Csc 2313 (lecture 4)
- 1. INTRODUCTION TO COMPUTER SYSTEM CSC 2313 LECTURE 4 Department of Maths and Computer-Science Faculty of Natural andApplied Science BY UMAR DANJUMA MAIWADA
- 2. OBJECTIVES Boolean Algebra, Functions, And Minimization 2
- 3. INTRODUCTION Binary logic consists of binary variables and logical operations.The variables are designated by letters of the alphabet such asA, B, C, x, y, z, etc., with each variable having two and only two distinct possible values: 1 and O.There are three basic logical operations:AND, OR, and NOT Logic gates are primarily implemented using diodes or transistors acting as electronic switches, but can also be constructed using vacuum tubes, electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements. 3
- 4. DIGITAL LOGIC DESIGN Digital (electronic) circuits are electronics that handle digital signals Digital techniques are useful because it is easier to get an electronic device to switch into one of a number of known states than to accurately reproduce a continuous range of values. They correspond to the false and true values of the Boolean domain respectively. 4
- 5. LOGIC GATES logic gate is an idealized or physical device implementing a Boolean function; that is, it performs a logical operation on one or more binary inputs, and produces a single binary output. With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing the construction of a physical model of all of Boolean logic, and therefore, all of the algorithms and mathematics that can be described with Boolean logic. 5
- 6. NOT GATE Inversion, negation”“NOT” symbol → Z= X' (Also called the complement) NOT gate is more commonly called an inverter The circle on the symbol is called a bubble, and is used in logic diagrams to indicate a logic negation between the external logic state and the internal logic state (1 to 0 or vice versa). On a circuit diagram it must be accompanied by a statement asserting that the positive logic convention or negative logic convention is being used (high voltage level = 1 or low voltage level = 0, respectively). 6
- 7. OR GATE The rest of the class relies on two-valued BooleanAlgebra, i.e. B = {0, 1}We use variables X,Y, Z,A, B, … and constants 0 and 1. **Note“0”and“1”are also called identity elements. Binary operators → “+” called “OR”. • “OR” symbol → Z=X+Y. 2-input gates 7 .
- 8. AND GATE AND symbol → Z=X.Y 2-input gates
- 9. XOR The output of a two input exclusive-OR is true only when the two input values are different, and false if they are equal, regardless of the value. If there are more than two inputs, the output of the distinctive-shape symbol is undefined.The output of the rectangular-shaped symbol is true if the number of true inputs is exactly one or exactly the number following the "=" in the qualifying symbol. 2-input gates 9
- 10. NAND GATE 10
- 11. NOR GATE Inverted 2- input gates 11
- 12. XNOR GATE Inverted 2- input gates
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- 18. DE MORGAN EQUIVALENT SYMBOLS By use of De Morgan's laws, an AND function is identical to an OR function with negated inputs and outputs. Likewise, an OR function is identical to an AND function with negated inputs and outputs.A NAND gate is equivalent to an OR gate with negated inputs, and a NOR gate is equivalent to anAND gate with negated inputs. This leads to an alternative set of symbols for basic gates that use the opposite core symbol (AND or OR) but with the inputs and outputs negated. Use of these alternative symbols can make logic circuit diagrams much clearer and help to show accidental connection of an active high output to an active low input or vice versa. De Morgan's theorem is most commonly used to implement logic gates as combinations of only NAND gates, or as combinations of only NOR gates, for economic reasons. 18
- 19. Boolean Algebra Theorems Purpose of Theorems TheTheorems & Huntington’s Postulates are key in our ability to reduce the number of literal (variables) used in a function and therefore reduce the number of gates required to implement a given function. Sometimes they are used to simply rearrange the expression so it is easier to implement. 19
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- 23. 23 Utilizing Demorgan’sTheorem NAND and NOR gates may be represented using the other’s base signal as shown below (“not” circle indicates complement):
- 24. 24 Transforming from one form to another requires only two steps: 1) Complement every input and output. 2) Swap OR and AND gates. Example: Design an XOR using only NAND gates. F (A, B) =A ⊕ B Solution: F (A, B) =A’.B + B’.A Apply conversion rules “Complement every input and output; Swap ORs and ANDs F (A, B) = ((A’.B)’ . (B’.A)’)’
- 25. BINARY CODED DECIMAL (BCD) A common way to represent the digits 0 - 9 is by the ten four-bit patterns There are six bit patterns (for example 1010) that are not used, and the question is what to do with them. The unused bit patterns might simply be ignored. If a decoder encounters one, perhaps as a result of an error in transmission or an error in encoding, it might return nothing, or might signal an output error. The unused patterns might be mapped into legal values. For example, the unused patterns might all be converted to 9, under the theory that they represent 10, 11, 12, 13, 14, or 15, and the closest digit is 9. They might be decoded as 2, 3, 4, 5, 6, or 7, by setting the initial bit to 0, under the theory that the first bit might have gotten corrupted.
- 26. 26 Digit Code 0 0 0 0 0 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 8 1 0 0 0 9 1 0 0 1 Binary Coded Decimal
- 27. ASCII TheAmerican Standard Code for Information Interchange Introduced by theAmerican National Standards Institute (ANSI) in 1963. ASCII is a seven-bit code, representing the 33 control characters and 95 printing characters (including space) The control characters are used to signal special conditions 27
- 29. 8 BITS ASCII characters follow a leading 0, and thus may be thought of as the “bottom half ” of a larger code on 8-bit context. The 128 characters represented by codes between HEX 80 and HEX FF (sometimes incorrectly called “highASCII” of “extendedASCII”) have been defined differently in different contexts. People now appreciate the need for interoperability of computer platforms, so more universal standards are coming into favor. Most people don’t want 32 more control characters (indeed, of the 33 control characters in 7-bitASCII, only about ten are regularly used in text). 29
- 31. REFERENCES Digital logic Design by Norman BalabanianUniversity of Florida Bradley Carlson SymbolTechnologies,Inc. Tinder,Richard F.(2000).Engineering digital design:Revised Second Edition. pp.317–319.ISBN 0-12-691295-5.Retrieved 2008-07-04. Bostock,Geoff (1988).Programmable logic devices:technology and applications. NewYork:McGraw-Hill.ISBN 978-0-07-006611-3.Retrieved 28 November 2012. 31
- 32. QUESTIONS??? THANK YOU FOR YOUR ATTENTION 32