Final Exam

Convert the IEEE 754 floating point number to decimal: 1|01111011|10000000000000000000

-0.09375

Using 8-bit signed magnitude arithmetic, find the sum of the two decimal numbers 13 & -19.

10000110 or 6

Generate a CRC syndrome for data 1001 and key 1011

1001110

Convert (111110101)2 to (_______)16

1F5 *maybe

How are sequential circuits different from combinational circuits?

Combinational Circuit is the type of circuit in which output is independent of time and only relies on the input present at that particular instant. On other hand Sequential circuit is the type of circuit where output not only relies on the current input but also depends on the previous output

An AND gate is the same as an OR gate with its inputs complemented_________(T/F)

F

T or F: Hamming codes, used for error detection and correction, are useful for burst errors (where we could reasonably expect multiple adjacent bits to be incorrect); Reed-Solomon coding is more useful for random errors (where one can reasonable expect errors to be rare events).

F

F(A,B,C) = m(0,2,4,5,6)

F = C' + AB'

Create the Kmaps and then simplify for the following function: F(x, y, z) = x'y'z' + x'yz' + xy'z' + xyz'

F = z'

Prove using truth table that yz + xyz' + x'y'z = xy + x'z

NEED TO GET ANSWER

Represent -0.75 in IEEE 754 single and double precision floating point representation.

NEED TO GET ANSWER

Assuming 2 complement 8 bit representation, consider the following addition: +70 = 01000110 +80 = 01010000 ----------------- 10010110 Is this correct? Why or why not?

No

How many bits are required to address a 1M * 8 main memory if a) Main memory is byte addressable? b) Main memory is word addressable?

20 20

How many bits are required to address a 4M * 16 main memory if a) Main memory is byte addressable? b) Main memory is word addressable?

23 22

Find the sum of 48 & -19 using 8-bit 1's complement.

29 or 00011101

What is the greatest magnitude negative number one can represent in 14-bit 2's complement code

2^13 - 1 = 8191

What is the largest positive number one can represent in 14-bit 2's complement code.

2^13 - 1 = 8191

Compute the Hamming distance of the following code: 0011010010111100 0000011110001111 0010010110101101 0001011010011110

4

Generate the HC for the data 01001011 a. using even parity check b. receiver receives 010111010110 bits of HC Check if this is correct using an even parity check.

a. Even: 010011010110 and odd: 010001011101 b. Even: 010011010110 (9th bit error) and odd: 010111110110 (6th bit error)

Check if the received info^n is correct using the key 1011 a. Received info^n = 1001110 b. Receive info^n = 100110

a. No error, R = 0 b. Error, R = 11

Info^n 1111101 has to be sent on a data link. The sender and receiver share a polynomial: x^3 + x^2 + 1 a. Generate the syndrome using CRC b. Show the actual message to be transmitted

a. R = 111 b. 1111101111

Given the truth table: a. Find the SOP b. Reduce using K map c. Draw the logic diagram for the reduced F d. NAND imp^n

a. SOP: m(4,5,12,13) b. - d. See 11.3 notes

Simplify the following functional expressions using Boolean algebra and its identities. List the identity used at each step. a)(ab + c + df)ef b) x + xy c) (xy'+ x'z)(wx'+ yz')

a. abef + cef + def b. x c. x'wz

Given the following table of 5 numeric values (listed both in hexadecimal and 8-bit binary representation). Answer the questions below, using the ID to identify the number you wish to specify as the correct answer. Num1 FF 1111 1111 Num2 07 0000 0111 Num3 F8 1111 1000 Num4 87 1000 0111 Num5 80 1000 0000 a. Which of the above 5 numbers is the largest 8-bit unsigned binary number b. Which of the above 5 numbers is equivalent to the same base 10 number when it is interpreted by a computer using signed-magnitude representation and a computer using 2's complement representation? c. Suppose Num 3 is interpreted as a 1's complement numbers. Which of the following numbers corresponds to the signed-magnitude representations of the same number? d. Which of the above 5 numbers is the largest 8-bit 2's complement? e. Which of the above 5 number is the smallest 8-bit 2's complement number? f. Which of the above 5 numbers is the smallest 8-bit signed magnitude number?

