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Unformatted text preview: ELEC 326: Homework 2 Kartik Mohanram Due in DH 3029 by 5pm on Friday, September 25th, 2009 Graded problems You may work together with others from the class to solve the problems on the homework, though each of
you has to submit solutions that are “individually” written up. Solutions that are “near” copies of each
other will not be awarded any credit. All solutions from past offerings are off limits under the honor code. 1. (From a past exam) This question is split into two parts: (a) What is the Boolean function that this circuit implements? What are its disadvantages? hm , '7’ (b) Draw the optimum CMOS network for the Boolean function. Hint: My current solution uses 10
transistors with only one level of logic, so these are good numbers to shoot for. 2. (From a past exam) Use exactly ONE INVERTER and any number of AND and OR gates to imple-
ment the XOR function of three inputs (1, b, and c. Note that you are given the inputs a, b, and c in
uncomplemented form only, i.e., a, b, and E, are not available. 3. Convert the following Boolean functions into CMOS networks. You are given the inputs a, b, and c
in uncomplemented form only, i.e., d, b, and E are not available and have to be generated if necessary.
Try to minimize the number of transistors and levels of logic that your solution requires. (a) f=a+5+a
(c) f=a5+dc 4. Implement the Boolean function shown in the K-map below using as few xor gates as possible. 5. Problem 2.18 from the textbook 6. (From a past exam) Give an example of a Boolean function which has more prime implicants than
minterms. (Draw a picture, showing the primes and minterms, rather than giving a Boolean formula
for the function.) 7. Prove or show a counter-example to each of the following statements: (a) If the pairwise product of all prime implicants of f is 0, then f has a unique minimum expression. (b) If a function f has a unique minimum sum-of-products expression, then f has a unique minimum
product-of-sums expression. 8. (From a past exam) The one—hot code on four inputs a, b, c, and d has exactly four valid input combi-
nations 1000, 0100, 0010, and 0001. (a) Derive the K-map for an error detection circuit for the one-hot code on four inputs, where only
single-bit, unidirectional 0 —> 1 errors can occur. In other words, your implementation must
detect (i.e., output 1) on all erroneous input combinations on a, b, c, and d (i) when a single bit
is in error and (ii) when these single-bit errors are unidirectional, i.e., they cause a 0 to become a
1. For example, 1100 is an erroneous input satisfying the single-bit, unidirectional 0 -—> 1 error condition. Warning: There are don’t cares in the K-map and they must be clearly identiﬁed in
the K-map for full credit. (b) Derive optimal sum—of—products and product—of—sums expressions for your implementation. (c) You are given the inputs 0., b, c, and d in uncomplemented form only, i.e., a, 3, J, and J are not
available and have to be generated if necessary. Draw the optimal transistor-level schematic for
the CMOS implementation. Please refresh your memory of what a CMOS circuit is before you
answer this question. Hint: My current solution uses 18 transistors with two levels of logic. 9. Anonymous course feedback. Please provide this on a separate page for easy access. I expect to
receive as many forms as there are students enrolled, so there is no skipping this exercise. (a) What do you like about the class to date? (b) What do you not like about the class to date?
(c) How prepared are you for the ﬁrst exam? ((1) Other suggestions. 10.
12. 13. Practice problems (not graded) . Draw the optimal (minimum number of transistors) CMOS network for the Boolean function F=(ab+bc+cd+da) Work out example (not problem) 2.10 from the textbook.
Work out example (not problem) 2.11 from the textbook. There are 1024 (hmm...) bottles of wine, of which 1 bottle is adulterated. It takes a day for the symptoms of consuming the adulterated wine to appear, and the only way to identify the adulterated
bottle is to have someone drink from it. If you have exactly one day to identify the adulterated bottle, what is the minimum number of volunteers that you will need to accomplish the task? (Hint: Try to
use binary numbers as identities.) When comparing the performance of gates implemented in different technologies or circuit styles, it is 4 important to not confuse the picture by including parameters such as local factors (i.e., capacitance), 14. fan-in, and fan-out. A uniform way of measuring gate delay so that technologies can be judged on an
equal footing is desirable. The defacto standard circuit for delay measurements is the ring oscillator,
which consists of an odd number of inverters connected in a circular chain. Due to the odd number
of inversions, the circuit does not have a stable operating point and oscillates. Determine the period,
i.e., time between two successive rising or falling transitions at the output of an inverter in the ring
oscillator as a function of the number of inverters n and the gate delay of an inverter A. What is the
frequency of the clock that can be tapped at the output (last inverter) of the ring oscillator? (Challenge problem from a past exam) Recent studies have indicated that a good diet should contain
adequate amount s of proteins (P), vitamins (V), fats (F), and cookies (C). An astronaut has to choose
from a menu of ﬁve different preparations with the following FDA—approved nutritional information labels.
(a) Preparation 1 contains V and P,
(b) Preparation 2 contains V and F,
(c) Preparation 3 contains P and F,
(d) Preparation 4 contains V, and
(e) Preparation 5 contains C.
Can the astronaut have a balanced diet with only two preparations? You can, of course, eye-ball the solution but there is no fun in that. What I would like to see is a Boolean formulation for this problem,
and a rational explanation for arriving at the solution. Hint: Think K—maps and covering minterms. ...
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This note was uploaded on 12/25/2009 for the course ECE 2300 at Cornell University (Engineering School).