digital electronics tutorial: sequential logic solutions

ELEC1041 – Tut Sequential 1

Digital Electronics

Tutorial: Sequential Logic

Solutions

Problem #1

In lecture, we presented an R-S latch based on cross-coupled NOR gates. It is also possible to construct an R’-S’ latch using cross-coupled NAND gates.

(a) Draw the R’-S’ latch, labeling the R’ and S’ inputs and the Q and Q’ outputs.

(b) Show the timing behavior across the four configurations of R’ and S’. Indicate on your timing diagram the behavior in entering and leaving the forbidden state when R’ = S’ = 0.

(c) Draw the state diagram that shows the complete input/output and state transition behavior of the R’-S’ latch.

(d) What is the characteristic equation of the R’-S’ latch.

(e) Draw a simple schematic for a gated R-S latch with an extra enable input, using NAND gates only.

Problem #1 Solution (1/5)

(a) Draw the R-S latch, labeling the R and S inputs and the Q and Q outputs.

(b) Show the timing behavior across the four configurations of R and S. Indicate on your timing diagram the behavior in entering and leaving the forbidden state when R=S=0.

Reset Set ForbiddenRace

Reset HoldHold ForbiddenSet

(c) Draw the state diagram that shows the complete input/output and state transition behavior of the R’-S’ latch.

Q Q0 1

Q Q1 0

Q Q1 1

Q Q0 0

SR=00SR=11S'R'=11

SR=11SR=01

SR=11SR=10

SR=00 SR=00

SR=01SR=10

SR=10 SR=01

possible oscillationbetween states 00 and 11

(d) What is the characteristic equation of the R-S latch.

S R Q(t) Q(t+)1 1 0 01 1 1 11 0 0 01 0 1 00 1 0 10 1 1 10 0 0 X0 0 1 X

not allowed characteristic equationQ(t+) = S + R Q(t)

1 0Q(t)

Break feedback path

(e) Draw a simple schematic for a gated R-S latch with an extra enable input, using NAND gates only.

Control when R and S inputs matter

The slightest glitch on R or S while enable is high could cause change in value stored

Enable

Problem #2

Consider a D-type storage element implemented in five different ways:

(a) D-latch (i.e., D wired to the S-input and D’ wired to the R-input of an R-S latch);

(b) Clock Enabled D-latch;

(c) Master-Slave Clock Enabled D-Flip-flop;

(d) Positive Edge-triggered Flip-flop;

(e) Negative Edge-triggered Flip-flop;

Complete the following timing charts indicating the behavior of these alternative storage elements. You can ignore set-up and hold time limitations (assume all constraints are meant):

Problem #2 Solution

(1) output from D-latch changes as D changes regardless of the clock.(2) output from clocked D-latch changes as D does only when the clock is high.(3) output from D M/S FF are the samples taken from the input at the falling edge of the clock.(4) output from positive edge FF changes relative to D at the rising edge of the clock.(5) output form negative edge FF are the samples taken from the input at the falling edge of the clock (same as part 3).

Problem #3

Complete the timing diagram for this circuit.

Problem #3 Solution

Complete the timing diagram for this circuit.

Toggles when T = 1 at the rising edge of the clock

Problem #4

Design a 4 bit counter that counts through the sequence 0111, 1000, 1001, 1010, 1011, 1100, 1101

Problem #4 Solution

Design a 4 bit counter that counts through the sequence 0111, 1000, 1001, 1010, 1011, 1100, 1101

DCBALOADCLKCLR

RCOQDQCQBQA

"0"“1"“1"“1"

Problem #5

You as a designer at XILION Micro Devices, have been asked to redesign part of your system to run at double the clock frequency. However, to maintain the compatibility with other parts of the system you like to keep the same clock frequency. What would you do to impress your Boss?

Discuss all timing constraint in your design

Problem #5 Solution

One way is to double of the clock frequency locally, or put it in another way make your flip flop to be both positive and negative edged triggered.

1. High Width of “CLK” should be larger than that required by the Flip FlopThis can be controlled by the number of invertors (odd) before the XNOR gate. 2. Period of “CLK” should be larger than the (set up time + propagation Delay) of flip flop.

