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1 28 The McGrawHill Companies, Inc. All rights reserved.
2 28 The McGrawHill Companies, Inc. All rights reserved. All or Nothing Gate Boolean Expression: A B = Y Truth Table (ee next slide) or AB = Y
3 28 The McGrawHill Companies, Inc. All rights reserved. Truth Table  AND Gate B A Y
4 28 The McGrawHill Companies, Inc. All rights reserved. QUIZ What is the output of the AND gate? L L? H H? L H? H L? Unique Output: Output HIGH only when all inputs are HIGH.
5 28 The McGrawHill Companies, Inc. All rights reserved. Any or All Gate Boolean Expression: A + B = Y Truth Table (ee next slide)
6 28 The McGrawHill Companies, Inc. All rights reserved. Truth Table  OR Gate B A Y
7 28 The McGrawHill Companies, Inc. All rights reserved. NOT Circuit Gives output that is not the same as the input. Boolean Expression: Y = A or Y = A Double inverting: A = A NOT gate inverts, or complements, or negates
8 28 The McGrawHill Companies, Inc. All rights reserved. NOT AND or inverted AND function. Boolean Expression: or A B = Y (A B)' = Y Truth Table (ee next slide)
9 28 The McGrawHill Companies, Inc. All rights reserved. Truth Table  NAND Gate B A AND NAND
10 28 The McGrawHill Companies, Inc. All rights reserved. NOT OR or Inverted OR Boolean Expression: A + B = Y or (A + B)' = Y Truth Table (ee next slide)
11 28 The McGrawHill Companies, Inc. All rights reserved. Truth Table  NOR Gate B A OR NOR
12 28 The McGrawHill Companies, Inc. All rights reserved. Known as Exclusive OR Gate Anything but not all Gate Boolean Expression: A B = Y Truth Table (ee next slide)
13 28 The McGrawHill Companies, Inc. All rights reserved. Truth Table  XOR Gate B A OR XOR
14 28 The McGrawHill Companies, Inc. All rights reserved. Known as the Exclusive NOR Gate The Inverted XOR Boolean Expression: A B = Y or (A B)' = Y or A B = Y Truth Table (ee next slide)
15 28 The McGrawHill Companies, Inc. All rights reserved. Truth Table  XNOR Gate B A XOR XNOR
16 28 The McGrawHill Companies, Inc. All rights reserved. The NAND as a Universal Gate Universal gate can be used in combination to create any other logic function. horting NAND inputs Example: Yields the NOT logic function A + B Equal to the OR logic function
17 28 The McGrawHill Companies, Inc. All rights reserved. Using Inverters to Convert Gates For example:
18 Combinational vs. equential logic circuits Combinatorial logic circuit : Output is determined by input at one moment equential logic circuit : Output is determined by input and current state of the logic circuit equential Logic Circuit input combinational logic circuit feedback memory element output=f(input,state) tate of memory element is determined by input and previous state of the element memory element: flipflops, latch 28 The McGrawHill Companies, Inc. All rights reserved.
19 28 The McGrawHill Companies, Inc. All rights reserved. Combinatorial vs. equential logic circuits Combinatorial logic circuits equential logic circuit
20 28 The McGrawHill Companies, Inc. All rights reserved. Tools of the Trade for olving Logic Problems Gate symbols Truth tables Boolean expressions Combinational(combinatorial) logic circuits: ANDOR pattern of gates from umofproducts Boolean expression such as: AB + CD = Y ANDOR pattern of gates ORAND pattern of gates from Productofsums Boolean expression such as: (A+B) (C+D) = Y ORAND pattern of gates
21 28 The McGrawHill Companies, Inc. All rights reserved. Logic Circuit From Boolean Expression Example: Draw the ANDOR logic diagram for the Boolean expression: AB + CD = Y tep : OR AB with CD tep 2: Add top AND gate tep 3: Add bottom AND gate
22 28 The McGrawHill Companies, Inc. All rights reserved. Boolean Algebra Boolean Postulates P X = or X = P2 = P3 = P4 + = P5 + = P6 = = P7 + = + =
23 28 The McGrawHill Companies, Inc. All rights reserved. Basic rules Boolean Algebra. X+=+X=X 2. X = X=X 3. X+=+X= 4. X = X= 5. X+X=X 6. X X=X 7. X X 8. X X 9. X X commutative law. X+Y=Y+X. XY=YX associate law 2. (X + Y) + Z = X + (Y + Z) 3. (XY) Z = X (YZ) distributive law 4. X (Y + Z) = XY + XZ 5. X + YZ = (X+Y)(X+Z) De Morgan's theorem 6. X Y X Y 7. XY X absorptive law 8. X + XY = X 9. X(X+Y) = X Y
