CDA-4101 Lecture 7 Notes

Basic simple digital elements How to combine these elements to do more useful things some mathematucal techniques for analyzing the combinations we start with "gates"

• a "gate" is a small electronic device that performs a simple electrical function
• gates are made from one or more transitors
• we will show how gates can be combined to do more complex functions

• physical properties of semiconductors and clever arrangements of them yield the basic functionality of a transitor.
• bipolar transitor (NPN), three connections to the outside
  • collector
  • base
  • emitter

• When V_in drops below a certain critical voltage (low), transitor turns "off" and its resistance is infinite
• the electrical flow will then go from +Vcc to V_out (path of least resistance)
• when V_in is high enough, the transition turns on and will conduct electricity
• electrical flow will then go from +Vcc to ground leaving V_out with low or nearly no voltage
• Thus, V_out is always the inverse of V_in, and we have an "inverter": if V_in is low, then V_out is high, if V_in is high, then V_out is low
• This inversion can happen very fast (from a few nanoseconds to even less)
• note that the resistor is used to prevent the transitor from burning out

• In order for Vcc to want to go to ground, it will require both transistors to be "on", i.e., V₁ and V₂ both set high
• If either V₁ or V₂ is low then V_out will be high

• Here, +Vcc will go to ground and V_out will be low if either V₁ or V₂ is set to high
• Thus, to get a high value on V_out it needs both V₁ and V₂ to be low

• These three transistor arrangements are the simplest three gates
• Because there is essentially two values "low" and "high", we can assign these to logical 0 and logical 1 respectively
• We can also create a table that defines the entire circuit's logical behavior

A X 0 1 1 0	A B X 0 0 1 0 1 1 1 0 1 1 1 0	A B X 0 0 1 0 1 0 1 0 0 1 1 0

• The AND and OR logical functions have corresponding gates as well.
• You can produce these by inverting the output of the NAND and NOR gates respectively

A B X 0 0 0 0 1 0 1 0 0 1 1 1	A B X 0 0 0 0 1 1 1 0 1 1 1 1

• Note that although these are the more common logical operators, they actually require more transitors than the NAND and NOR gates.
• Also, although the symbols for NAND and NOR explicitly show the inversion bubbles, it is the AND and the OR gates that will need the additional inverter transistor
• All of these gates can be made to have more than two inputs.

an algebra is any system of properties and relationshsips algebra, as you probably know it, is actually just the algebra of the real numbers there are many other mathematical systems, each with their own algebra in a digital world with only two values, we will need to study and use boolean algebra technically, we will be using a switching algebra which is very closely related to boolean algebra, but we will not worry about this difference

with real numbers, functions can have ranges and domains that span an (uncountably) infinite set of numbers. boolean functions are restricted to ranges and domains from the set { 0, 1 } Example: The NOT function 'f': f(0) = 1 f(1) = 0 Example: The AND function 'g': g(0,0) = 0 g(0,1) = 0 g(1,0) = 0 g(1,1) = 1

George Boole

• we can use variables just like the familiar algebra, though these take on only one of two values
• a variable can represent the voltage on the wire: high or low
• notice that because a function of 'n' variables has a finite number of possible combinations, we can enumerate them all
• a truth table is a tabular way to specify a boolean functions, e.g.,

  A f(A) 0 1 1 0

  A B g(A, B) 0 0 0 0 1 0 1 0 0 1 1 1

• You can directly implement a sum of products representation by:
  • pass all inputs through inverters (in case you need them)
  • have a multi-input AND gate for each product term
  • have a multi-input OR gate fed by all the AND gates
  • connect up correct inputs and/or their negations to the AND gates

  		    _      _      _
  		M = ABC + ABC + ABC + ABC
  	   

• Note that "dots" indicated where crossing lines are connected.

we've seen that NAND and NOR require less transitors also, it is sometimes more convenient to use only a single type of gate any circuit can be converted into a form that uses only NAND or only NOR gates a single gate type that can implement any boolean function is called a complete gate NAND and NOR are the only complete gates note that the set of NOT, AND and OR gates together can also form any boolean function

"In high school I took a little English, some science, some hubcaps and some wheel covers." - William James "Gates" Brown

NAND Completeness

NOR Completeness

• The straightforward implementation is not always the best
• conversion to only NANDs or NORs or reduction of the total number of gates will be cheaper, simpler and faster
• we need some techniques for taking arbitrary boolean functions and finding a logically equivalent, but differnt boolean function that uses fewer and/or different types of gates.
• we'll look at two techniques for finding simpler circuits:
  • Boolean algebra
  • Karnaugh Maps (not in the textbook)

• You can prove these identities by constructing their truth tables (this is a perfectly valid "proof")

Original Circuit

Reduction:

  F(X,Y,Z) = ((X + Y')Z) + (X'YZ')
           = XZ + Y'Z + X'YZ'         (distribute)

