---
# System prepended metadata

title: Techniki Cyfrowe - Sprawozdanie \#2
tags: [college]

---

---
title: Techniki Cyfrowe - Sprawozdanie \#2
description: Techniki Cyfrowe - Minimalizacja i praktyczna realizacja złożonych funkcji logicznych
tags: college
---

# Techniki Cyfrowe - Sprawozdanie \#2

## Minimalizacja i praktyczna realizacja złożonych funkcji logicznych

**Artur Majdak, Michał Dolaś**

### a) Korzystając z metody tablic Karnaugh'a, zminimalizować graficznie funkcję:

$$
Y(a, b, c, d) = ab\overline{c}\overline{d} + \overline{b}c\overline{d} + a\overline{c}d + a\overline{b}\overline{c} + bc\overline{d}
$$

$AB \backslash CD$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|0|0|0|1
**`01`**|0|0|0|1
**`11`**|1|1|0|1
**`10`**|1|1|0|1

$AB \backslash CD$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|0|0|0|1
**`01`**|0|0|0|1
**`11`**|![1_red]|![1_red]|0|1
**`10`**|![1_red]|![1_red]|0|1

$AB \backslash CD$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|0|0|0|![1_red]
**`01`**|0|0|0|![1_red]
**`11`**|1|1|0|![1_red]
**`10`**|1|1|0|![1_red]

$$
Y(a, b, c, d) = a\overline{c} + c\overline{d}
$$

![](https://i.imgur.com/hXiKBoP.gif)

[Układ z powyższego obrazka](https://drive.google.com/open?id=0BxYgq_SdG5b4dXpMQUoyRTFVMkE)

### b) Zaprojektować i przetestować w programie Multisim cyfrowy układ kombinacyjny zwiększający o jeden trzybitową nieujemną liczbę całkowitą.

Wejścia:
$A, B, C$

Wyjścia:
$X, Y, Z$

$A$|$B$|$C$||$X$|$Y$|$Z$
-|-|-|-|-|-|-
0|0|0||0|0|1
0|0|1||0|1|0
0|1|0||0|1|1
0|1|1||1|0|0
1|0|0||1|0|1
1|0|1||1|1|0
1|1|0||1|1|1
1|1|1||0|0|0

#### $X$

$A \backslash BC$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|0|0|1|0
**`1`**|1|1|0|1

$A \backslash BC$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|0|0|1|0
**`1`**|![1_red]|![1_red]|0|1

$A \backslash BC$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|0|0|![1_red]|0
**`1`**|1|1|0|1

$A \backslash BC$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|0|0|1|0
**`1`**|1|1|0|![1_red]

$X = \overline{A}BC + AB\overline{C} + A\overline{B}$

#### $Y$

$A \backslash CD$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|0|1|0|1
**`1`**|0|1|0|1

$A \backslash CD$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|0|![1_red]|0|1
**`1`**|0|![1_red]|0|1

$A \backslash CD$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|0|1|0|![1_red]
**`1`**|0|1|0|![1_red]

$Y = B\overline{C} + \overline{B}C$

$Y = B \; xor \; C$

#### $Z$

$A \backslash CD$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|1|0|0|1
**`1`**|1|0|0|1

$A \backslash CD$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`0`**|![1_red]|0|0|![1_red]
**`1`**|![1_red]|0|0|![1_red]

$Z = \overline{C}$

![](https://i.imgur.com/fr2tpKc.gif)

[Układ z powyższego obrazka](https://drive.google.com/open?id=0BxYgq_SdG5b4eVdZeE9hOVoxelk)

### c) Korzystając z tablic Karnaugh'a, opracować transkoder czterobitowych cyfr (od 0 do 9) na pojedynczy wyświetlacz siedmiosegmentowy. Zbudować i przetestować układ w programie Multisim.

