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Unformatted text preview: c A.H.Dixon CMPT 150: Week 3 (Jan 21 - 25, 2008) 16 6 SUM OF MINTERMS REPRESENTATION The product terms obtained from a function table, as described in the previous week’s notes has the property that every input variable occurs as a literal in the product. Such product terms are called minterms . A sum of products, where all product terms are minterms is called a sum of minterms . Sum of minterms representations are particularly important because they are unique to a given digitial system, except for the labeling of the inputs and the order in which the product terms appear. That is, if two Boolean expressions can be transformed to the same sum of minterms representation, then the two expressions are equivalent. Thus we have another way besides comparing function tables, for determining whether two Boolean expressions are equivalent. The sum of minterms representation is an example of a “canonical form”. This is a representation that is consider a “standard” representation for a given digital system. Because of its importance, an alternative notation can be used that eliminates the problem of variable labeling and the of the minterms, called Σ notation . For each minterm there is a binary sequence for which that minterm equals 1. Each such binary sequence can be interpreted as a base-2 integer. Since each integer corresponds to a row of the function table where the function equals 1, the list of all such integers corresponding to minterms, when given in ascending order, uniquely defines the function. Therefore that list can be used to represent the sum of minterms and this is expressed by the following notation where “Σ” denotes “sum of” and “ m ” suggests “minterms”: Σ m ( list of minterm integers in ascending order ) For example the function f ( a,b,c ) = a bc + ab c + abc can be represented in Σ notation by: f ( a,b,c ) = Σ m (3 , 4 , 7) 7 DIGITAL DESIGN The techniques and tools described to this point provide sufficient tools to illus- trate how to use a behavioral description to obtain a structural one; that is, a logic diagram. The general design process consists of the following steps: 1. Obtain a behavioral description from the informal problem description. Specif- ically: c A.H.Dixon CMPT 150: Week 3 (Jan 21 - 25, 2008) 17 (a) Obtain the entity definition by encoding the input and output alphabets of the problem into binary. The length of the binary sequences will determine the number of input and output ports. (b) Express the functional specification as a function table. 2. Obtain a Boolean sum of minterms expression from the function table. 3. Transform the sum of minterms expression for each output bit to meet any required constraints (eg., minimal sum of products, fewest literals, etc.) 4. Obtain a logic diagram from the transformed Boolean expressions Example: Design a circuit to compute the square of a non-negative integer less than 8....
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- Spring '08
- Boolean Algebra, Implicants, Implicants Algebraically, 00 00 0 w, 00 00 1 w