Consider a 16-bit binary floating point number representation system: + SE E EEE E m m m The first bit of the exponent is dedicated to its sign. Assume that the mantissa must start with a '1'. this system to answer the following questions: 1. What is the smallest (magnitude) number that can be represented with this system? 2. Calculate the relative round-off error introduced when this system tries to represent e = 2.718281828459045235360287

Consider a 16-bit binary floating point number representation system: + SE E EEE E m m m The first bit of the exponent is dedicated to its sign. Assume that the mantissa must start with a '1'. this system to answer the following questions: 1. What is the smallest (magnitude) number that can be represented with this system? 2. Calculate the relative round-off error introduced when this system tries to represent e = 2.718281828459045235360287

Systems Architecture

7th Edition

ISBN:9781305080195

Author:Stephen D. Burd

Publisher:Stephen D. Burd

Chapter8: Data And Network Communication Technology

Section: Chapter Questions

Problem 31VE

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Consider a hypothetical 8 bit floating point machine representation with a sign bit, a 3 bit exponent, and a 4 bit mantissa (se1e2e3b1b2b3b4), where the exponent bias is 3 (add 3 to exponent of number to form machine representation). Recall that actual mantissa has 5 bits, since the leading 1 is not stored on the machine. (a) What is the number ?≈2.718 in this 8-bit format? (b) What is the number that (10100111)2 represents in this 8-bit format? (c) What is the upper bound of the relative error when representing a real number in this 8-bit format?
Consider the following floating point representation where each number has 2exponent bits and 5 fraction bits. The exponent bias is 1. Round each number tohave 3 fraction bits. Apply round-to-even and round-toward-zero modes.a. 1 10 11100b. 0 00 00101c. 1 01 10101i. Can you see a pattern for rounding floating point numbers when thenumber of exponent bits are intact?
Consider a SEC code that protects 8-bit words with 4 parity bits. If we read the value0xABC, is there an error? If so, correct the error.Note: 0xABC = 1010 1011 1100Hint: Decode the 12-bit encoded data and find if there is an error.
a) A fixed-point binary number system has 5 whole bits and 3 fractional bits. What is the range of numbers in binary that can be represented in this format? What is the decimal value of the largest number that can be represented? b) A fixed-point binary number system has 4 whole bits and 12 fractional bits. Show the (4-hexit) hexadecimal representation of the following decimal numbers 0.375, - 1.33, and 1/40 in this finite format. Do not show recurring sequences. c) Assume a 16-bit floating point format designed as follows: 4-bit signed exponent and 12-bit fraction. Normalization is strictly enforced. What is the hexadecimal representation of the 16-bit floating point format for 1/40?
a) A digital sound recorder captures sound from the real-world, processes the audio and stores it in binary form. Using the principles of digital signal processing, practically explain how this phenomenon occurs. b) Use the two’s complement binary representation to represent each of the following integers as 10-bit binary numbers.i) -13710 ii) 14510 iii) -15810 iv) 16710 v) -11510 c) If an 8-bit binary number is used to represent an analog value in the range from 0 to 100kg, what does the binary value 011001002 represent?
In a Digital Logic examination, you are given two signed integers in different number systems. A= (64)16 B= -(103)8Determine their values in the decimal system AN, b) Find the value of n, the shortest word-length of a computer using two’s complement representation which can represent all these numbers. Find the largest and thesmallest numbers that can be represented. c) Give the n-bit two’s complement representations of the two numbers, where n isvalue determined in b.
Suppose A = +54 and B = -3 (both in base 10).Q1: What is the lowest number of rounds theoretically to complete the integer division using the optimized division algorithm?Q2: What is the resulting number in binary representation if we use 8 bits to represent it?Q3: What is the resulting number in FP decimal representation if we use the IEEE 754 standard for single precision? (form of this result should be in sign, true exponent in binary, IEEE-754 Exponent in binary and IEEE-754 exponent in decimal (base 10 number) )
Show Steps Please 20. Suppose we have just found yet another representation for floating point numbers. Using thisrepresentation, a 12-bit floating point number has 1 bit for the sign of the number, 4 bits forthe exponent and 7 bits for the mantissa, which is normalized as in the Simple Model so thatthe first digit to the right of the radix points must be a 1. Numbers in the exponent are in signed2's complement representation. No bias is used and there are no implied bits. Show therepresentation for the smallest positive number this machine can represent using the followingformat (simply fill in the squares provided). What decimal number does this equate to?
Suppose you have a 4-letter alphabet {A, B, C, D} and you wish to devise a binary code (with nospaces) for this alphabet such that no two words would be encoded the same. For example, if A = 1and B = 11, then AA and B would be encoded the same. It takes one second to transmit eachcharacter of the binary code (so, 101 would take 3 seconds and 110101 would take 6 seconds).(a) If each letter occurs with the same frequency, determine the most efficient code for the alphabetand explain why it is the most efficient.(b) Now suppose letter A occurs about 40% of the time, B and C both occur about 25% of thetime each, and C occurs about 10% of the time. Determine the most efficient code and explainwhy it is the most efficient.
Please explain step by step:Let e = 2.71828183 be the Euler’s number with 8 decimal digits of precision.Find the binary scientific notation of e with five bits after the binary point; i.e. find integer n and bits x1, x2, . . . , x5 such that e = 1.x1x2x3x4x5 × 2 n .Find the single-precision IEEE 754 representation of e. (Hint: Round the result of 223e to the closest integer like m. Convert m to its normalized binary scientific notation and round its fraction to 23 bits after the binary point. Finally, find the normalized binary scientific notation of m/2 23 by subtracting 23 from the power of two in the representation of m).
If an unsigned 8-bit integer were to be extended to a 16-bit one, the first 8 most significant bits (MSB) would be filled with ones, creating a signed 16-bit representation. Pick a solution from these two options: Is it true or false that...?
a) Determine the 32-bit binary representation for floating point numbers in the form± I.F x be where I is the interger part,F is the fraction,b is the base, ande is the exponent.Using this format, represent the number -1.00011011101 x 225 as a 32 bit floating point number. b) During the process of digital sampling, data can be lost due to aliasing. Analyse this problem and provide at least two possible solutions. Also, determine the rate at which a signal should be sampled, if the highest frequency is 60MHz. c) In your own words, discuss the relationship between the hexadicimal number system and the binary number system. Use this relationship to convert the following hexadecimal values to binary.i) 5BC916 ii) 6EF816 iii) 8A4D16 iv) 9ED516