ELEC1041 - Combinational Logic Examples - 1 Digital Electronics Tutorial: Combinational Logic Design Examples Solutions.

ELEC1041 - Combinational Logic Examples - 1

Digital Electronics

Tutorial: Combinational Logic Design Examples

Solutions

Problem #1

Develop a minimized Boolean implementation of a “ones count” circuit that works as follows. The subsystem has four binary inputs A, B, C, D, and generates a 3-bit output, XYZ. XYZ is 000 if none of the inputs are 1, 001 if one input is 1, 010 if two are one, 011 if three inputs are 1, and 100 if all four inputs are 1.

(a) Draw the truth tables for XYZ (A, B, C, D).

(b) Minimize the functions X, Y, Z using 4-variable K-maps. Write down the Boolean expressions for the minimized Sum of Products form of each function.

(c) Repeat the minimization process, this time deriving Product of Sums form.

Problem #1 Solution (1/6)

(a) Draw the truth tables for XYZ (A, B, C, D).

(b) Minimize the functions X, Y, Z using 4-variable K-maps. Write down the Boolean expressions for the minimized Sum of Products form of each function.

(c) Repeat the minimization process, this time deriving Product of Sums form.

Problem #2

Consider a combinational logic subsystem that performs a two-bit addition function. It has two 2-bit inputs A B and C D, and forms the 3-bit sum X Y Z.

(a) Draw the truth tables for XYZ(A,B,C,D).

(b) Minimize the functions using 4-variable K-maps to derive minimized Sum of Products forms.

(c) In your textbook and in class/lab we have introduced the Full Adder circuit. What is the relative performance to compute the resulting sum bits of the 2-bit adder compared to two full adders connected together? (Hint: which has the worst delay in terms of gates to pass through between the inputs and the final outputs, and how many gates is this?).

(a) Draw the truth tables for XYZ(A,B,C,D).

0 001 012 103 11

(b) Minimize the functions using 4-variable K-maps to derive minimized Sum of Products forms.

(c) In your textbook and in class/lab we have introduced the Full Adder circuit. What is the relative performance to compute the resulting sum bits of the 2-bit adder compared to two full adders connected together? (Hint: which has the worst delay in terms of gates to pass through between the inputs and the final outputs, and how many gates is this?).

The above circuit is better because there are only two gate levels. But the adder circuit can be used to build adder circuit of more bits. I can’t imagine doing a K-map to construct a adder circuit of 64 bits using the method above.

Problem #3

Show how to implement the full adder Sum(A, B, Cin) and Carry(A, B, Cin) in terms of:

(a) Two 8:1 multiplexers;

(b) Two 4:1 multiplexers;

(c) If you are limited to 2:1 multiplexers (and inverters) only, how would you use them to implement the full adder and how many 2:1 multiplexers would you need?

Ai Bi Cin Sum Cout0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

(a) Two 8:1 multiplexers (Sum);

Ai Bi Cin Sum0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 11 0 1 01 1 0 01 1 1 1

(a) Two 8:1 multiplexers (Carry);

Ai Bi Cin Cout0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1

(b) Two 4:1 multiplexers (Sum);

Ai Bi Cin Sum0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 11 0 1 01 1 0 01 1 1 1

(b) Two 4:1 multiplexers (Carry);

Ai Bi Cin Cout0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1

(c) 2:1 multiplexers (sum)

Ai Bi Cin Sum0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 11 0 1 01 1 0 01 1 1 1

(c) 2:1 multiplexers (Carry)

Ai Bi Cin Cout0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1

(c) 2:1 multiplexers (Sum) Ver 2 - USE the same carry

Ai Bi Cin Sum0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 11 0 1 01 1 0 01 1 1 1

Problem #4

Design a combinational logic subsystem with three inputs, I3, I2, I1, and two outputs, O1, O0, that behaves as follows. The outputs indicate the highest index of the inputs that is driven high. For example, if I3 is 0, I2 is 1, I1 is 1, then O1, O0 would be 10 (i.e., I2 is the highest input set to 1).