a. num 1 b. num 2 c. 1's complement = -7 and 2's complement = +8 d. num 2 e. num 5 f. num 1

Given the Boolean function F(x,y,z)=x'y + xyz' a)Derive an algebraic expression for the complement of F. Express in sum-of-products form. b)Show that FF' = 0. c)Show that F + F' =1.

a. xy' + xz + x'y' + y' + y'z b. 0 c. see photo

Using the CRC polynomial 1011, compute the CRC code word for the information word, 1011001. Check the division performed at the receiver.

codeword = 1011001011

If the receiver receives 100101111011 check for error using an even parity check

error on 6th bit, correct code: 100101011011

Find the 2's complement of 10011110.

01100010

Convert from binary to hex: a. 1100101000111111 b. 111100100011000.001111110101

a. CA3F b. F918.3F5

Convert from octal to decimal: a. 761 b. 1276.445

a. 497 b. 702.5722625

Express the negative value -22 as a 2's complement integer, using eight bits. Repeat it for 16 bits and 32bits. What does this illustrate with respect to the properties of sign extension as they pertain to 2's complement representation?

-22 = 11101010 -22 = 11111111 11101010 -22 = 11111111 11111111 11111111 11101010

Data: 10011010, even parity, generate the hamming code

100101011011

Suppose we want an error-correcting code that will allow all single-bit errors to be corrected for memory words of length 10. a)How many parity bits are necessary? b)Assuming we are using the Hamming algorithm presented in this chapter to design our error-correcting code, find the code word to represent the 10-bit information word: 1001100110.

10011010111000

Convert the hexadecimal number 𝐴𝐶12 to binary.

1010 1100 0001 0010

How many 256 * 8 RAM chips are needed to provide a memory capacity of 4096 bytes? a) How many bits will each address contain? b) How many lines must go to each chip? c) How many lines must be decoded for the chip select inputs? Specify the size of the decoder.

12 (4096 = 2^12) 8 (256 = 2^8) We have 16 chips so we need 4 lines to select the chip.

Using 8-bit signed magnitude arithmetic, find the sum of the decimal numbers 75 & 46. Verify your answer with the decimal equivalent.

121

Suppose that a 16M * 16 main memory is built using 512K * 8 RAM chips and memory is word addressable. a) How many RAM chips are necessary? b) How many RAM chips are there per memory word? c) How many address bits are needed for each RAM chip? d) How many banks will this memory have? e) How many address bits are needed for all of memory? f) If high-order interleaving is used, where would address 14 (which is E in hex) be located? g) Repeat this problem for low-order interleaving.

64 (32 rows of 2 columns) 2 512K = 2^19, so 19 bits 32 16M = 2^24, so 24 bits Bank 0 (000) Bank 14 if counting from 0, bank 15 if counting from 1

Simplify A'(A + B) + (B + AA)(A + B')

A + B

The truth table for Boolean expression is shown below. Write the Boolean expression in sum-of-products form.

F = x'y'z' + x'yz + xy'z + xyz'

Find the complement of the function F F = xy + xy'z + x'yz

F' = (x' + y') * (x' + y' + z') * (x + y' + z')

Assume you have the following truth tables for functions F1(w, x, y, z) and F2(w, x, y, z): a)Express F1 and F2 in sum-of-products form. b)Simplify each function c)Draw one logic circuit to implement the above two functions.

Last row: 1 1 1 1 1 1 a. F1 = w'x'y'z + w'x'yz + wx'y'z + wx'yz + wxy'z + wxyz' + wxyz F2 = w'x'y'z' + w'x'yz + wxy'z' + wxy'z + wxyz' + wxyz b. F1 = wx + x'z F2 = wx + w'x'z

Why is it important for Boolean Expressions to be minimized in the design of digital circuits?

Minimization is important since it reduces the cost and complexity of the associated circuit. It is clear from the above image that the minimized version of the expression takes a less number of logic gates and also reduces the complexity of the circuit substantially.

