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2 Short-ish Answer (2 points each unless otherwise noted) 1. What is the frequency of the periodic signal in the figure shown to the right? Be sure to include your units! (Note: ms means milliseconds = 10-3 seconds) 16 ms 4 ms frequency = = period 16 ms + 4 ms = = 50 Hz 20 ms You could have left your answer as 1/( ) Hertz if you wanted to. I just needed to see the units of Hertz to be sure that you knew what frequency was. 2. What is the duty cycle of the periodic signal from problem 1? Duty cycle = t h T * 100% = 4 ms 20 ms * 100% = 20% 3. True or False: The frequency of a periodic signal can be calculated from its duty cycle alone. The duty cycle is the ratio of the duration of a logic one to the duration of the period. This ratio removes the units of time from both the numerator and denominator leaving a unit-less ratio and therefore, the frequency cannot be calculated. If the frequency changes, then the period changes. If the period changes, then the duration of the logic one must change. 4. What is the most negative value that can be stored using a 10-bit 2's complement representation? The most negative 10-bit 2's complement value is To figure out its value, convert it to its positive counterpart by doing the bit-flipping thing. This gives us Using the standard unsigned binary method for converting to decimal gives us 2 9 = 512. This means that in 2's complement form represents 2 9 = 512. a.) (2 10 1) b.) (2 9 1) c.) (2 8 1) d.) (2 10 ) e.) (2 9 ) f.) (2 8 ) 5. What is the most positive value that can be stored using a 9-bit unsigned binary representation? The most positive 9-bit unsigned binary value is This is one less than the 10-bit unsigned binary value which equals 2 9. Therefore, equals a.) b.) c.) 2 9 d.) e.) f.) What is the minimum number of bits needed to represent using unsigned binary representation? Since the basic formula for the maximum value of an unsigned binary value is 2 n 1 for n bits, then all we should need to do is substitute a bunch of values for n to figure out the minimum number of bits to represent Starting at n=8, we get: Therefore, 10 bits is the minimum needed. n 2 n a.) 8 b.) 9 c.) 10 d.) 11 e.) 12 f.) 13

3 7. What is the minimum sampling rate needed in order to successfully capture frequencies up to 20,000 Hz in an analog signal? The minimum sampling rate is twice the highest frequency that is supposed to be captured. a.) 10,000 Hz b.) 20,000 Hz c.) 25,000 Hz d.) 30,000 Hz e.) 35,000 Hz f.) 40,000 Hz 8. For each of the following applications, what would be the optimum (best) binary representation, unsigned binary (UB), 2's complement (TC), IEEE 754 Floating Point (FP), or binary coded decimal (BCD)? Identify your answer in the blank to the left of each application. (2 points each) UB or BCD Student ID UB or FP Exam grade (Integers from 0 to 100 possibly added for average) FP Weight of an atom in grams 9. True or False: The 32-bit binary floating-point number is negative. MSB is the sign bit, and since it equals 1, it is a negative number. 10. Write the complete truth table for a 2-input NOR gate (3 points). Remember that a NOR is an OR gate with an inverted output. A B X Divide by 4. Leave your answer in binary. (Hint: There is a shortcut.) (3 points) Remember that a multiplication by 2 can be accomplished with a single left shift while a division by two can be accomplished with a single right shift. We're dividing, so each shift right is equivalent to a single division by 2. Therefore, since 4 = 2 2, you would need to shift right 2 times. We can test this by converting to decimal and then shifting right by 2 bit positions and converting it too to decimal. If the second value equals one fourth the first value, then it must have worked = = = = = It worked! 12. Circle the function that would first be performed in the following expression. A B ( C D + E)

4 Medium-ish Answer (4 points each unless otherwise noted) 13. Convert the floating-point number to its binary exponential format, e.g., x 2-12, (which, by the way, is not even close to the right answer). Sign bit Exponent Fraction = x = x 2 0 = Convert to decimal. (You may leave your answer in expanded form if you wish.) Point = = Draw the circuit exactly as it is represented by the Boolean expression ( A B) + ( A B). A B X 16. Convert to hexadecimal. First, let's create the conversion table between binary and hex. That table is shown to the right. Once the table has been created, divide the number to be converted into nibbles. The result is shown below: Notice that two leading zeros needed to be added. Each of these nibbles corresponds to a pattern from the table to the right. Now it just becomes a 1 to 1 conversion. 123CAB 16 Binary Hexadecimal A B C D E F

