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1 Save from: cs Logic design 1 st Class أستاذ المادة: د. عماد استاذة المادة: م.م ميساء

2 Contents Lectured One: Number system operation 1- Decimal numbers. 2- Binary numbers. 3- Octal numbers. 4- Hexadecimal numbers. Lectured Two: Binary arithmetic 1- Binary Addition. 2- Binary Subtraction. 3-1's and 2's Complement of Binary Number. 4- Hexadecimal Addition &Subtraction. 5- Octal Addition &Subtraction. 6- Gray Code. 7- Access 3 code. Lectured Three: Logic Gats 1- Set of Gats AND, OR, NOT, XOR, NOR, NAND, BUFFER. 2- HALF-ADDER. 3- FULL- ADDER. Lectured Four: Boolean Algebra &Logic Simplification 1- Rules of Boolean algebra. 2- Examples 3- Demorgan's theorems. 4- Example. 5- Sun of Product (SOP). 6- Product of Sum (POS). Lectured Five: Karnaugh map 1- Three variable karnaugh map. 2- Four variable karnaugh map. ٢

3 Lectured Six: Combinational Logic 1- The NAND Gate as a Universal Logic Element. 2- The NOR Gate as a Universal Logic Element. 3- Bit Parallel Adder. 4- Example. Lectured Seven: 1- Decoders. 2-Encoders. 3- Multiplexer. Lectured Eight: Flip-Flop 1- SR Flip-Flops. 2- D Flip-Flops. 3- JK Flip-Flops. References 1- Computer System Architecture Third Edition M. Morris Mano 2- Digital Fundamentals Eight Edition FLOYD ٣

4 Lectured One 1- Number Systems Operation: 1- Decimal Numbers. 2- Binary Numbers. 3- Octal Numbers. 4- Hexadecimal Numbers. 1- Decimal Numbers: In the decimal number system each of the ten digits (10digits), 0 through 9 (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). Decimal weight Example (1): (345) =10 2 * * *5=345= (345) Example (2): 23.5 = (23.5) 10 2* * *10-1 = =23.5 Where 10 0 =1 2- Binary Numbers: The binary number system its two digits a basetwo system. The two binary digits (bits) are 1 and 0 (1,0). Binary weight Weight value A- Binary to Decimal Conversion: *Binary number where 2 0 = = 2 6 * *1+2 4 *0+2 3 *1+2 2 *1+2 1 *0+2 0 *1 = =96+13=109 (109) 10 ٤

5 *The fractional binary number = 1* *2-2 +1*2-3 +1*2-4 = = (0.6875) 10 B- Decimal to Binary Conversion: 1- Convert a decimal whole number to binary using the repeated division by 2 method. 2- Convert a decimal fraction to binary using the repeated Multiplication by 2 method. Example (1): Number (58) 10 ==== (111010) mod LSB 2 29 == == == 0 ======= (111010) == == 1 0 == 1 MSB Example (2): Number (0.3125) 10 ====== (0101) 2 MSB LSB carry * * * * (0101) 2 ٥

6 3- Octal Numbers: The octal number system is composed of eight digits, which are 0, 1, 2, 3, 4, 5, 6, and 7. To count above 7, begin another column and start over: 10, 11, 12, 13, 14, 15, 16, and , 21, 22, 23, 24, 25, 26, and , 31, 37. A- Octal to Decimal conversion: Weight Octal number 2374 ==== (1276) 10 xample: (2374) 8 = 2*8 3 +3*8 2 +7*8 1 +4*8 0 = 2*512+3*64+7*8+4*1 = = (1276) 10 B- Decimal to Octal Conversion: Example: Decimal number (359) 10 ======= (547) mod LSB 8 44 == == 4 ==== (547) 8 0 == 5 MSB ٦

7 C- Octal to Binary Conversion: Octal digit can be represented by a 3-bit binary number. Octal digit binary Examples: (25) 8 (140) 8 (2 5) 8 (1 4 0) 8 (010101) 2 ( ) 2 D- Binary to Octal Conversion: Conversion binary number to octal number is start with right most group of three bits and moving from right to left. Examples: (110101) 2 ( ) (6 5) 8 (5 7 1) 8 (65) 8 (571) 8 ٧