Conditions:Clk period

Flip flop sample on both edges BCLOCK

Problem #6

Complete the count sequence table for the following shift register circuit.

Q1 Q2 Q3

Q4 Q1 Q2Q3Q4

0 0 0 0

Problem #6 Solution

Complete the count sequence table for the following shift register circuit.

Q1 Q2 Q3

Q4 Q1 Q2Q3Q4

01110110

00111011

00011101

00001110

01010000

00101000

10010100

11001010

Problem #7

RCO 12 13 14 15 0 1 2

Clear Load Count Inhibit

Consider the counter x74_163 from the xilinx library. Complete the timing diagram Below: (Note: LOAD and CLR are active low. P & T should be high to enable the counter

163RCO

Problem #7 Solution

RCO 12 13 14 15 0 1 2

Clear Load Count Inhibit

Indicates Undefined

P & T should be High to enable count

Problem #8

Design the logic for a 3-bit counter that follows the following sequence: 000, 111, 001, 110, 010, 101, 011, 100, 000 and repeats. Design the counter so when Reset is asserted, the counter enters the state 000.

Problem #9

Design the logic for a 3-bit counter that follows the following sequence: 001, 010, 100, 101, 110, 001, and repeat. Design the counter so that it is self-starting, i.e., whatever state it comes up in, it will eventually get into the sequence as shown above.

Problem #9 Solution

111 Q2 Q1 Q0 Q2 Q1 Q0

0 0 0 0 0 1

0 0 1 0 1 0

0 1 0 1 0 0

0 1 1 1 0 0

1 0 0 1 0 1

1 0 1 1 1 0

1 1 0 0 0 1

1 1 1 0 0 1

Q2 = Q2 Q1 + Q2Q1

Q1 = Q1 Q0

+Q0 = Q1 Q0 + Q2Q1

Problem #10

Consider 3 bit Johnson Counter below. Derive it state transition table and diagrams

Problem #10 Solution

0 0 0 0 0 10 0 1 0 1 10 1 0 1 0 00 1 1 1 1 01 0 0 0 0 01 0 1 0 1 01 1 0 1 0 01 1 1 1 1 0

Q2Q1Q0 Q2Q1Q0

Problem #11

Problem #12

0 0 x 00 0 x x0 0 x x0 1 x 0

0 0 x 11 0 x x1 1 x x0 0 x 0

0 1 x 00 1 x x1 0 x x1 0 x 0

0 1 x 01 1 x x1 0 x x0 1 x 0

Problem #13

Finite String Pattern Recognizer

A finite string recognizer has one input (X) and one output (Z).The output is asserted whenever the input sequence …010…has been observed, as long as the sequence 100 has never beenseen.

Step 1. Understanding the problem statement

Sample input/output behavior:

X: 00101010010…Z: 00010101000…

X: 11011010010…Z: 00000001000…

Step 2. Draw State Diagrams/ASM Charts for the strings that must be recognized. I.e., 010 and 100.

Moore State DiagramReset signal places FSM in S0

Outputs 1 Loops in State

S0 [0]

S1 [0]

S2 [0]

S3 [1]

S4 [0]

S5 [0]

S6 [0]

Exit conditions from state S3: have recognized …010 if next input is 0 then have …0100! if next input is 1 then have …0101 = …01 (state S2)

S0 [0]

S1 [0]

S2 [0]

S3 [1]

S4 [0]

S5 [0]

S6 [0]

Exit conditions from S1: recognizes strings of form …0 (no 1 seen) loop back to S1 if input is 0

S0 [0]

S1 [0]

S2 [0]

S3 [1]

S4 [0]

S5 [0]

S6 [0]

S2, S5 with incomplete transitions

S2 = …01; If next input is 1, then string could be prefix of (01)1(00) S4 handles just this case!S5 = …10; If next input is 1, then string could be prefix of (10)1(0) S2 handles just this case!