Design a combinational circuit with three inputs and one output. The output is '1' whenever the decimal value of the inputs is an even number. The output is '0' otherwise. Show truth table, SOP equation, Kmap, reduced SOP expression, and logic diagram

NEED TO GET ANSWER

Suppose we want an error correcting code that will allow all single-bit errors to be corrected for data of length 11. How many check bits are necessary? Suppose we are now working with data of length 8. The length of all code words will be 12. We now receive the following code word: 010111101110 Is this legal code word, assuming odd parity? If not, where is the error?

NEED TO GET ANSWER

Check if the received CRC code 1000110 is legal with key 1011

Not a legal key

Find the quotient and reminders for the following modulo 2 division: (111010111)2 / (10111)2

Q = 11001 R = 1000

Check for error at the receiver, assuming that incoming message is: 1111100111

R != 0, error during transmission

Design a 1-bit magnitude comparator. Show truth table, SOP equation, Kmap, reduced SOP expression and logic diagram.

SEE BOOK MARKED ANSWER

Implement the Boolean function F(A,B,C,D) = {1,3,4,11,12,13,14,15} using a 4X1 MUX. Show Kmap, equation for multiplexer inputs and block diagram.

SEE BOOK MARKED ANSWER

Given the Block diagram and truth table of: a. Flip Flop b. T Flip Flop c. JK Flip Flop

SEE NOTES FOR ANSWER

Given a B. function, draw the logic diagram and NAND imp F = (AB + A'B')(CD' + C'D)

See 10.27 notes for answer

Construct a truth table to evaluate the following: a)xyz + (xyz)' b)x(yz' + x'y)

See HW solution 3 for answer

T or F: A 2's complement integer representation includes more negative numbers than it does positive ones.

T

T or F: The largest value that a 60-bit unsigned binary integer can represent is (2^60 - 1)

T

What do we call a circuit that takes several inputs and their respective values to select one specific output line (n*1)? Name one important application for these devices.

The circuit is called multiplexer. A multiplexer is a circuit that accepts several input lines but give only one output line. Example of multiplexer is single pole multiposition switch. Multiplexers are used for various applications such as communication system, telephone network, computer memory. The efficiency can be increased by making use of multiplexer.

Design a circuit with three inputs (x, y, and z) representing the bits in a binary number, and three outputs (a, b, and c) also representing bits in a binary number. When the input is 1, 2, or 3, the binary output should be one lesser than the input. When the input is 4, 5, or 6, the binary output should be one greater than the input. When the input is 0, the output is 0, and when the input is 7, the output is 7. Show your truth table, all computations for simplification, and the final circuit.

a = x b = yz + xz + xy c = yz' + xy + xz'

Convert the following 2's complement binary numbers to decimal a)0110 b)1101 c)01101111 d)1101101100011100 e)1001110000000101

a. 6 b. -3 c. 111 d. -9444 e. -25595

Convert from binary to octal: a. 110010 b. 11111.01101

a. 62 b. 37.32

Convert from decimal to octal: a. 456 b. 117.16

a. 710 b. 165.12172

Convert from octal to hex: a. 136572 b. 177630.255254

a. BD7A b. FF98.56AB

Represent the following in 8-bit signed magnitude a)-15 b)-3 c)+5 d)+7

a. +15: 00001111 -15: 1000111 b. +3: 00000011 -3: 10000011 c. +5: 00000101 d. +7: 00000111

Convert the following to decimal: a. 1011110111000.....0 b. 010000101101001101000.....0

a. 0.09375 b. 105.625

The following binary numbers are 4-bit 2's complement binary numbers. Which of the following operations generate overflow? Justify your answers by translating the operands and results into decimal. a)0011+1100 b)0111+1111 c)1110+1000 d)0110+0010

a. 0001 = 1 b. 110 = 6 c. Overflow d. Overflow

Using arithmetic shifting, perform the following: a)double the value 000101012 b)quadruple the value 011101112 c)divide the value 110010102in half

a. 00101010 b. error c. 01100101

Represent the following decimal numbers in binary using 8-bit signed magnitude, one's complement, and two's complement representations: a) 60 b) -60 c) 20 d) -20

a. 00111100, 00111100, 00111100 b. 10111100, 11000011, 11000100 c. 00010100, 00010100, 00010100 d. 10010100, 11101011, 11101100