(a) Specify the function by filling out a complete truth table.

(b) Develop the minimized gate-level implementation using the K-map method.

(c) Develop an implementation using two 4:1 multiplexers.

(d) Compare your implementation for (b) and (c). Which is better and under what criterion?

(a) Specify the function by filling out a complete truth table.

I3 I2 I1 O1 O0

0 0 0 0 0

0 0 1 0 1

0 1 x 1 0

1 x x 1 1

(b) Develop the minimized gate-level implementation using the K-map method.

I3 I2 I1 O1 O0

0 0 0 0 0

0 0 1 0 1

0 1 x 1 0

1 x x 1 1

I3 I2 I1 O1 O0

0 0 0 0 0

0 0 1 0 1

0 1 x 1 0

1 x x 1 1

I3 I2 I1 O1 O0

0 0 0 0 0

0 0 1 0 1

0 1 x 1 0

1 x x 1 1

(d) Compare your implementation for (b) and (c). Which is better and under what criterion?

Problem #5

You are to design a simple combinational subsystem to the following specification. The system has the ability to pass its inputs directly to its outputs when a control input, S, is not asserted. It interchanges its inputs when the control input S is asserted. For example, given four inputs A, B, C, D and four outputs W, X, Y, Z, when S=0, WXYZ=ABCD and when S=1, WXYZ=BCDA.

Show how to implement this functionality using building blocks that are restricted to be 2:1 multiplexers and 2:1 demultiplexers. Draw your solution below, using black boxes for the mux/demux blocks.

Problem #5 Solution

Each output is tied to the output of a 2:1 Multiplexer. S is the select signal.

Problem #6

Your task is to design a combinational logic subsystem to decode a hexadecimal digit in the range of 0 through 9, A through F to drive a seven-segment display. The hexadecimal numerals are as follows:

Design a minimized implementation in PLA form. That is, look for common terms among the seven output functions.

BCD to 7–segmentcontrol signal

decoder

c0 c1 c2 c3 c4 c5 c6

A B C D

c5c4 c3

BCD to 7–segmentcontrol signal

decoder

c0 c1 c2 c3 c4 c5 c6

A B C D

c5c4 c3

Problem #6 Solution (3/5)BCDBCDACDBCDABDBCDACDABC

A B C D

F0 F1 F2 F3 F4 F5 F6

ABCABDABCABCABCABDABDACD ACDABC

Problem #6 Solution (4/5)BCDBCDACDBCDABDBCDACDABC

A B C D

F0 F1 F2 F3 F4 F5 F6

ABCABDABCABCABCABDABDACD ACDABC

18 terms

Problem #6 Solution (5/5)ABCDABCDABCDABCDABCDABCDABCDABCD

A B C D

F0 F1 F2 F3 F4 F5 F6

ABCDABCDABCDABCDABCDABCDABCDABCD

16 terms

Problem #7

Calendar subsystem

Determine number of days in a month (to control watch display)used in controlling the display of a wrist-watch LCD screen

inputs: month, leap year flagoutputs: number of days

integer number_of_days ( month, leap_year_flag) {switch (month) {

case 1: return (31);case 2: if (leap_year_flag == 1) then return (29) else return (28);case 3: return (31);case 4: return (30);case 5: return (31);case 6: return (30);case 7: return (31);case 8: return (31);case 9: return (30);case 10: return (31);case 11: return (30);case 12: return (31);default: return (0);

Calendar subsystem

Use software implementation to help understand the problem

leapmonth

28 29 30 31

month leap 28 29 30 310000 – – – – –0001 – 0 0 0 10010 0 1 0 0 00010 1 0 1 0 00011 – 0 0 0 10100 – 0 0 1 00101 – 0 0 0 10110 – 0 0 1 00111 – 0 0 0 11000 – 0 0 0 11001 – 0 0 1 01010 – 0 0 0 11011 – 0 0 1 01100 – 0 0 0 11101 – – – – –111– – – – – –

