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EE 231
Exam 1 September 24, 2008
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- You are tasked with designing a digital thermometer, which will display temperatures from -60o^ F to +140oF. (The system displays the temperature to the nearest degree, so you will not need to display fractions of a degree.) The numbers will be stored in a register in signed 2’s complement form. What is the minimum number of bits your register will need to hold numbers from -60 to +140? Explain your reasoning.
In 2’s compliment form, numbers range from − 2 N^ −^1 to 2 N^ −^1 − 1 , where N is the number of bits. For N = 8, the range is -128 to +127. For N = 9 the range is -256 to +255. Thus, you need a 9 bit register.
- Convert the following decimal number to hexadecimal. You only need to keep two digits to the right of the decimal point: (73.28) 10
Quot Rem Int Frac 73 / 16 = 4 R 9 0.28 x 16 = 4 + 0. 4 / 16 = 0 R 4 0.48 x 16 = 7 + 0.
To get the integer part, read the remainder column going up: (73) 10 = (49) 16. To get the fractional part, read the integer column going down: (0.28) 10 = (0.47) 16. (Actually, the last fractional digit should be rounded up, since 0.68 is greater than 1/2.) (73.28) 10 = (49.48) 16.
- For this problem, assume all numbers are held in 8-bit registers in a digital system, and the numbers are represented in 2’s complement form.
(a) Convert (+87) 10 to a 2’s complement 8-bit hex number.
Quot Rem 87 / 16 = 5 R 7 5 / 16 = 0 R 5 (+87) 10 = (57) 16
(b) Convert (−95) 10 to a 2’s complement 8-bit hex number. First, find the hex representation for +95: Quot Rem 95 / 16 = 5 R 15 5 / 16 = 0 R 5
(+95) 10 = (5F ) 16 Next, take the 2’s complement of 5F to make it negative: Ones’ complement of 5F is A0, add 1 to ones’ complement to get two’s complement: A0 + 1 = A1. (−95) 10 = (A1) 16
(c) Use the results from (a) and (b) to perform the following operation using 2’s complements: (+87) 10 − (−95) 10
(57) 16 − (A1) 16 = (57) 16 + (5F ) 16 = (B6) 16
(d) Convert the answer to of Part (c) to its decimal equivalent. Because (B6) 16 is negative, take its 2’s complement, convert that number to decimal, and put a minus sign in front: Ones’ complement of B6 is 49, add 1 to ones’ complement to get two’s complement: 49 + 1 = 4A. Convert 4A to decimal: 4 x 16 + 10 x 1 = 74. (B6) 16 = (−74) 10
- Use Boolean algebra to simplify the following expressions to a minimum number of literals:
(a) (xy + yz′^ + x′z)(x + z) (xy + yz′^ + x′z)(x + z) = xy + xyz′^ + x′z + xyz + yz′z + x′z (xy + yz′^ + x′z)(x + z) = xy + xyz′^ + x′z + xyz + x′z (xy + yz′^ + x′z)(x + z) = xy + xyz′^ + xyz + x′z (xy + yz′^ + x′z)(x + z) = xy(1 + z′^ + z) + x′z (xy + yz′^ + x′z)(x + z) = xy + x′z (xy + yz′^ + x′z)(x + z) = xy + x′z