Equivalent Circuit

• There are less "levels" in the equivalent circuit, so there will be less latency time for the signals to propogate

  Identity:
    (AB) + (CD) = (A+C)(A+D)(B+C)(B+D)       ("adding out")

  Follows from:
    (AB) + (CD) = (AB + C)(AB + D)           (distributive law)
    (AB) + (CD) = ((A+C)(B+C))((A+D)(B+D))   (distributive law)

  Generalizes to:
    (X1 X2 ... Xn) + (Y1 Y2 ... Yn)
    = (X1+Y1)(X1+Y2)...(X1+Yn)(X2+Y1)(X2+Y2)...(Xn+Y1)...(Xn+Yn)

Another Reduction:

  F(X,Y,Z) = ((X + Y')Z) + (X'YZ')
           = (X + Y' + X')(X + Y' + Y)(X + Y' + Z')(Z + X')(Z + Y)(Z + Z')
           =       1           1      (X + Y' + Z')(Z + X')(Z + Y)   1
           = (X + Y' + Z')(Z + X')(Z + Y)

Another Equivalent Circuit

Karnaugh Map of 2 Variables

Karnaugh Map of 3 Variables

Karnaugh Map of 4 Variables

• a graphical representation of a logic function's truth table
• small number in boxes is the row of the truth table (assuming truth table ordered by their binary number equivalents)
• boxes are filled in with the function's value from that row of the truth table.
• the non-sequential ordering of the varibale values in the rows and columns is very important: it ensures that adjacent cells differ by only one bit
• adjacency of cells "wraps around" on both the left/right and top/bottom

X Y Z F(X,Y,Z) 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1

F(X,Y,Z) = X'Y'Z + X'YZ' + XY'Z + XYZ note that each '1' in the box represents one of the product terms in a sum of products representation for the function we can reduce clutter in the map by only writing the 1's and assuming empty cells are zeroes

First look at the last two product terms in the formula: XY'Z (cell 5) and XYZ (cell 7) Algebraically, we can simplify: XY'Z + XYZ = (XZ)Y' + (XZ)Y = (XZ)(Y + Y') = XZ effectively, the value of Y has no impact on the truth values for these two rows of the truth table: regardless of Y's value, when X=1 and Z=1 the function is '1'. This is graphically represented in the Karnaugh map by the adjacent 1's in cells 5 and 7

Cell 2 of the Karnaugh map has no adjacent 1's (diagonals do not count), so its corresponding product term X'YZ' cannot be reduced. Cells 1 and 5 are considered adjacent (wrap around) so this indicates that we can simplify its two product terms: X'Y'Z + XY'Z = Y'Z In fact the Karnaugh map will give us a complete equivalent form of the function if we simply take care of all the 1's An equivalent formula from the Karnaugh map is: F(X,Y,Z) = X'Y'Z + X'YZ' + XY'Z + XYZ = X'YZ' + XZ + Y'Z

• Fill in the Karnaugh map from the truth table or any other representation of the function.
• group 1's that are adjacent and make the groupings as large as possible
• dimensions of resulting groupings must be rectangles whose dimensions are powers of two (e.g., 1x1, 1x2, 1x4, 2x4, 4x4, etc.)
• ensure all 1's are contained in at least one grouping, though note that they can appear in more than one grouping
• if within the grouping a variable always has value 1, then the variable will appear in the product term.
• if within the grouping a variable always has value 0, then the complement of that variable will appear in the product term.
• if within the grouping a variable take on both 0 and 1 values, then that variable will not appear in the product term
• add (OR) all the product terms from each grouping together

Prime numbers from 0 to 15 are: 1, 2, 3, 5, 7, 11, 13 F(W,X,Y,Z) = W'X'Y'Z + W'X'YZ' + W'X'YZ + W'XY'Z + W'XYZ + WX'YZ + WXY'Z

there are 4 groupings F(W,X,Y,Z) = W'Z + XY'Z + X'YZ + W'X'Y

• You can focus on the zeroes of a Karnaugh map instead of the ones.
• You just have to make sure that you complement the results you get.
• don't worry about doing this though

• Often in digital design, there are combinations of input that we might know are impossible or that we really just don't care about what value the function takes for these inputs
• we can exploit this knowledge in Karnaugh maps to help us get a minimal boolean function

with a BCD representation, the values 10 through 15 aree not possible, so the primes are: 1, 2, 3, 5, 7 we don't care whether the function is 1 or 0 for any values greater than 9 F(W,X,Y,Z) = W'X'Y'Z + W'X'YZ' + W'X'YZ + W'XY'Z + W'XYZ

we have choices about which groupings to take groupings containing only "don't cares" shouldn't be included just ensure that set of groupings selected "cover" all 1's F(W,X,Y,Z) = W'Z + X'Y