8 4- Hexadecimal Numbers: The hexadecimal number system has a base of sixteen; it is composed of 16 digits and alphabetic characters. Decimal Binary Hexadecimal A B C D E F ٨

9 A- Binary to Hexadecimal conversion: 4-bit groups, starting at the right-most bit. Example: ( ) 2 ======= (CA57) C A 5 7 B- Hexadecimal to Binary Conversion: Example: (10A4) 16 ========= ( ) A C- Hexadecimal to Decimal Conversion: By to method * First method: Example: (A85) 16 ==== (2693) Convert to binary number. 2- Convert from binary number to decimal number. A = 2 11 * *0+2 9 *1+2 8 *0+2 7 *1+2 6 *0+2 5 *0+2 4 *0+2 3 *0+2 2 *1+2 1 *0+2 0 *1= = =2693= (2693) 10 * Second method: Example: (E5) 16 ======== (229) 10 (E5) 16 =E* *16 0 =14*16+5*1=224+5=229= (229) 10 ٩

10 D- Decimal to Hexadecimal Conversion: Example: Convert the decimal number 650 to hexadecimal by repeated division by 16. (650) 10 ===== (28A) 16 Mod LSD ====== A 16 2 ====== 8 MSD 2 8 A LSD = (28A) 8 0 ====== 2 MSD ١٠

11 Lectured Two 2-Binary Arithmetic: 1- Binary Addition. 2- Binary Subtraction. 3- Binary Multiplication. 4- Binary Division. 1- Binary Addition: The four basic rules for adding binary digits (bits) are as follows. 0+0=0 Sum of 0 with a carry 0 0+1=1 Sum of 1 with a carry 0 1+0=1 Sum of 1 with a carry 0 1+1=1 0 Sum of 0 with a carry 1 Examples: Binary Subtraction: The four basic rules for subtracting are as follows. 0-0=0 1-1=0 1-0=1 0-1=1 0-1 with a borrow of 1 ١١

12 Examples: 's And 2's Complement of Binary Number: The 1's complement and the 2's complement of binary number are important because they permit the representation of negative numbers. Binary Number 'sComplement o 1 0 2's Complement of a binary number is found by adding 1 to the LSB of the 1's Complement. 2's Complement= (1's Complement) +1 Binary number 'scomplement Add 's complement ١٢

13 In decimal number complement such as: 0==== 9 7==== 2 6==== 3 9==== 0 4==== 5 1==== 8 Signed Numbers: Signed binary number consists of both sign and magnitude information. 0 positive numbers The sign bit 1 negative numbers sign bit magnitude bits Example: Express the decimal number - 39 as an 8-bit number in the sign-magnitude, 1's complement, and 2's complement forms. Solution: 1- Write the 8-bit number for 's complement Add 1 1 sign bit negative = - 39 ١٣

14 4- Hexadecimal Addition & Subtraction: Hexadecimal Addition: 2A7 2AB 2B BE 5C2 AF Hexadecimal subtraction: CA2 47C - A1B - 2BE 287 1BE 5- Octal Addition & Subtraction: Binary Coded Decimal (BCD): Binary coded decimal means that each decimal digit, 0 through 9, is represented by a binary code of four bits. The (BCD) Code: The code is a type of (BCD) code. The indicates the binary weights of the four bits (2 3, 2 2, 2 1, 2 0 ). ١٤

15 6 -The Gray Code: Example: Convert binary to Gray Binary ==== Gray ==== Convert Binary to Gray Gray ==== Binary ==== Decimal Binary Gray ١٥

16 7- Excess -3 Code: Addition three to any number in decimal number or binary number such as in table. Decimal BCD Excess -3 Excess -3 Gray ١٦

17 Logic Gats: 1- Set of Gets Lectured Three ١٧

X Y C S 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 Figure (1-a) Logic diagram for half adder Figure (1-b) Truth table for half adder Half- Adder

18 2- Half Adder: The basic digital arithmetic circuit is the addition of two binary digits. Input variables of a half-adder call augends & addend bits. The output variables the sum & carry. X Y C S Figure (1-a) Logic diagram for half adder Figure (1-b) Truth table for half adder Half- Adder questions: S=XY+XY S=X (+)Y C=X*Y 3-Full-Adder: A full - adder is a combinational circuit that forms the arithmetic sum of three input bits. It consists of three inputs &two outputs. Figure (2-a) Logic diagram for full adder ١٨

