Why digital? Overview. Number Systems. Binary to Decimal conversion

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1 Why digital? Overview It has the following advantages over analog. It can be processed and transmitted efficiently and reliably. It can be stored and retrieved with greater accuracy. Noise level does not affect much. Digital takes only two voltage levels either Low or High. It is because of switching devices we use like Diode, Transistors and relays. Number Systems inary rithmetic inary Codes Logic Gates oolean lgebra Karnaugh Map Combinational Circuit Design / 59 / 59 Number Systems ecause of two levels, binary numbers are used in digital systems. High means 5 V. Low means 0 V. In positive logic, High In negative logic Each binary digit is called a bit. Low 0 High 0 Low inary to Decimal conversion The right most bit of the binary number is the LS (least significant bit) whose weight is 0 =. The left most bit is the MS (most significant bit) whose weight depends on the size of binary number. Example: 00 = = = 0 In a fractional number, the left most bit is the MS whose weight is = = = / 59 4 / 59 Decimal to inary conversion systematic method of converting decimal to binary is repeated division by. Example : Convert 4 to binary Remainder 4 0 LS MS * - Stop when the whole number quotient is zero. Decimal fractions to inary Decimal fractions can be converted to binary by repeated multiplication by. Example : Convert 0.65 to binary. Carry 0.65 =.5 MS 0.5 = =.0 LS * - Continue to the desired number of decimal places or stop when the fractional part is all zeros = = / 59 6 / 59 Hexadecimal Numbers. It has sixteen characters.. It is very easy to convert between binary and hexadecimal. 3. When we program in machine language such as writing an instruction in microprocessor, it is easy to write in hexadecimal system Decimal inary Hexadecimal Decimal inary Hexadecimal C D E F 7 / 59 inary to Hexadecimal conversion reak the binary number in 4-bit groups, starting at the right most bit and replace each 4-bit group with the equivalent hexadecimal symbol. Example : convert 0000 and 0 to hexadecimal. 00 }{{} 00 }{{} = }{{} }{{} F 0 = F 6 * - two zeros have been added to complete a 4-bit group 8 / 59

2 Hexadecimal to inary conversion Replace each hexadecimal symbol with a 4-bit binary number. Example: Convert 0 6 and C 6 }{{} }{{} = 0000 }{{} C }{{} C 6 = Octal Numbers. It has eight digits.. Like hexadecimal number system, it provides a convenient way to express binary numbers. 3. It is very easy to convert between octal and binary. Decimal inary Octal However, it is used less frequently than hexadecimal. 9 / 59 0 / 59 inary to Octal conversion. reak the binary number in 3-bit groups, starting at the right most bit and replace each 3-bit group with the equivalent octal digit.. If there are not three bits in the left most group, add enough zeros to complete a 3-bit group. Example: Convert 0000 and 00 }{{} 0 }{{} = }{{} }{{} 0 6 Octal to inary conversion Replace each octal digit with a 3-bit binary number. Example: Convert 5 8 and 50 8 }{{} }{{} = 0 }{{} 5 }{{} = = 6 8 *- one zero has been added to complete a 3-bit group. / 59 / 59 inary ddition = = + 0 = + = 0 and carry to the next column Carrying a is equivalent to adding to that column. Example : dd 4 and 0 in binary 4 0 = = = 4 0 inary Subtraction 0-0 = = and borrow from the next column - 0 = - = 0 orrowing a from a column is equivalent to subtracting from that column. Example : Subtract 5 from 8 binary 8 0 = = = / 59 4 / 59 s and s complements Finding s complement s and s complements are mainly used to represent negative numbers. Finding s complement: s complement of a binary number is found by changing all s to 0s and all 0s to s. Example : inary number 0 0 s complement The s complement of a binary number is found by adding to the LS of the s component. Example : Find the s complement of inary number 0 0 s complement dd s complement 5 / 59 6 / 59

