"Digital Electronic Technology 6th Edition"
Logic function and a method described in 2.5
For the introduction of chestnut:
If the logical variables as input to the calculation result as an output, the input variable when the value is determined, then the output value will be. Thus, but a functional relationship between input and output. This function is called logic function written:
the Y = F. (A, B, C ...)
due to the variable and an output (function) of the values 0 and 1 only two states, so we are binary logic discussed function.
1. Description The method of logic function
The method commonly used logic functions described logic truth table, the logic function of formula (referred to as a logical expression or function formula), logic diagram, a waveform chart, hardware description language Karnaugh map and the like.
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(1) Logic Truth Table
The values corresponding to all input variables to find out an output value, tabulated, a truth table can be obtained.
(2) logic function formula
The logical relationship between the output and input and written, or other non-modular arithmetic, logic algebraic i.e., to give the desired functional logic.
(3) logic diagram
The logic function wherein each variable logic AND, OR, and other non-represented by graphical symbols, description will be drawn as a function of the logic of FIG.
(4) the waveform of FIG.
If the output value corresponding to the value of the input variable logic function every possible line up chronological order, to obtain a waveform diagram of the logic function described. This is also referred to as a timing waveform diagram of FIG.
If a waveform diagram to describe the logic function (1), the input variable table is simply given in the corresponding output variable values arrayed in chronological order together, you can get the desired waveform diagram!
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Conversion between the various methods described
(1) logic functions and truth table conversion formula
Converting into a logic functional truth table:
By the embodiment may be summarized by the write logic truth table functional general manner, that is:
① manipulation logic function truth table to identify those combinations of input variables Y = the value 1.
② each input variable value corresponding to a combination of product terms, wherein the original value of a variable is written, the value 0 is written counter variable.
③ adding these product terms, i.e. of formula Y to obtain logic functions.
Converted to the logic function truth table:
(2) conversion of a logical functional logic diagram of the
Given functional logic into corresponding logic diagram from, as long as the priority order to connect them with Boolean logic symbols press graphic symbols instead of the logical operation function formula, can be obtained in the required logic to FIG.
And when switching from a given logic diagram corresponding functional logic, as long as the input to outputs from the logic of FIG stepwise write output logic of each type of graphical symbols, logic seeking can be obtained at the output functional a.
(3) conversion of the waveform diagram and truth table
When seeking a waveform diagram corresponding to a truth table of the logic function from known, first need to find the value for each input variable and the time period from the output function on the waveform, then these inputs, the output value corresponding list, He was asked for the truth table.
When converting a waveform diagram of the truth table, the truth table simply all the input variables and the output variable value corresponding to arranged so drawn as the horizontal axis time waveform is obtained in a waveform of FIG ask .
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Two standard forms (1) of logic functions
Before the standard form about the logic functions, to introduce the concept of minimum and maximum term items, and then introduce "a sum of the minimum" and "maximum term product of" logic function of two standard forms.
1. The minimum entry
In the n variable logic function, comprising m product term if the factors of n, and the n variable or variables are the original form of the inverse variable m is one, m is said minterm for the group variable.
For example: minterm A, B, C are three variables A'B'C ', A'B'C, A'BC', A'BC, AB'C ', AB'C, ABC', ABC co eight (i.e., 2 3). minterm variable n should be 2 n number.
From the definition of the smallest items may prove that it has the following important properties:
① Any input variable values must be a minimum entry, and only a 1 value the smallest item.
② the sum of all minimum term is 1.
③ product of any two minterms 0.
④ properties and having two adjacent minterms may be combined into one and the elimination of one pair of factors.
If only one of the two smallest item of different factors, both said to have a minimum term adjacency.
For example: A'BC 'and ABC' minimum two different items are only the first factor, they have adjacent property. Both minterm addition timing can be incorporated into one and the elimination of a pair of different factors:
A'BC '+ the ABC' = (A '+ A) the BC' = the BC '
2. The maximum term
In the n variable logic function, when the M is the sum of n variables, and these variables are the n original anti-variable or variables in the form of a M, M is the term called the maximum set of variables.
