## ECE 275: Representations of logic systems

Vikas Dhiman
[email protected]
TA: Pascal Francis-Mezger
[email protected]

## Basic gates

AND gate
$$L_{\text{AND}}(x_1, x_2) = x_1 \cdot x_2$$
OR gate
$$L_{\text{OR}}(x_1, x_2) = x_1 + x_2$$

NOT gate
$$L_{\text{NOT}}(x_1) = \bar{x}_1$$
Source:Verilog 3rd, Brown and Vranesic 2014

## ANSI symbols

AND gate
$$L_{\text{AND}}(x_1, x_2) = x_1 \cdot x_2$$
OR gate
$$L_{\text{OR}}(x_1, x_2) = x_1 + x_2$$

NOT gate
$$L_{\text{NOT}}(x_1) = \bar{x}_1$$

## AND Gate

Circuit (an implentation)

Source:Verilog 3rd, Brown and Vranesic 2014
Truth table
Row num$$x_1$$ $$x_2$$ $$x_1 \cdot x_2$$
0000
1010
2100
3111
Boolean expression
$$f = x_1 \cdot x_2 = x_1x_2$$
ANSI Symbol

## OR Gate

Circuit (an implentation)

Source:Verilog 3rd, Brown and Vranesic 2014
Truth table
Row num$$x_1$$ $$x_2$$ $$x_1 + x_2$$
0001
1011
2101
3110
Boolean expression
$$f = x_1 + x_2$$
ANSI Symbol

## NOT Gate

Circuit (an implentation)

Source:Verilog 3rd, Brown and Vranesic 2014
Truth table
Row num$$x_1$$ $$\bar{x}_1$$
001
110
Boolean expression
$$f = \bar{x}_1$$
ANSI Symbol

## Network to truth table

$$f = \overline{(A\bar{B} + C)} D$$

#### Truth table

Row numABCD$$\bar{B}$$ghkf
0000010010
1000110011
2001010100
3001110100
4010000010
5010100011
6011000100
7011100100
8100011100
9100111100
10101011100
11101111100
12110000010
13110100011
14111000100
15111100100

## Truth table to Minterms

Row numABCDf
000000
100011
200100
300110
401000
501011
601100
701110
810000
910010
1010100
1110110
1211000
1311011
1411100
1511110

#### Minterms

List of rows where function is 1

$$f = m_1 + m_5 + m_{13}$$

$$f = \sum m(1, 5, 13)$$

## Minterms to Sum of products

#### Minterms

$$f = \sum m(1, 5, 13)$$

 $$m_0$$ $$\triangleq \bar{A} \bar{B} \bar{C} \bar{D}$$ $$m_1$$ $$\triangleq \bar{A} \bar{B} \bar{C} D$$ $$m_2$$ $$\triangleq \bar{A} \bar{B} C \bar{D}$$ $$m_3$$ $$\triangleq \bar{A} \bar{B} C D$$ $$m_4$$ $$\triangleq \bar{A} \bar{B} \bar{C} \bar{D}$$ $$m_5$$ $$\triangleq \bar{A} \bar{B} \bar{C} D$$ $$m_6$$ $$\triangleq \bar{A} \bar{B} C \bar{D}$$ $$m_7$$ $$\triangleq \bar{A} \bar{B} C D$$ $$m_8$$ $$\triangleq A \bar{B} \bar{C} \bar{D}$$ $$m_9$$ $$\triangleq A \bar{B} \bar{C} D$$ $$m_{10}$$ $$\triangleq A \bar{B} C \bar{D}$$ $$m_{11}$$ $$\triangleq A \bar{B} C D$$ $$m_{12}$$ $$\triangleq A B \bar{C} \bar{D}$$ $$m_{13}$$ $$\triangleq A B \bar{C} D$$ $$m_{14}$$ $$\triangleq A B C \bar{D}$$ $$m_{15}$$ $$\triangleq A B C D$$

#### Sum of products

$$f = \bar{A} \bar{B} \bar{C} D + \bar{A} \bar{B} \bar{C} D + A B \bar{C} D$$

## Truth table to Maxterms

Row numABCDf
000000
100011
200100
300110
401000
501011
601100
701110
810000
910010
1010100
1110110
1211000
1311011
1411100
1511110

