language Programming Overview of second half on Digital Logic DigSim Demo UMC, CMSC33, Richard Chang Good-bye ssembly Language What a pain! Understand
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationDigital Circuit And Logic Design I. Lecture 4
Digital Circuit And Logic Design I Lecture 4 Outline Combinational Logic Design Principles (2) 1. Combinational-circuit minimization 2. Karnaugh maps 3. Quine-McCluskey procedure Panupong Sornkhom, 2005/2
More informationCHAPTER 5 KARNAUGH MAPS
CHAPTER 5 1/36 KARNAUGH MAPS This chapter in the book includes: Objectives Study Guide 5.1 Minimum Forms of Switching Functions 5.2 Two- and Three-Variable Karnaugh Maps 5.3 Four-Variable Karnaugh Maps
More informationNumber System conversions
Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number
More informationLogic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 2 Circuit Optimization
Logic and omputer Design Fundamentals hapter 2 ombinational Logic ircuits Part 2 ircuit Optimization harles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active in View Show mode)
More information9.1. Unit 9. Implementing Combinational Functions with Karnaugh Maps or Memories
. Unit Implementing Combinational Functions with Karnaugh Maps or Memories . Outcomes I can use Karnaugh maps to synthesize combinational functions with several outputs I can determine the appropriate
More informationWorking with Combinational Logic. Design example: 2x2-bit multiplier
Working with ombinational Logic Simplification two-level simplification exploiting don t cares algorithm for simplification Logic realization two-level logic and canonical forms realized with NNs and NORs
More informationComputer Organization I. Lecture 13: Design of Combinational Logic Circuits
Computer Organization I Lecture 13: Design of Combinational Logic Circuits Overview The optimization of multiple-level circuits Mapping Technology Verification Objectives To know how to optimize the multiple-level
More informationCSE 140: Components and Design Techniques for Digital Systems
Lecture 4: Four Input K-Maps CSE 4: Components and Design Techniques for Digital Systems CK Cheng Dept. of Computer Science and Engineering University of California, San Diego Outlines Boolean Algebra
More informationLecture 4: Four Input K-Maps
Lecture 4: Four Input K-Maps CSE 4: Components and Design Techniques for Digital Systems Fall 24 CK Cheng Dept. of Computer Science and Engineering University of California, San Diego Outlines Boolean
More informationIntroduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps
Introduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology
More informationPhiladelphia University Student Name: Student Number:
Philadelphia University Student Name: Student Number: Faculty of Engineering Serial Number: Final Exam, Second Semester: 2015/2016 Dept. of Computer Engineering Course Title: Logic Circuits Date: 08/06/2016
More informationOutcomes. Spiral 1 / Unit 5. Logic Function Synthesis KARNAUGH MAPS. Karnaugh Maps
-. -. Spiral / Unit Mark Redekopp Outcomes I know the difference between combinational and sequential logic and can name examples of each. I understand latency, throughput, and at least technique to improve
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-2700: Digital Logic Design Winter Notes - Unit 2
ECE-7: Digital Logic Design Winter 8 Notes - Unit OPTIMIZED IMPLEMENTATION OF LOGIC FUNCTIONS BASIC TECHNIQUES: We can alas minimie logic unctions using the Boolean theorems. Hoever, more poerul methods
More informationTextbook: Digital Design, 3 rd. Edition M. Morris Mano
: 25/5/ P-/70 Tetbook: Digital Design, 3 rd. Edition M. Morris Mano Prentice-Hall, Inc. : INSTRUCTOR : CHING-LUNG SU E-mail: kevinsu@yuntech.edu.tw Chapter 3 25/5/ P-2/70 Chapter 3 Gate-Level Minimization
More informationWeek-I. Combinational Logic & Circuits
Week-I Combinational Logic & Circuits Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and
More informationELEC Digital Logic Circuits Fall 2014 Sequential Circuits (Chapter 6) Finite State Machines (Ch. 7-10)
ELEC 2200-002 Digital Logic Circuits Fall 2014 Sequential Circuits (Chapter 6) Finite State Machines (Ch. 7-10) Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering
More information14:332:231 DIGITAL LOGIC DESIGN. Combinational Circuit Synthesis
:: DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering all Lecture #: Combinational Circuit Synthesis I Combinational Circuit Synthesis Recall: Combinational circuit
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationDigital Logic Appendix A
Digital Logic Appendix A Boolean Algebra Gates Combinatorial Circuits Sequential Circuits 1 Boolean Algebra George Boole ideas 1854 Claude Shannon, apply to circuit design, 1938 Describe digital circuitry
