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### Transcript of Logika kombinasi

EE207: Digital Systems I, Semester I 2003/2004

CHAPTER 3 -ii: Combinational Logic Design Design Procedure, Encoders/Decoders (Sections 3.4 3.6)

Overviewn n n

Design Procedure Code Converters Binary Decodersn n

n n

Binary Encoders Priority Encoders

Expansion Circuit implementation

Combinational Circuit Designn

n

Design of a combinational circuit is the development of a circuit from a description of its function. Starts with a problem specification and produces a logic diagram or set of boolean equations that represent the circuit.

Design Procedure1. 2.

3.

4. 5.

Determine the required number of inputs and outputs and assign variables to them. Derive the truth table that defines the required relationship between inputs and outputs. Obtain and simplify the Boolean function (Kmaps, algebraic manipulation, CAD tools, ). Consider any design constraints (area, delay, power, available libraries, etc). Draw the logic diagram. Verify the correctness of the design.

Design Examplen

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

More Examples - Code Convertersn

Code Converters transform/convert information from one code to another:n BCD-to-Excess-3n

Code Converter Converter

n BCD-to-Seven-Segmentn

Useful in some cases for digital arithmetic Used to display numeric info on 7 segment displays

BCD-to-Excess-3 Code Convertern

n

n

Design a circuit that converts a binarycoded-decimal (BCD) codeword to its corresponding excess-3 codeword. Excess-3 code: Given a decimal digit n, its corresponding excess-3 codeword (n+3)2 Example: n=5 n+3=8 1000excess-3 n=0 n+3=3 0011excess-3 We need 4 input variables (A,B,C,D) and 4 output functions W(A,B,C,D), X(A,B,C,D), Y(A,B,C,D), and Z(A,B,C,D).

BCD-to-Excess-3 Converter (cont.)

The truth table relating the input and output variables is shown below Note that the outputs for inputs 1010 through 1111 are don't cares (n shown here).

Maps for BCD-to-Excess-3 Code Converter

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

BCD-to-Excess-3 Converter (cont.

Another Code Converter Example: BCD-to-Seven-Segment Convertern

Seven-segment display:n7

LEDs (light emitting diodes), each one controlled by an input a n 1 means on, 0 means off f b n Display digit 3? gSet a, b, c, d, g to 1 n Set e, f to 0n

e d

c

BCD-to-Seven-Segment Convertern n

n

Input is a 4-bit BCD code 4 inputs (w, x, y, z). Output is a 7-bit code (a,b,c,d,e,f,g) that allows for the decimal equivalent to be displayed. a Example: fn Input:

0000BCD n Output: 1111110 (a=b=c=d=e=f=1, g=0)

g

b

e d

c

BCD-to-Seven-Segment (cont.) Truth TableDigit 0 1 2 3 4 5 6 7 ?? wxyz 0000 0001 0010 0011 0100 0101 0110 0111 abcdefg 1111110 0110000 1101101 1111001 0110011 1011011 X011111 11100X0 Digit 8 9 wxyz 1000 1001 1010 1011 1100 1101 1110 1111 abcdefg 1111111 111X011 XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX XXXXXXX

Decodersn

A combinational circuit that converts binary information from n coded inputs to a maximum 2n decoded outputs n-to- 2n decoder n-to-m decoder, m = 2n Examples: BCD-to-7-segment decoder, where n=4 and m=7

n n

ecoders (cont.)

to-4 Decoder

to-4 Active Low Decoder

data data

3-to-8 Decoder

3-to-8 Decoder (cont.)n n n n n

Three inputs, A0, A1, A2, are decoded into eight outputs, D0 through D7 Each output Di represents one of the minterms of the 3 input variables. Di = 1 when the binary number A2A1A0 = i Shorthand: Di = mi The output variables are mutually exclusive; exactly one output has the value 1 at any time, and the other seven are 0.

Implementing Boolean functions using decodersn n

n

Any combinational circuit can be constructed using decoders and OR gates! Why? Here is an example: Implement a full adder circuit with a decoder and two OR gates. Recall full adder equations, and let X, Y, and Z be the inputs:n n

n

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

S(X,Y,Z) = X+Y+Z = m(1,2,4,7) C (X,Y,Z) = m(3, 5, 6, 7).

Implementing a Binary Adder Using a DecoderS(X,Y,Z) = Sm(1,2,4,7) C(X,Y,Z) = Sm(3,5,6,7)

Decoder 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 4to-16 decoders).

3-to-8 decoder using two 2-to-4 decoders

4-input tree decoder

Encodersn

n

An encoder is a digital circuit that performs the inverse operation of a decoder. An encoder has 2n input lines and n output lines. The output lines generate the binary equivalent of the input line whose value is 1.

Encoders (cont.)

Encoder Example

n

Example: 8-to-3 binary encoder (octal-to-binary)

A0 = D1 + D 3 + D5 + D7 A1 = D2 + D 3 + D6 + D7 A2 = D4 + D 5 + D6 + D7

Encoder Example (cont.)

Simple Encoder Design Issuesn

There are two ambiguities associated with the design of a simple encoder:1.

2.

Only one input can be active at any given time. If two inputs are active simultaneously, the output produces an undefined combination (for example, if D3 and D6 are 1 simultaneously, the output of the encoder will be 111. An output with all 0's can be generated when all the inputs are 0's,or when D0 is equal to 1.

Priority Encodersn n

n

Solves the ambiguities mentioned above. Multiple asserted inputs are allowed; one has priority over all others. Separate indication of no asserted inputs.

Example: 4-to-2 Priority Encoder Truth Table

4-to-2 Priority Encoder (cont.)n n

n

The operation of the priority encoder is such that: If two or more inputs are equal to 1 at the same time, the input in the highestnumbered position will take precedence. A valid output indicator, designated by V, is set to 1 only when one or more inputs are equal to 1. V = D3 + D2 + D1 + D0 by inspection.

Example: 4-to-2 Priority Encoder K-Maps

Example: 4-to-2 Priority Encoder Logic Diagram

8-to-3 Priority Encoder

A Matrix of switches = KeypadC0 C1 C2 C3

1 4 7 0

2 5 8 A

3 6 9 B

F E D C

R0 R1 R2 R3

Keypad Decoder IC - EncoderCOL. 4-bit

1 4 7 0

2 5 8 A

3 6 9 B

F E D CROW 4-bit

4-bit Binary (encoded

Priority Interrupt Encoder SchematicInterrupting DevicesDevice A Device B Device C Device D IntRq Req(1:0)

Interrupt Encoder

Microprocessor

Priority Encoding - Interrupt RequestsInterrupting Device A B C D 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 Req (1:0) IntRq 00 0 00 1 01 1 01 1 10 1

Exercise: Complete this table?