1) Number Conversion, Complement Systems (20 points)

a) Convert 93.87510 to binary, octal and hexadecimal. Show all steps.  (3 points)

b) Obtain the binary 1’s and 2’s complement representation of the following numbers: -2910  and +4110 . Represent your answer using the least number of bits possible.  Show all work.  (4 points)

c) What is the range of values that can be represented in an 8-bit 2’s complement system?  Express your answer in both decimal and binary forms.   (3 points)

d)      Represent the decimal numbers +14, -33, -49 in a signed-magnitude system.  For each, specify how many bits are needed.   (6 points)

e)       Which of the number system schemas is most practical for arithmetic?  Explain and give at least two benefits of using this schema.  (4 points)

2) Arithmetic and Boolean Algebra (20 points)

Through parts a-c, assume A = 2316, B = 4916, and C = 7616 .

a)           Represent A, B, C, -A, -B, and -C in binary 2’s complement using the fewest number of bits possible.  (6 points)

b)           Perform the following computations: C – A, B + C, and -B - C.  In each case, specify whether overflow occurs.  Show all work.  (4 points)

c)           Explain the two ways to check for overflow in a 2’s complement system.  Can overflow occur when adding two numbers of different signs?  (2 points)

d)            Given the following circuits, find the equation for F and use boolean algebra to write it in simplest form.  (4 points)

Circuit 1

Circuit 2

e)           Given the Boolean expression of F = A XOR B:

i)     Represent the above expression using only NAND gates and draw the equivalent NAND circuit.  (2 points)

ii)    Represent the above expression using only NOR gates and draw the equivalent NOR circuit.  (2 points)

3)  Consider the function F = ΣW,X,Y,Z  (1, 3, 5, 6, 7, 8, 12, 14) (25 points)

a) Determine the truth table based on the minterm expression above then draw the K-map and find the minimal sum-of-products expression.  (5 points)

b) Find the product-of-sums expression (using any method of choice).  (5 points)

c) Draw the logic circuit implementing the SOP expression found in part a.  (3 points)

d) Find the input pair(s) where a transition between the two would produce a static-1 hazard (using part a).  (5 points)

e) Assuming all gate delays are equal, draw the approximate timing diagrams that identify the glitch from either input pairs.  (5 points)

f)  Provide the expression of an equivalent logic function in which the timing hazard is eliminated.  (2 points)

4) Multiplexers, Functional Blocks, Floating Point (20 points)

Consider the function F (A, B, C, D) given as a maxterm list (M0, M1, M2, M3, M6, M7, M8, M9, M12, M14, M15).  As a reminder, the maxterms correspond to the binary inputs ABCD where the function  equals 0.

a) Implement this function with a 16:1 multiplexer.  (4 points)

b) Implement this function with a 4:1 multiplexer, and any additional gates necessary.  Show all steps.  (4

points)

c) An 8:1 multiplexer has inputs A, B, and C connected to selection bits S2 (MSB), S1, and S0,                     respectively.  Data input bits 0-7 are connected in the following order from input bit0 to input bit7: D’, 0, 0, 1, D’, 1, D, 0.  Determine the truth table of ABCD that this multiplexer implements.  (6 points)

d) The table below represents a priority encoder.  What is the order of priority in the following inputs? Explain your reasoning. (3 points)

e) The number below is represented in IEEE-754 Floating point format. What do each of these three fields represent (no need to computer the number)?  Explain the tradeoff in floating point between precision and range.  (3 points)

5) Design problem (15 points).

Grey code is a numbering system based on two consecutive values differing by just 1-bit.  In practice,    this scheme has the advantage of preventing false outputs due to simultaneous bit changes and is seen in applications such as electromechanical switches, optical sensors, and cable TV.  For example, a 3-bit grey code number g2g1g0 would correspond to the binary number b2b1b0 in the table below:

a)   Write the product-of-sum expressions for G2, G1, and G0.  No need to simplify! (6 points)

b)   Derive the minimal sum-of-product expressions for each bit using K-maps.  (5 points)