ELEC1303 - Lecture 14

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ELEC1303 - Lecture 13
Arithmetic Circuits - Slides

Binary Addition

• Four basic rules for adding two bits $X$ and $Y$ are as follows:

• $S$ is the sum and $C$ is the carry
  • When there is a carry of 1 to the next column, you have a situation in which three bits are being added
    
• Typically, you should write the carry as part of your addition

Half Adder

• Adds 2 input bits
  
• Outputs 2 bits:
  • Carry bit $-$ $C_{out}$
  • Sum bit $-$ $S$
    $S=A\oplus B$
    $C_{out}=AB$

Full Adder

• Adds 3 bits
  
• Sum bit is 1 if the # of 1’s in the input is odd
• Carry bit is 1 if there are two-three “1” inputs
  $S=A\oplus B\oplus C_{in}$
  $C_{out}=AB+A{C}_{in}+B{C}_{in}$

The Ripple Carry Adder

• Can build $N$-bit adder using identical smaller full adders

Binary Subtraction

• When subtracting 2 bits ($X-Y$), we get the difference ($D=X-Y$) and the borrow-out $B$, with $B=1$ only for $X<Y$

• When subtracting a 1 from a 0, you have to borrow a “1” from the next column to the left

Half Subtractor

• Subtracts 2 input bits
  
• Outputs 2 bits:
  • Difference bit $-$ $D=X-Y$
  • Borrow bit $-$ $B_{out}=1$ if $X<Y$
    $D=X\oplus Y$
    $B_{out}=X^{\prime }Y$

Full Subtractor

• Has 3 input bits
  
  $D=X\oplus Y\oplus B_{in}$
  $B_{out}=X^{\prime }{B}_{in}+X^{\prime }Y+Y{B}_{in}$

The Ripple-Borrow Subtractor

• Same as earlier but using multiple full subtractors

Combined Addition/Subtraction

• Use 0 and 1 to indicate if a number is positive/negative
  e.g. 0 0110 $=$ +4${}_{10}$, 1 0100 $=$ -4${}_{10}$
• This is a signed number system
  • The following notes relate directly to using this system

Converting from binary to decimal:

Two’s Complement Representation

• 2’s complement of a binary number is formed by taking 1’s complement of the number and adding 1 to the least significant bit
  • Alternate way: complement bits to the left of the rightmost 1
• 1’s complement negates numbers by complementing each bit of the number

2’s complement = 1’s complement +1

Taking the 2’s complement of a binary number negates it (changes its sign)

• Summing the complement of a number with itself should equal 0

Converting 2’s complement binary to decimal:

• Same process as earlier, but minus away the sign bit

Two’s Complement Addition

• To find A + B, you just have to:
  • Do unsigned addition on A and B, including their sign bits
  • Ignore any carry out originating from MSB

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ELEC1303 - Lecture 15