24 28 The McGrawHill Companies, Inc. All rights reserved. Boolean Expressions umofproducts form: A B + C D = Y Productofsums form: (A + B) (C + D) = Y
25 28 The McGrawHill Companies, Inc. All rights reserved. Boolean Expressions Minterm A term of a boolean function in sum of products form composed of all input variables For f(w,x,y,z) W XY Z WXYZ minterms W XY Maxterm A term of a boolean function in product of sums form composed of all input variables For f(w,x,y,z) W X Z Not minterms ( W X Y Z ) ( W X Y Z ) maxterms ( W X Y ) ( W X Z ) Not maxterms
26 28 The McGrawHill Companies, Inc. All rights reserved. Boolean Expression from Truth Table Write the Boolean expression that describes the logic in this truth table. Truth Table Input Output ABC Y tep : Focus only on the truth table lines with outputs of. tep 2: AND the inputs for these two lines and logically OR the ANDed groups. A B C + A B C = Y Minterm Boolean expression: A B C + A B C = Y
27 Truth Table From Boolean Expressions Fill in a truth table from a minterm Boolean Expression. Minterm Boolean expression: A B C + A B C + A B C = Y tep : Place three s in output column. tep 2: Place five s in blanks in output column of truth table. Truth Table Input Output ABC Y 28 The McGrawHill Companies, Inc. All rights reserved.
28 28 The McGrawHill Companies, Inc. All rights reserved. Truth Table from Boolean Expressions Fill in a truth table from a Boolean Expression. Boolean expression: A B + A B C = Y tep : Place single output column for term with three variables. tep 2: Place two s in output column for term with two variables. tep 3: Fill in s. Truth Table Input Output ABC Y
29 28 The McGrawHill Companies, Inc. All rights reserved. Boolean Expression Minterm expression of 3 Variables x y z f minterm symbol f ( x, y, z) f ( x, y, z) x y z x y z x y z x y z x y z x y z x y z x y z m(,, 3, 5, 7) f ( x, y, z) m(2, 4, 6) m(,, 3, 5, 7) m m m 2 m 3 m 4 m 5 m 6 m 7 m(2, 4, 6) x y z f ( x, y, z) m(,, 3, 5, 7) xyz xyz xyz xyz f ( x, y, z) m(2, 4, 6) xyz xyz xyz xyz xyz xyz x y z xyz x y z x y z x y z x y z xyz xyz xyz xyz xyz xyz xyz
30 Boolean Expression 28 The McGrawHill Companies, Inc. All rights reserved. Minterm expression of 4 Variables a b c d minterm symbol a b c d minterm symbol m abcd abcd m abcd m 2 abcd m 3 abcd m 4 abcd m 5 abcd m 6 abcd m 7 abcd abcd abcd abcd abcd abcd abcd abcd m 8 m 9 m m m 2 m 3 m 4 m 5 Example f ( a, b, c, d ) m(,, 5, 9,,5) abcd abcd abcd abcd abcd abcd
31 Boolean Expression 28 The McGrawHill Companies, Inc. All rights reserved. Maxterm expression of 3 Variables x y z f maxterm symbol Example x x x x x x x x y z y z y y z y z y z y z z y z M M M 2 M 3 M 4 M 5 M 6 M 7 f ( x, y, z) M(,, 3, 5, 7) ( x y z)( x y z)( x y z)( x y z)( x y z)
32 28 The McGrawHill Companies, Inc. All rights reserved. 7) 6, (, 7) 6, (, 5) 4, 3, 2, (, 5) 4, 3, 2, (, ),, ( m M M m c b a f 4,5) 2,3, (, ) )( )( )( )( ( 4,5) 2,3, (, ),, ( M c b a c b a c b a c b a c b a abc abc abc abc abc abc abc abc abc abc abc abc abc abc abc m c b a f (,6,7) ) )( )( ( (,6,7) ),, ( M c b a c b a c b a abc abc abc abc abc abc abc abc abc m c b a f 4,5) 3, 2, (, 4,5) 3, 2, (, (,6,7) (,6,7) ),, ( m M M m c b a f Negation of minterms becomes maxterms Negation of maxterms becomes minterms OP functions and PO functions
33 28 The McGrawHill Companies, Inc. All rights reserved. implify Boolean Expression Karnaugh Maps(Kmap) A Kmap is a diagram made up of squares, with each square representing one minterm of the function is to be minimized. The simplest expression is a Boolean expression with minimum number of terms and the smallest possible number of literals in each term. This expression produces a circuit with minimum number of gates and with minimum number of inputs to each gate. The simplest expression is not unique.
