Arithmetic Logic Units (ALU)

• Lecture 4 covers Arithmetic Logic Units (ALU)

Topics

• ALU Overview
• Function Unit
• Instruction Decoder
• Output Multiplexor
• Adders and Subtractors

What is an ALU?

• An ALU performs arithmetic and logic operations
• It is multi operational and combinational logic circuit
• It uses selection lines to choose the operation performed

Arithmetic Logic Unit Diagram

• For example, with Value 1 = 4 and Value 2 = 5, an ALU performs the ADD operation and produces a Result = 9

The ALU Components

• Function unit: Contains logic blocks to carry out each operation simultaneously
• Controller: Selects the required output based on the instruction
• Instruction: A binary 'word' that tells the ALU what to do
• Switch: Obeys the controller signals and connects the output to the required function

ALU Arithmetic and Logic Functions

• Output examples include:
  • AND
  • OR
  • NOT
  • XOR
  • Addition
  • Subtraction
• Instruction Decoder
  • Instruction decoder functions as a combinatorial logic circuit (CLC)
  • CLC circuits depend only on current inputs for their outputs at any given time
  • Converts 'n' input lines to a maximum of 2ⁿ output lines
• Multiple output lines can be chosen via ctrl1 to ctrl4
• Only one output line is active ('on') for each instruction
• 'n' instruction lines equal 2ⁿ control lines, impacting decoder size, where decoder size is n * 2ⁿ
• Output Multiplexor
  • Collections of AND gates connect to an OR gate, creating a multiplexer logic circuit
  • Functions on a many-to-one principle
  • Selects outputs using decoder outputs

Half Adder (HA)

• Two-bit adder circuit, adding two single binary digits
• Has 2 inputs (x and y) and 2 outputs (sum and carry)

Boolean Expressions for Half Adder

• Sum(s) = x'y + xy' = (x XOR y)
• Carry(c) = xy

Full Adder (FA)

• Three-bit adder circuit i.e. 3 inputs (x, y, and carry-in), and 2 outputs (sum and carry)
• Constructed using two Half Adders and an OR gate

ALU Components

• Adders and Subtractors
• Full Adder Circuit: Generates sum and carry outputs

Multi Bit ALU

• Connects multiple single-bit ALUs together
• The 'carry out' of each ALU links to the 'carry in' of the next ALU

Subtraction

• Performed by adding A to the 2's complement form of B
• S = A + (2's Complement of B) which expands to A + (B' + 1)
• Convert B to -B in 2's complement by inverting each bit and adding 1

Add / Subtract Circuit

• Addition and subtraction circuits combine by using Controlled Inversion
• A control input manages circuit behavior to add or subtract.

Controlled Inversion Truth Table

• Input Control is the CONTROL, and Input B is the Data
• With Input Control is 0, then Output F equates to Input B; where applicable the Output is referred to as "Output F=B"
• With Input Control is 1, then Output F equates to NOT Input B; where applicable the Output is referred to as "Output F=B'"

Controlled Inversion

• When Control = 0, then F = B
• When Control = 1, then F = B
• Truth table matches XOR

Circuit for Add / Subtract Unit

• When Input Control is 0, then Output F = B
• When Input Control is 1, then Output F = B' (NOT B)

Control of the Add / Subtract Unit

• The Control input determines the operation
  • Control = 0 indicates Addition
    • Carry-in is 0
    • Output is A plus B plus 0
  • Control = 1 indicates Subtraction
    • Carry-in is 1
    • Output is A plus (inverse of B plus 1), or A plus (-B)