设计实现Y=X^2的组合逻辑电路

小咪咪 2023-06-03 09:56 22阅读 0赞

(15分)设计能实现函数![Y=X^\{2\}][Y_X_2]的组合电路, X是三位二进制数![X\_\{2\}X\_\{1\}X\_\{0\}][X_2_X_1_X_0], 其取值范围是0~5的正整数. 要求用与非门实现(要有设计和化简过程, 不必画出逻辑图)

思维过程梳理:

X的范围是0~5 (000, 001, 010, 011,100, 101), ![Y=X^\{2\}][Y_X_2]的范围是0~25(00000, 00001, 00010, 00011, ..., 11001), 需要五位二进制数来表示. -> 设Y的五位二进制数变量从高位到低位依次为 ![Y\_\{4\}][Y_4]、![Y\_\{3\}][Y_3]、![Y\_\{2\}][Y_2]、![Y\_\{1\}][Y_1]、![Y\_\{0\}][Y_0]. ->

根据![Y=X^\{2\}][Y_X_2] ,X与Y的逻辑对应关系为:

![X\_\{2\}X\_\{1\}X\_\{0\}=000, Y\_\{4\}Y\_\{3\}Y\_\{2\}Y\_\{1\}Y\_\{0\}=00000;][X_2_X_1_X_0_000_ Y_4_Y_3_Y_2_Y_1_Y_0_00000]

![X\_\{2\}X\_\{1\}X\_\{0\}=001, Y\_\{4\}Y\_\{3\}Y\_\{2\}Y\_\{1\}Y\_\{0\}=00001;][X_2_X_1_X_0_001_ Y_4_Y_3_Y_2_Y_1_Y_0_00001]

![X\_\{2\}X\_\{1\}X\_\{0\}=010, Y\_\{4\}Y\_\{3\}Y\_\{2\}Y\_\{1\}Y\_\{0\}=00100;][X_2_X_1_X_0_010_ Y_4_Y_3_Y_2_Y_1_Y_0_00100]

![X\_\{2\}X\_\{1\}X\_\{0\}=011, Y\_\{4\}Y\_\{3\}Y\_\{2\}Y\_\{1\}Y\_\{0\}=01001;][X_2_X_1_X_0_011_ Y_4_Y_3_Y_2_Y_1_Y_0_01001]

![X\_\{2\}X\_\{1\}X\_\{0\}=100, Y\_\{4\}Y\_\{3\}Y\_\{2\}Y\_\{1\}Y\_\{0\}=10000;][X_2_X_1_X_0_100_ Y_4_Y_3_Y_2_Y_1_Y_0_10000]

![X\_\{2\}X\_\{1\}X\_\{0\}=101, Y\_\{4\}Y\_\{3\}Y\_\{2\}Y\_\{1\}Y\_\{0\}=11001;][X_2_X_1_X_0_101_ Y_4_Y_3_Y_2_Y_1_Y_0_11001]

根据上述逻辑关系可得到图示真值表.

注意 ![X\_\{2\}X\_\{1\}X\_\{0\}= 110][X_2_X_1_X_0_ 110], ![X\_\{2\}X\_\{1\}X\_\{0\}=111][X_2_X_1_X_0_111] 为无关项, 在填充卡诺图时要用"![\\times][times]"填充.

在化简卡诺图后, ![Y\_\{0\}=X\_\{0\}][Y_0_X_0], ![Y\_\{1\}=0][Y_1_0], ![Y\_\{4\}=X\_\{2\}][Y_4_X_2]这三位输出直接对应连接输入即可, 而![Y\_\{2\}][Y_2]和![Y\_\{3\}][Y_3]的表达式须用与非门实现.

对于给定表达式![Y=X][Y_X], 我们可以通过 ![Y=X \\cdot X][Y_X _cdot X] \-> 加重叠率 -> 摩根定律 来转换为与非式.