3 Subtraction Using s complement Example : Subtract 5 from 8. Find s complement of binary number corresponding to 5 and add with binary number of 8. s complement of 00 is Example : Subtract 8 from Discard carry and the answer is 00. Since there is no carry, the answer = -( s complement of 0) = / 59 inary Coded Decimal (CD) CD means that each decimal digit, 0 through 9, is represented by a binary code of four bits. It is very easy to convert from CD to decimal and vice versa. Only 0 combinations are used. The remaining 6 combinations are invalid. Decimal Digit CD pplications: Seven segment displays use CD codes. Example: Digital clocks, digital meters. 8 / 59 CD - Conversion Example: Convert 5 and 00 to CD (a) Example: Convert the CD codes 0000 and 0000 to decimal. 00 }{{} 00 }{{} 9 5 (c) (b) 0 }{{} 000 }{{} 7 (d) Logic Gates - NOT Gate It is also called as an inverter. F = F 0 0 = High, 0= Low 9 / 59 0 / 59 Logic Gates - ND Gate Logic Gates - OR Gate F = F = + For a input ND Gate, output F is High iff and are High. For a input OR Gate, output F is High if either or is High or both and are high = High, 0= Low F / 59 = High, 0= Low F / 59 Logic Gates - NND Gate The NND Gate operation is opposite that of ND Gate. Logic Gates - NOR Gate The NOR Gate operation is opposite that of OR Gate. F = F = + For a input NND Gate, output F is Low iff and are High. For a input NOR Gate, output F is High iff and are Low. F F = High, 0= Low 3 / 59 = High, 0= Low 4 / 59

4 Logic Gates - Exclusive-OR Gate Logic Gates - Exclusive-NOR Gate Exclusive-NOR is opposite that of Exclusive-OR. F = = + F = = + The output of an exclusive-or gate is High only when the two inputs are at opposite logic levels. The output of an exclusive-nor gate is High only when the two inputs are at the same level. F F = High, 0= Low 5 / 59 = High, 0= Low 6 / 59 NND - Universal Gate. Invert (NOT) NOR - Universal Gate. Invert (NOT) F = F =. ND Gate. ND Gate 3. OR Gate F = F = F = = + 3. OR Gate F = + F = + = F = + 7 / 59 8 / 59 Logic Circuits oolean lgebra ny system/subsystem/arithmetic circuit can be expressed as functional blocks containing logic gates. There are two types of logic circuits.. Combinational logic : Outputs depend only on the present inputs. Example: Logic gates, Code converters. Sequential logic : Outputs not only depend on the present inputs but also on past inputs. Example: Counter George oole, an English mathematician, developed this algebra in 854. That is why it is named after him. oolean algebra provides a mathematics of digital systems. oolean algebra is a systematic way of analyzing and designing the logic circuits. ll digital systems are sequential logic circuits. In EE0, we design combinational logic circuits. 9 / / 59 Laws of oolean lgebra Rules of oolean lgebra. Commutative laws + = + =. ssociative laws + ( + C) = ( + ) + C (C) = ()C + 0 = + = 0 = 0 = + = + = = = 0 = + = + = + ( + )( + C) = + C 3. Distributive Law ( + C) = + C,, or C can represent a single variable or a combination of variables. 3 / 59 3 / 59

5 DeMorgan s Theorems Example. The complement of a product of variables is equal to the sum of the complements of the variables. = +. The complement of a sum of variables is equal to the product of the complements of the variables. + = Use DeMorgan s theorems to simplify the following expressions.. ( + C). ( + )(C + D) Solution:. ( + C) = + + C = + C. ( + )(C + D) = ( + )+(C + D) = +C D = +C D 33 / / 59 Simplification using oolean lgebra Check yourself. oolean algebra is used to simplify a particular expression to its simplest form.. simplified boolean expression uses the fewest gates possible to implement the given expression. Example: ( + )C + C C + C + C C + C( + ) C + C ( + )C Simplify. + C + C C + C ( + )( + C) + C + + C + C + C + + C + C + C + C + C 35 / / 59 Standard form of oolean expressions Implementation of SOP Example: + CD ny oolean expressions can be converted into either of the following two forms.. Sum of Products (SOP) Form Examples C + C + C + C. Product of Sums (POS) Form Examples ( + + C)( + + C) ( + ) C D CD + CD 37 / / 59 Implementation of POS Karnaugh Map Example : ( + )( + C + D) C D ( + ) ( + C + D) ( + )( + C + D) Karnaugh map provides a systematic way of minimizing the boolean expressions. If properly used, it will produce the simplest form possible.. It is an array of cells in which each cell represents a binary value of the input variables.. The number of cells in a Karnaugh map is equal to the total number of possible input variable combinations. 3. For variables, no of cells is = 4. For 3, 3 = 8 and so on. 4. The cells are arranged in such a way that there is a single variable change (djacency) between adjacent cells. 39 / / 59

