Digital Logic Circuits 1 Introduction DIGITAL LOGIC CIRCUITS
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Digital Logic Circuits 1 Introduction DIGITAL LOGIC CIRCUITS Logic Gates Boolean Algebra Map Specification Combinational Circuits Flip-Flops Sequential Circuits Memory Components Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Logic Gates BASIC LOGIC BLOCK - GATE Binary Digital Input Signal . . . Gate Binary Digital Output Signal Types of Basic Logic Blocks - Combinational Logic Blocks whose output logic value depends only on the input logic values - Sequential Logic Blocks whose output logic value depends on the input values and the state (stored information) of the blocks Functions of Gates can be described by - Truth Table - Boolean Function - Karnaugh Map Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Logic Gates COMBINATIONAL GATES Name AND OR Symbol A B Function X X X=A • B or X = AB X=A+B I A X X=A Buffer A X X=A NAND A B NOR XOR A B A Exclusive OR B XNOR A Exclusive NOR or Equivalence B Computer Organization X X = (AB)’ X X = (A + B)’ X X=A B or X = A’B + AB’ X X = (A B)’ or X = A’B’+ AB Truth Table A 0 0 1 1 A 0 0 1 1 B 0 1 0 1 A 0 1 B 0 1 0 1 X 0 0 0 1 X 0 1 1 1 X 1 0 X 0 1 X 1 1 1 0 X 1 0 0 0 X 0 1 1 0 X 1 0 0 1 Computer Architectures Lab
Digital Logic Circuits 1 Boolean Algebra LOGIC CIRCUIT DESIGN x 0 0 1 1 Truth Table Boolean Function Logic Diagram Computer Organization y 0 0 1 1 z 0 1 0 1 F 0 1 0 0 1 1 F = x + y’z x y F z Computer Architectures Lab
Digital Logic Circuits 1 Boolean Algebra BASIC IDENTITIES OF BOOLEAN ALGEBRA [1] x + 0 = x [3] x + 1 = 1 [5] x + x = x [7] x + x’ = 1 [9] x + y = y + x [11] x + (y + z) = (x + y) + z [13] x(y + z) = xy +xz [15] (x + y)’ = x’y’ [17] (x’)’ = x [2] x • 0 = 0 [4] x • 1 = x [6] x • x = x [8] x • X’ = 0 [10] xy = yx [12] x(yz) = (xy)z [14] x + yz = (x + y)(x + z) [16] (xy)’ = x’ + y’ [15] and [16] : De Morgan’s Theorem Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Boolean Algebra EQUIVALENT CIRCUITS Many different logic diagrams are possible for a given Function F = ABC + ABC’ + A’C = AB(C + C’) + A’C = AB • 1 + A’C = AB + A’C (1) . . . . …… (1) [13]. . …. (2) [7] [4]. . . …. (3) A B C F (2) A B F C (3) A B F C Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Boolean Algebra COMPLEMENT OF FUNCTIONS A, B, . . . , Z, a, b, . . . , z A’, B’, . . . , Z’, a’, b’, . . . , z’ (p + q)’ - Replace all the operators with their respective complementary operators AND OR OR AND - Basically, extensive applications of the De. Morgan’s theorem (x 1 + x 2 +. . . + xn )’ x 1’x 2’. . . xn’ (x 1 x 2. . . xn)' x 1' + x 2' +. . . + xn' Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Map Simplification SIMPLIFICATION Boolean Function Truth Table Unique Many different expressions exist Karnaugh Map(K-map) is a simple procedure for simplifying Boolean expressions. Truth Table Karnaugh Map Boolean function Computer Organization Simplified Boolean Function Computer Architectures Lab