Formalize the problem Encoding:binary number for month: 4 bits4 wires for 28, 29, 30, and 31

one-hot – only one true at any time

Block diagram:

month leap 28 29 30 310000 – – – – –0001 – 0 0 0 10010 0 1 0 0 00010 1 0 1 0 00011 – 0 0 0 10100 – 0 0 1 00101 – 0 0 0 10110 – 0 0 1 00111 – 0 0 0 11000 – 0 0 0 11001 – 0 0 1 01010 – 0 0 0 11011 – 0 0 1 01100 – 0 0 0 11101 – – – – –111– – – – – –

Choose implementation target

and perform mapping Discrete gates

Can translate to S-o-P or P-o-S

m8’ m4’ m2 m1’ leap’

m8’ m4’ m2 m1’ leap

m8’ m4 m1’ + m8 m1

m8’ m1 + m8 m1’

Problem #8

Leap year flag

Determine value of leap year flag given the yearFor years after 1582 (Gregorian calendar reformation), leap years are all the years divisible by 4, except that years divisible by 100 are not leap years, but years divisible by 400 are leap years.

Leap year flag

Encoding the year:binary – easy for divisible by 4, but difficult for 100 and 400 (not powers

of 2)BCD – easy for 100, but more difficult for 4, what about 400?

Parts:construct a circuit that determines if the year is divisible by 4construct a circuit that determines if the year is divisible by 100construct a circuit that determines if the year is divisible by 400combine the results of the previous three steps to yield the leap year flag

Divisible-by-4 circuit

BCD coded year YM8 YM4 YM2 YM1 – YH8 YH4 YH2 YH1 – YT8 YT4 YT2 YT1 – YO8 YO4

YO2 YO1

Only need to look at low-order two digits of the year all years ending in 00, 04, 08, 12, 16, 20, etc. are divisible by 4 if tens digit is even, then divisible by 4 if ones digit is 0, 4, or 8 if tens digit is odd, then divisible by 4 if the ones digit is 2 or 6.

Translates into the following Boolean expression (where YT1 is the year's tens digit low-order bit, YO8 is the high-order bit of year's ones digit, etc.):

YT1’ (YO8’ YO4’ YO2’ YO1’ + YO8’ YO4 YO2’ YO1’ + YO8 YO4’ YO2’ YO1’ ) + YT1 (YO8’ YO4’ YO2 YO1’ + YO8’ YO4 YO2 YO1’ )

Digits with values of 10 to 15 will never occur, simplify further to yield:

YT1’ YO2’ YO1’ + YT1 YO2 YO1’

Divisible-by-100 and divisible-by-400 circuits

Divisible-by-100 just requires checking that all bits of two low-order digits are all 0:

YT8’ YT4’ YT2’ YT1’ • YO8’ YO4’ YO2’ YO1’

Divisible-by-400 combines the divisible-by-4 (applied to the thousands and hundreds digits) and divisible-by-100 circuits

(YM1’ YH2’ YH1’ + YM1 YH2 YH1’) • (YT8’ YT4’ YT2’ YT1’ • YO8’ YO4’ YO2’ YO1’ )

Combining to determine leap year flag

Label results of previous three circuits: D4, D100, and D400

leap_year_flag = D4 (D100 • D400’ ) ’

= D4 • D100’ + D4 • D400

= D4 • D100’ + D400

Implementation of leap year flag

Problem #9

Consider the following functions, which are five different functions over the inputs A,B,C,D.

(1) F(A,B,C) = m(1,2,6,7)

(2) F(A,B,C,D) = m(0,1,3,9,11,12,14,15)

(3) F’(A,B,C,D) = m(2,4,5,6,7,8,10,13)

(4) F(A,B,C,D) = (A B C + A’ B’) (C + D)

(5) F(A,B,C,D) = (A + B + C) (A + B + C’ + D) (A + B’ + C + D’) (A’ + B’)

a) you were to implement these in a single PLA structure with four inputs, five outputs, and an unlimited number of product terms, how many unique product terms are there in this PLA implementation

b) If you are trying to maximize the number of shared product terms across the five functions, rather than minimizing the literal count for each function independently, how many unique terms do you obtain? Draw the new K-maps with your selection of implicants that minimizes the number of unique terms across all five functions

Consider the following functions, which are five different functions over the inputs A,B,C,D.

