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EECS 270 Fall 2014, Lecture 25 Page 1 of 7 What we’ve done and working some problems… Today we’re going into review mode. I want to touch on all the material we’ve covered, smooth a few things out, and work a few problems as a group. The “big” topics we’ve covered are: Logic and logic rules. Recall you are to have the following theorems memorized: Commutativity, Associativity, Distrbutivity and DeMorgan’s Law. You should be able to use the rest of the theorems and axioms (found at http://www.eecs.umich.edu/courses/eecs270/Misc/Axioms.pdf) Computer arithmetic and number representation. Using binary numbers, addition and subtraction. o Two’s complement numbers, binary representation of fractions, signed-magnitude. Including range of representation and overflow. o Full adders, half-adders, ripple-carry adders, carry-look ahead adders. o Subtractors, absolute value. o Question: Can you get overflow when taking the absolute value of a two’s complement number? A signed magnitude number? An unsigned number? Combinational logic delay o Finding the worst-case (longest) delay for a circuit. Glitches. Designing circuits to minimize delay. MSI devices. o Combinational: MUX, adder, decoder, encoder, priority encoder. o Sequential: Counter (modulo vs. saturating, up/down), register, shift-register, memory. Memory o Memory types (dynamic vs. static, volatile vs non-volatile), memory organization (square memory converted to something more useful). o Using memory (in a data path mainly) Verilog o Combinational logic, sequential logic, state machines. Reg vs. wire. o Writing and instantiating modules. Being able to convert Verilog to a circuit diagram and back. Sequential logic devices o How to build (SR latch, SR latch with enable, D latch, D flip-flop) o Non-ideal issues (setup and hold time of a flip-flop, oscillation of a latch, how they are related) State machines o How to design with a state-transition diagram o Converting from a state-transition diagram to gates and flip-flops and reversing that. EECS 270 Fall 2014, Lecture 25 Page 2 of 7 o Encoding SMs (One’s hot vs. minimal), minimizing SMs. Datapath and control o How to design datapath and control. Generally we’ve had the datapath and written the controller for it, but.. o Specific applications: One-bus systems (tri-states, reading, timing issues); Basic computer (what makes it a computer vs. just a state machine). Underlying implementation options. o CMOS: Design a circuit with CMOS. Figure out what a CMOS circuit is implementing. o FPGA: Idea of memory as logic. Understanding as to why. Ability to design with it. o Understanding of levels of abstraction below us as a general thing (read lecture 15) Mealy Machines o Timing issues, pros and cons. Minimization o Kmaps, Quine-McClusky, implication tables. Parity and error correction o Hamming distance. Various schemes to detect and correct errors. Lab: o NES controller, traffic light, Robbie and Renee, calculator, up/down counter. You should review them. Misc. o Logic synchronization: Two (or three) flip-flops. Why they work (most of the time). When they won’t work (data too fast compared to our clock) and why. o Tri-state devices: How to use them. How not to use them. Making them out of CMOS. Questions 1. Design a non-inverting tri-state buffer out of CMOS. EECS 270 Fall 2014, Lecture 25 Page 3 of 7 2. Multiple choice. a. A 6-bit 2’s complement number can represent values from _________ to _________. b. ______ is the only negative value that has the same 3-bit representation as both a signed magnitude and 2’s complement number. c. A 6-input NAND gate has ____________ minterms d. If F=∑A,B,C,D (0,2,3,4,5) then F has __________ maxterms. e. DRAM and SRAM both are volatile / are dynamic / store their data in a capacitor. f. Write 23.36 as a decimal number: _____________. g. When comparing a 32-bit ripple-carry adder and a 32-bit carry-lookahead adder, you would expect the carry-lookahead adder to have more / less / exactly the same delay and use more / fewer / exactly the same number of gates. h. Say you want to develop an encoding that allows you to send 2-bit messages (one of {00,01,10,11}) over a wire. Say you wish to encode that information so that one bit of error can be detected. How many bits would you need, at a minimum, to encode each of these messages allowing for one bit of detection? 3 / 4 / 5 / 6 / 10 i. If A↑B=C and B↑C=A then ↑ is the Boolean operator OR / AND / NOR / NAND/ XOR. EECS 270 Fall 2014, Lecture 25 Page 4 of 7 3. Reduce the number of states in the state transition diagram to as few states as. Show your work and draw the reduced state diagram. [10] X X’ X’ B Z=0 A Z=1 X’ X X X X’ C Z=0 D Z=0 X’ X X’ X F Z=1 X E Z=0 A B C D E F A B C D EECS 270 Fall 2014, Lecture 25 Page 5 of 7 4. Fill the boxes in both 8x2 memories and the switch matrix below to implement the following logic functions: X=((A*B)+(C*!B))⊕(D*E) Y=A+B+C Clearly indicate what each Px is connected to in the table below leaving any values that don’t matter blank. [12] P0 P1 A P2 P3 P4 P5 P6 P7 P8 P9 EECS 270 Fall 2014, Lecture 25 Page 6 of 7 5. A common task is to count the number of 1s in a string. Let X[6:0] be the input and C[2:0] be the output. C should be the number of 1s in X. So if X=”1100011” C should be “100” or 4 because there are 4 1s in X. Design a circuit which computes C given X. You may only use the following components: 1-bit full adders, 2-to-4 decoders, 2-to-1 MUXes and 3-input AND gates. To get full credit, you must use as few devices as possible. Any solution that uses more than 10 devices will get zero points. EECS 270 Fall 2014, Lecture 25 Page 7 of 7 6. Using a K-map find all the minimal sum-of-rpodcuts of F= Σ(PQRS) (1,2,4,5,6,7,8,9,10,12,13,14). Place a star (*) next to each distinguished 1 in your K-map. 7. Using only standard gates (AND, OR, NOT, XOR), D flip-flops and bubbles as well as a single up counter with a synchronous reset, design a device which generates a clock with 12 times the period of the input clock and has a duty cycle of 1/3. The output should be glitch free and your answer should be clear and neat.