19 X FA s Y c Z Figure (2-b) Block diagram for full adder Inputs Out puts X Y Z C S Figure (2-c) Truth table for full adder Full - Adder questions: S=x (+) y (+) z C=XY+ (XZ (+) YZ) C=X*Y+ (X (+) Y) Z ١٩

20 Lecture Four Boolean Algebra &Logic Simplification: 1-Rules of Boolean algebra: 1- A+0=A 2- A+1=1 3- A*0=0 4- A*1=A 5- A+A=A 6- A+A=1 7- A*A=A 8- A*A=0 9- A=A ====== Demoragan's theorems 10- A+BA=A 11- A+AB=A+B 12- (A+B)(A+C)=A+BC 2- Examples: Example 1: F = X + ý Determine the truth table and logic diagram X Y Z F Truth table Logic diagram ٢٠

21 Example 2: AB+ A (B+C)+ B(B+C) 1- AB+AB+AC+BB+BC 2- AB+AB+AC+B+BC 3- AB+AC+B+BC 4- AB+AC+B 5- B+AC Example 3: F=ABC+ABĆ+ĂC F= AB(C+Ć) +ĂC F= AB+ĂC ٢١

22 Example 4: 3- Demorgan's theorems: Demorgan's theorems ٢٢

23 4- Example: Example 1: a- (A+B)+C = (A+B) C = (A+B)C b- (A+B) +CD = (A+B) CD =(A B) (C+D) = A B (C+D) c- (A+B) C D + E + F = ((A+B) C D) (E+ F) = (A+B+C+D) (E F) = ( A+B+C+D) E F 5- Sum Of Products (SOP): X=AB+BCD+AC Example: a- AB+B(CD+EF)=AB+BCD+BEF b- (A+B)(B+C+D)=AB+AC+AD+BB+BC+BD c- (A+B)+C=(A+B)*C =(A+B)C=AC+BC ٢٣

24 6- Product Of Sum(POS): (A+B)(B+C+D)(A+C) Example: SOP A B X F A B A B Example: POS A B X F A+ B A+ B ٢٤

25 Lectured Five Karnaugh map: 1- Three variable karnaugh map. ٢٥

26 2- Four variable karnaugh map. ٢٦

27 Combinational Logic: Lectured Six 1-The NAND Gate as a Universal Logic Element: Figure (3) NAND Gates ٢٧

28 2-The NOR Gate as a Universal Logic Element: Figure (4) NOR Gates ٢٨

29 3- Bit Parallel Adder: A group of four bits is a nibble. A basic 4-bit parallel adder is implementation with four full adder stages. Figure (5) 4-bit parallel adder Figure (6) Symbol Logic ٢٩

30 4- Example: Draw the 4-bit parallel adder, find the sum and output carry for the addition of the following two 4-bit numbers if the input carry (C n-1 ) is 0: A4A3A2A1=1010 and B4B3B2B1=1011 Solution: For n=1 A1=0, B1=1, C n-1 =0 =1, and C1=0 For n=2 A2=1, B2=1, C n-1 =0 =0, and C2=1 For n=3 A3=0, B3=0, C n-1 =1 =1, and C3=0 For n=4 A4=1, B4=1, C n-1 =0 =0, and C4=1 ٣٠

Lectured Seven Decoders & encoders: 1- Decoder: A decoders is combinational circuit that converts binary information form the n coded inputs to a maximum of 2 n unique outputs.

31 Lectured Seven Decoders & encoders: 1- Decoder: A decoders is combinational circuit that converts binary information form the n coded inputs to a maximum of 2 n unique outputs. That decoders are called n-to-m line decoders where m <=2 n. The logic diagram of a 3-to-8 line decoder is three data inputs,a0,a1,and A2 are decoded into eight out puts, each out puts representing one of the combinations of the three binary input variables. This decoder is a binary to octal conversion. Figure (7) 3-to-8 line decoder ٣١

Enable Inputs Outputs E A2 A1 A0 D7 D6 D5 D4 D3 D2 D1 D0 0 X X X 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0

32 Enable Inputs Outputs E A2 A1 A0 D7 D6 D5 D4 D3 D2 D1 D0 0 X X X Truth table for 3-to-8 line decoder Figure (8) 2-to-4 line decoder ٣٢

