Digital-electronics 简明教程
3 Variable K-Map in Digital Electronics
K-Map 或 Karnaugh Map 是一种图形化方法，用于简化布尔函数中的复杂代数表达式。此方法避免使用复杂定理和等式运算。卡诺图基本上是真值表的特殊形式，可以轻松绘出参数的值并得出简化的布尔表达式。
A K-Map or Karnaugh Map is a graphical method that used for simplifying the complex algebraic expressions in Boolean functions. This method avoid the use of complex theorems and equations manipulations. A K-Map is basically a special form of a truth table that can easily map out the values of parameters and gives a simplified Boolean expression.
卡诺图方法最适合具有两个到四个变量的布尔函数。然而，它也能用于具有五个或六个变量的布尔函数，但随着函数的变量增加，运算过程也会变得更加困难。
K-Map method is best suited for such Boolean functions that have two to four variables. However, it can be used for Boolean functions having five or six variables as well, but its process becomes more difficult with the increased number of variables in the function.
因此，在实践中，我们通常使用两变量卡诺图、三变量卡诺图和四变量卡诺图。但有时，五变量卡诺图和六变量卡诺图也用于导出布尔表达式。
Therefore, in practice, we mostly use Two-Variable K-Map, Three-Variable K-Map, and Four-Variable K-Map. But, sometimes, the Five-Variable K-Map and Six-Variable K-Map are also used to derive the Boolean expressions.
在这里，我们将介绍 3 Variable K-Map 及其在复杂布尔函数简化中的应用。
Here, we will discuss the 3 Variable K-Map and its application to simplify a complex Boolean function.
Three-Variable K-Map
我们可以使用 K-Map 来简化一个三变量布尔函数。一个三变量布尔函数（A、B、C）可以用标准的乘积和（SOP）形式表示，它可以有八个可能的组合，如下所示：
We can use the K-Map to simplify a Boolean function of three-variables. A Boolean function in three variables (A, B, C) can be expressed in the standard sum of product (SOP) form that can have total eight possible combinations, which are as follows −
\mathrm{（A’B’C’），（A’B’C），（A’BC’），（A’BC），（AB’C’），（AB’C），（ABC’），（ABC}）
\mathrm{(A’B’C'), (A’B’C), (A’BC'), (A’BC), (AB’C'), (AB’C), (ABC'), (ABC)}
我们可以用 m0、m1、m2、m3、m4、m5、m6 和 m7 来分别表示这些组合。这些项中的每一个都被称为 min-term 。在这些组合中，A 称为 MSB（最高有效位），C 称为 LSB（最低有效位）。
We can designate each of these combinations by m0, m1, m2, m3, m4, m5, m6, and m7 respectively. Each of these terms are called a min-term. In these combinations, A is called MSB (Most Significant Bit) and C is called LSB (Least Significant Bit).
使用和的积（POS）形式，三变量布尔表达式的八个可能的组合如下：
In terms of POS (Product of Sum) form, the eight possible combinations of the three variables Boolean expression are as follows −
\mathrm{（A+B+C），（A+B+C’），（A+B’+C），（A+B’+C’），（A’+B+C），（A’+B+C’），（A’+B’+C），（A’+B’+C’}）
\mathrm{(A+B+C), (A+B+C'), (A+B'+C), (A+B'+C'), (A'+B+C), (A'+B+C'), (A'+B'+C), (A'+B'+C')}
这些组合中的每一个常常分别用 M0、M1、M2、M3、M4、M5、M6 和 M7 来表示。这些项中的每一个都被称为最大项。类似于最小项，A 称为 MSB（最高有效位），C 称为 LSB（最低有效位）。
Each one of these combinations are often designated by M0, M1, M2, M3, M4, M5, M6, and M7 respectively. Each of these terms is called a maxterm. Similar to the minterm, A is called MSB (Most Significant Bit) and C is called LSB (Least Significant Bit).
因此，一个三变量卡诺图有八个（23）个方格或单元格，卡诺图上的每个方格都表示最大项或最小项，如下图所示。
Therefore, a three variable K-Map has eight (23) squares or cells, and each square on the K-Map represents a minterm of a maxterm as shown in the following figure.
在此，每个单元格右下角的小数字表示最小项或最大项的名称。
Here, the small number on the bottom right corner of each cell indicates the minterm or maxterm designation.
卡诺图顶部的二进制数表示每列变量 B 和 C 的状态。卡诺图左侧每一行的二进制数表示该行变量 A 的状态。
The binary numbers along the top of the K-Map indicates the condition of variables B and C for each column. The binary number along the left side of the map against each row represents the condition of the variable A for that row.
例如，上图中第四列顶部的二进制数 10 表明变量 B 以非补码形式出现，变量 C 以补码形式出现在该列的所有最小项中。卡诺图第一行左侧的二进制数 0 表明变量 A 以补码形式出现在所有最小项中，卡诺图第二行左侧的二进制数 1 表明变量 A 以非补码形式出现在所有最小项中。
For example, the binary number of 10 on the top of the fourth column in the above figure indicates that the variable B appears in non-complimented form and the variable C appears in complimented form in all the minterms in that column. The binary number 0 on the left of the first row on the K-map indicates that the variable A appears in its complimented form in all the minterms, and the binary number 1 on the left of the second row on the K-Map indicates that the variable A appears in its non-complimented form in all the minterms.
另外要注意的是，卡诺图顶部的二进制数不是按照正常的二进制顺序排列，而是按照格雷码排列。格雷码在卡诺图中的使用确保了两个物理相邻的单元格实际上是相邻的，这意味着它们的最小项或最大项仅相差一个变量。
Also, note that the binary numbers on top of the K-map are not in the normal binary order, but they are actually in the Gray code. The use of Gray code in K-map ensures that the two physically adjacent cells are actually adjacent which means their minterms or maxterms differs by one variable only.
Numerical Example
将以下三变量布尔表达式映射到卡诺图上。
Map the following three-variable Boolean expression on K-Map.
\mathrm{f \: = \: \overline{A} \: \overline{B} \: C \: + \: A \: \overline{B} \: C \: + \: \overline{A} \: B \: \overline{C} \: + \: A \: \overline{B} \: \overline{C} \: + \: A \: B \: C}
Solution
在给定的布尔表达式中，最小项为：
In the given Boolean expression, the minterms are −
\mathrm{\overline{A} \: \overline{B} \: C \: = \: 001; \: A \: \overline{B} \: C \: = \: 101; \: \overline{A} \: B \: \overline{C} \: = \: 010; \: A \: \overline{B} \: \overline{C} \: = \: 100; \: ABC \: = \: 111}
因此，
Therefore,
\mathrm{m_{1} \: = \: \overline{A} \: \overline{B} \: C \: = \: 001}
\mathrm{m_{5} \: = \: A \: \overline{B} \: C \: = \: 101}
\mathrm{m_{2} \: = \: \overline{A} \: B \: \overline{C} \: = \: 010}
\mathrm{m_{4} \: = \: A \: \overline{B} \: \overline{C} \: = \: 100}
\mathrm{m_{7} \: = \: ABC \: = \: 111}
因此，表达式如下：
Hence, the expression is given by,
\mathrm{f \: = \: \sum \: m (1, \: 5, \: 2, \: 4, \: 7)}
该表达式的卡诺图如下图所示：
The K-map of this expression is shown in the following figure −