6 Karnaugh Map for 3-variables C C C C C C C C C Notice here, the cells are arranged so that there is a single variable change between adjacent cells. It is called as djacency. For example, 00 cell is adjacent to 0, 000, and cell is not adjacent 00,, 00, and 0. Mapping of SOP in Karnaugh Map Example: Example : C + C + C + C C / 59 4 / 59 Simplification - Karnaugh Map fter mapping Sum of Products (SOP) in Karnaugh Map,. Group the s. For grouping use the following procedure.. group must contain either,, 4, 8 or 6 cells. ( powers of two). lways include the largest possible number of s in a group..3 Each cell in a group must be adjacent to one or more cells in that same group..4 Each on the map must be included in at least one group. The s already in a group can be included in another group.. Determine the minimum product term for each group.. For -variable map.. -cell group gives a -variable term.. -cell group gives a -variable term..3 4-cell group gives a value.. For 3-variable map.. -cell group gives a 3-variable term.. -cell group gives a -variable term..3 4-cell group gives a -variable term..4 8-cell group gives a value 3. Sum all the terms to form the minimum SOP expression. 43 / 59 Example Let us simplify C + C + C + C C Minimum SOP expression 0 44 / 59 Check yourself SOP expression from Truth Table Simplify C + C + C + C C Minimum SOP expression 0 + C Consider the truth table for three input variables logic circuit. The output F is true for the following combinations. C, C, C and C. These terms are called minterms. C F F = C + C + C + C It is called as Sum of Products (SOP) expression. F = m(,, 6, 7) m indicates minterm. 45 / / 59 POS expression from Truth Table Consider the same truth table. The output F is false for the following combinations. C, C, C and C. C F F = C + C + C + C y DeMorgan s theorems, F = (++C)(++C)(++C)(++C) It is called as Product of Sums (POS) expression. F = M(0, 3, 4, 5) Check yourself SimplifyF = m(, 3, 4, 6) to minimum SOP. C Minimum SOP expression 0 F = C + C M indicates maxterm. 47 / / 59

7 Check yourself SimplifyF = M(0,, 5, 7) to minimum POS. C F = C + C Minimum POS expression If simplified, F = ( + C)( + C) F = + C + C + C C F = C + C same minimum SOP expression... (see previous slide) Don t Care Conditions In practice, in some applications the function is not specified for certain combinations of the variables. In most applications, we simply don t care what value is assigned to them. These combinations are called don t care conditions. These conditions will provide further simplification of the oolean expression. Example: Map the oolean function F = (, 3, 4, 6) that has the don t care conditions. d = (0, ) C X X 49 / / 59 Combinational Circuit Design Design procedure involves the following steps.. From the specifications given, identify the number of inputs and outputs.. Construct the truth table that defines the required relationship between inputs and outputs 3. Map the minterms into Karnaugh map and obtain the simplified oolean expression for each output. 4. Draw the logic diagram and verify it manually. Design - Example Let us design a circuit with three inputs and one output for the following condition. The output is when the binary value of the inputs is an even number. Solution: Let us name the inputs as,, and C and the output F. Truth Table C F / 59 5 / 59 Design - Example (contd...) K- map C F = C Logic diagram Decoder decoder is a digital circuit that detects the presence of a specified combination of bits on its inputs and indicates the presence of that code by a specified output level. It has N input lines and N output lines pplications: Decoders are used in computer to select input/output devices. CD-to-7-Segment Decoder C F 53 / / 59 -to-4-decoder Let us design. It has inputs and 4 outputs. Depending on the input configuration, decoder will produce high at specified output. Input Output D 0 D D D D 0 = D = D = D 3 = -to-4-decoder - Logic Circuit D 0 = D = D = D 3 = 55 / / 59

8 Encoder Decimal-to-CD Encoder It converts familiar symbols like decimal to coded form like binary or CD. It is just opposite to Decoder. It has N input lines and N output lines. pplication: Keyboard encoder : The ten decimal digits on the keyboard must be encoded for processing by the logic circuitry. When one of the keys pressed, the decimal digit is encoded to the corresponding CD code It has ten inputs (one for each decimal digit) and four outputs corresponding to the CD code. 0 3 Decimal CD Digit / / 59 Decimal-to-CD Encoder - Logic Circuit 3 0 LS MS 59 / 59

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