Chapter 2, Logic Design

• Combinational Circuits
  • Intro
    • A combinational circuit is a circuit where the output is completely determined by the input
    • Consider the following functions

    • int foo(int x) {
          return (x+3); 
      }
      
      int bar (int x) {
          return (x + (rand() % 4));
      }
      	   

    • Intput to a circuit, output from a circuit is know as a signal
    • The time it takes for a signal to travel through a circuit and have the values changed is called propagation delay
    • Look at this simulation in tkgate
      • 
      • In this simulation, I have 4 and gates, (make->gate->and)
      • An LED (make->IO->LED)
      • An Switch (make->IO->Switch)
      • I connected the wires, this is sometimes a pain in tk-gate
      • I renamed the wires so that they are in order, to do this
        • right click on the wire
        • select properties
        • change the Net Name field
        • Click on OK
      • Run the simulation
        • Double click on each wire you want to trace
        • Select Run from the Simulate menu
        • Click on the switch to see things change.
      • This simulation is here
    • We can describe the behavior of a circuit with n inputs as a table
      • There are 2ⁿ rows
      • Each row represents a specific input value
      • Each column represents a different quantity.

      • Input		Output
        a	b	a AND b
        0	0	0
        0	1	0
        1	0	0
        1	1	1

      • Note, in this table there are 2 inputs, and 2²=4 rows.
      • 
    • We can make a table for a 4 bit odd parity test
      • There are four inputs i₀, i₁, i₂, i₃
      • There is one output, 0 if the parity of the input is not odd and 1 if the parity is even.
      • We might represent this graphicaly :
      • 
      • we could write an expression o₄( i₀, i₁, i₂, i₃)
      • The table would be

        Input				Output
        i0	i1	i2	i3	o4
        0	0	0	0	0
        0	0	0	1	1
        0	0	1	0	1
        0	0	1	1	0
        0	1	0	0	1
        0	1	0	1	0
        0	1	1	0	0
        0	1	1	1	1
        1	0	0	0	1
        1	0	0	1	0
        1	0	1	0	0
        1	0	1	1	1
        1	1	0	0	0
        1	1	0	1	1
        1	1	1	0	1
        1	1	1	1	0

      • We could build a two bit odd parity checker o₂( i₀,i₁)

        • Input		Output
          i0	i1	o2
          0	0	0
          0	1	1
          1	0	1
          1	1	0

        • 
      • From this, we could construct a 4 bit odd parity tester.
      • 
    • Some notes on the images above
      • All inputs and outputs are labeled
      • Arrows show the flow of data in the circuit
      • These are called circuit diagrams.
      • I draw these with a package called xifg.
      • Xfig is available on cslab41.
    • 
    • 
    • This was constructed by:
      • I added four switches
      • I added three XOR gates
      • I added an led bar
        • Right click on the led
        • Select properties
        • select port
        • Edit the signal
        • Change the bit width to 5
      • I added a wire merge
        • make->io->wiremerge
        • right click, add input (twice)
        • Right click, properties->port->
        • edit the output (5 bits out)
    • the source

    ---

  • Gates and Boolean Expressions
    • The basic building blocks computing devices are logic gates.
      • There are four basic gates we deal with:

        Operation Name	Symbol	Table	Boolean Expression
        and		Input Output i0 i1 o 0 0 0 0 1 0 1 0 0 1 1 1	o=i0*i1
        or		Input Output i0 i1 o 0 0 0 0 1 1 1 0 1 1 1 1	o=i0+i1
        not		Input Output i0 o 0 1 1 0
        xor		Input Output i0 i1 o 0 0 0 0 1 1 1 1 1 1 0 0

      • Boolean expressions are equations in boolean algebra.
      • The table is for each operation is called a truth table
      • xor is a combination of the first three (as we will see) and is more complex when counting gates.
      • not is more simmple than the two above it, so it is generaly noted counted when counting gates.
    • Two bit odd parity is really just an xor.
    • What is the (circuit, boolean expression) for four bit odd parity?

    • Using only and, or, and not gates, create an exclusive or circuit.

    • Let us think about adding two bits
      • We have two outputs from such a function
        • Carry
        • Sum

        • Input		Output
          i0	i1	s	c
          0	0	0	0
          0	1	1	0
          1	0	1	0
          1	1	0	1

        • Notice, the sum is an XOR and the carry is an and.
        • s = i₀*i₁
        • 
        • 
        • Note the black dot at the intersection of lines.
        • This is called a half adder.
      • When we do addition, we normally have the two bits we are adding and the carry in from the previous two bits.
        • 
        • We really need a circuit to take three inputs
          • The two input bits
          • The carry in.
        • And produces two outputs (sum, carry out)
        • Consider this circuit
        • 
        • This is called a full adder