Cst131 Assignment 1-07
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CST 131 Assignment 1
1. Write a 2-3 pages description on a figure that has greatly contributed to the computing field. Please describe the person’s contribution in detail. Answer :
Charles Babbage (1791-1871), a computer pioneer, designed the first automatic computing engines and widely known as the Father of Computers was born 26 December 1791 in London. He invented computers but failed to build them. The father of Charles was a rich man, so it was possible for Charles to receive instruction from several elite schools and teachers during the course of his elementary education. He was about eight when he had to move to a country school to recover from a dangerous fever. Then, he joined King Edward VI Grammar School in Totnes, South Devon, a thriving comprehensive school that's still operative today, but his fragile health status forced him back to private teaching for a period. He finally joined a 30 student closed number academy managed by Reverend Stephen Freeman. The academy had a big library, where Babbage used to study mathematics by himself, and learned to love it. Babbage arrived at Trinity College, Cambridge in October 1810. He had a big culture because he knew Lagrange, Leibniz, Lacroix, Simpson and he was seriously disappointed about the math programs available at Cambridge. So he with J.Herschel, G.Peacock, and other friends, decided to form the Analytical Society.
When, in 1812, Babbage transferred to Peterhouse, Cambridge where he was the best mathematician; but he failed to graduate with honours. He received an honorary degree later,
1
CST 131 Assignment 1
without even being examinated, in 1814. Babbage sought a method by which mathematical tables could be calculated mechanically, removing the high rate of human error. Three different factors seem to have influenced him: a dislike of untidiness; his experience working on logarithmic tables; and existing work on calculating machines carried out by Wilhelm Schickard, Blaise Pascal, and Gottfried Leibniz. He first discussed the principles of a calculating engine in a letter to Sir Humphry Davy in 1822.
Figure 1.1: Difference Engine Babbage's machines were among the first mechanical computers, although they were not actually completed, largely because of funding problems and personality issues. He directed the building of some steam powered machines that achieved some success, suggesting that calculations could be mechanized. Although Babbage's machines were mechanical and unwieldy, their basic architecture was very similar to a modern computer. The data and program memory were separated, operation was instruction based, the control unit could make conditional jumps and the machine had a separate
2
CST 131 Assignment 1
Figure 1.2 Analytical Engine Design of computers Babbage was inspired by Napier's logarithm tables and Napier's logs and bones. He began to design a difference engine in 1821 which was a very large and complicated machine intended for doing logarithmic calculations automatically. The device was based on the principle that the difference between certain values of the expression at a certain stage becomes constant. But it was difficult to make the machine parts accurate enough to prevent errors to occur, using the technology available at that time. Encouraged by the work of difference engine, he began design of another machine the analytical machine, which could carry out many different types of calculations. Analytical engine used the concepts of Automatic Loom and Difference Engine.
The Analytical machine had five units- input, output, store, mill, and control. Store was used for storing numbers and Mill was used to do the calculations by rotation of gears and wheels. Control unit did the job of supervision of all other units. Working with him was a mathematician Lady Ada Lovelace, who provided the input material in the form of programs. Thus, Lady Ada Lovelace became known to the world as the first computer programmer. She coded these programs on punch cards. Her contribution was so valuable that a language called "Ada" was named after her.
3
CST 131 Assignment 1
The two concepts implemented in this machine, which revolutionized the Computer era in the following years are
1. The machine was able to compare quantities and then decide which of the sequences to follow 2. The results of a calculation could alter the numbers and instructions already stored in the machine.
Babbage devoted his last 37 years for the development of the analytical engine. His machine was too ahead of the time of development. Lack of better technology leaded to incomplete work of Babbage that time and he passed away in 1871 before he had completed his work.
4
CST 131 Assignment 1
2. Implement the following Boolean function with a multiplexer. (a) F (A, B, C, D) = ∑ (0, 2, 5, 7, 11, 14) Answer: Step 1) A truth table is drawn for 8x1 multiplexer F (A, B, C, D) = ∑(0, 2, 5, 7, 11, 14)
A
B
C
D
F
0
0
0
0
1
0
0
0
1
0
0
0
1
0
1
0
0
1
1
0
0
1
0
0
0
0
1
0
1
1
0
1
1
0
0
0
1
1
1
1
1
0
0
0
0
1
0
0
1
0
1
0
1
0
0
1
0
1
1
1
1
1
0
0
0
1
1
0
1
0
1
1
1
0
1
1
1
1
1
0
Table 2.1 Truth table
Step2) The inputs are determined as D',D',D,D,0,D,0,D based on the table below:
5
CST 131 Assignment 1
0
1
D
0
0
0
1
1
1
1
0
D'
Table 2.2 Step 3) The inputs are identified as
D' = 0, 1, 7 D = 2, 3, 5 0 = 4, 6
Step 4) The diagram of the multiplexer implemented shown. S1 S B A C 5 3 1 7 D 0 6 4 2 S2 0X 8 1 MUX