Final State Diagram

S0 [0]

S1 [0]

S2 [0]

S3 [1]

S4 [0]

S5 [0]

S6 [0]

Finite String RecognizerReview of Process:

• Write down sample inputs and outputs to understand specification

• Write down sequences of states and transitions for the sequences to be recognized

• Add missing transitions; reuse states as much as possible

• Verify I/O behavior of your state diagram to insure it functions like the specification

Problem #14

Finite String Pattern RecognizerA finite string recognizer has one input (X) and two output (Z1 & Z2).The output Z1 is asserted whenever the input sequence …010…has been observed, as long as the sequence 100 has never beenseen. The output Z2 is asserted whenever the input sequence …100…has been observedNote that once Z2 = 1 has occurred, Z1 = 1 can never occur, but never vice versa

Step 1. Understanding the problem statement Sample input/output behavior:

X : 0010101001000…Z1: 0001010100000…Z2: 0000000010010…

X : 110110100100…Z1: 000000010000…Z2: 000000001001…

Step 2. Draw State Diagrams/ASM Charts for the strings that must be recognized. I.e., 010 and 100.

Moore State DiagramReset signal places FSM in S0

Output Z1= 1

S0 [00]

S1 [00]

S2 [00]

S3 [Z1=1]

S4 [00]

S5 [00]

S6 [Z2=1]

Output Z2= 1

Exit conditions from state S3: have recognized …010 if next input is 0 then have …0100! if next input is 1 then have …0101 = …01 (state S2)

S0 [00]

S1 [00]

S2 [00]

S3 [Z1=1]

S4 [00]

S5 [00]

S6 [Z2=1]

S0 [00]

S1 [00]

S2 [00]

S3 [Z1=1]

S4 [00]

S5 [00]

S6 [Z2=1]

S2, S5 with incomplete transitions

S2 = …01; If next input is 1, then string could be prefix of (01)1(00) S4 handles just this case!S5 = …10; If next input is 1, then string could be prefix of (10)1(0) S2 handles just this case!

S0 [00]

S1 [00]

S2 [00]

S3 [Z1=1]

S4 [00]

S5 [00]

S6 [Z2=1]

S6 = …100; If next input is 1, then string could be prefix of (100)1(00) S4 handles just this case!

Final State Diagram

S0 [00]

S1 [00]

S2 [00]

S3 [Z1=1]

S4 [00]

S5 [00]

S6 [Z2=1]

BUT IS IT? Remember S3 should never be entered again, so S5 should be avoided

S6 = …100; If next input is 1, then string could be prefix of (100)1(00) Draw additional states for the string that must be recognized after S6 has been entered i.e. 100. S0

S1 [00]

S2 [00]

S3 [Z1=1]

S4 [00]

S5 [00]

S6 [Z2=1]

S7 [00]

S8 [00]

S1 [00]

S2 [00]

S3 [Z1=1]

S4 [00]

S5 [00]

S6 [Z2=1]

S7 [00]

S8 [00]

S9 [00]

S1 [00]

S2 [00]

S3 [Z1=1]

S4 [00]

S5 [00]

S6 [Z2=1]

S7 [00]

S8 [00]

S9 [00]

Problem #15

N + 2 converter

A sequential network has one input X and two outputs S and V. X represent a four bit binary number N, which is input least significantbit first. S represents a four bit binary number equal to N + 2, which is output least significant bit first. At the time the fourth input is sampled,V = 1, in N + 2 is too large to be represented by four bits; otherwise V = 0.

Derive a Mealy state graph and table with a minimum number of states

Example: N +2 Serial ConverterAssume numbers are +ve (0N15)

(N) (N +2)0 0000 0010 01 0001 0011 02 0010 0100 03 0011 0101 04 0100 0110 05 0101 0111 06 0110 1000 07 0111 1001 08 1000 1010 09 1001 1011 010 1010 1100 011 1011 1101 012 1100 1110 013 1101 1111 014 1110 0000 115 1111 0001 1

Conversion Process

Bits are presented in bit serial fashionstarting with the least significant bit

Single input X, Two output S, V

N3N2N1N0

Problem #15 Solution (ver 1) (2/3)

