ECE2072 - Digital systems - S2 2022 Mid Semester Test
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Mid Semester Test
ECE2072 - Digital systems - S2 2022
Question 1 Not yet answered Marked out of 6.00 |
A logic function F has three inputs A, B and C. F has the following truth table:
A B C F
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 X
1 0 0 1
1 0 1 0
1 1 0 X
1 1 1 X
Express F(A,B,C) as a Sum of Products Canonical Form.
F = Σ m(
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
)+Σ d(
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
)
Express F(A,B,C) as a Product of Sums Canonical Form.
F = Π M(
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
) . Π D(
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
)
Use any minimisation technique to find the minimum Sum of Products for F:
12, 13, 14, 15, none of these
12, 13, 14, 15, none of these
12, 13, 14, 15, none of these
12, 13, 14, 15, none of these
Use standard SoP form: Order your literals within terms alphabetically and list fewer literal terms before longer. No spaces, no brackets and use ' for inversion and no punctuation. Do not use any symbol for AND. Eg AB+B'C+AB'C. Use 0 for unused terms.
How many product terms are there?
F = + + +
Use any minimisation technique to find the minimum Product of Sums for F:
Use standard PoS form: Order your literals within terms alphabetically and list fewer literal terms before longer. No spaces, use + for OR and ' for inversion. No additional brackets should be entered - they are already provided below. Eg (C)(B'+C')(A+B'+C). Use 1 for unused terms.
How many sum terms are there?
F= () () () ()
Question 2 Not yet answered Marked out of 6.00 |
This question refers to the 2's complement representation using 4 bits. Complete the following table:
Decimal representation
-7
2's complement representation (show all 4 bits)
1010
011
Suppose a 2's complement adder with 4 bit operands produces the sum output of 1110 and also indicates that an overflow has
occurred. What would be the correct result using a 5 bit 2's complement representation? (Your answer must use 5 characters which are either 0 or 1 each and no other characters): What is this in decimal (base 10)?
Suppose a 2's complement adder of 4 bit operands produces the sum output of 0110 and also indicates that an overflow has occurred. What would be the correct result using a 5 bit 2's complement representation? (Your answer must use 5 characters which are either 0 or 1 each and no other characters): What is this in decimal (base 10)?
Question 3 Not yet answered Marked out of 6.00 |
H is a logic function with 4 inputs A, B, C, D. We have expressed H as follows:
H = (A+B'+C)(B'+C+D)(A+B+C+D')
Write H in the Product of Sums canonical form:
H(A,B,C,D) = ∏ M(
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, none of these
) . ∏ D(
)0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, none of these
Question 4 Not yet answered Marked out of 6.00 |
The following Verilog HDL code describes the logic function Out. Note that the primitive gates nandand nor are used. These have the first port as the output and the next two are the inputs.
module VerilogToSoPQ(Out,A, B, C, D);
output Out;
input A, B, C, D;
wire y, z;
nor nor1(y, A, B); // y is the output
nand nand1(z,C, D); // z is the output
nand nand2(Out,y, z); //Out is the output
endmodule
Hint: Draw out the gates and wires on a sheet of paper with labels on the wires.
Write Out as a minimum SoP Boolean expression.
Use the standard SoP form: Order your literals within terms alphabetically and list fewer literal terms before longer. No spaces, no brackets and use ' for inversion and no punctuation. Do not use any symbol for AND. Eg AB+B'C+AB'C Enter 0 for unused terms. If you have more than 4 terms, just enter the first 4.
How many terms are there?
Out = + + +
Question 5 Not yet answered Marked out of 8.00 |
Implement the following logic function F that depends on 4 inputs A, B, C and D. You are to use a single 4:1 multiplexer with inputs I0, I1, I2 and I3, select lines S1 and S0. These operate conventionally, so for example S1 S0 = 10 selects the value on I2 to appear on the
output F of the multiplexer. Use no other logic or gates to implement F.
Implement the logic function F = AC+B'C+BC'D
To make this problem simpler, assume that S1 is connected to B.
Enter your answers by completing the connections below, where each answer is just one of 0,1,A, B, C, D. Inversion is not permitted in this question.
I0 =
I1 =
I2 =
I3 =
S1 = B
S0 =
Question 6 Not yet answered Marked out of 6.00 |
Add extra product terms to the following logic function G with inputs A, B, C and D to remove all static hazards for single input changes:
G = B'D'+ABC'+A'BD
Number of extra product terms required:
Each variable within a product term must be in alphabetical order. Order the product terms with fewer literals first. Within each
group of the same number of literals, order them alphabetically with A before A' etc. If there are fewer than 3 product terms fill the remaining ones with 0. If there are more than 3 just list the first 3. Note that G is ordered this way above in the question.
I suggest you write down your extra product terms on paper and then order them.
Enter your ordered list of extra product terms needed to remove hazards here:
Question 7 Not yet answered Marked out of 8.00 |
This question refers to the circuit described by the following Verilog module. Note that the primitive gates xor and not are used. These have the first port as the output and the next port(s) are inputs.
module Qtiming(Out, A,B,C);
output Out;
input A,B,C;
wire x,y,t;
xor xor1(x, A, t); // x is the output
xor xor2(y, B, x); // y is the output
xor xor3(Out, y, C); // Out is the output
not inv(t, Out); // t is the output
endmodule
Hint: Draw a logic circuit on a sheet of paper with gates and labelled wires.
For each combination of A, B, C determine whether the circuit is stable.
ABC=000
ABC=001
ABC=010
ABC=011
ABC=100
ABC=101
ABC=110
ABC=111
Suppose each gate has the same delay of a "gate delay", what is the period of oscillation when the circuit is unstable? delays.
Question 8 Not yet answered Marked out of 8.00 |
A logic function G has 4 inputs A, B, C, D. It is shown here in the PoS canonical form:
G(A,B,C,D) = ΠM(1,4,6,7,12) . ΠD(13,15)
On your own paper, use a K-map to find the minimum Sum of Products. From this identify all of the Essential Prime Implicants. How many essential prime implicants are there?
Use standard ordering of SoP terms: List the essential prime implicants in increasing order of number of literals and then
alphabetical order. Use alphabetical order of variables within each implicant. Any boxes that are left over, enter a 0. If there are more than 5 enter just the first 5.
Do not upload your K-map.
Not yet answered Marked out of 6.00 |
Refer to the following subset of laws of Boolean Algebra.
In the proof below complete the steps that include quoting the appropriate laws above.
Note that commutative and associative laws of AND and OR (6, 6D, 7, 7D) are not listed above and can be used without quoting
them. The variables that can be used are A, B, C, .. Z. The AND operator must not to be written in your answers, so for example AB must be used instead of A • B. Inversion must use ' and OR must use +. Brackets ( ) should only be used for sum terms and inversion where necessary only, eg (A+B)(C+D), and (A(B+C)D)'. Do not include any spaces or extra characters. Quoting unnecessary laws
deducts marks.
AC+AB+B'C
tick the laws needed proceed to the next line :
1 1D 2 2D 3 3D 4 5 5D 6 6D 8 8D
=(B+B')AC+AB+B'C
Now simplify as much as possible by applying these laws only: 1D, 8 and 2 in any order any number of times.
=
Consider another expression now:
(B+C)(B+C')
Apply only the law 8D and no others, reduce the number of literals by 1 (from 4 above to 3 next line):
=
tick which of the following laws are required for the fewest step simplification to the next line
1 1D 2 2D 3 3D 4 5 5D 6 6D 8 8D
= B
2023-09-06