Journal of Computers - Academy Publisher

1974 JOURNAL OF COMPUTERS, VOL. 6, NO. 9, SEPTEMBER 2011
equals to 1, otherwise equals to 0. All values are showm
in Fig.3.
B. Analysis for Fig.1
Q1 Q0
' '
Q1 Q0
Fig.3 All values for equations(8)
By the proposed method, the analysing process for the
3-bit synchronous counter shown in Fig.1 includes main
three steps.
Step 1: The standard SOPs
Each state transition equation is converted to standard
SOPs, taking 2 Q′ as an example, the converting processes
are as shown in Equation.9, this processes can also be
realized by Karnaugh Maps[19,20].
Q′ = Q Q Q + Q Q
2 2 1 0 2 1
= QQQ + QQ( Q + Q)
2 1 0 2 1 0 0
= QQQ + QQQ + QQQ
= ∑ m (010,110,111)
= ∑ m (2,6,7)
2 1 0 2 1 0 2 1 0
The conversion to obtain standard SOPs By similar
processes, the equations Q′ and 1
0 Q′ can be obtained as
shown in equations(10).
Q′ = QQQ + Q QQ + Q QQ
1 2 1 0 2 1 0 2 1 0
= ∑ m(
001,010,011)
= ∑ m(
1,2,3)
Q′ = Q QQ + Q QQ + Q QQ + Q Q Q
0 2 1 0 2 1 0 2 1 0 2 1 0
= ∑ m(
000, 001,100, 110)
= ∑ m(
01,4,6 , )
Step 2: The truth table
(9)
(10)
(11)
We develop a truth table for the equations(9), (10) and
(11), eight possible combinations of binary values are
listed in the medium columns, the decimal digit
corresponding to each binary value is listed in the left
column. According to the principle for the value of next
state, the present state values that make the next state 2 Q′
equal to 1 are 010(2), 110(6), and 111(7). For each of
these values, a 1 is filled in each corresponding position
in the first column of three right columns.
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Using the same principle, when the present state values
are 001(1), 010(2), and 011(3), the next state Q′ equal
1
to 1, a 1 is filled in each corresponding position in the
second column of right columns. When the present state
values are 000(0); 001(1); 100(4); and 110(6), the next
state 0 Q′ equals to 1, a 1 is filled in each corresponding
position in the third column of right columns. After all 1
are filled, the view of truth table is shown as Tab.3.
Tab. 3 The truth table
Q2 Q1 Q0
' ' '
Q 2 Q 1 Q 0
All the remaining positions in right columns are placed
by a 0, we can get the same truth table as Tab. 1.
The third step is same as the present method, detailed
analysis is no longer given here.
IV. CONCLUSIONS AND DISCUSSIONS
This paper deduces the principle for obtaining next
state values in state transition equations. Based on the
principle, a modified method has been proposed to
analyse synchronous counters constructed with flip-flops.
The method can develop the truth table directly from state
transition equations with SOPs. This method facilitates
analysis of synchronous counters constructed with Flip-
Flops by eliminating large number of Boolean
calculations.
The number of state variables n is three in the example,
if there have more state variables, the calculation will be
increased, such as n=4, the number of minterm is 16 and
the state transition equation is 4, it is needed 64 times
next state equation calculations to accomplish the truth
table. Clearly, the advantages are more obvious with the
increase of n. Certainly, if n is enough large, for example,
n=10, although the proposed method is still more rapid
and convenient than present one, the obtaining for
minterm is very complex, so we advise that one should
analyze with computer.
Obviously, the method is also suitable for synchronous
counters constructed with other FFs, such as D FFs, and
so on.
ACKNOWLEDGMENT
The authors thank their colleagues at College of
Communication Engineering for fruitful discussions. This
work was supported by Natural Science Foundation
Project of CQ CSTC under contract no: 2010BB2240.