(1) F(A,B,C) = m(1,2,6,7)

(2) F(A,B,C,D) = m(0,1,3,9,11,12,14,15)

(3) F’(A,B,C,D) = m(2,4,5,6,7,8,10,13)

(4) F(A,B,C,D) = (A B C + A’ B’) (C + D)

(5) F(A,B,C,D) = (A’ + B + C) (A + B + C’ + D) (A + B’ + C + D’) (A’ + B’)

a) you were to implement these in a single PLA structure with four inputs, five outputs, and an unlimited number of product terms, how many unique product terms are there in this PLA implementation

F(A,B,C)= BC’+AB+A’B’C

F(A,B,C,D)=ABD’+ACD+B’D+A’B’C’

F’(A,B,C,D)=A’B+BC’D+AB’D’+B’CD’

F(A,B,C,D)=A’B’D+A’B’C+ABC

F(A,B,C,D)=A’C’D’+A’B’D+A’BC+AB’C

From Revision Tutorial

17 unique terms

b) If you are trying to maximize the number of shared product terms across the five functions, rather than minimizing the literal count for each function independently, how many unique terms do you obtain? Draw the new K-maps with your selection of implicants that minimizes the number of unique terms across all five functions

1. BC’ + AB + A’B’C => ABD' + ABD + A'BC' + ABC' + A'B'C

2. ABD’ + ACD + B’D + A’B’C’ => ABD' + ACD + AB'D + A'B'D + B'CD + B'C'D + A'B'C'

3. A’B + BC’D + AB’D’ + B’CD’ => A'BC + A'BC' + A'BD + A'BD' + BC'D + AB'D' + B'CD'

4. A’B’D + A’B’C + ABC => A’B’D + A’B’C + ABC

5. A’C’D’+A’B’D+A’BC+AB’C => A’C’D’+A’B’D+A’BC+AB’C

shared terms:

A'B'C is shared between 1 and 4. A’BC’ is shared between 1 and 3.

ABD' is shared between 1 and 2. A’B’D is shared between 2, 4 and 5.

A’BC is shared between 3 and 5.

5 shared terms +[22 – (4 2) – (1 3)] => 11 unique terms across all five functions

NOTE: PLA Minimization is always carried out by CAD Tools

Problem #10

Consider the following Boolean function in Product of Sums form:

F(A,B,C,D) = (A + B’ + D) (A’ + B’ + D) (B’ + C’ + D’) (A’ + C + D) (A’ + C’ + D)

Show how to implement this function with an 8:1 multiplexer, with A,B,C on the control inputs and D, its complement, and the constants 0 and 1 available as data inputs. Show your work:

F(A,B,C,D)

S2 S1 S0

F(A,B,C,D)

S2 S1 S0

01234567

F(A,B,C,D)

S2 S1 S0

11D0DDD)

Problem #11

Consider a variation on the calendar combinational subsystem that works as follows. Given the inputs MONTH (1-12), DAY (1-31), and LEAP_YEAR flag, the subsystem generates the output DAY_OF_YEAR (1-365 or 366). In this problem, you will design the subsystem to the block diagram level only.

(a) One block maps the month into a day offset into the year. Identify the inputs and outputs and their bit widths. Use any formal specification method you wish (e.g., truth tables, ROM contents, equations, hardware description language, etc.) to describe the function of this block.

