Logika kombinasi

EE207: Digital Systems I, EE207: Digital Systems I, Semester I 2003/2004Semester I 2003/2004

CHAPTER 3 CHAPTER 3 --ii:ii:Combinational Logic Design Combinational Logic Design ––

Design Procedure, Encoders/DecodersDesign Procedure, Encoders/Decoders(Sections 3.4 (Sections 3.4 –– 3.6)3.6)

30-Apr-09 2Chapter 3-ii: Combinational Logic Design (3.4 - 3.6)

OverviewOverviewnn Design ProcedureDesign Procedurenn Code ConvertersCode Convertersnn Binary DecodersBinary Decoders

nn ExpansionExpansionnn Circuit implementationCircuit implementation

nn Binary EncodersBinary Encodersnn Priority EncodersPriority Encoders


Combinational Circuit DesignCombinational Circuit Design

nn DesignDesign of a combinational circuit is the of a combinational circuit is the development of a circuit from a development of a circuit from a description of its function.description of its function.

nn Starts with a problem specification and Starts with a problem specification and produces a logic diagram or set of produces a logic diagram or set of boolean equations that represent the boolean equations that represent the circuit.circuit.


Design ProcedureDesign Procedure1.1. Determine the required number of inputs Determine the required number of inputs

and outputs and assign variables to them.and outputs and assign variables to them.2.2. Derive the truth table that defines the Derive the truth table that defines the

required relationship between inputs and required relationship between inputs and outputs.outputs.

3.3. Obtain and Obtain and simplifysimplify the Boolean function (Kthe Boolean function (K--maps, algebraic manipulation, CAD tools, maps, algebraic manipulation, CAD tools, ……). ). Consider any design constraints (area, delay, Consider any design constraints (area, delay, power, available libraries, etc).power, available libraries, etc).

4.4. Draw the logic diagram.Draw the logic diagram.5.5. VerifyVerify the correctness of the design.the correctness of the design.


Design ExampleDesign Example

nn Design a combinational circuit with 4 Design a combinational circuit with 4 inputs that generates a 1 when the # of inputs that generates a 1 when the # of 1s equals the # of 0s. Use only 21s equals the # of 0s. Use only 2--input input NOR gatesNOR gates

……


More Examples More Examples -- Code ConvertersCode Converters

nn Code Converters transform/convert Code Converters transform/convert information from one code to another:information from one code to another:nn BCDBCD--toto--ExcessExcess--3 Code Converter3 Code Converter

nn Useful in some cases for digital arithmeticUseful in some cases for digital arithmeticnn BCDBCD--toto--SevenSeven--Segment ConverterSegment Converter

nn Used to display numeric info on 7 segment Used to display numeric info on 7 segment displaysdisplays


BCDBCD--toto--ExcessExcess--3 Code Converter3 Code Converter

nn Design a circuit that converts a binaryDesign a circuit that converts a binary--codedcoded--decimal (BCD) codeword to its decimal (BCD) codeword to its corresponding excesscorresponding excess--3 codeword.3 codeword.

nn ExcessExcess--3 code: Given a decimal digit 3 code: Given a decimal digit nn, its , its corresponding excesscorresponding excess--3 codeword 3 codeword (n+3)(n+3)22Example: Example:

n=5 n=5 àà n+3=8 n+3=8 àà 10001000excessexcess--33n=0 n=0 àà n+3=3 n+3=3 àà 00110011excessexcess--33nn We need 4 input variables (A,B,C,D) and 4 We need 4 input variables (A,B,C,D) and 4

output functions W(A,B,C,D), X(A,B,C,D), output functions W(A,B,C,D), X(A,B,C,D), Y(A,B,C,D), and Z(A,B,C,D).Y(A,B,C,D), and Z(A,B,C,D).


BCDBCD--toto--ExcessExcess--3 Converter (cont.)3 Converter (cont.)nn The truth table relating the input and output variables is shownThe truth table relating the input and output variables is shown below.below.nn Note that the outputs for inputs 1010 through 1111 are Note that the outputs for inputs 1010 through 1111 are don't caredon't cares (not s (not

shown here).shown here).


Maps for BCD-to-Excess-3 Code Converter

The KThe K--maps for are constructed using the don't care termsmaps for are constructed using the don't care terms


BCDBCD--toto--ExcessExcess--3 Converter (cont.)3 Converter (cont.)