33 Enable Inputs Outputs E A1 A0 D0 D1 D2 D X X Truth table for 2-to-4 line decoder 2- Encoder: An encoder is a digit circuit that performs the inverse operation of a decoder. An encoder has 2 n (or less) input lines and n output lines. An encoder is the octal to binary encoder. It has eight inputs, one for each of the octal digits, and three outputs that generate the corresponding binary number. Ao = D1+D3+D5+D7 A1 = D2+D3+D6+D7 A2 = D4+D5+D6+D7 (Implementation in three OR gates) Inputs Outputs D7 D6 D5 D4 D3 D2 D1 D0 A2 A1 A Truth table for octal to binary encoder ٣٣

3- Multiplexers: A multiplexer is a combinational circuit that receiver binary information form one of 2 n input data lines and directs it to a single out put line.

34 3- Multiplexers: A multiplexer is a combinational circuit that receiver binary information form one of 2 n input data lines and directs it to a single out put line. The selection of a particular input data line for the output is determined by a set of selection inputs. A 2 n - to- 1, A 4-to-1. Multiplexer is called Data Selector. Figure (9) 4-to-1 line multiplexer Inputs Outputs 0 0 Y1 0 1 Y2 1 0 Y3 1 1 Y4 Truth table for 4-to-1 multiplexer ٣٤

Lectured Eight Flip-Flop: The storage elements employed in clocked sequential circuits are called flip-flops. A flip-flops is a binary cell capable of storing one bit of information.

35 Lectured Eight Flip-Flop: The storage elements employed in clocked sequential circuits are called flip-flops. A flip-flops is a binary cell capable of storing one bit of information. It has two outputs, one for the normal value and one for the complement value of the bit stored in it. Type of flip-flops: 1- SR flip-flops. 2- D flip-flops. 3- JK flip-flops. 1- SR FLIP-FLOPS: Figure (10) Logic diagram for SR flip-flop ٣٥

Inputs Outputs S R Q Q Comments 0 0 1 1 Invalid condition 0 1 1 0 Latch set 1 0 0 1 Latch reset 1 1 N.C N.

36 Inputs Outputs S R Q Q Comments Invalid condition Latch set Latch reset 1 1 N.C N.C No change Truth table for SR flip-flop 2- D FLIP-FLOPS: FIGURE (11) D Flip-flop Inputs Outputs D CLK Q Q Comments Set(stor1) Reset(stor0) Truth table for D flip-flop ٣٦

37 3- J K FLIP-FLOPS: Figure (12) JK Flip-flop Inputs Outputs J K CLK Q Q Comments 0 0 Q0 Q 0 No change Reset Set 1 1 Q 0 Q0 Toggle Truth table for JK flip-flop ٣٧

38 Answer these questions: Q1- Convert the following: 1- (CF8E) 16 to ( ) (1725) 10 to ( ) ( ) 10 to ( ) 2 4- (7526) 8 to ( ) 10 5-(2591) 10) to ( ) (B2F8) 16 to ( ) 10 Q2- Perform the following: 1- (2AB) 16 (317) (101101) 2 (1110) 2 3- (6410) 8 (324) 8 4- (2CF) 16 (FDB) (4732) 8 + (4611) 8 Q3- Express the decimal number -98, -68 as 8-bit number in the sign-magnitude, 1'S and 2'S Complement. Q4- Design Full-Adder circuit. Q5- Design Half-Adder circuit. Q6- Draw the 4-bit parallel adder, find the sum and output carry for the addition of the following two 4-bit numbers if the input carry (C n-1 ) is 0: A4A3A2A1=1011 and B4B3B2B1=0111. Q7- use K- map to minimize the following SOP expression and convert to POS in K- map. F(A,B,C,D)= 2,3,4,5,6,7,9,12,13,14,15 Q8- Design 3-to-8 lines decoders. OR Design Binary-to-Octal line decoder. ٣٨

39 Q9- Design 2- to 4 lines decoders Q10- Design block diagram of quadruple 2-to-1 line multiplexer. Q11- Design SR flip-flop and explain function. Q12- Design D flip-flop and explain function. Q13- Design JK flip-flop and explain function. Q14- Design Octal-to- Binary line encoder. Q15- Difference between SR and JK flip-flop. ٣٩

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION