1. Dr. Clincy Professor of CS
CS Chapter 2
Dr. Clincy
Professor of CS
Dr. Clincy
Lecture 3

2. Multiplication in base 2 – dealing with negative numbers
By hand – signed case – best to use 2’s complement
If both numbers are negative, perform as if both numbers are positive
If one is negative and one number is positive, see below – extend out left-most bit
Dr. Clincy
Dr. Clincy
Lecture 2

3. How does the computer multiply integers (shifting) ?
Computer doesn’t actually multiply – it adds and shifts
Dr. Clincy
Dr. Clincy
Lecture 3

4. Examples of Integer Multiplication by 2
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5. Another Example of Integer Multiplication by 2
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6. Examples of Integer Division by 2
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7. Booth’s Algorithm – Faster 2’s Complement Multiplication
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8. Booth’s Algorithm Concept
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9. Booth’s Algorithm Concept
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10. Booth’s Algorithm Standard Approach Booth’s Algorithm Approach
This is where 2’s complement comes into play
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11. Booth’s Algorithm
Only consider the first 16 bits – ignore beyond the 16th bit
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12. Character Codes
Calculations aren’t useful until their results can be displayed in a manner that is meaningful to people.
We also need to store the results of calculations, and provide a means for data input.
Thus, human-understandable characters must be converted to computer-understandable bit patterns using some sort of character encoding scheme.
Dr. Clincy

13. Character Codes
Binary-coded decimal (BCD) was one of these early codes. It was used by IBM mainframes in the 1950s and 1960s.
In 1964, BCD was extended to an 8-bit code, Extended Binary-Coded Decimal Interchange Code (EBCDIC).
Until recently, ASCII was the dominant character code outside the IBM mainframe world.
Many of today’s systems embrace Unicode, a 16-bit system that can encode the characters of every language in the world
Dr. Clincy

14. ASCII
Dr. Clincy

15. EBCDIC
Dr. Clincy

16. Character Codes
Calculations aren’t useful until their results can be displayed in a manner that is meaningful to people.
We also need to store the results of calculations, and provide a means for data input.
Thus, human-understandable characters must be converted to computer-understandable bit patterns using some sort of character encoding scheme.
Dr. Clincy
Lecture 9 and 10

17. Character Codes
Binary-coded decimal (BCD) was one of these early codes. It was used by IBM mainframes in the 1950s and 1960s.
In 1964, BCD was extended to an 8-bit code, Extended Binary-Coded Decimal Interchange Code (EBCDIC).
Until recently, ASCII was the dominant character code outside the IBM mainframe world.
Many of today’s systems embrace Unicode, a 16-bit system that can encode the characters of every language in the world
Dr. Clincy
Lecture 9 and 10

18. Error Detection Vs Error Correction
Error Detection – determining if an error occurred or not
Error Correction – given an error has occurred, correcting the error – much more complex than error detection
(Method 1) Fix the error by retransmitting the frame that is in error - Rx ask the Tx to retransmit
(Method 2) Forward error control – fixing the error on the Rx side (complex)
Dr. Clincy
Lecture 9 and 10

19. Error Control Techniques
Trade off between detection probability and processing requirements
Approaches
Detection
Simple Parity (Double Parity)
Cyclic Redundancy Checksum
Error correction
Hamming Approach
Dr. Clincy
Lecture 9 and 10

20. Error Detection: Single Parity
Bit added to each character to make all bits add up to an even number (even parity) or odd number (odd parity)
Good for detecting single-bit errors only
High overhead (one extra bit per 7-bit character=12.5%)
Examples of parity bits
What’s the disadvantage ?: what happens for 2 bit errors ? .
Dr. Clincy
Dr. Clincy
Lecture 9 and 10
Lecture 20 20

21. Do Some Examples w/students
Even,
Odd,
Odd,
Even, 11111
Dr. Clincy
Dr. Clincy
Lecture 9 and 10
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22. Parity Bit Concept
Given the word: – add “parity bit”
Even Parity: even # of 1’s:
Odd Parity: odd # of 1’s:
What’s the disadvantage ?: what happens for 2 bit errors ?
Another Example:
Encode 1100 in (a), even parity added in (b), Error in AC in (c)
NOW, do we see how we can use the Parity Bit Concept to Detect Errors
Dr. Clincy
Lecture

23. Error Detection: Double Parity
Uses horizontal and longitudinal parity
Individual characters are grouped together in a block
Each row (character) would have a parity bit
Each column would have a parity bit
The group of parity bits formed by the rows and columns
Parity bits (B)
If 1 bit is in error, (B) catches it, if 2 bits are in error, (A) catches it
Parity bits (A)
Dr. Clincy
Lecture

24. Do some examples w/students
Even
111101
101010
110011
Odd
11101
11010
11000
Even
110001
100010
110011
Odd
10001
11010
11100
Dr. Clincy
Lecture