5 17. Use any method you wish to prove rule 10: A + A B = A + B. Show all steps. The easiest way is to prove it using a truth table: A B A A B A + A B A + B List two benefits of a digital circuit that uses fewer gates. Lower chip count results in cheaper circuit faster performance lower power consumption higher reliability (fewer things to break) smaller circuit board 19. In the space to the right, create the truth table for the circuit shown below. (5 points) A B C X A B C ^A A B A C X = ^A + A B + A C Write the Boolean expression for the circuit shown in the previous problem. Do not simplify! X = A + A B + A C 21. If an 8-bit binary number is used to represent an analog value in the range from -25 to 250, how large an analog value does a single binary increment represent? In other words, if the binary number is incremented by one, how much change in the analog range is represented? (Leave your answer in the form of a fraction.) A single increment = range/number of increments = (250 ( 25))/(2 8 1) = 275/255 = units/increment You could have left your answer in any of the last three forms.

6 22. Use DeMorgan's Theorem to distribute the inverse of the expression to the individual input terms. Do not simplify! (A B) + C + D (A B) + C + D (A B) C D (A + B) C D First, distribute inverse across OR. This can work across all three inputs, (A B), C, and D. Next, distribute inverse across the A B product. This gives us the final answer. Note the importance of the parenthesis. Without them, the order of precedence would AND the inverses of B, C, and D first which would be wrong. 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.) b.) True (A B) A = A B Answer: (A B) A Apply DeMorgans (A + B) A Apply Associative Law A (A + B) Apply Distributive Law A A + A B Anything AND'd w/inverse = A B Anything OR'd w/0 = itself A B Answer is TRUE A (A + B) + (A C + A C) = A + B Answer: False A (A + B) + (A C + A C) Pull A from last two products A (A + B) + A (C + C) Anything OR'd w/inverse = 1 A (A + B) + A 1 Anything AND'd w/1 = itself A (A + B) + A Apply Distributive Law A A + A B + A Anything AND'd w/inverse = A B + A Anything OR'd w/0 = itself A B + A = A + B Apply Rule 11

7 c.) A B + B C(B + C) = B (A + C) Answer: True A B + B C(B + C) Multiply B C through (B + C) A B + B C B + B C C A B + B B C + B C C A B + B C + B C Apply Associative Law to B C B Anything AND'd w/itself = itself Anything OR'd w/itself = itself A B + B C = B (A + C) Pull out the B 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. (7 points per row) Decimal 2's complement binary Signed magnitude binary Unsigned binary Unsigned BCD X X For the first row, begin by breaking 60 into its power-of-two components. To do this, we successively pull out the largest powers of 2 that we can from 60 beginning with 2 5 = 32. The result we get is: 60 = = Therefore, the unsigned binary representation of 60 has bits 2, 3, 4, and 5 set: = Since 60 is a positive number, the 2's complement and signed magnitude representations are the same. To convert from decimal to BCD, simply take each decimal digit, i.e., '6' and '0', and convert it to the associated BCD nibble which is the same as the hexadecimal nibbles from 0 to 9. Therefore, the BCD value is '6' = 0110 and '0' = 0000 For the second row, since the MSB of the 2's complement value equals 0, then is positive. Since this is a positive value, the unsigned binary and signed magnitude representations are the same.

8 To convert to decimal, add up the powers of two represented by the positions containing ones = = To convert from decimal 98 to BCD, simply take each decimal digit, i.e., '9' and '8', and convert it to the associated BCD nibble which is the same as the hexadecimal nibbles from 0 to 9. Therefore, the BCD value is '9' = 1001 and '8' = 1000 In the last row, we can put an X's in the unsigned binary column and the BCD column since neither of these representations allow for negative values in eight bits. To convert from decimal to any of the other signed binary representation, we must first figure out what the unsigned binary representation of the positive value of +45 is. To do this, we successively pull out the largest powers of 2 that we can from 45 beginning with 2 5 = 32. The result we get is: 45 = = Therefore, the unsigned binary representation of 45 has bits 0, 2, 3, and 5 set: = To convert this to 45 in 2's complement, we can use the short cut which says starting on the right side, copy all digits up to and including the first 1 we come to. After that, invert the remaining bits to the left. The work below shows what happens with the values in red being copied straight down and the values in blue inverted. 45: : Therefore, the 2's complement representation of 45 is To convert to signed magnitude, simply take the original value for +45 and flip the MSB. This gives us the signed magnitude representation of 45 is

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