Figure 2.1 the diagram of the multiplexer implemented
6
CST 131 Assignment 1
2. Implement the following Boolean function with a multiplexer.
(b) F (A, B, C, D) = П (3, 8, 12)
Answer: Step 1) A truth table is drawn for 8x1 multiplexer F (A, B, C, D) = П(3,8,12)=F (A,B,C,D)=(0,1,2,4,5,6,7,9,10,11,13,14,15)
A
B
C
D
F
0
0
0
0
1
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
0
1
0
0
1
0
1
0
1
1
0
1
1
0
1
0
1
1
1
1
1
0
0
0
0
1
0
0
1
1
1
0
1
0
1
1
0
1
1
1
1
1
0
0
0
1
1
0
1
1
1
1
1
0
1
1
1
1
1
1
Table 2.3 truth table
Step2) The inputs are determined as D',D',D,D,0,D,0,D based on the table below:
7
CST 131 Assignment 1
0
1
D
0
0
0
1
1
1
1
0
D'
Table 2.4 Step 3) The inputs are identified as
D = 4, 6 D'= 1 1 = 0, 2,3,57
Step 4) The diagram of the multiplexer implemented shown. S1 S B A C 5 3 7 D 1 0 6 4 2 S2 0X 8 1 MUX
8
CST 131 Assignment 1
Figure 2.2 the diagram of the multiplexer implemented
3.
Design a half-subtractor circuit with inputs x and y and outputs Diff and Bout. The circuit subtracts the bits x–y and places the difference in Diff and the
X Y Bout Diff
Figure 3.1: Half Subtractor.
X 0 1 0 1
Y 0 0 1 1
DIFF 0 1 1 0
BORROW 0 0 1 0
Table 1: Truth Table
DIFFERENCE = XY + XY BORROW = XY
9
CST 131 Assignment 1
The half subtractor is a combinational circuit. It is used to perform subtraction of two bits. There are two input bits in the half subtractor (X and Y) and two output bits, a difference DIFF = (X-Y) and a Borrow.
10
1. Write a 2-3 pages description on a figure that has greatly contributed to the computing field. Please describe the person’s contribution in detail. Answer :
Charles Babbage (1791-1871), a computer pioneer, designed the first automatic computing engines and widely known as the Father of Computers was born 26 December 1791 in London. He invented computers but failed to build them. The father of Charles was a rich man, so it was possible for Charles to receive instruction from several elite schools and teachers during the course of his elementary education. He was about eight when he had to move to a country school to recover from a dangerous fever. Then, he joined King Edward VI Grammar School in Totnes, South Devon, a thriving comprehensive school that's still operative today, but his fragile health status forced him back to private teaching for a period. He finally joined a 30 student closed number academy managed by Reverend Stephen Freeman. The academy had a big library, where Babbage used to study mathematics by himself, and learned to love it. Babbage arrived at Trinity College, Cambridge in October 1810. He had a big culture because he knew Lagrange, Leibniz, Lacroix, Simpson and he was seriously disappointed about the math programs available at Cambridge. So he with J.Herschel, G.Peacock, and other friends, decided to form the Analytical Society.
When, in 1812, Babbage transferred to Peterhouse, Cambridge where he was the best mathematician; but he failed to graduate with honours. He received an honorary degree later,
1
CST 131 Assignment 1
without even being examinated, in 1814. Babbage sought a method by which mathematical tables could be calculated mechanically, removing the high rate of human error. Three different factors seem to have influenced him: a dislike of untidiness; his experience working on logarithmic tables; and existing work on calculating machines carried out by Wilhelm Schickard, Blaise Pascal, and Gottfried Leibniz. He first discussed the principles of a calculating engine in a letter to Sir Humphry Davy in 1822.
Figure 1.1: Difference Engine Babbage's machines were among the first mechanical computers, although they were not actually completed, largely because of funding problems and personality issues. He directed the building of some steam powered machines that achieved some success, suggesting that calculations could be mechanized. Although Babbage's machines were mechanical and unwieldy, their basic architecture was very similar to a modern computer. The data and program memory were separated, operation was instruction based, the control unit could make conditional jumps and the machine had a separate
2
CST 131 Assignment 1
Figure 1.2 Analytical Engine Design of computers Babbage was inspired by Napier's logarithm tables and Napier's logs and bones. He began to design a difference engine in 1821 which was a very large and complicated machine intended for doing logarithmic calculations automatically. The device was based on the principle that the difference between certain values of the expression at a certain stage becomes constant. But it was difficult to make the machine parts accurate enough to prevent errors to occur, using the technology available at that time. Encouraged by the work of difference engine, he began design of another machine the analytical machine, which could carry out many different types of calculations. Analytical engine used the concepts of Automatic Loom and Difference Engine.