![](https://i.imgur.com/x2jEvaB.png)

$X_{3}$|$X_{2}$|$X_{1}$|$X_{0}$||$A$|$B$|$C$|$D$|$E$|$F$|$G$
-|-|-|-|-|-|-|-|-|-|-|-
0|0|0|0||1|1|1|1|1|1|0
0|0|0|1||0|1|1|0|0|0|0
0|0|1|0||1|1|0|1|1|0|1
0|0|1|1||1|1|1|1|0|0|1
0|1|0|0||0|1|1|0|0|1|1
0|1|0|1||1|0|1|1|0|1|1
0|1|1|0||1|0|1|1|1|1|1
0|1|1|1||1|1|1|0|0|0|0
1|0|0|0||1|1|1|1|1|1|1
1|0|0|1||1|1|1|1|0|1|1
1|0|1|0||X|X|X|X|X|X|X
1|0|1|1||X|X|X|X|X|X|X
1|1|0|0||X|X|X|X|X|X|X
1|1|0|1||X|X|X|X|X|X|X
1|1|1|0||X|X|X|X|X|X|X
1|1|1|1||X|X|X|X|X|X|X

#### $A$

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|1|1
**`01`**|0|1|1|1
**`11`**|X|X|X|X
**`10`**|1|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|1|1
**`01`**|0|1|1|1
**`11`**|![x_red]|![x_red]|![x_red]|![x_red]
**`10`**|![1_red]|![1_red]|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|![1_red]|![1_red]
**`01`**|0|1|![1_red]|![1_red]
**`11`**|X|X|![x_red]|![x_red]
**`10`**|1|1|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|![1_red]|0|1|![1_red]
**`01`**|0|1|1|1
**`11`**|X|X|X|X
**`10`**|![1_red]|1|X|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|1|1
**`01`**|0|![1_red]|![1_red]|1
**`11`**|X|![x_red]|![x_red]|X
**`10`**|1|1|X|X

$A = X_{3} + X_{1} + \overline{X_{2}}\;\overline{X_{0}} + X_{2}X_{0}$

$A = X_{3} + X_{1} + \overline{X_{2} \; xor \; X_{0}}$

#### $B$

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|1|1|1
**`01`**|1|0|1|0
**`11`**|X|X|X|X
**`10`**|1|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|1|1|1
**`01`**|1|0|1|0
**`11`**|![x_red]|![x_red]|![x_red]|![x_red]
**`10`**|![1_red]|![1_red]|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|![1_red]|![1_red]|![1_red]|![1_red]
**`01`**|1|0|1|0
**`11`**|X|X|X|X
**`10`**|![1_red]|![1_red]|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|![1_red]|1|1|1
**`01`**|![1_red]|0|1|0
**`11`**|![x_red]|X|X|X
**`10`**|![1_red]|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|1|![1_red]|1
**`01`**|1|0|![1_red]|0
**`11`**|X|X|![x_red]|X
**`10`**|1|1|![x_red]|X

$B = X_{3} + \overline{X_{2}} + \overline{X_{1}}\;\overline{X_{0}} + X_{1}X_{0}$

$B = X_{3} + \overline{X_{2}} + \overline{X_{1} \; xor \; X_{0}}$

#### $C$

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|1|1|0
**`01`**|1|1|1|1
**`11`**|X|X|X|X
**`10`**|1|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|1|1|0
**`01`**|1|1|1|1
**`11`**|![x_red]|![x_red]|![x_red]|![x_red]
**`10`**|![1_red]|![1_red]|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|![1_red]|![1_red]|1|0
**`01`**|![1_red]|![1_red]|1|1
**`11`**|![x_red]|![x_red]|X|X
**`10`**|![1_red]|![1_red]|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|![1_red]|![1_red]|0
**`01`**|1|![1_red]|![1_red]|1
**`11`**|X|![x_red]|![x_red]|X
**`10`**|1|![1_red]|![x_red]|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|1|1|0
**`01`**|![1_red]|![1_red]|![1_red]|![1_red]
**`11`**|![x_red]|![x_red]|![x_red]|![x_red]
**`10`**|1|1|X|X

