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2 Short-ish Answer (2 points each unless otherwise noted) 1. Calculate the frequency of a periodic pulse train with a period of 1 millisecond (1x10-3 seconds). Frequency = 1 1 Period = 1x10-3 seconds = 1 khz 2. True or False: A signal s frequency can be calculated from its duty cycle alone. The duty cycle is a ratio of the pulse width to the period. Although the period depends on the frequency, it is a ratio, and the frequency is not involved in the final answer. Therefore, the answer is FALSE. 3. A digital signal with a duty cycle of 100%: (select the best answer) a.) has a constant frequency b.) is not possible c.) is a constant logic 0 d.) is a constant logic 1 e.) never stops pulsing f.) none of the above Although a digital signal with a duty cycle implies that the signal has a frequency, a duty cycle of 100% is a constant logic 1. Therefore, the best answer is D. 4. For each of the following binary representations, what is the smallest/lowest value that can be represented using 10 bits. (2 points each) a.) unsigned binary: the lowest value is = 0 10 b.) 2's complement: the lowest value is = (2 (10-1) ) = c.) signed magnitude: the lowest value is = (2 (10-1) 1) = How many bits does each of the following terms represent? (1 point each) a.) nibble 4 b.) byte 8 c.) word 16 or What is the minimum number of bits needed to represent in signed magnitude representation? You can do this a number of ways. First, you could look at the binary value for which is Notice that the most significant 1 is in the 8 th bit position. Since signed magnitude requires a sign bit, then needs to be written Therefore, 9 bits are needed. The other way you could do it is to use the formula to calculate the maximum value of a signed magnitude number and apply it to different numbers of bits. 5 bits = = 15 6 bits = = 31 7 bits = = 63 8 bits = = bits = = 255 This shows that 9 bits will do the trick.

3 7. True or False: The number is a valid BCD number. First, let's create the conversion table between binary, hex, and BCD noting that the last 6 bit patterns do not have BCD equivalents. Binary Hexadecimal BCD A B C D E F Next, divide the bit pattern from the problem into nibbles so we can see what the conversion to BCD would be Each of these nibbles has a corresponding BCD value, so the bit pattern is a valid BCD number. Therefore the answer is TRUE. 8. True or False: The 8-bit value has the same value in both signed magnitude and 2's complement form. The binary representation for a positive number is the same for all binary representations. Therefore the answer is TRUE. 9. Write the complete truth table for a 2-input NAND gate. The following two questions are based on the 8-bit binary addition shown below Carry out A B X True or False: If the addition above is considered 8-bit 2 s complement, an overflow has occurred. An overflow in 2's complement is indicated when two numbers with the same sign are added and result has a different sign. This is not the case with the above example, so the answer is FALSE.

4 11. True or False: If the addition above is considered 8-bit unsigned, an overflow has occurred. An overflow in unsigned binary is indicated when a carry occurs out of the MSB. This has happened in the above example, so the answer is TRUE. 12. True or false: There is an algorithm that allows us to add BCD numbers without converting. Yes, two BCD numbers can be added. If the addition of two nibbles results in either an illegal BCD value or a carry, adding six (the nibble 0110) to the result will fix the problem. Therefore, the answer is TRUE. 13. How many possible combinations of ones and zeros do 5 boolean variables have? a.) 15 b.) 64 c.) 32 d.) 16 e.) 31 f.) None of the above The number of possible combinations of ones and zeros with 5 bits is 2 5 = 32, answer C. 14. How many positions must the number be shifted left in order to multiply it by 16? Each shift left is equivalent to a single multiplication by 2. Therefore, since 16 = 2 4, you would need to shift left 4 times. We can test this by converting to decimal and then shifting it left by 4 bit positions and converting that result to decimal. If the second value equals 16 times the first value, then it must have worked = = = = = It worked! Medium-ish Answer (5 points each) 15. Convert the floating-point number to its binary exponential format, e.g., x 2-12, (which, by the way, is not the answer). Sign bit Exponent Fraction = x = x Convert to decimal. Point = = 5.75

5 17. Draw the circuit exactly as it is represented by the Boolean expression A (C + B C). A B C X 18. In the space to the right, create the truth table for the circuit shown below. A B C A B X A B + C A B + C A B C A B A B + C A B + C Convert to hexadecimal. Break the binary number into nibbles starting at the right side of the number, then use the table from problem 7 to convert the nibbles to hexadecimal. Notice that three leading zeros had to be added to make the first nibble. The answer is 135A4D A 4 D 20. List two benefits of a digital circuit that uses fewer gates. The benefits discussed in class included cheaper, faster, smaller circuit board, runs cooler, and more reliable. 21. If an 8-bit binary number is used to represent an analog value in the range from 25 o to 115 o, what does the binary value represent? (Leave your answer in the form of a fraction.) Digital value = = Minimum value = -25 o Maximum value = 115 o Range = Maximum value Minimum value = 115 ( 25) = 140 Number of increments = = 255 (max min) ( ) (2 n 1) binary value + min = ( = 29.9o 1)

6 22. What is the duty cycle of the signal shown to the right? Duty cycle = t h T * 100% = * 100% = 67% 4 seconds 2 seconds Longer Answers (Points vary per problem) 23. Mark each boolean expression as true or false depending on whether the right and left sides of the equal sign are equivalent. Show all of your work to receive partial credit for incorrect answers. (3 points each) a.) A B B = A Answer: False A (B B) (Using Associative Law) A 0 (Using Rule 8) 0 (Using Rule 3) b.) ABC + A + B + C = 1 Answer: True A B C + A B C (Using DeMorgan's Theorem) 1 (Using Rule 6) c.) (A + B)(B + A) = B Answer: True A B + A A + B B + B A (Using "F-O-I-L") A B B B + B A (Using Rule 8) A B + B B + B A (Using Rule 1) A B + B + B A (Using Rule 7) B + A B + B A (Using Commutative Law) B + B (A + A) (Using Distributive Law) B + B 1 (Using Rule 6) B + B (Using Rule 4) B (Using Rule 5)

7 24. Fill in the blank cells of the table below with the correct numeric format. For cells representing binary values, only 8-bit values are allowed! If a value for a cell is invalid or cannot be represented in that format, write "X". Use your scrap paper to do your work. (2 points per cell) Decimal 2's complement binary Signed magnitude binary Unsigned binary X (no negatives) 150 X (out of range) X (out of range)

Points missed: Student's Name: Total score: /100 points East Tennessee State University Department of Computer and Information Sciences CSCI 2150 (Tarnoff) Computer Organization TEST 1 for Spring Semester,

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