The Analytical machine had five units- input, output, store, mill, and control. Store was used for storing numbers and Mill was used to do the calculations by rotation of gears and wheels. Control unit did the job of supervision of all other units. Working with him was a mathematician Lady Ada Lovelace, who provided the input material in the form of programs. Thus, Lady Ada Lovelace became known to the world as the first computer programmer. She coded these programs on punch cards. Her contribution was so valuable that a language called "Ada" was named after her.
3
CST 131 Assignment 1
The two concepts implemented in this machine, which revolutionized the Computer era in the following years are
1. The machine was able to compare quantities and then decide which of the sequences to follow 2. The results of a calculation could alter the numbers and instructions already stored in the machine.
Babbage devoted his last 37 years for the development of the analytical engine. His machine was too ahead of the time of development. Lack of better technology leaded to incomplete work of Babbage that time and he passed away in 1871 before he had completed his work.
4
CST 131 Assignment 1
2. Implement the following Boolean function with a multiplexer. (a) F (A, B, C, D) = ∑ (0, 2, 5, 7, 11, 14) Answer: Step 1) A truth table is drawn for 8x1 multiplexer F (A, B, C, D) = ∑(0, 2, 5, 7, 11, 14)
A
B
C
D
F
0
0
0
0
1
0
0
0
1
0
0
0
1
0
1
0
0
1
1
0
0
1
0
0
0
0
1
0
1
1
0
1
1
0
0
0
1
1
1
1
1
0
0
0
0
1
0
0
1
0
1
0
1
0
0
1
0
1
1
1
1
1
0
0
0
1
1
0
1
0
1
1
1
0
1
1
1
1
1
0
Table 2.1 Truth table
Step2) The inputs are determined as D',D',D,D,0,D,0,D based on the table below:
5
CST 131 Assignment 1
0
1
D
0
0
0
1
1
1
1
0
D'
Table 2.2 Step 3) The inputs are identified as
D' = 0, 1, 7 D = 2, 3, 5 0 = 4, 6
Step 4) The diagram of the multiplexer implemented shown. S1 S B A C 5 3 1 7 D 0 6 4 2 S2 0X 8 1 MUX
Figure 2.1 the diagram of the multiplexer implemented
6
CST 131 Assignment 1
2. Implement the following Boolean function with a multiplexer.
(b) F (A, B, C, D) = П (3, 8, 12)
Answer: Step 1) A truth table is drawn for 8x1 multiplexer F (A, B, C, D) = П(3,8,12)=F (A,B,C,D)=(0,1,2,4,5,6,7,9,10,11,13,14,15)
A
B
C
D
F
0
0
0
0
1
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
0
1
0
0
1
0
1
0
1
1
0
1
1
0
1
0
1
1
1
1
1
0
0
0
0
1
0
0
1
1
1
0
1
0
1
1
0
1
1
1
1
1
0
0
0
1
1
0
1
1
1
1
1
0
1
1
1
1
1
1
Table 2.3 truth table
Step2) The inputs are determined as D',D',D,D,0,D,0,D based on the table below:
7
CST 131 Assignment 1
0
1
D
0
0
0
1
1
1
1
0
D'
Table 2.4 Step 3) The inputs are identified as
D = 4, 6 D'= 1 1 = 0, 2,3,57
Step 4) The diagram of the multiplexer implemented shown. S1 S B A C 5 3 7 D 1 0 6 4 2 S2 0X 8 1 MUX
8
CST 131 Assignment 1
Figure 2.2 the diagram of the multiplexer implemented
3.
Design a half-subtractor circuit with inputs x and y and outputs Diff and Bout. The circuit subtracts the bits x–y and places the difference in Diff and the
X Y Bout Diff
Figure 3.1: Half Subtractor.
X 0 1 0 1
Y 0 0 1 1
DIFF 0 1 1 0
BORROW 0 0 1 0
Table 1: Truth Table
DIFFERENCE = XY + XY BORROW = XY
9
CST 131 Assignment 1
The half subtractor is a combinational circuit. It is used to perform subtraction of two bits. There are two input bits in the half subtractor (X and Y) and two output bits, a difference DIFF = (X-Y) and a Borrow.
10
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