Represent the following decimal numbers in binary using 8-bit signed magnitude, one's complement, and two's complement: a) 77 b) -42 c) 119 d) -107

a. 01001101, 01001101, 01001101 b. 10101010, 11010101, 11010110 c. 01110111, 01110111, 01110111 d. 11101011, 10010100, 10010101

Show how each of the following floating-point values would be stored using IEEE-754 double precision (be sure to indicate the sign bit, the exponent, and the significand fields): a) 12.5 b) 0.75

a. 0|100000000010|1001000.....0 b. 0|01111111110|1000000.....0

show the representation of -16 assume 8-bit representation using: a. signed-magnitude representation b. signed-1's complement c. signed-2's complement

a. 10010000 b. 11101111 c. NEED TO GET ANSWER

Convert to binary: a. 156 b. 12.34375

a. 10011100 b. 1100.01011

Convert from hex to binary: a. AA23 b. DB23.879A

a. 1010101000100011 b. 1101101100100011 1000011110011010

Convert the following decimal fractions to binary with a maximum of six places to the right of the binary point: a) 25.84375 b) 57.55 c) 80.90625 d) 84.874023

a. 11001.11011 b. 111001.100011 c. 1010000.11101 d. 1010100.110111

Perform the following using shifting in 8-bit representations: a. (50)10 * (4)10 b. (-102)10/(2)10

a. 11001000 b. NEED TO GET ANSWER

Find the quotients and remainders for the following division problems modulo 2. A)10101112÷11012 B)10111112÷111012

a. 1101 R = 110 b. 111 R = 1100

Convert the following decimal fractions to binary with a maximum of six places to the right of the binary point: a) 26.78125 b) 194.03125 c) 298.796875 d) 16.1240234375

a. 11010.11001 b. 11000010.00001 c. 100101010.110011 d. 10000.000111

Convert from octal to binary: a. 654 b. 7654.123

a. 110101100 b. 111110101100.011010011

Perform the following base conversions using subtraction or division-remainder: a) 45810= b) 67710= c) 151810= d) 440110=

a. 121222 b. 10202 c. 4266 d. 6030

Convert to decimal: a. 10010011 b. 11010.11001

a. 147 b. 26.78125

What decimal value does the 8-bit binary number 10011110 have if: a) it is interpreted as an unsigned number? b) It is on a computer using signed-magnitude representation? c) It is on a computer using one's complement representation? d) It is on a computer using two's complement representation?

a. 158 b. -30 c. -97 d. -98

Show how each of the following floating-point values would be stored using IEEE-754 single precision (be sure to indicate the sign bit, the exponent, and the significand fields): a) -1.5 b) 26.625

a. 1|01111111|10000000.....0 b. 0|10000011|1010101.....0

Convert to IEEE 754 precision: a. -176.375 b. 347.625 c. 639.6875

a. 1|10000110|0100000110.....0| b. 0|1000011|010110111010.....0| c. 0|10001000|00111111110110.....0|

Perform the following base conversions using subtractions or division-remainder: a) 58810= b) 225410= c) 65210= d) 310410=

a. 210210 b. 33004 c. 1621 d. 4228

Convert from decimal to hex: a. 766 b. 1982.5001

a. 2FE b. 7BE.80068

Convert from hex to octal: a. 34BC b. ABC3.FF23

a. 32274 b. 125703.776214

Convert from hex to decimal: a. AA12 b. 23FE.AA

a. 43538 b. 9214.6640625

Write a simplified expression for the Boolean function define by the following Kmaps:

wy' + y'z + w'xy + x'yz'

Using DeMorgan's Law, write an expression for the complement of F if F(w,x,y,z) = xyz'(y'z + x)'+ (w'yz+x').

x' + y' + z + ((y'z = x)((w + y' + z')x))

Find the truth table that describes the following circuit:

x: 00001111 y: 00110011 z: 01010101 F: 11001001

Using the basic identities of Boolean algebra, show that: xy + x'z + yz = xy + x'z

xy + x'z