![Y\_\{2\}=X\_\{1\} \\cdot \\overline\{X\_\{0\}\} =X\_\{1\} \\cdot \\overline\{X\_\{0\}\} + 0 =\\overline\{ \\overline\{ X\_\{1\} \\cdot \\overline\{X\_\{0\}\} + 0 \} \} =\\overline\{ \\overline\{X\_\{1\} \\cdot \\overline\{X\_\{0\}\}\} \\cdot \\overline\{0\} \} =\\overline\{ \\overline\{X\_\{1\} \\cdot \\overline\{X\_\{0\}\}\} \\cdot 1 \} =\\overline\{ \\overline\{X\_\{1\} \\cdot \\overline\{X\_\{0\} \\cdot 1\}\} \\cdot 1 \}.][Y_2_X_1_ _cdot _overline_X_0_ _X_1_ _cdot _overline_X_0_ _ 0 _overline_ _overline_ X_1_ _cdot _overline_X_0_ _ 0 _ _ _overline_ _overline_X_1_ _cdot _overline_X_0_ _cdot _overline_0_ _ _overline_ _overline_X_1_ _cdot _overline_X_0_ _cdot 1 _ _overline_ _overline_X_1_ _cdot _overline_X_0_ _cdot 1_ _cdot 1 _.]

![Y\_\{3\}=X\_\{2\}X\_\{0\}+X\_\{1\}X\_\{0\} =\\overline\{ \\overline\{ X\_\{2\}X\_\{0\}+X\_\{1\}X\_\{0\} \} \} =\\overline\{ \\overline\{X\_\{2\}X\_\{0\} \} \\cdot \\overline\{ X\_\{1\}X\_\{0\} \} \}.][Y_3_X_2_X_0_X_1_X_0_ _overline_ _overline_ X_2_X_0_X_1_X_0_ _ _ _overline_ _overline_X_2_X_0_ _ _cdot _overline_ X_1_X_0_ _ _.]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MjA0ODQ2Mw_size_16_color_FFFFFF_t_70][]