Present State S0 S1S2 S3 S4 S5

Next StateOutput

SVX=0 S1 S2 S4 S4 S0 S0

X=1 S1 S3 S4 S5 S0 S0

X=0 00 10 00 10 00 10

X=1 10 00

10 00 10 01

0/00,1/10

S10/10 1/00

0/00,1/10

1/000/10

S40/00,1/10

S50/10,1/01

ROM-based Implementation

Circuit Level Realization74175 = 4 x positive edge triggered D FFs

In ROM-based designs, no need to consider state assignment

converter ROMX Q2 Q1 Q0

V D2 D1 D0

D CB A

10 \Reset

X 175Z

Truth Table/ROM I/Os

X 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1

Q2 0 0 0 0 1 1 1

0 0 0 0 1 1 1 1

Q1 0 0 1 1 0 0 1

0 0 1 1 0 0 1 1

Q0 0 1 0 1 0 1 0

0 1 0 1 0 1 0 1

ROM AddressD2 0 0 1 1 0 0 X

0 0 1 1 0 0 X

D1 0 1 0 0 0 0 X

0 1 0 0 0 0 X X

D0 1 0 0 0 0 0 X

1 10 1 0 0 XX

ROM OutputsS 0 1 0 1 0 1 X

1 0 0 1 0 1 X X

V 0 0 0 0 0 0 X

0 0 0 0 0 0 X X

0 1 1 1 X X X X X

N3N2N1N0 + 0 0 1 0---------------- N0

1 bit adder

D QCoQCoD

Reset S0[0,0]

S2[0,1]

S3[0,1]

S1[1,0]

Data pathController

PS NS OutputsQ1Q0 Q1

+ Q0+ Sb Sc

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

= Q1 XOR Q0

Sb = Q1 Q0

Sc = Q1

S = A XOR B XOR CinCo = AB + (A + B) Cin

= Q1 XOR Q0

Sb = Q1 Q0

Sc = Q1

1 bit adder

D QCoQ

D Q D QQ0 Q1

Problem #16

N - 2 converter

A sequential network has one input X and two outputs S and V. X represent a four bit binary number N, which is input least significantbit first. S represents a four bit binary number equal to N - 2, which is output least significant bit first. At the time the fourth input is sampled,V = 1, in N - 2 is too small to be represented by four bits; otherwise V = 0.

Derive a Mealy state graph and table with a minimum number of states

Example: N -2 Serial Converter

Conversion Process

Bits are presented in bit serial fashionstarting with the least significant bit

Single input X, Two output S, V

(N) (N - 2)0 0000 1110 11 0001 1111 12 0010 0000 03 0011 0001 04 0100 0010 05 0101 0011 06 0110 0100 07 0111 0101 08 1000 0110 09 1001 0111 010 1010 1000 011 1011 1001 012 1100 1010 013 1101 1011 014 1110 1100 015 1111 1101 0

N3N2N1N0

Assume numbers are +ve (0N15)

Present State S0 S1 S2 S3 S4 S5

Next StateOutput

SVX=0 S1 S2 S4 S5 S0 S0

X=1 S1 S3 S5 S5 S0

X=0 00 10 10 00 11 00

X=1 10 00

00 10 00 10

0/00,1/10

S10/10 1/00

0/00,1/10

S40/11,1/00

S50/00,1/10

ROM-based Implementation

Truth Table/ROM I/Os

Circuit Level Realization74175 = 4 x positive edge triggered D FFs

In ROM-based designs, no need to consider state assignment

converter ROMX Q2 Q1 Q0

V D2 D1 D0

D CB A

10 \Reset

X 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

Q2 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

Q1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

Q0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

ROM AddressD2 0 0 1 1 0 0 X X 0 0 1 1 1 0 X

D1 0 1 0 0 0 0 X X 0 1 0 0 1 0 X X

D0 1 0 0 1 0 0 X X 1 1 1 10 0 X X

ROM OutputsS 0 1 1 0 1 0 X X 1 0 0 1 0 1 X X

V 0 0 0 0 1 0 X X 0 0 0 0 0 0 X X

N3N2N1N0 - 0 0 1 0---------------- X0

1 bit subtractor

D QBoQCoD

Reset S0[0,0]

S2[0,1]

S3[0,1]

S1[1,0]

Data pathController

PS NS OutputsQ1Q0 Q1

+ Q0+ Sb Sc

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

= Q1 XOR Q0

Sb = Q1 Q0

Sc = Q1

S = A XOR B XOR BinBo = A’B + (A’ + B) Bin

= Q1 XOR Q0

= Q0 Sb = Q1 Q0

Sc = Q1

1 bit subtractor

D QCoQ

D Q D QQ0 Q1

Problem #17

Problem #17 Solution version 1 (1/4)

We relax the requirement for undefined states toGo to zero, however, we still require them to go to

a valid state

101 010110 010111 100undefined states end uo in valid states

Problem #18

Consider the design of a sequence detector finite state machine that will assert a 1 when the current input equals the just previously seen input.