Another Code Converter Example:Another Code Converter Example:BCDBCD--toto--SevenSeven--Segment ConverterSegment Converter

nn SevenSeven--segment display:segment display:nn 7 LEDs (light emitting diodes), each one 7 LEDs (light emitting diodes), each one

controlled by an inputcontrolled by an inputnn 1 means “on”, 0 means “off”1 means “on”, 0 means “off”nn Display digit “3”?Display digit “3”?

nn Set a, b, c, d, g to 1Set a, b, c, d, g to 1nn Set e, f to 0Set e, f to 0

d

a

b

c e

f g


BCDBCD--toto--SevenSeven--Segment ConverterSegment Converter

nn Input is a 4Input is a 4--bit BCD code bit BCD code àà 4 inputs (w, 4 inputs (w, x, y, z).x, y, z).

nn Output is a 7Output is a 7--bit code (a,b,c,d,e,f,g) that bit code (a,b,c,d,e,f,g) that allows for the decimal equivalent to be allows for the decimal equivalent to be displayed.displayed.

nn Example: Example: nn Input: 0000Input: 0000BCDBCDnn Output: 1111110 Output: 1111110

(a=b=c=d=e=f=1, g=0)(a=b=c=d=e=f=1, g=0)d

a

b

c e

f g


BCDBCD--toto--SevenSeven--Segment (cont.)Segment (cont.)Truth TableTruth Table

11100X011100X00111011177X011111X0111110110011066101101110110110101010155011001101100110100010044111100111110010011001133110110111011010010001022011000001100000001000111111111011111100000000000abcdefgabcdefgwxyzwxyzDigitDigit abcdefgabcdefgwxyzwxyzDigitDigit

XXXXXXXXXXXXXX11111111XXXXXXXXXXXXXX11101110XXXXXXXXXXXXXX11011101XXXXXXXXXXXXXX11001100XXXXXXXXXXXXXX10111011XXXXXXXXXXXXXX10101010

111X011111X0111001100199111111111111111000100088

??


DecodersDecoders

nn A combinational circuit that converts A combinational circuit that converts binary information from binary information from nn coded inputs coded inputs to a maximum 2to a maximum 2n n decoded outputs decoded outputs àà nn--toto-- 22nn decoderdecoder

nn nn--toto--mm decoder, decoder, m m == 22nn

nn Examples: BCDExamples: BCD--toto--77--segment decoder, segment decoder, where where n=4n=4 and and m=7m=7


Decoders (cont.)


2-to-4 Decoder


2-to-4 Active Low Decoder


33--toto--8 Decoder8 Decoder

addressaddress

datadata


33--toto--8 Decoder (cont.)8 Decoder (cont.)nn Three inputs, AThree inputs, A00, A, A11, A, A22, are decoded into , are decoded into

eight outputs, Deight outputs, D0 0 through Dthrough D77

nn Each output DEach output Dii represents one of the represents one of the minterms of the 3 input variables.minterms of the 3 input variables.

nn DDii = 1 when the binary number A= 1 when the binary number A22AA11AA00 = = iinn Shorthand: DShorthand: Dii = m= mii

nn The output variables are The output variables are mutually exclusivemutually exclusive; ; exactly one output has the value 1 at any time, exactly one output has the value 1 at any time, and the other seven are 0.and the other seven are 0.


nn AnyAny combinational circuit can be constructed combinational circuit can be constructed using decoders and OR gates! Why?using decoders and OR gates! Why?

nn Here is an example:Here is an example:Implement a full adder circuit with a decoder Implement a full adder circuit with a decoder and two OR gates.and two OR gates.

nn Recall full adder equations, and let X, Y, and Z Recall full adder equations, and let X, Y, and Z be the inputs:be the inputs:nn S(X,Y,Z) = X+Y+Z = S(X,Y,Z) = X+Y+Z = ΣΣm(1,2,4,7) m(1,2,4,7) nn CC (X,Y,Z) = (X,Y,Z) = ΣΣm(3, 5, 6, 7).m(3, 5, 6, 7).

nn Since there are 3 inputs and a total of 8 Since there are 3 inputs and a total of 8 minterms, we need a 3minterms, we need a 3--toto--8 decoder.8 decoder.