#### Maxterms

List of rows where function is 0

$f = M_0 \cdot M_2 \cdot M_3 \cdot M_4 \cdot M_6 \cdot M_7 \\ \cdot M_8 \cdot M_9 \cdot M_{10} \cdot M_{11} \cdot M_{12} \cdot M_{14} \\ \cdot M_{15}$

$$f = \prod M(0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15)$$

## Maxterms to products of sums

#### Maxterms

$$f = \prod M(0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15)$$

 $$M_0$$ $$\triangleq A + B + C + D$$ $$M_1$$ $$\triangleq A + B + C + \bar{D}$$ $$M_2$$ $$\triangleq A + B + \bar{C} + D$$ $$M_3$$ $$\triangleq A + B + \bar{C} + \bar{D}$$ $$M_4$$ $$\triangleq A + \bar{B} + C + D$$ $$M_5$$ $$\triangleq A + \bar{B} + C + \bar{D}$$ $$M_6$$ $$\triangleq A + \bar{B} + \bar{C} + D$$ $$M_7$$ $$\triangleq A + \bar{B} + \bar{C} + \bar{D}$$ $$M_8$$ $$\triangleq\bar{A} + B + \bar{C} + D$$ $$M_9$$ $$\triangleq\bar{A} + B + \bar{C} + \bar{D}$$ $$M_{10}$$ $$\triangleq\bar{A} + B + \bar{C} + D$$ $$M_{11}$$ $$\triangleq\bar{A} + B + \bar{C} + \bar{D}$$ $$M_{12}$$ $$\triangleq\bar{A} + \bar{B} + \bar{C} + D$$ $$M_{13}$$ $$\triangleq\bar{A} + \bar{B} + \bar{C} + \bar{D}$$ $$M_{14}$$ $$\triangleq\bar{A} + \bar{B} + \bar{C} + D$$ $$M_{15}$$ $$\triangleq\bar{A} + \bar{B} + \bar{C} + \bar{D}$$

#### Products of sums

\begin{multline} f = (A + B + C + D)( A + B + \bar{C} + D ) \dots \\ (\bar{A} + \bar{B} + \bar{C} + D) (\bar{A} + \bar{B} + \bar{C} + \bar{D}) \end{multline}

## VENN Diagrams for gates

NOT Venn Diagram
$$f = \bar{x}_1$$; $$f = \bar{x}_2$$
AND Venn Diagram
$$f = x_1 \cdot x_2$$
OR Venn Diagram
$$f = x_1 + x_2$$

## Karnaugh maps for gates

AND K-map
$$f = x_1 \cdot x_2$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$00
$$x_2$$01
OR K-map
$$f = x_1 + x_2$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$01
$$x_2$$11

NOT K-map
$$f = \bar{x}_1$$
$$\bar{x}_1$$ $$x_1$$
10

## Truth table to K-map

Row numABCDf
000000
100011
200100
300110
401000
501011
601100
701110
810000
910010
1010100
1110110
1211000
1311011
1411100
1511110

#### K-map

$$\bar{A}$$ $$A$$
$$\bar{B}$$ $$B$$ $$\bar{B}$$
$$\bar{C}$$$$\bar{D}$$ $$m_0$$ $$m_4$$ $$m_{12}$$ $$m_8$$
$$D$$ $$m_1$$ $$m_5$$ $$m_{13}$$ $$m_9$$
$$C$$ $$m_3$$ $$m_7$$ $$m_{15}$$ $$m_{11}$$
$$\bar{D}$$ $$m_2$$ $$m_6$$ $$m_{14}$$ $$m_{10}$$

## Truth table to K-map

Row numABCDf
000000
100011
200100
300110
401000
501011
601100
701110
810000
910010
1010100
1110110
1211000
1311011
1411100
1511110

#### K-map

$$\bar{A}$$ $$A$$
$$\bar{B}$$ $$B$$ $$\bar{B}$$
$$\bar{C}$$$$\bar{D}$$ 0 0 0 0
$$D$$ 1 1 1 0
$$C$$ 0 0 0 0
$$\bar{D}$$ 0 0 0 0

## K-map to expression

#### K-map

$$\bar{A}$$ $$A$$
$$\bar{B}$$ $$B$$ $$\bar{B}$$
$$\bar{C}$$$$\bar{D}$$ 0 0 0 0
$$D$$ 1 1 1 0
$$C$$ 0 0 0 0
$$\bar{D}$$ 0 0 0 0

#### Expression

$$f = \bar{A}\bar{C}D + B\bar{C}D$$

## Expression to truth table

#### Expression

$$f = \bar{A}\bar{C}D + B\bar{C}D$$

#### Truth table

Row numABCD $$\bar{A}\bar{C}D$$ $$B\bar{C}D$$ f
0 0000000
1 0001101
2 0010000
3 0011000
4 0100000
5 0101111
6 0110000
7 0111000
8 1000000
9 1001000
101010000
111011000
121100000
131101011
141110000
151111000

Row numABCDf
000000
100011
200100
300110
401000
501011
601100
701110
810000
910010
1010100
1110110
1211000
1311011
1411100
1511110

Row numABCDf
000000
100011
200100
300110
401000
501011
601100
701110
810000
910010
1010100
1110110
1211000
1311011
1411100
1511110