More informationLecture 7: Karnaugh Map, Don t Cares
EE210: Switching Systems Lecture 7: Karnaugh Map, Don t Cares Prof. YingLi Tian Feb. 28, 2019 Department of Electrical Engineering The City College of New York The City University of New York (CUNY) 1
More informationCHAPTER 7. Exercises 17/ / /2 2 0
CHAPTER 7 Exercises E7. (a) For the whole part, we have: Quotient Remainders 23/2 /2 5 5/2 2 2/2 0 /2 0 Reading the remainders in reverse order, we obtain: 23 0 = 0 2 For the fractional part we have 2
More informationEE 209 Spiral 1 Exam Solutions Name:
EE 29 Spiral Exam Solutions Name:.) Answer the following questions as True or False a.) A 4-to- multiplexer requires at least 4 select lines: true / false b.) An 8-to- mux and no other logic can be used
More informationCombinatorial Logic Design Principles
Combinatorial Logic Design Principles ECGR2181 Chapter 4 Notes Logic System Design I 4-1 Boolean algebra a.k.a. switching algebra deals with boolean values -- 0, 1 Positive-logic convention analog voltages
More informationLecture 4: More Boolean Algebra
Lecture 4: More Boolean Algebra Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University ENGIN2
More informationCh 7. Finite State Machines. VII - Finite State Machines Contemporary Logic Design 1
Ch 7. Finite State Machines VII - Finite State Machines Contemporary Logic esign 1 Finite State Machines Sequential circuits primitive sequential elements combinational logic Models for representing sequential
More informationCh 2. Combinational Logic. II - Combinational Logic Contemporary Logic Design 1
Ch 2. Combinational Logic II - Combinational Logic Contemporary Logic Design 1 Combinational logic Define The kind of digital system whose output behavior depends only on the current inputs memoryless:
More informationECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering
ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering Boolean Algebra Boolean Algebra A Boolean algebra is defined with: A set of
More informationContents. Chapter 3 Combinational Circuits Page 1 of 36
Chapter 3 Combinational Circuits Page of 36 Contents Combinational Circuits...2 3. Analysis of Combinational Circuits...3 3.. Using a Truth Table...3 3..2 Using a Boolean Function...6 3.2 Synthesis of
More informationFaculty of Engineering. FINAL EXAMINATION FALL 2008 (December2008) ANSWER KEY
1 McGill University Faculty of Engineering DIGITAL SYSTEM DESIGN ECSE-323 FINAL EXAMINATION FALL 2008 (December2008) ANSWER KEY STUDENT NAME McGILL I.D. NUMBER Examiner: Prof. J. Clark Signature: Associate
More information3. Complete the following table of equivalent values. Use binary numbers with a sign bit and 7 bits for the value
EGC22 Digital Logic Fundamental Additional Practice Problems. Complete the following table of equivalent values. Binary. Octal 35.77 33.23.875 29.99 27 9 64 Hexadecimal B.3 D.FD B.4C 2. Calculate the following
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev
E&CE 223 Digital Circuits & Systems Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean Algebra & Logic Gates Major topics Boolean algebra NAND & NOR gates Boolean
More informationEECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive
EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive March 30, 2010 John Wawrzynek Spring 2010 EECS150 - Lec19-cl1 Page 1 Boolean Algebra I (Representations of Combinational
More informationCMSC 313 Lecture 16 Announcement: no office hours today. Good-bye Assembly Language Programming Overview of second half on Digital Logic DigSim Demo
CMSC 33 Lecture 6 nnouncement: no office hours today. Good-bye ssembly Language Programming Overview of second half on Digital Logic DigSim Demo UMC, CMSC33, Richard Chang Good-bye ssembly
More informationECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2
ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 Instructor: Andrew B. Kahng (lecture) Email: abk@ucsd.edu Telephone: 858-822-4884 office, 858-353-0550 cell Office: 3802
More information14:332:231 DIGITAL LOGIC DESIGN
:: DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall Lecture #: Combinational Circuit Synthesis II hat if we have input variables? V = V = of Example with variables
More informationELEC Digital Logic Circuits Fall 2014 Logic Minimization (Chapter 3)
ELE 2200-002 Digital Logic ircuits Fall 204 Logic Minimization (hapter 3) Vishwani D. grawal James J. Danaher Professor Department of Electrical and omputer Engineering uburn University, uburn, L 36849
More informationBoolean Algebra. Digital Logic Appendix A. Postulates, Identities in Boolean Algebra How can I manipulate expressions?
