Chapter 4: Arithmetic for Computers Part 2

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Chapter 4 Arithmetic for Computers(Part 2)

• CS 447
• Jason Bakos

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Logic/Arithmetic

• From the truth table for the mux, we can use
  sum-of-products to derive the logic equation
• With sum-of-products, for each 1 row for each
  output, we AND together all the inputs (inverting
  the input 0s), then OR all the row products
• To make it simpler, lets add dont cares to
  the table

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Logic/Arithmetic

• This gives us the following equation
• (A and (not D)) or (B and D)
• We dont need the inputs for the dont cares in
  our partial products
• This is one way to simplify our logic equation
• Other ways include propositional calculus,
  Karnaugh Maps, and the Quine-McCluskey algorithm

4
Logic/Arithmetic

• Here is a (crude) digital logic design for the
  2-to-1 mux
• Note that multiple muxes can be assembled in
  stages to implement multiple-input muxes

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Logic/Arithmetic

• For the adder, lets minimize the logic using a
  Karnaugh Map
• For CarryOut, we need 23 entries
• We can minimize this to
• CarryOutABCarryInBCarryInC

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Logic/Arithmetic

• Theres no way to minimize this equation, so we
  need the full sum of products
• Sum(not A)(not B)CarryIn ABCarryIn (not
  A)BCarryIn A(not B)CarryIn

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Logic/Arithmetic

• In order to implement subtraction, we can invert
  the B input to the adder and set CarryIn to be 1
• This can be implemented with a mux select B or
  not B (call this input Binvert)
• Now we can build a 1-bit ALU using an AND, OR,
  addition, and subtraction operation
• We can perform the AND, OR, and ADD in parallel
  and switch the results with a 4-input mux
  (Operation will be our D-input)
• To make the adder a subtractor, well need to
  have to set Binvert and CarryIn to 1