Chapter 4 Digital Logic and Circuit Design

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Chapter 4 Digital Logic and Circuit Design
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Two's Compliment

• 1 bit for sign, 7 bits to represent data
• Thus what's the range of values?

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Where We're At

• Boolean logic
• Statements that can be either true or false
• Gates
• Electronic devices which implement a boolean
  expression
• Now let's take one step further to build actual
  circuits in a computer...

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Circuit Types

• Combinatorial
• Output depends only upon input
• Addition/subtraction circuits
• Sequential
• Output depends on input and the current state
  of the circuit.
• Computer Memory
• We will focus only on combinatorial circuits

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How Do We Design Circuits?

• Many techniques
• We will look at one called
• The Sum Of Products Algorithm

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Sum Of Products Algorithm

• Step 1 Construct the truth table for the circuit
• Step 2 Group sub-expressions with ANDs and NOTs
• Step 3 Group sub-expressions with Ors
• Step 4 Draw the diagram

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Example

• Design a circuit which takes two bits as input,
  and outputs the result of adding the two bits
  together
• Or design a two-bit addition circuit

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Step 1 Truth Table
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Step 2 Subexpression Grouping

• For each output line, do the following
• For each row that contains a true value
• Write boolean expression for that row
• Repeat for each output line

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Step 2 (cont)

• For output Y we have one row where there is a 1,
  when a1 and b1
• Boolean expression

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Step 2 (cont)

• For output Z we have two rows which contain a 1
• A1, B0 and A0, B1
• Two boolean expressions

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Step 3 Group Subexpressions

• For each subexpression created in step 2, OR them
  together
• For Z input
• For Y input

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Step 4 Draw The Diagram

• For each expression from step 3, draw the
  corresponding circuit diagram

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Final Result
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Control Circuits

• The brain circuits of the computer
• Make decisions based upon inputs
• We will look at two types
• Multiplexors
• Decoders

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Multiplexors

• Sometimes called a MUX
• Selects exactly one of its input lines based upon
  its selector lines
• If a MUX has n selector lines, then it has 2n
  input lines

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Decoders

• Outputs a 1 on one of its output lines
• Line chosen by its input lines
• If a decoder has n input lines, it has 2n output
  lines
• If a decoder has N input lines and M output
  lines, it is sometimes called a N to M decoder

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And now a test...

• If a decoder has 2n output lines, how many
  selector lines does it have?
• If a multiplexor has N selector lines, how many
  output lines does it have?

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Design A Circuit

• Using the SOP algorithm design a 2 to 4 decoder
• How many input lines?
• How many output lines?