1. CS151 Introduction to Digital Design
Chapter 1: Digital Systems and Information
Lecture 4
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Decimal Codes
Digital systems manipulate not only binary numbers, but also many other discrete elements of information.
Discrete elements can be represented by binary codes.
For n-bit binary code, we have 2n distinct combinations of 1’s and 0’s, with each combination representing one element of the set that is being coded. Each element can be assigned a unique binary bit combination.
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Decimal Codes (Cont.)
Examples:
How many elements can be coded using a 2-bit binary code? Give the different combinations.
A set of 4 elements.
The different combinations are: 00,01,10 and 11.
How many bits of code does a set of 8 elements require?
3-bit code (000, 001, 010, 011, 100, 101, 110, 111)
How many bits of code does a set of 16 elements require?
4-bit code (0000, 0001, 0010, …, 1111)
The bit combinations of an n-bit code can be determined from the count in binary from 0 to 2n -1
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Decimal Codes (Cont.)
What if the number of elements in the set is not a power of 2?
Example:
How many bits of code does a set of 10 elements require?
4 bits. However 6 out of the 16 possible combinations will be unused.
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5. Binary Coded Decimal (BCD)
Each decimal digit is represented by 4 bits.
A number with n decimal digits requires 4n bits in BCD
Example: (396)10 =
( )BCD
A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9.
The binary combinations 1010 through 1111 are not used and have no meaning in BCD.
Digit
BCD Code
0000 5
0101 1
0001 6
0110 2
0010 7
0111 3
0011 8
1000 4
0100 9
1001
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6. Binary Coded Decimal (BCD)
Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE. 
1310 = (This is conversion) 
13  0001|0011 (This is coding)
 1310 = = ( )BCD
The BCD value has (2 x 4 = 8 bits) while the equivalent binary number needs only 4 bits.
 BCD needs more bits than its equivalent binary representation, why is it still important?
Because people think in decimal!
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7. Putting It All Together
BCD not very efficient
Used in early computers (40s, 50s)
Used to encode numbers for seven-segment displays.
Easier to read?
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BCD Addition
Remember! Each digit does not exceed 9  the sum cannot be greater than 9+9+1= 19.
Case 1:
Case 2:
0001 1
0101 5
(0) (0) 6
0110 6
0101 5
(0) (1) 1
WRONG!
Case 3:
1000 8
1001 9
(1) (1) 7
Note that for cases 2 and 3, adding a factor of 6 (0110) gives us the correct result.
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BCD Addition
BCD addition is therefore performed as follows
1) Add the two BCD digits together using normal binary addition
2) Check if correction is needed
a) 4-bit sum is in range of 1010 to 1111
b) carry out of MSB = 1
3) If correction is required, add 0110 to 4-bit sum to get the correct result; BCD carry out = 1
We need to add (0110)2 for any result larger than (1001)2.
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10. Final answer (two digits) Created by: Ms.Amany AlSaleh
BCD Addition (Cont.)
Remember that We need to add (0110)2 for any result larger than (1001)2.
Case 1: 9
1001
No correction is needed
Case 2: 11
1011
+0110
Add (0110)2 because result is larger than 10012
10001
0001 | 0001
Final answer (two digits)
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11. Final answer (two digits) Created by: Ms.Amany AlSaleh
BCD Addition (Cont.)
Given a BCD code, we use binary arithmetic to add the digits: 8
1000
Eight +5
+0101
Plus 5 13
1101
is 13 (> 9)
Note that the result is MORE THAN 9, so must be represented by two digits!
To correct the digit, add 6 8 +5
1000
Eight
+0101
Plus 5 13
1101
is 13 (> 9)
+0110
so add 6
carry = 1
0011
leaving 3 + cy
0001 | 0011
Final answer (two digits)
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BCD Addition (Cont.)
Add (2905)BCD to (1897)BCD showing carries and digit corrections. 1 1 1
0100
1 0010
1010
1100
+ 0110
+ 0110
+ 0110
0100
Result: 4 8 2
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Alphanumeric Codes
Binary codes used to represent alphabetic and numeric characters
Computers process not only numbers, but also letters (e.g.. Student name), and symbols (e.g. *, {, &,…etc).
We need to encode the letters of the alphabet, in addition to symbols and numbers.
This code needs to be standard, so that computers can communicate.
Two most common are:
ASCII, 7 bit code, 128 symbols
EBCDIC, 8 bit code, 256 symbols
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ASCII Character Codes
American Standard Code for Information Interchange (ASCII).
ASCII is a 7-bit code, frequently used with an 8th bit for error detection (more about that in a bit).
This code is a popular code used to represent information sent as character-based data.
It uses 7-bits to code 128 characters.
94 printing characters.
26 uppercase letters
26 lowercase letters
10 numbers (0,1,..,9)
32 special symbols ..)
34 Non-printing characters
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15. ASCII Character Codes (Cont.)
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ASCII Character Codes
Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return)
Other non-printing characters are used for information separating (e.g. file and record separators FS and RS) and communication control (e.g. STX and ETX start and end text areas).
ASCII is a 7-bit code, but computers manipulate a singe 8-bit byte ASCII characters are stored one-per-byte with the MSB set to 0.
This extra bit is sometimes used for special purposes; e.g. additional characters (like Greek) recognized by printers, or could be used for error detection.
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17. ASCII Codes and Data Transmission
Transmission susceptible to noise
How to keep data transmission accurate?
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Parity Bit
Parity codes are formed by concatenating a parity bit, P to each code word of C.
In an odd-parity code, the parity bit is specified so that the total number of ones is odd.
In an even-parity code, the parity bit is specified so that the total number of ones is even.
Information Bits P 
Added even parity bit 
Added odd parity bit
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Parity Bit
Error Detection Scenarios where even parity is used
Send ‘A’
Two 1’s parity bit = 0 Receive
Three 1’s  error
Receive
Five 1’s  error
Receive
Four 1’s  ? Even# ?
Sender
Receiver
Odd number of errors is detected
Even number of errors is undetected
Control Characters:
Error  Retransmission send NAK
No Error  Receiver sends Back ACK
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Parity Code Example
Concatenate a parity bit to the ASCII code for the characters 0, X, and = to produce both odd-parity and even-parity codes.
Character
ASCII
Odd-Parity ASCII
Even-Parity ASCII X =
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UNICODE
UNICODE extends ASCII to 65,536 universal characters codes
For encoding characters in world languages
Available in many modern applications
2 byte (16-bit) code words
See Reading Supplement – Unicode on the Companion Website
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Gray Codes
Gray code is not a number system.
It is an alternate way to represent four bit data
Only one bit changes from one decimal digit to the next
Useful for reducing errors in communication.
Can be scaled to larger numbers.
Digit
Binary
Gray Code
0000 1
0001 2
0010
0011 3 4
0100
0110 5
0101
0111 6 7 8
1000
1100 9
1001
1101 10
1010
1111 11
1011
1110 12 13 14 15
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Gray Codes (Cont.)
Gray codes are minimum change codes
From one numeric representation to the next, only one bit changes
Primary use is in numeric input encoding apps. where we expect non-random input values changes (I.e. value n to either n-1 or n+1)
Milling machine table position
Rotary shaft position
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