Slide 1Slide 2Relating Boolean Algebra to Digital DesignExample: Seat Belt Warning Light SystemBoolean Algebra NotationBoolean Algebra PropertiesBoolean Algebra: Additional PropertiesSlide 8Multiple-Output CircuitsMultiple-Output Example: BCD to 7-Segment Converter1Microprocessors vs. Custom Digital CircuitsWhy would anyone ever need to design new digital circuits?Microprocessors are readily available, inexpensive, easy to program, and easy to reprogramMicroprocessors are sometimes:• Too slow;• Too big;• Consume too much power;• Too costlyDesigners that work with digital phenomena often buy an off-the-shelf microprocessor and program it.2Combinational Logic CircuitsLogic gates – building blocks of logic circuitsA digital circuit whose output depends solely on the present combination of input values is called a combinational circuit AND OR NOTBoolean AlgebraBoolean algebra is a branch of mathematics that uses variables whose values can only be 1 or 0 (“true” or “false”, respectively) and whose operators, like AND, OR, NOT, operate on such variables and return 1 or 0.We can build circuits by doing math3Relating Boolean Algebra to Digital Design01yxxyF10FxTransistorCircuits01xyFyxLogic GatesTruth Tables4Example: Seat Belt Warning Light System•Design circuit for warning light•Sensors–s=1: seat belt fastened–k=1: key inserted–p=1: person in seat•Capture Boolean equation–person in seat, and seat belt not fastened, and key inserted•Convert equation to circuit•Notice –Boolean algebra enables easy capture as equation and conversion to circuitw = p AND NOT(s) AND kkpswBeltWarn5Boolean Algebra Notation•By defining logic gates based on Boolean algebra, we can use algebraic methods to manipulate circuits–So let’s learn some Boolean algebraic methods•Start with notation: Writing a AND b, a OR b, and NOT(a) is cumbersome–Use symbols: a * b, a + b, and a’ (in fact, a * b can be just ab). •Original: w = (p AND NOT(s) AND k) OR t •New: w = ps’k + t–Spoken as “w equals p and s prime and k, or t”–Or even just “w equals p s prime k, or t”–s’ known as “complement of s”•While symbols come from regular algebra, don’t say “times” or “plus”Boolean algebra precedence, highest precedence first. Symbol Name Description ( ) Parentheses Evaluate expressions nested in parentheses first ’ NOT Evaluate from left to right * AND Evaluate from left to right + OR Evaluate from left to right6Boolean Algebra Properties• Commutativea + b = b + aa * b = b * a• Distributivea * (b + c) = a * b + a * ca + (b * c) = (a + b) * (a + c)• Associative(a + b) + c = a + (b + c)(a * b) * c = a * (b * c)• Identity0 + a = a + 0 = a1 * a = a * 1 = a• Complementa + a’ = 1a * a’ = 07Boolean Algebra: Additional Properties• Null elementsa + 1 = 1a * 0 = 0• Idempotent Lawa + a = aa * a = a•Involution Law(a’)’ = a• De Morgan’s Law(a + b)’ = a’ b’ (a b)’ = a’ + b’Example: Simplification of an automatic sliding door systemf = h c’ + h’ p c’f = c’ (h + p)8Boolean FunctionsBoolean function is a mapping of each possible combination of input values to either 0 or 1.Boolean function can be represented as an equation, a circuit, and as a truth table.F = a b + a’ F = a’ b’ + a’ b + a bConverting a truth table to an equationFor any function, there may be many equivalent equations, and many equivalent circuits, but there is only one truth table!9Multiple-Output Circuits•Many circuits have more than one output•Can give each a separate circuit, or can share gates•Ex: F = ab + c’, G = ab + bcabcFG(a)abcFG(b)Option 1: Separate circuitsOption 2: Shared gates10Multiple-Output Example: BCD to 7-Segment Convertera = w’x’y’z’ + w’x’yz’ + w’x’yz + w’xy’z + w’xyz’ + w’xyz + wx’y’z’ + wx’y’zabcdefg = 1111110 0110000 1101101afbdgec(b)(a)b = w’x’y’z’ + w’x’y’z + w’x’yz’ + w’x’yz + w’xy’z’ + w’xyz + wx’y’z’ +