Digital Logic Circuits 1 Combinational Logic Circuits COMBINATIONAL LOGIC CIRCUITS x 0 0 1 1 Half Adder Full Adder x y 0 0 0 1 0 1 1 1 1 cn-1 cn 0 0 1 0 0 0 1 1 1 x y cn-1 Computer Organization y 0 1 c 0 0 0 1 s 0 1 1 0 0 1 y 0 1 x 1 0 s = xy’ + x’y =x y y 0 0 x 0 1 c = xy s 0 1 1 0 y x x y c s y 0 0 0 1 c n-1 1 1 0 1 cn x 0 1 s 1 0 c n-1 1 0 cn = xy + xcn-1+ ycn-1 = xy + (x y)cn-1 s = x’y’cn-1+x’yc’n-1+xy’c’n-1+xycn-1 = x y cn-1 = (x y) cn-1 S cn Computer Architectures Lab
Digital Logic Circuits 1 Combinational Logic Circuits COMBINATIONAL LOGIC CIRCUITS Other Combinational Circuits etc Computer Organization Multiplexer Encoder Decoder Computer Architectures Lab
Digital Logic Circuits 1 Combinational Logic Circuits MULTIPLEXER 4 -to-1 Multiplexer Select S 1 S 0 0 1 1 Output Y I 0 I 1 I 2 I 3 I 0 I 1 I 2 Y I 3 S 0 S 1 Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Combinational Logic Circuits ENCODER/DECODER Octal-to-Binary Encoder D 1 D 2 A 0 D 3 D 4 D 5 D 6 D 7 A 2 A 1 2 -to-4 Decoder E 0 0 1 A 1 0 0 1 1 d A 0 0 1 d D 0 0 1 1 Computer Organization D 0 D 1 1 0 1 1 1 D 2 1 1 0 1 1 D 3 1 1 1 0 1 A 0 D 1 D 2 A 1 E D 3 Computer Architectures Lab
Digital Logic Circuits 1 Sequential Circuits SEQUENTIAL CIRCUITS - Registers A 0 A 1 Q A 2 Q D C A 3 Q D C D C Clock I 0 Shift Registers Serial Input I 1 D Q C I 2 I 3 D Q C Serial Output Clock Bidirectional Shift Register with Parallel Load A 0 A 1 A 2 A 3 Q Q D C 4 x 1 MUX Clock S 0 S 1 Seria. I I 0 Computer Organization Input D C D C 4 x 1 MUX I 1 4 x 1 MUX I 2 Serial I 3 Input Computer Architectures Lab
Digital Logic Circuits 1 Memory Components MEMORY COMPONENTS 0 Logical Organization words (byte, or n bytes) Random Access Memory N-1 - Each word has a unique address - Access to a word requires the same time independent of the location of the word - Organization n data input lines k address lines Read 2 k Words (n bits/word) Write n data output lines Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Memory Components READ ONLY MEMORY(ROM) Characteristics - Perform read operation only, write operation is not possible - Information stored in a ROM is made permanent during production, and cannot be changed Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Question • The following memory units are specified by the number of words times the number of bits per word. How many address lines and input-output data lines are needed in each case? • (a)2 K x 16. • (b)64 M x 8. • (c)16 G x 32. Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 Question • Specify the number of bytes that can be stored in the memories listed in he following memory units: • (a)2 K x 16. • (b)64 M x 8. • (c)16 G x 32. Computer Organization Computer Architectures Lab
Digital Logic Circuits 1 REPRESENTATION OF NUMBERS Decimal Binary Octal Hexadecimal 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 00 01 02 03 04 05 06 07 10 11 12 13 14 15 16 17 0 1 2 3 4 5 6 7 8 9 A B C D E F Binary, octal, and hexadecimal conversion 1 2 7 5 4 3 1 0 1 1 0 0 0 1 1 Computer Organization Octal Binary Hexa Computer Architectures Lab
Digital Logic Circuits 1 Questions • Convert the following binary numbers to decimal: – 10011100 – 00110110 • Convert the following decimal numbers to binary : – 70 – 160 • Convert the following binary numbers to Hexadecimal: – 1100 1010 0011 – 0110 1000 1100 0000 Computer Organization Computer Architectures Lab
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