(a) Draw as simple state diagrams for a MEALY MACHINE and a MOORE MACHINE implementation as you can (minimization is not necessary). The MEALY MACHINE should have fewer states. Briefly explain why.

(b) If the Mealy Machine is implemented as a SYNCHRONOUS MEALY MACHINE, draw the timing diagram for the example input/output sequence described above.

(c) If the timing behaviors are different for the MOORE, MEALY, and SYNCHRONOUS MEALY machines, explain the reason why.

Problem #18 Solution(1/4)

Sample input/output sequence:

001110101100

010110000101

(a) Draw as simple state diagrams for a MEALY MACHINE and a MOORE MACHINE implementation as you can (minimization is not necessary). The MEALY MACHINE should have fewer states. Briefly explain why.

Since the output of Mealy Machines depend on both the current state and the current input, they don’t need as many states to represent input/output combinations.

S2[0] S4

0S4[1]

(b) If the Mealy Machine is implemented as a SYNCHRONOUS MEALY MACHINE, draw the timing diagram for the example input/output sequence described above.

001110101100

010110000101

(c) If the timing behaviors are different for the MOORE, MEALY, and SYNCHRONOUS MEALY machines, explain the reason why.

The timing behaviors of a Moore Machine and a Synchronous Mealy Machine are the same.

The Mealy Machine will behave differently because the output changes as soon as the input changes rather than at the next clock cycle.

S1S0 S1 S2 S2 S2 S1 S2 S1 S2 S2 S1

Problem #19

Consider a 3-bit counter that behaves as follows. The counter has a mode input M. When M is true, the counter counts in the sequence 0, 2, 4, 6, 1, 3, 5, 7, 0 and repeat. When M is false, the counter counts in the sequence 0, 1, 6, 7, 2, 3, 4, 5, 0 and repeat. The M input can change at anytime to cause the counter to change into either mode.

(a) Complete the Encoded STATE TRANSITION TABLE for this counter.

(b) Use the K-maps below to find the minimized two-level implementation of the counter’s next state functions.

(a) Complete the Encoded STATE TRANSITION TABLE for this counter.

M = 0: 0, 1, 6, 7, 2, 3, 4, 5, 0 and repeat

M = 1: 0, 2, 4, 6, 1, 3, 5, 7, 0 and repeat

M’ Q0’ Q2 + M Q1 Q2’ + M’ Q0 Q2’ + M Q1’ Q2

M Q1’ + Q0 Q1’ Q2’ + M’ Q1 Q2 + M’ Q0’ Q1

M’Q0’ + Q0’ Q1 Q2 + M Q0 Q2’ + M Q0 Q1’

Problem #20

QAQBQCQD

163RCO

switch

Figure below illustrates the use a Single-Pole Single-Throw (SPST) switch to increment the count value in the counter.Every throw of the switch causes a low-to-high transition on the clock input (CLK) which in turn causes the counter to increment by 1.However, we have noticed that each throw of the switch causesmultiple increments in the count value. Furthermore multipleincrements are equal to each other and changes from one throw of the switch to the next.a) Explain Why? b) How to do rectify the problem

Switch

a) This problem is due to bouncing effect in in a mechanical switch. The switch does not make a clean contact at-once. It bounces several times from its final position before coming to rest. This causes several low-to-high transition at the clk signal causing multiple count increments.

switch

\switch

/switch

switch

b) This problem can be overcome by using a debouncing circuitry shown below. Here we use a Single-Pole Double-Throw (SPDT) switch to increment the count value in the counter. In spite of mechanical switch making several contacts, since the switch after the throw to one position never bounces back all the way to the opposite position the S-R latch will either be in set or hold state and its clk output stays constant after the first transition.

Initial position

digital electronics tutorial: sequential logic solutions

gated rs latch

rs latchb clock

configurations of r

d changes

clocked dlatch changes

timing behavior

timing diagram

d ms ff

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