Digital Logic Appendix A Gates Combinatorial Circuits Sequential Circuits Other operations NAND A NAND B = NOT ( A ANDB) = AB NOR A NOR B = NOT ( A ORB) = A + B Truth tables What is the result of the operation
More informationCSE 140 Midterm I - Solution
CSE 140 Midterm I - Solution 1. Answer the following questions given the logic circuit below. (15 points) a. (5 points) How many CMOS transistors does the given (unsimplified) circuit have. b. (6 points)
More informationLecture 2 Review on Digital Logic (Part 1)
Lecture 2 Review on Digital Logic (Part 1) Xuan Silvia Zhang Washington University in St. Louis http://classes.engineering.wustl.edu/ese461/ Grading Engagement 5% Review Quiz 10% Homework 10% Labs 40%
More informationLogic Design I (17.341) Fall Lecture Outline
Logic Design I (17.341) Fall 2011 Lecture Outline Class # 06 October 24, 2011 Dohn Bowden 1 Today s Lecture Administrative Main Logic Topic Homework 2 Course Admin 3 Administrative Admin for tonight Syllabus
More informationL2: Combinational Logic Design (Construction and Boolean Algebra)
L2: Combinational Logic Design (Construction and Boolean Algebra) Acknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. Borriello, Contemporary Logic Design (second edition), Pearson
More informationCombinational Logic (mostly review!)
ombinational Logic (mostly review!)! Logic functions, truth tables, and switches " NOT, N, OR, NN, NOR, OR,... " Minimal set! xioms and theorems of oolean algebra " Proofs by re-writing " Proofs by perfect
More informationSystems I: Computer Organization and Architecture
Systems I: Computer Organization and Architecture Lecture 6 - Combinational Logic Introduction A combinational circuit consists of input variables, logic gates, and output variables. The logic gates accept
More informationMidterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes
Page 1 of 10 School of Computer Science 60-265-01 Computer Architecture and Digital Design Winter 2009 Semester Midterm Examination # 1 Wednesday, February 25, 2009 Student Name: First Name Family Name
More informationAdministrative Notes. Chapter 2 <9>
Administrative Notes Note: New homework instructions starting with HW03 Homework is due at the beginning of class Homework must be organized, legible (messy is not), and stapled to be graded Chapter 2
More informationAdvanced Digital Design with the Verilog HDL, Second Edition Michael D. Ciletti Prentice Hall, Pearson Education, 2011
Problem 2-1 Recall that a minterm is a cube in which every variable appears. A Boolean expression in SOP form is canonical if every cube in the expression has a unique representation in which all of the
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates
Digital Circuits & Systems Lecture Transparencies (Boolean lgebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean lgebra & Logic Gates Major topics Boolean algebra NND & NOR gates Boolean algebra
More informationLogic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 1 Gate Circuits and Boolean Equations
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hperlinks are active in
More informationCOM111 Introduction to Computer Engineering (Fall ) NOTES 6 -- page 1 of 12
COM111 Introduction to Computer Engineering (Fall 2006-2007) NOTES 6 -- page 1 of 12 Karnaugh Maps In this lecture, we will discuss Karnaugh maps (K-maps) more formally than last time and discuss a more
More informationKarnaugh Map & Boolean Expression Simplification
Karnaugh Map & Boolean Expression Simplification Mapping a Standard POS Expression For a Standard POS expression, a 0 is placed in the cell corresponding to the product term (maxterm) present in the expression.
More informationSequential Circuit Design
Sequential Circuit esign esign Procedure. Specification 2. Formulation Obtain a state diagram or state table 3. State Assignment Assign binary codes to the states 4. Flip-Flop Input Equation etermination
More informationELEC Digital Logic Circuits Fall 2015 Logic Minimization (Chapter 3)
ELE 2200-002 igital Logic ircuits Fall 205 Logic Minimization (hapter 3) Vishwani. grawal James J. anaher Professor epartment of Electrical and omputer Engineering uburn University, uburn, L 36849 http://www.eng.auburn.edu/~vagrawal
More informationENGG 1203 Tutorial_9 - Review. Boolean Algebra. Simplifying Logic Circuits. Combinational Logic. 1. Combinational & Sequential Logic
ENGG 1203 Tutorial_9 - Review Boolean Algebra 1. Combinational & Sequential Logic 2. Computer Systems 3. Electronic Circuits 4. Signals, Systems, and Control Remark : Multiple Choice Questions : ** Check
More informationENGG 1203 Tutorial - 2 Recall Lab 2 - e.g. 4 input XOR. Parity checking (for interest) Recall : Simplification methods. Recall : Time Delay
ENGG 23 Tutorial - 2 Recall Lab 2 - e.g. 4 input XOR Parity checking (for interest) Parity bit Parity checking Error detection, eg. Data can be Corrupted Even parity total number of s is even Odd parity
More informationinflow outflow Part I. Regular tasks for MAE598/494 Task 1
MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the
More information