1. 1 Combinational Logic Design Digital Computer Logic Kashif Bashir http://www.taleem.greatnow.com http://www.taleem.greatnow.com Email: kashif@pafkiet.edu.pk

2. 2 Module Outline Combinational Circuits Design Topics Analysis Procedure Design Procedure Decoders Encoders Multiplexers Binary Adders Binary Subtraction Binary Adder-subtractors Binary Multipliers Decimal Arithmetic

3. 3 Combinational Circuits The outputs are a function of the present set of inputs only The inside of a combinational circuit is made of logic gates Combinational logic circuits are important components of digital systems Each output can be thought of as a function of all the inputs – if there are m outputs and n inputs then there are m boolean functions, one describing each output

4. 4 Combinational Circuits Combinational Logic Circuit m Outputsn Inputs

5. 5 Design Topics - Design Hierarchy Uses divide and conquer approach Break the circuit into smaller pieces called blocks If a block is too large break it up into still smaller blocks At each level or hierarchy we have a representation of the blocks such as block diagram, gates, circuits Benefits Reusability of blocks Quicker design completion Easier to verify and isolate problems

6. 6 Figure 3-2

7. 7 Diagrams Representing the Hierarchy for Figure 3-2

8. 8 Design Topics - Computer Aided Design Digital Design of real world systems has become too complex to be done by hand alone Digital design tools or software helps in creating design and verify it Uses Hardware Description languages such as VHDL, Verilog Computer Aided Design Tools exist for: Schematic Capture Simulation of Design Verification Synthesis Etc.

9. 9 Design Topics - Top Down Design Ideally deign is top down which means the design is specified at very high levels of abstraction as text along with constraints on cost, performance size, reliability, etc. Then the design is repeatedly divided into smaller and smaller blocks and implemented with these blocks In order to obtain reusability and to make maximum reuse of predefined modules, often portions of the design are performed bottom-up

10. 10 Analysis Procedure Derivation of Boolean Functions Derivation of the Truth Table Logic Simulation

11. 11 Design Procedure 1. From the specifications of the circuit, determine the required number of inputs and outputs and assigned a letter symbol to each 2. Derive the truth table that defined the relationship between inputs and putouts 3. Obtain the simplified Boolean functions for each output as a function f the input variables 4. Draw the logic diagram 5. Verify the correctness of the design

12. 12 Some Design Examples BCD to Excess-3 code converter BCD to Seven Segment Decoder

13. 13 BCD to Excess-3 code converter

14. 14 BCD to Excess-3 code converter K-Maps

15. 15 BCD to Excess-3 code converter Logic Diagram

16. 16 Decoders n to 2 n decoder - a combinational logic circuit that converts binary information from the n coded inputs to a maximum of 2 n unique outputs Also called the n-to-m line deciders for example: 2-to-4 line decoder 3-to-8 line decoder Decoder Combinational Logic Circuit n Inputs 2 n Outputs

17. 17 Decoder Diagrams

18. 18 Decoder with Enable Input

19. 19 Decoder Expansion Can use smaller decoders to build bigger decoders. e.g. using two 2- 4 decoders we can build a 3-8 decoder or using two 3-8 decoders we can build a 4- 16 decoder

20. 20 Combinational Circuit Implementation using Decoder A n-input decoder generates the minterms corresponding to a boolean function of n- variables Can use the outputs of a decoder along with an OR gate to implement a Boolean function in SOP form

21. 21 Combinational Circuit Implementation using Decoder

22. 22 Encoders Performs the inverse operation of a decoder Has 2 n or fewer input lines and n output lines The output generates the binary code corresponding to the input value Encoder Combinational Logic Circuit n Outputs2 n Inputs

23. 23 Examples of Encoders Octal to Binary Encoder Priority Encoder

24. 24 Priority Encoder

25. 25 Multiplexers A multiplexer is a combinational circuit that selects binary information from one of many input lines and directs the information to a single output line MULTIPLEXER Combinational Logic Circuit 1 Output2 n Inputs n Selection Lines

26. 26 4-to-1-Line Multiplexer MULTIPLEXER Combinational Logic Circuit Output4 Inputs S0S0 S1S1 D0D0 D1D1 D2D2 D3D3 S1S0S1S0 Function table Output 0 0101 1010 1 D0D0 D1D1 D2D2 D3D3

27. 27 Building MUXes

28. 28 Building MUXes A MUX can be built using transmission gates Mutiplexer blocks can be combined in parallel with common selection and enable lines to perform selection on multi-bit quantities

29. 29 Implementing Boolean Functions Using MUX Any Boolean function of n-variables can be implemented using a MUX with n selection lines A more efficient method is using a MUX with n-1 selection lines. Let us take a look at each method

30. 30 Example of Implementing Boolean Functions Using MUX

31. 31 Example of Implementing Boolean Functions Using MUX

32. 32 Demultiplexers Performs the inverse operation of a multiplexer A combinational circuit that receives input from a single line and transmits it to one of 2 n possible output lines The selection of the specific output is controlled by the bit combination of n selection lines Same as a Decoder with an enable – how?