Introduction to Computer Design. Standard Forms for Boolean Functions. Sums and Products. Standard Forms for Boolean Functions (cont ) CMPT-150
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1 CMPT- Itroducto to Computer Desg SFU Harbour Cetre Sprg 7 Lecture : Ja. 6 7 Stadard orms or boolea uctos Sum o Products Product o Sums Stadard Forms or Boolea Fuctos (cot ) It s useul to spec Boolea uctos a orm that: Allows comparso or equalt. Has a correspodece to the truth tables Stadard (caocal) Forms commo usage: Sum o Products Product o Sums Stadard Forms or Boolea Fuctos A boolea ucto ca be descrbed uquel b a truth table but several was as a boolea epresso leads to deret logc crcuts. ( ) = ( ) ( ) = ( ) ( ) () Sums ad Products Lteral: A varable or a complemet o a varable. Eamples: A B Product term: a lteral or a cojucto (AND) o lterals. Eamples: Sum o Products: a product term or a dsjucto (OR) o several product terms. Sum term: a lteral or a dsjucto (OR) o lterals. Eamples: Product o Sums: a sum term or a dsjucto (OR) o several product terms. ( )( ) ( )( ) ( )
2 Sums ad Products (cot ) For each ucto below spec t s a Sum o Products orm or Product o Sums orm or ether. = = ( )( ) = ( ) = ( ) = Materms Materm: a sum term whch ever varable appears eactl oce ether complemeted or ot complemeted. Eample: For varables there are 8 possble materms: For a materm there s a uque assgmet o s ad s to the varables such that the value o the materm s. Note: ever materm the varables appear the same order (usuall lecographc). Mterms Mterm: a product term whch ever varable appears eactl oce ether complemeted or ot complemeted. Eample: For varables there are 8 possble mterms: For a mterm there s a uque assgmet o s ad s to the varables such that the value o the mterm s. Note: ever mterm the varables appear the same order (usuall lecographc). Q: How ma mterms are there or varables? Idces o Mterms ad Materms Istead o wrtg the mterms/materms we ca order them ad reer to a mterm/materm b ts de the order. Ide (decmal) 6 7 Ide (bar) Mterm Materm
3 Idces o Mterms ad Materms (cot ) The de bar base dcates whch varables are ther complemeted orm: For a mterm the th bt s the the correspodg varable s complemeted otherwse ot complemeted. For a materm the th bt s the the correspodg varable s ot complemeted otherwse complemeted. Eample: For the our varables wrte the mterm ad the materm that correspod to the de 9. 9 = () The mterm wth de s deoted b m. The materm wth de s deoted b M. Truth Tables or Both Mterms o Materms o varables varables m m m m M M M M Each colum the materm ucto table s the complemet o the colum the mterm ucto table sce M s the complemet o m. Mterm ad Materm Relatoshp Revew: B De-Morga s laws = ad = Two varables eample: m ad M = = M s the complemet o m ad vce-versa. De-Morga s laws ca be geeraled to varables: L = L ad L = L For a umber o varables ad a de we have: m = = M ad M m Observatos I the truth tables: Each mterm has eactl oe preset the terms. Each materm has eactl oe preset the terms. We ca mplemet a ucto b "ORg" the mterms correspodg to "" etres the ucto table. These are called the mterms o the ucto. We ca mplemet a ucto b "ANDg" the materms correspodg to "" etres the ucto table. These are called the materms o the ucto. Ths gves us two stadard (caocal) orms: Sum o Mterms (SOM) Product o Materms (POM) or statg a Boolea ucto.
4 Sum o Mterms: Eample We ca mplemet a ucto b "ORg" the mterms correspodg to "" etres the ucto table. Eample: wrte () SOM orm. Ide () Short orm b dces: ( ) = (67 ) 6 7 The Number o Levels a Crcut Revew: Gve two crcuts that mplemet the same ucto whch oe s better? Possble measure: the umber o levels a crcut. Product o Materms: Eample We ca mplemet a ucto b "ANDg" the materms correspodg to "" etres the ucto table. Eample: wrte () POM orm. Ide () Short orm b dces: = ( ) 6 7 Propagato Dela Propagato/gate dela: the elapsed tme rom the momet the puts o a gate are stable utl the outputs are stable. NOT Gate (verter) OUTPUT INPUT. HIGH HIGH. Volts A Ā LOW.... Volts LOW A Ā tme t t tme t Propagato dela: t pd = t - t
5 Propagato Dela (cot ) The dela o a path rom a put sgal to a output sgal s the sum o delas o the gates alog that path. The dela o a crcut s the mamum dela o a o paths rom a put sgal to a output sgal the crcut. D D D t pd (D ) =.s t pd (D ) =.s D D levels crcut t pd (D ) =.s The logest path t pd (D )t pd (D )t pd (D )t pd (D )=.9s I all the gates have roughl the same dela the the total dela s proportoal to the mamum umber o gates alog a path rom put to output = the umber o levels. Smplg Sum o Products Ever ucto has at least oe represetato as Sum o Products: Sum o Mterms. Other represetatos as Sum o Products possbl smpler mght also est ad ca be oud usg Boolea dettes. Eample: ( ) = ( = ) = = (Dstrbutve) (Complemet) (Idett) Sum o Products ad -Level Gates Sum o Products represetato elds a -level gate* Level : AND gates or the Products o the puts. Level : OR gates or the Sum o Products. Eample: ( ) = Smlarl Product o Sums also elds a -level gate* * Assumg or ever put ts complemet s also a put. Summar Stadard orms or boolea uctos: Sum o Products Product o Sums Ever boolea ucto ca be represeted as a Sum o Mterms ad as a Sum o Materms. These represetatos correspod to -level crcuts that eld a small propagato dela. Boolea Algebra ca be used to d smpler orms as sum o products or product sums lead to smpler -level crcuts. Ca we atta the smplest epresso? Is there ol oe mmum cost crcut?
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