34 28 The McGrawHill Companies, Inc. All rights reserved. implify Boolean Expression Drawing and grouping Karnaugh Maps(Kmap) Minterms in a Kmap must be arranged in a sequence, and the sequence is that only one bit changes in value from one adjacent column to the next. It utilizes the basic rule 7 of Boolean algebra. X+X = hape of a mintermgroup has to be a square or a rectangle. The number of minterms in a group must be multiple of 2 n. Group the adjacent minterms or groups. The bigger the better!
35 28 The McGrawHill Companies, Inc. All rights reserved. implify Boolean Expression Drawing Karnaugh Maps(Kmap) F ( A, B, C) m(,2,3,4,5,7) AB C 2 3 F ( A, B, C, D) m(,2,4,5,7,9, 2, 3, 5) AB CD Only one bit is different 4= = = 3=
36 28 The McGrawHill Companies, Inc. All rights reserved. implify Boolean Expression F ( A, B, C, D) m(4,5,7,2,3,5) AB CD implified as:. Group 4,5,2,3 and 7,5 BC + BCD 2. Group 4,5,2,3 and 5,7,3,5 BC + BD Which one is more simple? Can 4,5,7,2,3,5 be grouped?
37 28 The McGrawHill Companies, Inc. All rights reserved. implify Boolean Expression Don t care terms: terms that can be either or F( A, B, C, D) m(,2,3,4,5,) d(,7,9,5) F( A, B, C, D) m(,2,3,4,6,8,) d(,2,4) m(,2,3,4,8,9,) F( A, B, C, D) d(,5,6,7,,2) CD AB x x x x CD AB x x x CD AB x x x x x x F AB CD AC F D AB F A B
38 28 The McGrawHill Companies, Inc. All rights reserved. implify Boolean Expression f ( w, x, y, z) wx wxy wyz w yz wxy z wx( y y )( z z) wxy( z z) w( x x) yz w( x x) yz wxy z m(, 3, 5, 6, 7, 2, 3, 4, 5) yz wx f ( w, x, y, z) wx wz xy
39 28 The McGrawHill Companies, Inc. All rights reserved. Encoder Encoder is a combinational circuit that converts binary information from a maximum of 2 n input lines to n output lines. This output lines generate a binary code corresponding to the input lines. The encoder detects the active input. 4 2 encoder input output D 3 D 2 D D D 3 D 2 D D B B B B B D2 D3, B D D3
40 28 The McGrawHill Companies, Inc. All rights reserved. 8 3 encoder input output D 7 D 6 D 5 D 4 D 3 D 2 D D B 2 B B D D D D B D D D D B D D D D B D 7 D 6 D 5 D 4 D 3 D 2 D D B B B 2
41 Example of commercial IC 28 The McGrawHill Companies, Inc. All rights reserved.
42 28 The McGrawHill Companies, Inc. All rights reserved. Multiplexer A multiplexer is a combinational circuit that selects binary information from one of many input lines and directs it to a single output line. There are 2 n input lines and n selection lines whose bit combinations determine which input is selected. D D D 2 D 3 8x multiplexer x8 demultiplexer D D D 2 D 3 D 4 D 4 D 5 D 5 D 6 D 6 D 7 (sender) (receiver) D elect signals elect signals
43 28 The McGrawHill Companies, Inc. All rights reserved. 4 multiplexer election lines output F F D D D 2 D 3 D D D2 D3 D D D 2 F D 3
44 28 The McGrawHill Companies, Inc. All rights reserved. 8 multiplexer election lines output 2 F D D D 2 D 3 D 4 D 5 D 6 D 7 D D D 2 D 3 F 2 D 4 D 5 D 6 D D D D D D D D D F
45 28 The McGrawHill Companies, Inc. All rights reserved. Decoder A decoder is a combinational logic circuit that converts binary information from n input lines to maximum 2 n unique output lines. Decoders performs the inverse operation of encoders. 2 4 decoder/demultiplexer B A input output B A Y 3 Y 2 Y Y Y Y Y 2 Y Y 2 BA BA Y Y 3 BA BA Y 3
46 28 The McGrawHill Companies, Inc. All rights reserved. Commercial IC s are constructed with NAND gates. input output B A B A Y 3 Y 2 Y Y Y Y Y 2 Y Y 2 BA BA Y Y 3 BA BA Y 3
