楊伏夷

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數位系統 Digital Systems

Department of Computer Science and Information Engineering, Chaoyang University of Technology

朝陽科技大學資工系

Speaker: Fuw-Yi Yang

楊伏夷

伏夷非征番,

道德經察政章(Chapter 58)

伏

者潛藏也

道紀章(Chapter 14) 道無形象, 視之不可見者曰

夷

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Text Book: Digital Design 4th Ed.

Chap 1 Digital Systems and Binary Numbers

1.1 Digital Systems

1.2 Binary Numbers-- Table 1.1

1.3 Number-Base Conversions-- Example 1.1~1.4 1.4 Octal and Hexadecimal Numbers -- Table 1.2 1.5 Complements -- Example 1.5~1.8

1.6 Signed Binary Numbers -- Table 1.3 1.7 Binary Codes -- Table 1.4~1.7

1.8 Binary Storage and Registers -- Figure 1.1~1.2 1.9 Binary Logic -- Table 1.8, Figure 1.3~1.6

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Chap 1 1.2 Binary Numbers

In general, a number expressed in a base-r system has coefficients multiplied by powers of r:

a_n⋅rⁿ+a_n-1⋅r^n-1+…+a₁⋅r¹+a₀+a_-1⋅r^-1+a_-2⋅r^-2+…+a_-m⋅r^-m r is called base or radix.

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In general, a number expressed in a base-r system has coefficients multiplied by powers of r:

a_n⋅rⁿ+a_n-1⋅r^n-1+…+a₁⋅r¹+a₀+a_-1⋅r^-1+a_-2⋅r^-2+…+a_-m⋅r^-m r is called base or radix.

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Chap 1 1.2 Binary Numbers

是否注意到指數為負之值呢?

Ex: 2^-1, 2^-3,2^-4

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Chap 1 1.3 Number-Base Conversions

Example1.1 Convert decimal 41 to binary, (41)₁₀= (?)₂ (41)_D= (?)_B Example1.2 (153)₁₀= (?)₈

Example1.3 (0.6875)₁₀= (?)₂ Example1.4 (0.513)₁₀= (?)₈

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Chap 1 1.4 Octal and Hexadecimal Numbers

See Table 1.2

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Chap 1 1.4 Octal and Hexadecimal Numbers

See Table 1.2

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Chap 1 1.5 Complements

Diminished Radix Complement

Given a number N in base r having n digits, the (r - 1)’s complement of N is defined as (rⁿ - 1) - N.

The 1’s complement of 1011000 is 0100111 Radix Complement

Given a number N in base r having n digits, the r’s complement of N is defined as

rⁿ - N for N

≠

0 and as 0 for N = 0 . The 2’s complement of 1011000 is 0101000

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Chap 1 1.5 Complements— Subtraction with Complements

The subtraction of two n-digit unsigned numbers M - N in base r can be done as follows:

1. M + (rⁿ - N ), note that (rⁿ - N ) is r’s complement of N.

2. If M ≥ N, the sum will produce an end carry rⁿ, which can be discarded; what is left is the result M - N.

3. If M < N, the sum does not produce an end carry and is equal to rⁿ - (N - M), which is r’s complement of

(N - M). Take the r’s complement of the sum and place a negative sign in front.

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Example 1.5 Using 10’s complement, subtract 72532 - 3250.

1. M = 72532, N = 3250, 10’s complement of N = 96750 2.

3. answer: 69282

sum addend augend

....

169282 96750 72532

←←

←

+ ←

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Example 1.6 Using 10’s complement, subtract 3250 - 72532.

1. M = 3250, N = 72532, 10’s complement of N = 27468 2.

3. answer: -(100000 - 30718) = -69282

30718

27468

03250

+

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Example 1.7 Using 2’s complement,

subtract 1010100 - 1000011.

1. M = 1010100,

N = 1000011, 2’s complement of N = 0111101 2.

3. answer: 0010001

10010001

0111101

1010100

+

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Example 1.7' Using 2’s complement,

subtract 1000011 - 1010100.

1. M = 1000011,

3. answer: - (10000000 - 1101111) = -0010001

1101111

0101100

1000011

+

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Example 1.8 Using 1’s complement,

subtract 1010100 - 1000011.

1. M = 1010100,

3. answer: 0010001 (rⁿ carry, call end-around carry)

10010000

0111100

1010100

+

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Example 1.8' Using 1’s complement,

subtract 1000011 - 1010100.

1. M = 1000011,

3. answer: - (1111111 - 1101110) = -0010001

1101110

0101011

1000011

+

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Chap 1 1.6 Signed Binary Numbers

Next table shows signed binary numbers

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Arithmetic addition Arithmetic subtraction See next table

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Arithmetic addition with comparison:

The addition of two numbers in the signed magnitude system follows the rules of ordinary arithmetic.

If the signed are the same, we add the two magnitudes and give the sum the common sign.

If the signed are different, we subtract the smaller magnitude from the larger and give the difference the sign of the larger magnitude. EX. (+25) + (-38) = -(38 - 25) = -13

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Arithmetic addition without comparison:

The addition of two signed binary numbers with negative numbers represented in signed 2’s complement form is obtained from the addition of the two numbers, including their signed bits. A carry out of the signed bit position is discarded (note that the 4th case).

See examples in next page.

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Arithmetic addition without comparison:

11101101 11110011 11111010 19

13 06 11111001

11110011 00000110 07

13 06

00000111 00001101 11111010 07

13 06 00010011

00001101 00000110 19

13 06

+ + +

−

− − +

+ +

− + −

+ + + +

− + +

+ + +

+ +

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Chap 1 1.7 Binary Codes

BCD (Binary-Coded Decimal) Code Table 1.4 Decimal codes Table 1.5

(4 different Codes for the Decimal Digits) Gray code Table 1.6

ASCII character code Table 1.7 Error Detecting code

See next tables 1.4~1.7

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BCD Code

Decimal codes Gray code

ASCII character code Error Detecting code See next tables

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BCD Code

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BCD Code

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BCD Code

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Chap 1 1.8 Binary Storage and Registers

A register is a group of binary cells.

A register with n cells can store any discrete quantity of information that contains n bits.

A digital system is characterized by its registers and the components that perform data processing.

In digital systems, a register transfer operation is a basic operation that consists of a transfer of binary information from one set of registers into another set of registers.

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Chap 1 1.9 Binary Logic

Binary logic consists of binary variables and a set of logical operations.

There are three basic logical operations:

AND, OR, and NOT.

AND, OR, NOT: Table 1.8 Binary signals: Figure 1.3

Logic circuit: Figures 1.4~1.6 Ex 1.35, 1.36

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Binary logic consists of binary variables and a set of logical operations.

There are three basic logical operations:

AND, OR, and NOT.

See truth table 1.8 next page.

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