(b) You may assume any width binary adder you may require. Indicate how the adder is composed with the block of part (a) and any other blocks or inputs to compute the correct output. Be sure to describe how you deal with the LEAP_YEAR input

(a) One block maps the month into a day offset into the year. Identify the inputs and outputs and their bit widths. Use any formal specification method you wish (e.g., truth tables, ROM contents, equations, hardware description language, etc.) to describe the function of this blockl m

d8 d7 d6 d5 d4 d3 d2 d1 d0

x 0 0 0 0 1 0 0 0 0 0 0 0 0 0

x 0 0 0 1 0 0 0 0 0 1 1 1 1 1

0 0 0 0 1 1 0 0 0 1 1 1 0 1 1

1 0 0 0 1 1 0 0 0 1 1 1 1 0 0

0 0 0 1 0 0 0 0 1 0 1 1 0 1 0

1 0 0 1 0 0 0 0 1 0 1 1 0 1 1

Day Offset

(a) One block maps the month into a day offset into the year. Identify the inputs and outputs and their bit widths. Use any formal specification method you wish (e.g., truth tables, ROM contents, equations, hardware description language, etc.) to describe the function of this block

ROM26 x 10

m4m3m2m1m0

d8d7d6d5d4d3d2d1d0

(b) You may assume any width binary adder you may require. Indicate how the adder is composed with the block of part (a) and any other blocks or inputs to compute the correct output. Be sure to describe how you deal with the LEAP_YEAR input

Since leap has been taken care of in part a, you can just add whatever output from that block with day, which will give you the correct day of the year.

d8d7d6d5d4d3d2d1d0

ROM26 x 10

m4m3m2m1m0

d8d7d6d5d4d3d2d1d0

9-bitAdder

day8 – day0

day8day7day6day5day4day3day2day1day0

day8 – day5 are always 0

Problem #12

Design a two-bit comparator with the following inputs and outputs:

Inputs: Numbers N1 and N2 to be compared

N1 = A B

N2 = C D

Outputs: LT, GT, EQ

LT = 1 when AB < CD example (A B = 01) < (C D = 10)

GT = 1 when AB > CD example (A B = 11) > (C D = 10)

EQ = 1 when AB = CD example (A B = 01) = (C D = 01)

we'll need a 4-variable Karnaugh map for each of the 3 output functions

block diagram

LTEQGT

A B < C DA B = C DA B > C D

A B C D LT EQ GT0 0 0 0 0 1 0

0 1 1 0 01 0 1 0 01 1 1 0 0

0 1 0 0 0 0 10 1 0 1 01 0 1 0 01 1 1 0 0

1 0 0 0 0 0 10 1 0 0 11 0 0 1 01 1 1 0 0

1 1 0 0 0 0 10 1 0 0 11 0 0 0 11 1 0 1 0

andtruth table

A' B' D + A' C + B' C D

B C' D' + A C' + A B D'

K-map for EQK-map for LT K-map for GT

= (A xnor C) • (B xnor D)

LT and GT are similar (flip A/C and B/D)

A'B'C'D' + A'BC'D + ABCD + AB'CD’

two alternativeimplementations of EQwith and without XOR

XNOR is implemented with at least 3 simple gates

A B C D

Problem #13

Design a 2X2 bit multiplier:

Inputs: Numbers N1 and N2 to be multiplied

N1 = A1 A0

N2 = B1 B0

Outputs: products: P8, P4, P2, P0

P0 = Product with weighting 20 = 1

Example: 11 A1A0 11 B1B0 11 B0(AA0) 11 B1(AA0) 1001 P8 P4 P2 P0

block diagramand

truth table

4-variable K-mapfor each of the 4output functions

A2 A1 B2 B1 P8 P4 P2 P10 0 0 0 0 0 0 0

0 1 0 0 0 01 0 0 0 0 01 1 0 0 0 0

0 1 0 0 0 0 0 00 1 0 0 0 11 0 0 0 1 01 1 0 0 1 1

1 0 0 0 0 0 0 00 1 0 0 1 01 0 0 1 0 01 1 0 1 1 0

1 1 0 0 0 0 0 00 1 0 0 1 11 0 0 1 1 01 1 1 0 0 1

P1P2P4P8

A1A2B1B2

K-map for P8 K-map for P4

K-map for P2 K-map for P1

B2 P8 = A2A1B2B1

P4= A2B2B1'+ A2A1'B2

P2 = A2'A1B2+ A1B2B1'+ A2B2'B1+ A2A1'B1

P1 = A1B1

Problem #14

Design a BCD increment by 1:

