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Rekha.AFaculty

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Binary addition

0+0 =0

0+1=0

1+1=10=0+carry of 1 to the next position

1+1+1=11=1+carry of 1 to the next position

Illustration:

1001 11.011

+ 1111 +10.110

------------------ -------------------

11000 110.001

---------------------- ---------------------

Digital Arithmetic

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Ones complement representation

In a binary number if we replace each 1 by 0 and each 0 by 1, we

obtain binary number which is known as the ones complement.

Example: write the 1’s complement of the binary number 101011

1’s complement is 010100

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Two’s complement Representation

If 1 is added to the 1’s complement of a binary number, the resulting

number is known as the two’s complement of the binary number.

Example: Find the two’s complement of the number 10111001

1’s complement of the number: 01000110

01000110

+ 1

-----------------------------------

01000111 2’s complement

---------------------------------

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Subtraction of the Binary Numbers

• Binary subtraction can be performed by adding the two’s complement of the subtrahend to the minuend

• If a carry is generated discard the carry.

• If the final carry is 1, the answer is positive, i.e the minuend is greater than the subtrahend.

• If the final carry is 0, the answer is negative , i.e the minuend is smaller than the subtrahend and the answer is in the two’s complement form.

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Perform Binary subtraction

(i) Subtract 7 from 5

Binary representation of 5: 0101

Binary representation of 7: 0111

1’s complement of 7 : 1000

2’s complement of 7: 1001

0101

+ 1001

--------------------

1110 Carry is 0, so result is a negative number

----------------- and is in two’s complement form.

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Multiplication of Binary Numbers

The multiplication of binary numbers is done similar to the decimal number .

Multiplication for unsigned numbers

1001

x 1011

---------

1001

1001

0000

1001

------------------

1100011

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Multiplication of signed numbers

• Case 1(when both numbers are negative)

When Two numbers are negative, they will be in the two’s complement form.

The two numbers are multiplied.

The product is kept a positive number and given the sign bit of 0.

• Case 2(when one is positive and other is negative)

The negative number is converted to its two’s complement form.

Multiply the numbers.

Since the product has to be negative , the product is changed to 2’s complement form and given the sign bit 1.

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A combinational circuit consists of logic gates whose outputs at any time are determined from the present combination of the input.

A combinational logic gates react to the values of the signals at their inputs and produce the value of the output signal, transforming the binary information from the given input data to a required output data.

The block diagram of the combinational circuit

COMBINATIONAL LOGIC

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• Sequential circuits employ storage elements in addition to the logic gates

• Their outputs are a function of the present inputs and the state of the storage elements and also on the past input.

• The state of the storage element is a function of previous inputs.

The block diagram of the sequential circuit

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Two main types of sequential circuits

• Asynchronous sequential circuit

A sequential circuit whose behavior depends upon the sequence in which the input signals change is referred to as asynchronous sequential circuit.

The output will be affected whenever the input change.

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• Synchronous sequential circuit

Sequential circuit whose behaviour can be defined from the knowledge of its signals at discrete instants of time is referred to as synchronous sequential circuit.

The memory elements are affected only at discrete instant of time.

The timing is achieved by a device called system clock which generates a periodic train of clock pulses.

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Input

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Analysis procedure

• The first step is to make sure that the given circuit is combinational circuit or sequential.

• Proceed to obtain the output Boolean function or truth table.

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Design Procedure

• From the specifications of the circuit, determine the required number of inputs and outputs and assign a symbol to each.

• Derive the truth table that defines the required relationship between inputs and outputs.

• Obtain the simplified Boolean functions for each output as a function of the input variables.

• Draw the logic diagram and verify the correctness of the design(manually or by simulation)

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Design a combinational circuit to output the 2’s complement of a 4 bit binary number.

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The Minterm expressions are

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The karnaugh maps can now be drawn

00

01

11

10

AB

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Design a combinational circuit to convert BCD to Excess-3

The truth table for BCD to Excess-3

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Implementation of BCD to Excess 3

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• A binary adder- subtractor is a combinational circuit that performs the arithmetic operations of addition and subtraction of binary numbers

• A combinational circuits that performs the addition of two bits is called half adder.

• A combinational circuits that performs the addition of three bits is called full adder.

BINARY ADDER-SUBTRACTOR

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HALF ADDER

Block Diagram of Half Adder Truth Table of Half Adder

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Implementation of Half Adder

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FULL ADDER

• A Full adder is a combinational circuit that performs the arithmetic sum of three bits.

• It Consists of three inputs and two outputs.

Block Diagram

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Truth Table of Full Adder

S = x’y’z + x’yz’ + xy’z’ + xyzC = xy + xz + yz

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K- Map for Full Adder

yz

x 00 01 11 10

0

1

S = x’y’z + x’yz’ + xy’z’ + xyz

m0 m1

1

m3 m2

1

m4

1

m5 m7

1

m6

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K- map for Full Adder

yz

x 00 01 11 10

0

1

C = yz + xz + xy

1

1 1 1

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Implementation of full adder in sum of product form

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Implementation of full adder using 2 half adders

S

C

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Four Bit Binary Ripple Adder

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Half Subtractor

A Half subtractor has two inputs and two outputs.

Block diagram of half subtractor Truth table of half subtractor

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Implementation of Half subtractor using gates