Implementing Boolean functionsImplementing Boolean functionsusing decodersusing decoders


Implementing a Binary Adder Using a Decoder

S(X,Y,Z) = Sm(1,2,4,7)

C(X,Y,Z) = Sm(3,5,6,7)


Decoder ExpansionsDecoder ExpansionsLarger decoders can be constructed using a number of smaller ones.

-> HIERARCHICAL design!Example:A 6-to-64 decoder can be designed using four 4-to-16 and one 2-to-4 decoders. How? (Hint: Use the 2-to-4 decoder to generate the enable signals to the four 4-to-16 decoders).


33--toto--8 decoder using two 28 decoder using two 2--toto--4 decoders4 decoders


44--input tree decoderinput tree decoder


EncodersEncoders

nn An encoder is a digital circuit that An encoder is a digital circuit that performs the inverse operation of a performs the inverse operation of a decoder. An encoder has 2decoder. An encoder has 2nn input lines input lines and and nn output lines. output lines.

nn The output lines generate the binary The output lines generate the binary equivalent of the input line whose value equivalent of the input line whose value is 1.is 1.


Encoders (cont.)Encoders (cont.)


Encoder ExampleEncoder Examplenn Example: 8Example: 8--toto--3 binary encoder (octal3 binary encoder (octal--toto--binary)binary)

A0 = D1 + D3 + D5 + D7A1 = D2 + D3 + D6 + D7A2 = D4 + D5 + D6 + D7


Encoder Example (cont.)Encoder Example (cont.)


Simple Encoder Design IssuesSimple Encoder Design Issuesnn There are two ambiguities associated with There are two ambiguities associated with

the design of a simple encoder:the design of a simple encoder:1.1. Only one input can be active at any given time. If Only one input can be active at any given time. If

two inputs are active simultaneously, the output two inputs are active simultaneously, the output produces an undefined combination (for example, produces an undefined combination (for example, if Dif D33 and Dand D66 are 1 simultaneously, the output of are 1 simultaneously, the output of the encoder will be 111.the encoder will be 111.

2.2. An output with all 0's can be generated when all An output with all 0's can be generated when all the inputs are 0's,or when Dthe inputs are 0's,or when D00 is equal to 1.is equal to 1.


Priority EncodersPriority Encoders

nn Solves the ambiguities mentioned above.Solves the ambiguities mentioned above.nn Multiple asserted inputs are allowed; Multiple asserted inputs are allowed;

one has priority over all others.one has priority over all others.nn Separate indication of no asserted Separate indication of no asserted

inputs.inputs.


Example: 4Example: 4--toto--2 Priority Encoder2 Priority EncoderTruth TableTruth Table


44--toto--2 Priority Encoder (cont.)2 Priority Encoder (cont.)nn The operation of the priority encoder is The operation of the priority encoder is

such that:such that:nn If two or more inputs are equal to 1 at If two or more inputs are equal to 1 at

the same time, the input in the highestthe same time, the input in the highest--numbered position will take precedence.numbered position will take precedence.

nn A A valid output indicatorvalid output indicator, designated by , designated by V, is set to 1 only when one or more V, is set to 1 only when one or more inputs are equal to 1. V = Dinputs are equal to 1. V = D33 + D+ D22 + D+ D11 + + DD00 by inspection.by inspection.


Example: 4Example: 4--toto--2 Priority Encoder2 Priority EncoderKK--MapsMaps


Example: 4Example: 4--toto--2 Priority Encoder2 Priority EncoderLogic DiagramLogic Diagram


88--toto--3 Priority Encoder3 Priority Encoder


A Matrix of switches = KeypadA Matrix of switches = KeypadC0 C1 C2 C3

R0

R1

R2

R3

1 2 3 F

4 5 6 E

7 8 9 D

0 A B C


Keypad Decoder IC Keypad Decoder IC -- EncoderEncoder

1 2 3 F

4 5 6 E

7 8 9 D

0 A B C

COL.4-bit

ROW4-bit

4-bitBinary

(encoded)


Priority Interrupt Encoder Priority Interrupt Encoder SchematicSchematic

Device A

Device B

Device C

Device D

MicroprocessorInterruptEncoder

Req(1:0)

IntRq

InterruptingDevices


Priority Encoding Priority Encoding -- Interrupt Interrupt RequestsRequests

Interrupting DeviceInterrupting Device

111 01 000001100110 10 111110000110 10 100110000110 00 011000000000 00 000000000

IntRqIntRqReqReq (1:0)(1:0)DDCCBBAA

Exercise: Complete this table?

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