For example: the maximum term of three variables A, B, C are (A '+ B' + C '), (A' + B '+ C), (A' + B + C '), (A' + B + C), (A + B ' + C'), (A + B '+ C), (A + B + C'), (A + B + C) a total of eight (i.e., 2 3). For n variables, there are 2 n largest items. It found: The maximum number of items and the minimum number of items is equal to the variable n.
The term also defines the maximum can be obtained Its main properties:
① there must be a maximum value of items in any of the input variables, and only a maximum entry is 0.
② all the product of the maximum term is zero.
③ a sum of any two is a maximum.
④ there is only one different than the product of the two largest items is equal to the sum of each of the same variables.
If the maximum table entry and two comparative minimum entry, and the entry can be found in the largest term minimum the following relationship:
For example: m0 = A'B'C ', the m0' = (A'B'C ')' = A + B + C = M0
3. The form and minterms logic function
First, a given functional logic and a plurality of product terms into the form, also known as "sum of products" form. Then, using the basic formula of A + A '= 1 in each product term missing complement factor so that it can be formalized or with the standard form and the minterms.
Product Form maximum Paragraph (4) Logic Functions
Use logic algebra formulas and basic theorems, first of all we must be able to type into any logical function of several polynomial multiplication or form (also known as "and the product of" form). Then using the basic formula AA '= 0 each missing variable polynomial filled, on the functional form may be with or into the form of the product of the largest item.
2.6 Simplification.pdf Logic Function
In addition functional product terms can not be reduced, and the multiplication factor in each can not be reduced when the function formula of the most simplest form.
Object simplification logic function is to eliminate redundant product term and product term of each factor in excess, to give the simplest form of the functional logic. Commonly used simplified methods are simple formulation method, map Simplification Act as well as applicable to the preparation of QM method of computer-aided analysis program.
1. Jane formulaic method
Simple principles of law is formulated repeatedly logic algebraic equations and the basic formula used product terms erasing redundant and redundant functional factors, in order to achieve most simple functional form.
(1) and term method
Using the formula AB + AB '= A two may be combined into one, and to eliminate B and B' on this factor, and, according to the theorem, is substituted, A and B may be any complex logical expression.
(2) Absorption Method
Using the formula A + AB = A AB items may be erased. A and B may likewise be any complex logical expression.
(3) Method canceling terms
Using the formula AB + A'C + BC = AB + A'C and AB + A'C + BCD = AB + A'C the BC or BCD entries erased. Wherein A, B, C, D can be any complex logical expression.
(4) Method extinction factor
Using the formula A + A'B = A + B may be the A'B A 'eliminated. A, B can be any complex logical expression.
(5) process with item
① basic formula A + A = A writing may be repeated in a particular logic function according to the formulas, and sometimes can be obtained even simpler method of simplification.
② The basic formula of A + A '= 1 can be multiplied by an item (A + A in the formula of'), and then split into two separately combined with other items, can sometimes be more simple simplification result.
The integrated use of both together:
2. map Simplification Act
Thinking: Since any logic function can be expanded in the form of a sum of the minimum, then combined using the method of minterm simplifying logic functions, is a kind applicable to any logic function, and having a fixed and a method of operating steps Jane method.
Karnaugh map simplification method is a kind of simplified method based on merging the smallest item! ! !
(1) Introduction Karnaugh map:
Definition: all minterms respective small blocks with a variable n represents, and has a logical adjacency minterms geometrically arranged adjacent positions can together, the resulting graph is called Carnot variable n minterm Fig.
FIG follows: is two to five variable Karnaugh map minterm. Pattern on both sides of the 0 and 1 are denoted minimizing entry corresponding to small squares of a value of a variable. At the same time, the 0 and 1 binary number corresponding to the decimal number is the size corresponding to the smallest number of items.
In order to ensure the geometrical position of FIG adjacent minterms also have logical adjacency, which does not work as a digital natural binary number in ascending order of the earth and must be arranged in the drawing way to ensure minimal adjacent two only one variable terms are different.
It can also be seen from the figure, in any row or column of smallest item across different only one variable, they also have a logical adjacency. Thus, from the geometric position of the Karnaugh map should be considered as upper and lower, left and right closed figures.
