Calculating circuits

Comparator - compares two sets of values
  a. equality detector - equal or not equal - cluster of XOR gates.
  b. magnitude detector - which value is larger or if equal.


  1-bit comparator - 5 gates.
  Found at https://www.electronics-tutorials.ws
  Another online electronics tutorial site.
  
  4-bit comparator - 53 gates.

  Multi-bit comparators more complex. 

  Alternative :
    Take twos-comp of B
    Add A and -B
      resulting in positive, 0, or negative value.
    Test CPU flags Z and N
      Uses fewer gates but may take longer.

      Also adder more flexible circuit.

  Comparator - much faster, but more expensive customized circuit.

  Add/subtract - 'cheaper' uses existing general purpose adder circuits 
                 and software.

Shifter.
  a. Shifts all bits making up a memory unit.
  b. Multiply or divide by 2
  c. Daisy chained.
  d. LSL(R) ASL(R) ROL(R)

    Logical shift -
      zero shifted in
      bit shifted out ignored
      software and hardware implementations differ.

       

    Roll shift - ROR ROL - unsigned multiply/divide
      Variation of circular and logical shifts, adds use of carry/borrow bit.
      Shifted in and out between register and carry flag 
       Or bit shifted out is shifted in at other end. fairly rare. 
      * Unsigned integer mul/divide.

       

    Arithmetic shift - signed multiply/divide
      zero shifted in (or carry flag on some systems)
      high bit (sign) kept as is
      2nd highest bit -> overflow
      # If carry flag zeroed before shift, 
        circular shift can be used for arithmetic shift.

       

    Barrel shifter.
      provides the ability to multiple shifts in a single action.
      Commonly used in hardware floating point math to align significands
        in single clock cycle.
 
Logical shift



Calculating Circuits - Adder
  a. Half-adder - adds 1 bit from each number together






Graphic License



A
B
|
Sum
Carry
 
 
|
XOR
AND

-
-
-
-
-
0
0
|
0
0
0
1
|
1
0
1
0
|
1
0
 
1
1
|
0
1




  b. Full-adder - adds bits with carry from lower bits.
     i. Half adders daisy chained.





Graphic License



A
B
Cin
|
Sum
Cout
0
0
0
|
0
0
0
1
0
|
1
0
1
0
0
|
1
0
1
1
0
|
0
1
0
0
1
|
1
0
0
1
1
|
0
1
1
0
1
|
0
1
1
1
1
|
1
1



A series of full-adders are daisy chained together (16, 32, 64) to create a
numeric adder.

Adder circuit complexity is linear. 32 bit adder requires 32 full adders.