Inputs: BCD Number N

N = I8 I4 I2 I0

Outputs: products: O8, O4, O2, O0

O8 O4 O2 O0 = I8 I4 I2 I0 + 1

Example: 0011 I8 I4 I2 I0 0100 O8 O4 O2 O0 O8 O4 O2 O0

I8 I4 I2 I1 O8 O4 O2 O10 0 0 0 0 0 0 10 0 0 1 0 0 1 00 0 1 0 0 0 1 10 0 1 1 0 1 0 00 1 0 0 0 1 0 10 1 0 1 0 1 1 00 1 1 0 0 1 1 10 1 1 1 1 0 0 01 0 0 0 1 0 0 11 0 0 1 0 0 0 01 0 1 0 X X X X1 0 1 1 X X X X1 1 0 0 X X X X1 1 0 1 X X X X1 1 1 0 X X X X1 1 1 1 X X X Xblock diagram

andtruth table

4-variable K-map for each of the 4 output functions

O1O2O4O8

I1I2I4I8

O8 = I4 I2 I1 + I8 I1'

O4 = I4 I2' + I4 I1' + I4’ I2 I1

O2 = I8’ I2’ I1 + I2 I1'

O1 = I1'

Problem #15

Analyse the behavior of the Circuit below when Input A changes from one logic state to another.

A B C D

F is not always 0pulse 3 gate-delays wide

Problem #15 Solutions (1/1)

pulse shaping circuit (Momentary Changes in outputs) A' xnor A = 0 delays matter in function

0 1 0 1

d d d1 0 1 0

1 0d 3d

FA B C D

D remains high forthree gate delays after

A changes from low to high

D remains low forthree gate delays after

A changes from high to low

Problem #16

Analyse the circuit below for static hazard

Static-1 hazard

static-0 hazard static-1 hazard

Due to a literal and its complement momentarily taking on the same value through different paths with different delays and reconverging we get Static 1 Hazard

1 0 12d 3d

Problem #17

Analyse the pulse shaping circuit below

open switch A B

An Oscillator

initially undefined

close switch

open switch

open switch A B

0 1U 0

U = Undefined

U 02d 3d

Problem #18

Which of the components below cab be used to build an inverter?

1A A’

A xor 1 = A’

0A A’

A xnor 0 = A’

1A A’

A nand 1 = A’

0A A’

A nor 0 = A’

Problem #19

Consider the Equation:

Z= A’ B’ C’ D + A’ B’ C D’ + A’ B C’ D’ + A’ B C D + A B C’ D + A B C D’ + A B’ C’ D’ + A B’ C D

Use Shanon’s decomposition to implement this using 2-1 multiplexers.

Z= A’ B’ C’ D + A’ B’ C D’ + A’ B C’ D’ + A’ B C D + A B C’ D + A B C D’ + A B’ C’ D’ + A B’ C D

Z = A’ [B (C’ D’ + C D) + B’ (C’ D + C D’)] + A [B (C’ D + C D’) + B’ (C’ D’ + C D)]

= A’ [B (C xor D)’ + B’ (C xor D)] + A [B (C xor D) + B’ (C xor D)’]

= A’ [( B xor C xor D)] + A [( B xor C xor D)’]

= A xor B xor C xor D

Z= A xor B xor C xor D

Decompose around A:

Z = A’ (B xor C xor D) + A (B xor C xor D)’

Now Decompose (B xor C xor D) around B:

B xor C xor D = B’ (C xor D) + B (C xor D)’

Z( B xor C xor D)

ZC xor D

Now Decompose (C xor D) around C:

C xor D = C’ D + B D’

ELEC1041 - Combinational Logic Examples - 1 Digital Electronics Tutorial: Combinational Logic Design Examples Solutions.

combinational logic

sum x y

truth tables

resulting sum bits

functions x

adder circuit

binary inputs

boolean expressions

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