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