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Number Systems & Conversions Digital Systems and Switching Circuits Wednesday, Aug 26 CEC 220 Digital Circuit Design What is the difference between analog and digital systems/signals? Analog – Continuous Natural Phenomena (Pressure, Temperature, Speed … ) Difficult to realize processing using electronics Digital – Discrete Binary Digit Signal Processing as Bit unit Easy to realize processing using electronics High performance due to Integrated Circuit Technology Slide 3 of 16
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CEC 220 Digital Circuit DesignNumber Systems & Conversions
Wednesday, Aug 26 CEC 220 Digital Circuit Design Slide 1 of 16
Number Systems & Conversions
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Number Systems and Conversions Digital Systems and Switching Circuits Number Systems and Conversion Binary, Hexadecimal, and Octal representations Examples
Slide 2 of 16
Number Systems & ConversionsDigital Systems and Switching Circuits
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• What is the difference between analog and digital systems/signals?
• Analog – Continuous Natural Phenomena (Pressure, Temperature, Speed…) Difficult to realize processing using electronics
• Digital – Discrete Binary Digit Signal Processing as Bit unit Easy to realize processing using electronics High performance due to Integrated Circuit Technology
Slide 3 of 16
Number Systems & ConversionsDigital Systems and Switching Circuits
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• What is a binary signal?• Binary = Two values (0, 1)
• Each digit is referred to as a “bit”• Number representation with only two values (0, 1)• Can be implemented with simple electronics devices • For Example
• Voltage: High = (1) and Low = (0)• Switch: On = (1) and Off = (0)
Slide 4 of 16
Number Systems & ConversionsDigital Systems and Switching Circuits
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• What is a switching circuit?• Combinational Circuit:
• Outputs depend on only present inputs, not on past inputs
• Sequential Circuit:• Outputs depend on both present inputs and past inputs• Sequential circuits have “memory” !!!!!
Slide 5 of 16
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
4 3 2 1 0 1 2 3( . )RN a a a a a a a a
2 1 0 1 29 10 5 10 3 10 7 10 8 10
21011.11
Decimal:(base 10)
Binary:(base 2)
Base “R”
10953.78
3 2 1 0 1 21 2 0 2 1 2 1 2 1 2 1 2 1 18 0 2 12 4
10
311 11.754
4 3 2 1 04 3 2 1 0
1 2 31 2 3
a R a R a R a R a R
a R a R a R
Slide 6 of 16
9 100 5 10 3 1 7 /10 8 /100 900 50 3 7 /10 8 /100
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Examples Base 8 to Decimal
Base 16 to Decimal
8147.3
10103.375
2 1 0 11 8 4 8 7 8 3 8 364 32 78
162A F 2 1 010 16 2 16 15 16
102607
2560 32 15
Slide 7 of 16
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Conversion of Decimal to base R
1 2 1 01 2 1 0 1 2 1 0
n nn n n nR
N a a a a a a R a R a R a R a R
1 2 11 2 1 1
n nn n
N a R a R a R a QR
2 3 111 3 2 2
n nn n
Q a R a R a R a QR
3 421 3 3
n nn n
Q a R a R a QR
.
.
.
0remainder a
1remainder a
2remainder a
Quotient
Slide 8 of 16
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Example: Convert Decimal 53 (i.e., base 10) to base 2 (i.e., binary)
532
262
132
62
32
12
rem. = 1 = a0rem. = 0 = a1
rem. = 1 = a2
rem. = 0 = a3
rem. = 1 = a4
0 rem. = 1 = a5
210 11010153
Slide 9 of 16
How would I check this result?
10 253 ? ?
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Conversion of a Decimal fraction to base R
1 2 30 m RF a a a a
1 2 12 3 11 1
mma R a R a RF R Fa a
11 22
232
mma R aRa FR aF
2 3 333m
mF R a aa R F
.
.
.
1 2 31 2 3
mma R a R a R a R
Slide 10 of 16
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Example: Convert Decimal 0.625 to base 2 (i.e., binary)
.625 21.250
F
1 .250 20.500
F
2 .500 21.000
F
10 20.625 0.101
Slide 11 of 16
10 20.625 0.???
1( 1)a 2( 0)a 3( 1)a
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Example: Convert 231.34 to base 7 First convert to decimal
Convert the integer portion (4510 = ???7)
Convert the decimal portion ( .7510 = .???7)
104 75.45431431623.231
457
67
0 rem. = 6
rem. = 3
.75
5 7
( ).25
.75
5 7
( ).25
.25
1 7
( ).75
.25
1 7
( ).75
4 10 7231.3 45.75 63.5151
Slide 12 of 16
…
763
70.5151
How would I check this result?
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
Binary000001010011100101110111
QUESTION:How many binary “bits” do we need to represent a single Octal digit?
Slide 13 of 16
Octal01234567
Binary Hex Decimal0000 0 00001 1 10010 2 20011 3 30100 4 40101 5 50110 6 60111 7 71000 8 81001 9 91010 A 101011 B 111100 C 121101 D 131110 E 141111 F 15
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
Binary Hex Decimal
0000 0 00001 1 10010 2 20011 3 30100 4 40101 5 50110 6 60111 7 71000 8 81001 9 91010 A 101011 B 111100 C 121101 D 131110 E 141111 F 15
QUESTION:How many binary “bits” do we need to represent a single Hexadecimal digit?
Slide 14 of 16
C
Number Systems & ConversionsNumber Systems and Conversion
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Converting to/from Binary, Hex, and Octal An example of converting Binary to Octal
An example of converting Binary to Hexadecimal
Slide 15 of 16
511 2 7
1001101.0101112 = 115.278
D4 5
1001101.0101112 = 4D.5C16
Next Lesson
Wednesday, Aug 26 CEC 220 Digital Circuit Design
• Binary Arithmetic• Representation of Negative Numbers
Slide 16 of 16