47 28 The McGrawHill Companies, Inc. All rights reserved. 2 4 decoders with enable inputs Most decoders include one or more enable inputs to control the circuit operation. A decoder operates when the enable input is equal to. A decoder constructed with NAND gates operates when the enable input is equal to. B A E input output E B A Y 3 Y 2 Y Y Y Y Y 2 Y Y 2 EBA EB A Y Y 3 EBA EBA Y 3
48 28 The McGrawHill Companies, Inc. All rights reserved. 3 8 decoder input output C B A Y 7 Y 6 Y 5 Y 4 Y 3 Y 2 Y Y Y Y 4 CBA, CBA, Y Y 5 CBA, CBA, Y Y 2 6 CBA, CBA, Y Y 3 7 CBA CBA C B A Y Y Y 2 Y 3 Y 4 Y 5 Y 6 Y 7
49 Demultiplexer 28 The McGrawHill Companies, Inc. All rights reserved. Demultiplexer is combinational circuit that receives information from a single input line and directs it to one of 2 n possible output lines. A decoder with enable input can function as a demultiplexer. A B 2x4 Decoder Y Y Y 2 Y 3 E x4 Demultiplexer Y Y Y 2 Y 3 E A B 2 4 decoder 4 demultiplexer
50 28 The McGrawHill Companies, Inc. All rights reserved. 4 cases of onebit addition X Y + C input output X Y C Y X C Y X XY XY X Y C HA C X Y : sum C : carry Half Adder
51 28 The McGrawHill Companies, Inc. All rights reserved. Full Adder Used for adding binary place values other than the s place Input Output Logic ymbol: C in A B Full Adder (sum) C (carry out) Logic Diagram:
52 28 The McGrawHill Companies, Inc. All rights reserved. Full Adder 8 cases of onebit addition C in X + Y C out input output X Y C in C out C XYC in X ( YC in X ( Y C XYC YC in X Y C out C C in in XYC in in in in ( XY XY ) ( X Y ) XY XYC ) X ( YC ) X ( Y C XYC in XY ( C in in in ) XY C in in C XYC YC in in XYC ) in ) in
53 28 The McGrawHill Companies, Inc. All rights reserved. Full Adder X Y C in X Y FA HA C out C in C out X Y C in C out C ( X Y) in XY X Y HA HA C in C out
54 28 The McGrawHill Companies, Inc. All rights reserved. Parallel Adding Use half adder for LD Use full adder for other digits A 2 A A + B 2 B B
55 Parallel Adder + UM appears here Parallel adders are available in IC form. s place uses halfadder 2s, 4s, 8s places use full adders The The McGrawHill Companies, Inc. Inc. All All rights reserved.
56 28 The McGrawHill Companies, Inc. All rights reserved. Paralleladder/subtracter X 3 Y 3 X 2 Y 2 X Y X Y FA FA FA FA C 3 C 2 C C Parallel adder X 3 Y 3 X 2 Y 2 X Y X Y (sign) :Add :ub FA C 3 FA C 2 FA C FA C C Parallel adder/subtracter
57 28 The McGrawHill Companies, Inc. All rights reserved. Highspeed adder The FA s must wait until their right FA s generate carries, i.e., a parallel adder has the delay time it takes to propagate the carry through the full adder. A highspeed adder(carry Lookahead Adder(CLA)) A CLA employs the carrylookahead generator that does not have any delay time for propagating the carry. C ) For 4bit CLA C C G PC out Ci XiYi ( Xi Yi Ci i i i i C where G i X i Y i P i X i Y G: generate, P: propagate G P C 2 G PC G PG P P C 3 G2 P2 C2 G2 P2 G PG P P C) G2 P2 G 2 4 G3 PC 3 3 G3 PG 3 2 P3 P2 G P3 P2 PG P3 P2 PP C C C i X i Y i C i P C i ( P PG P P P C i 2 i
58 28 The McGrawHill Companies, Inc. All rights reserved. Highspeed adder XY Partial Full Adder (PFA) G P C X 3 Y 3 X 2 Y 2 X Y X Y 3 G P C 2 G P C G P C G P C G 3 P 3 C 3 G 2 P 2 C 2 G P C G P Carry Lookahead Logic C
59 28 The McGrawHill Companies, Inc. All rights reserved. Highspeed adder 6bit CLA with four 4bit CLA s P G P P P G G G 3 2 P C 4 G3 PC 3 3 G3 PG 3 2 P3 P2 G P3 P2 PG P3 P2 PP C 3 P3 G2 P3 P2 G P3 P2 P G X 25 Y 25 X 8 Y 8 X 47 Y 47 X 3 Y 3 4Bit Adder 4Bit Adder 4Bit Adder 4Bit Adder 25 G P C 8 G P C 47 G P C 3 G P C G 3 P 3 C 3 G 2 P 2 C 2 G P C G P Carry Lookahead Logic G G P G C
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