After the variable> = 5, only the geometry of the adjacent two-dimensional space to represent a logical adjacency longer enough. For example: five variable d illustrated in FIG minterm Karnaugh map, in addition to the geometric position having the minimum line adjacent to the logical adjacency to FIG double vertical axis on the left and right symmetrical positions minterm also it has a logical adjacency.
Since any logic function can be expressed in the form of a minimum number of items and, then naturally you can try using a Karnaugh map to represent any logic function. Specific method is: first logic function minterm and into the form, and then fill a position corresponding to the smallest item on the Karnaugh map, 0 is filled in the remaining positions, it means that the obtained logic function Karnaugh map. That is, any logic function is equal to a Karnaugh map its fill those minterms 1.
(2) Logic Function Karnaugh map simplification
Using simple logic function Karnaugh map method called Karnaugh map method or simple graphic method. When the simplification is based on the basic principles of adjacency with the smallest items can be combined, and elimination of different factors. Since adjacent the geometric position of the adjacent logically consistent, and thus from the Karnaugh map can intuitively identify those entries having minimum adjacency Simplification and combined in the Karnaugh map.
1. Principles of consolidation smallest item
If two adjacent minimum term can be merged into one and eliminate one pair of factors. Combined results in only common factor. .
FIG. (A) and (b) drawn in two adjacent minterm several possible cases. For example: FIG. (A), A'BC (m1) and ABC (m7) adjacent, it can be combined into;
A'BC the ABC + = (A '+ A) = the BC the BC
after combined A and A' one pair factor disappeared, leaving only common factor B and C.
If four neighboring items and arranged in a minimum rectangle group may be combined into one factor and two pairs of erasing. Combined results contain only common factor.
For example: FIG. (D) of, A'BC'D (m5), A'BCD ( m7), ABC'D (m13) and ABCD (m15) adjacent, it can be combined. The combined to give:
A'BC'D A'BCD + + + ABCD ABC'D
= A'BD (C + C ') the ABD + (C + C')
= the BD (A + A ') = the BD
visible, combined after elimination of the a, a 'and C, C' factor of two pairs, only four common factors B and D. minterm
Summary:
The combined minterm general rule: If there is 2 ** n smallest item adjacent (n = 1,2, ...) and are arranged in a rectangular group, they may be combined into one, and to eliminate a factor of n. The combined result of these factors includes only common minterm.
2. Karnaugh map method steps:
(1) into the function and form of minterms
(2) shown Karnaugh map represents the logic function
(3) items can be combined to find the minimum
product term post (4) Select simplification. Principle selected:
① These product terms shall include all minterms function formula (Karnaugh map should cover all 1)
minimum number of product terms used ②. I.e. items can be combined set of minimum rectangle composed of a minimum number.
③ Each product term comprising a minimum of a factor. That is, each rectangle be combined minterm group should contain as many minterms.
3. The logic function having independent and simplification items
(1) constraint term, regardless of any of items and logic functions of the formula
Constraints items:
A special case of the first value of the input variable is not arbitrary, the value of the input variable is called constraint limits. We logic function having such characteristics is called logic functions with constraints, while the set of input variables to a set of logical variables is called a binding.
Any of:
second special case is the function value is 0 or 1 can be made at some value of the input variable does not affect the function of the circuit. Under these variables values, the minimum value equal to those of any of items 1 is referred to. The presence of any of the logic function is also called incompletely defined logical function.
Note: Because the value of the constraint term is equal to the value of the input variable 1 is not allowed, so the value of the constraint term is always 0 ,;
and any of the different functions during the operation, there may appear to make any term value of the input variable 1 value.
Functional logic of unrelated items:
We will be bound by any terms and items unrelated items collectively referred to as logic function formula. Here the word "do not care" refers to whether these minimum functional item into a logic does not matter, you can write can also be deleted.
Application (2) independent of simplifying logic function items in the formula
Note: 1. Add items should function independently of the formula in the minimum possible number of items (including the original entry and the smallest item that have been written independent) logical adjacency.
2. When the minimum combined item, whether the item regardless Karnaugh map as 1 (i.e., the function that the formula contains minterms) or as 0 (i.e., that does not include this function formula minterms) treated should give minterm adjacent rectangle combination maximum and minimum number of rectangles composition principle.
To solve this problem by drawing a Karnaugh map :::
