1. Materials on the Exam Introduction Data Representation in Computer Systems Boolean Algebra Digital Logic MARIE: An Introduction to a Simple Computer Until first of Page 193.

2. Introduction Please go through the chapter’s one slide

3. Data Representation in Computer Systems This is Chapter 2 Concentrate on solving all the examples

4. Signed Binary Numbers Although there is only one way to represent +9, there are three different ways to represent (-9) In signed-magnitude representation 10001001 In signed-1’sComplement representation 11110110 In signed-2’sComplement representation 11110111

5. In signed-magnitude representation The Addition of two numbers in the signed- magnitude system follows the rules of ordinary arithmetic sign. If the signs are the same, we add the two magnitudes and give the sum the common sign. If the signs are different, we subtract the smaller magnitude from the larger and give the result the sign of the larger magnitude

6. One’s Complement To understand the Ones Complement Mathematics, you should solve the examples on page 53

7. Two Complement Solve the examples on page 55,56 Important Rule on page 55 in Bold sentence. To understand this rule, please go through a table on page 63 table 2.2. very important

8. T/F question Typical T/F question in the exam as follows When a carry occurs during the arithmetic operation this means an overflow is occurred T/F Adding +4 to +6 in four bit binary system will result in overflow T/F 3 bits for magnitude and 1 bit for sign this result in overflow and the result is error To answer this question you have to know the rules for overflow

9. Floating Point Representation 14 bit model See page 63,64,65 The idea of adding Floating point is the same idea in normal mathematics (unify the exponent)

10. Boolean Algebra It’s all about AND, OR, Not You should know the truth table for them page 111 You have to know that before designing a circuit you have to make mathematical model for it. As your model is simple, your circuit will be simple also and less cost

11. Figure 02.01 Simple Model Floating-Point Representation Boolean Algebra To simplify the models (Boolean Algebra) Use the table 3.5 page 113 Study the examples on page 114,115 See the logic gates symbols at page 119 You can design simple circuits as in page 120 You should know XOR exclusive, you have to make a truth table

12. XOR Gate

13. Boolean Algebra Study the pages of 124,125

14. Designing a Circuit The first point for designing a digital circuit are Describe your input, all combinations Determine the out put, could be one ore more Then construct the truth table Simplify the output using algebra simplification (this step is only to simplify the design)

15. Binary n-to-2 n Decoders A binary decoder has n inputs and 2 n outputs. Only one output is active at any one time, corresponding to the input value

16. 2-to-4 Binary Decoder A 2 to 4 decoder consists of two inputs and four outputs, truth table and symbols of which is shown below

17. 2-to-4 Binary Decoder Truth Table2 to 4 decoder

18. 2-to-4 Binary Decoder

19. Q28 Find Truth Table

20. Flip-Flops All the digital circuits we studied in the past are called combinational circuits, the following circuits are called sequential circuits and they used a lock to operate. And they consider the main unit of memory design.

21. Flip-Flops SR-flip flop on page 133

22. Figure 03.21 a) Actual SR Flip-Flop b) Characteristic Table for the SR Flip-Flop See Characteristic table figure 3.21 (b) page 133 SR-flip flip

23. Figure 03.23 a) D Flip-Flop b) D Flip-Flop Characteristic Table c) D Flip-Flop as a Modified SR Flip-Flop D-flip flop

24. Figure 03.22 a) JK Flip-Flop b) JK Characteristic Table c) JK Flip-Flop as a Modified SR Flip-Flop JK flip-flop

25. Mealy Model Mealy machine’s outputs are a function of it’s current state and it’s input We will see an example on D flip flop

26. Analyze this sircuit

27. The Solution First we have to write the equations 1.A(t+1)=Ax+BX 2.B(t+1)=A`x 3.Y=(A+B)X Second :draw the state table base on the equations you have

28. State Table

29. Second Form of state table

30. State Diagram

31. Q41 Complete the truth table

32. Q41 complete the truth table ABXA(t+1)B(t+1) 000?? 001?? 010?? 011?? 100?? 101?? 110?? 111??

33. Q41 /solution JA=(A+X)` =A`X` ;;;; KA=JA B=AB` ABXJAKAD 000110 001000 010110 011000 100111 101111 110110 111110

34. Q41 solution ABXJAKADA(t+1)B(t+1) 00011010 00100000 01011010 01100000 10011101 10111101 11011000 11111000