$C = X_{3} + \overline{X_{1}} + X_{0} + X_{2}$

#### $D$

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|1|1
**`01`**|0|1|0|1
**`11`**|X|X|X|X
**`10`**|1|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|1|1
**`01`**|0|1|0|1
**`11`**|![x_red]|![x_red]|![x_red]|![x_red]
**`10`**|![1_red]|![1_red]|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|1|![1_red]
**`01`**|0|1|0|![1_red]
**`11`**|X|X|X|![x_red]
**`10`**|1|1|X|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|![1_red]|![1_red]
**`01`**|0|1|0|1
**`11`**|X|X|X|X
**`10`**|1|1|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|![1_red]|0|1|![1_red]
**`01`**|0|1|0|1
**`11`**|X|X|X|X
**`10`**|![1_red]|1|X|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|1|1
**`01`**|0|![1_red]|0|1
**`11`**|X|![x_red]|X|X
**`10`**|1|1|X|X

$D = X_{3} + X_{1}\overline{X_{0}} + \overline{X_{2}}X_{1} + \overline{X_{2}}\;\overline{X_{0}} + X_{2}\overline{X_{1}}X_{0}$

#### $E$

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|0|1
**`01`**|0|0|0|1
**`11`**|X|X|X|X
**`10`**|1|0|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|![0_red]|![0_red]|1
**`01`**|0|![0_red]|![0_red]|1
**`11`**|X|![x_red]|![x_red]|X
**`10`**|1|![0_red]|![x_red]|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|0|1
**`01`**|![0_red]|![0_red]|0|1
**`11`**|![x_red]|![x_red]|X|X
**`10`**|1|0|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|0|1
**`01`**|0|0|0|1
**`11`**|X|X|![x_red]|![x_red]
**`10`**|1|0|![x_red]|![x_red]

$E = \overline{X_{0} + X_{2}\overline{X_{1}} + X_{3}X_{1}}$

#### $F$

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|0|0
**`01`**|1|1|0|1
**`11`**|X|X|X|X
**`10`**|1|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|0|0
**`01`**|1|1|0|1
**`11`**|![x_red]|![x_red]|![x_red]|![x_red]
**`10`**|![1_red]|![1_red]|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|![1_red]|0|0|0
**`01`**|![1_red]|1|0|1
**`11`**|![x_red]|X|X|X
**`10`**|![1_red]|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|0|0
**`01`**|![1_red]|1|0|![1_red]
**`11`**|![x_red]|X|X|![x_red]
**`10`**|1|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|1|0|0|0
**`01`**|![1_red]|![1_red]|0|1
**`11`**|![x_red]|![x_red]|X|X
**`10`**|1|1|X|X

$F = X_{3} + \overline{X_{1}}\;\overline{X_{0}} + X_{2}\overline{X_{0}} + X_{2}\overline{X_{1}}$

#### $G$

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|0|0|1|1
**`01`**|1|1|0|1
**`11`**|X|X|X|X
**`10`**|1|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|0|0|1|1
**`01`**|1|1|0|1
**`11`**|![x_red]|![x_red]|![x_red]|![x_red]
**`10`**|![1_red]|![1_red]|![x_red]|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|0|0|1|![1_red]
**`01`**|1|1|0|![1_red]
**`11`**|X|X|X|![x_red]
**`10`**|1|1|X|![x_red]

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|0|0|1|1
**`01`**|![1_red]|![1_red]|0|1
**`11`**|![x_red]|![x_red]|X|X
**`10`**|1|1|X|X

$X_{3}X_{2} \backslash X_{1}X_{0}$|**`00`**|**`01`**|**`11`**|**`10`**
-|-|-|-|-
**`00`**|0|0|![1_red]|![1_red]
**`01`**|1|1|0|1
**`11`**|X|X|X|X
**`10`**|1|1|X|X

$G = X_{3} + X_{1}\overline{X_{0}} + X_{2}\overline{X_{1}} + \overline{X_{3}}\;\overline{X_{2}}X_{1}$

![](https://i.imgur.com/ZEnO2fT.gif)

[Układ z powyższego obrazka](https://drive.google.com/open?id=1SiJbw5pLdLFP9YkpiyBj4azDFJ6wsSLO)

[0_red]: https://i.imgur.com/K9NVaj7.png "0"
[1_red]: https://i.imgur.com/YSOIjc5.png "1"
[x_red]: https://i.imgur.com/dFAXEzP.png "X"