## Index

 STA & SI Chapter1 Chapter2 Chapter3 Chapter4 Chapter5 Chapter6 Chapter7 Chapter8 Introduction Static Timing Analysis Clock Advance STA Signal Integrity EDA Tools Timing Models Other Topics

 Extraction & DFM Chapter1 Chapter2 Chapter3 Chapter4 Chapter5 Chapter6 Introduction Parasitic Interconnect Corner (RC Corner) Manufacturing Effects and Their Modeling Dielectric Layer Process Variation Other Topic

## Saturday, December 7, 2013

### DIGITAL BASIC - 1.2 : DIGITAL ARITHMETIC

 Index Chapter1 Chapter2 Chapter3 Chapter4 Digital Background Semiconductor Background CMOS Processing

 1.1 1.2 1.3a 1.3b 1.4 1.5 1.6 Number System Digital Arithmetic Logic Gates Logic Gates Combinational Circuits Multiplex (MUX)

Binary Arithmetic

Below are the basics which most of us know very well. But still for reference point of view I have mentioned everything into a table followed by examples.

 Binary addition Binary subtraction Binary multiplication Binary Division 0+ 0 =0 0 – 0 = 0 0 x 1 = 0 0 ÷ 1 = 0 0 + 1 =1 1 – 0 = 1 1 x 0 = 0 1 ÷ 1 = 1 1 + 0 = 1 1 – 1 = 0 0 x 0 = 0 1 + 1 = 10 1 in (10) is Carry bit Carry it to the next higher order column 0 – 1 = 10 1 in (10) is Borrow bit Carries from to the next higher order column 1 x 1 = 1

 Hex addition rule Subtraction F + 1 = 10 10 – 1 = F F + F = 1E A – 1 = 9 F + F + 1 = 1F 1 + 1 = 2 9 + 1 = A

Best way to do hexadecimal arithmetic (Subtraction | Addition | Multiplication |Division) is –
• First convert the number into decimal equivalents
• Perform the operation
• Convert back from decimal to hexadecimal
Note: Above method can be used for any number system.

There is a difference in binary addition and BCD addition. In binary maximum possible number is 1111 but in BCD, it is 1001. When the binary sum is equal to or less than 1001 (without a carry), corresponding BCD digit is correct. However, when binary sum is greater than or equal to 1010, the result is an invalid BCD digit. The addition of 6 = (0110)2  to the binary sum converts it to the correct digit & also produces a carry as required. This is because the difference between a carry in the most significant bit position of the binary sum & a decimal carry differ by 16 - 10 = 6
Example 6:  Add 184 & 576 in BCD

 BCD carry 1 1 0001 1000 0100 184 +0101 0111 0110 +576 Binary sum 0111 10000 1010 Add 6 0110 0110 BCD sum 0111 0110 0000 760

Boolean properties

 AND function X. 0 = 0 0. X = 0 X. 1 = X 1. X = X OR function X + 0 = X 0 + X = X X + 1 = 1 1 + X = 1 Commutative laws x. y = y. x x + y = y + x Distributive laws x(y +z) = x.y + x.z x + y. z = ( x+y) (x + z) Associative laws x(y.z) = (x. y) z x + ( y + z) = (x + y) +z Absorption laws x + xy =  x x(x + y) = x x + x'y = x+ y x(x' + y) = xy Demorgan’s laws (x + y)' = x'. y' (x. y)' = x' + y' Duality Principle x + x = x           =>               x .x = x                         by duality x + 1 = 1          =>                x. 0 = 0                        by duality x + xy = x         =>                x(x + y) = x                  by duality x + y = y + x     =>                xy = yx                        by duality x + (y+ z) = (x + y) + z     =>     x(yz) = (xy)z            by duality

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3. In binary subtraction 0-1 should be 1

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6. 0 minus 1 should be one ! please correct that.