[Y_X_2]: https://private.codecogs.com/gif.latex?Y%3DX%5E%7B2%7D
[X_2_X_1_X_0]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D
[Y_4]: https://private.codecogs.com/gif.latex?Y_%7B4%7D
[Y_3]: https://private.codecogs.com/gif.latex?Y_%7B3%7D
[Y_2]: https://private.codecogs.com/gif.latex?Y_%7B2%7D
[Y_1]: https://private.codecogs.com/gif.latex?Y_%7B1%7D
[Y_0]: https://private.codecogs.com/gif.latex?Y_%7B0%7D
[X_2_X_1_X_0_000_ Y_4_Y_3_Y_2_Y_1_Y_0_00000]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D%3D000%2C%20Y_%7B4%7DY_%7B3%7DY_%7B2%7DY_%7B1%7DY_%7B0%7D%3D00000%3B
[X_2_X_1_X_0_001_ Y_4_Y_3_Y_2_Y_1_Y_0_00001]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D%3D001%2C%20Y_%7B4%7DY_%7B3%7DY_%7B2%7DY_%7B1%7DY_%7B0%7D%3D00001%3B
[X_2_X_1_X_0_010_ Y_4_Y_3_Y_2_Y_1_Y_0_00100]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D%3D010%2C%20Y_%7B4%7DY_%7B3%7DY_%7B2%7DY_%7B1%7DY_%7B0%7D%3D00100%3B
[X_2_X_1_X_0_011_ Y_4_Y_3_Y_2_Y_1_Y_0_01001]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D%3D011%2C%20Y_%7B4%7DY_%7B3%7DY_%7B2%7DY_%7B1%7DY_%7B0%7D%3D01001%3B
[X_2_X_1_X_0_100_ Y_4_Y_3_Y_2_Y_1_Y_0_10000]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D%3D100%2C%20Y_%7B4%7DY_%7B3%7DY_%7B2%7DY_%7B1%7DY_%7B0%7D%3D10000%3B
[X_2_X_1_X_0_101_ Y_4_Y_3_Y_2_Y_1_Y_0_11001]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D%3D101%2C%20Y_%7B4%7DY_%7B3%7DY_%7B2%7DY_%7B1%7DY_%7B0%7D%3D11001%3B
[X_2_X_1_X_0_ 110]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D%3D%20110
[X_2_X_1_X_0_111]: https://private.codecogs.com/gif.latex?X_%7B2%7DX_%7B1%7DX_%7B0%7D%3D111
[times]: https://private.codecogs.com/gif.latex?%5Ctimes
[Y_0_X_0]: https://private.codecogs.com/gif.latex?Y_%7B0%7D%3DX_%7B0%7D
[Y_1_0]: https://private.codecogs.com/gif.latex?Y_%7B1%7D%3D0
[Y_4_X_2]: https://private.codecogs.com/gif.latex?Y_%7B4%7D%3DX_%7B2%7D
[Y_X]: https://private.codecogs.com/gif.latex?Y%3DX
[Y_X _cdot X]: https://private.codecogs.com/gif.latex?Y%3DX%20%5Ccdot%20X
[Y_2_X_1_ _cdot _overline_X_0_ _X_1_ _cdot _overline_X_0_ _ 0 _overline_ _overline_ X_1_ _cdot _overline_X_0_ _ 0 _ _ _overline_ _overline_X_1_ _cdot _overline_X_0_ _cdot _overline_0_ _ _overline_ _overline_X_1_ _cdot _overline_X_0_ _cdot 1 _ _overline_ _overline_X_1_ _cdot _overline_X_0_ _cdot 1_ _cdot 1 _.]: https://private.codecogs.com/gif.latex?Y_%7B2%7D%3DX_%7B1%7D%20%5Ccdot%20%5Coverline%7BX_%7B0%7D%7D%20%3DX_%7B1%7D%20%5Ccdot%20%5Coverline%7BX_%7B0%7D%7D%20&plus;%200%20%3D%5Coverline%7B%20%5Coverline%7B%20X_%7B1%7D%20%5Ccdot%20%5Coverline%7BX_%7B0%7D%7D%20&plus;%200%20%7D%20%7D%20%3D%5Coverline%7B%20%5Coverline%7BX_%7B1%7D%20%5Ccdot%20%5Coverline%7BX_%7B0%7D%7D%7D%20%5Ccdot%20%5Coverline%7B0%7D%20%7D%20%3D%5Coverline%7B%20%5Coverline%7BX_%7B1%7D%20%5Ccdot%20%5Coverline%7BX_%7B0%7D%7D%7D%20%5Ccdot%201%20%7D%20%3D%5Coverline%7B%20%5Coverline%7BX_%7B1%7D%20%5Ccdot%20%5Coverline%7BX_%7B0%7D%20%5Ccdot%201%7D%7D%20%5Ccdot%201%20%7D.
[Y_3_X_2_X_0_X_1_X_0_ _overline_ _overline_ X_2_X_0_X_1_X_0_ _ _ _overline_ _overline_X_2_X_0_ _ _cdot _overline_ X_1_X_0_ _ _.]: https://private.codecogs.com/gif.latex?Y_%7B3%7D%3DX_%7B2%7DX_%7B0%7D&plus;X_%7B1%7DX_%7B0%7D%20%3D%5Coverline%7B%20%5Coverline%7B%20X_%7B2%7DX_%7B0%7D&plus;X_%7B1%7DX_%7B0%7D%20%7D%20%7D%20%3D%5Coverline%7B%20%5Coverline%7BX_%7B2%7DX_%7B0%7D%20%7D%20%5Ccdot%20%5Coverline%7B%20X_%7B1%7DX_%7B0%7D%20%7D%20%7D.
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MjA0ODQ2Mw_size_16_color_FFFFFF_t_70]: /images/20230601/d4ff3e725795471389bd162bab78b522.png