Operational Trans conductance Amplifiers:Karnaugh Maps
Karnaugh Maps
Logic functions can be visually expressed using a Karnaugh map, which is simply a different way of representing a truth table, as exemplified for four variables in Figure 26.1(a). For the case of four variables, for example, a Karnaugh map consists of 16 cells; that is, 16 small squares as shown in Figure 26.1(a). Here, two-bit numbers along the horizontal line above the squares show the values of x1 and x2, and two-bit binary numbers along the vertical line on the left of the squares show the values of x3 and x4. The top left cell in Figure 26.1(a) has 1 inside for x1 = x2 = x3 = x4 = 0. Also, the cell in the second row and the second column from the left has d inside. This means f = d (i.e., don’t-care) for x1 = 0, x2 = 1, x3 = 0, and x4 = 1. The binary numbers that express variables are arranged in such a way that binary numbers for any two cells that are horizontally or vertically adjacent differ in only one bit position. Also, the two numbers in each row in the first and last columns differ in only one bit position and are interpreted to be adjacent. Also, the two numbers in each column in the top and bottom rows are similarly interpreted to be adjacent. Thus, the four cells in the top row are interpreted to be adjacent to the four cells in the bottom row in each column. The four cells in the first column are interpreted to be adjacent to the four cells in the last column in each row. With this arrangement of cells and this interpretation, a Karnaugh map is more than a concise representation of a truth table; it can express many important algebraic concepts, as we will see later. A Karnaugh map is a two-dimensional representation of the 16 cells on the surface of a torus, as shown in Figure 26.1(b), where the two ends of the map are connected vertically and horizontally.
Figure 26.2 shows the correspondence between the cells in the map in Figure 26.2(a) and the rows in the truth table in Figure 26.2(b). Notice that the rows in the truth table are not shown in consecutive order in the Karnaugh map. The Karnaugh map labeled with variable letters, instead of with binary numbers, shown in Figure 26.2(c), is also often used. Although a 1 or 0 shows the function’s value corresponding to a particular cell, 0 is often not shown in each cell. Cells that contain 1’s are called 1-cells (similarly, 0-cells).
Patterns of Karnaugh maps for two and three variables are shown in Figures 26.3(a) and (b), respectively. As we extend this treatment to the cases of 5 or more variables, the maps, which will be explained in a later subsection, become increasingly complicated.
A rectangular loop that consists of 2i 1-cells without including any 0-cells expresses a product of literals for any i, where i ≥ 1. For example, the square loop consisting four 1-cells in Figure 26.4(a) represents the
Two Completely Specified Functions to Express an Incompletely Specified Function Suppose we have an incompletely specified function f, as shown in Figure 26.5(a), i.e., a function that has some don’t-cares. This incompletely specified function f can be expressed alternatively with two completely specified functions, f ON and f OFF, shown in Figure 26.5(b) and (c), respectively.
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