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Computer Science
Posted 2 months ago
Using T flip flops, design a 
3
-
bit synchronous counter to count in the followingsequence: 
0
,
 
2
,
 
4
,
 
6
,
 
0
.
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Answer from Sia
Posted 2 months ago
To design a 3-bit synchronous counter using T flip-flops that counts in the sequence 0, 2, 4, 6, 0, we need to understand how T flip-flops work and how to set them up to count in the desired sequence. A T flip-flop toggles its output on every clock cycle if the T input (toggle input) is high (1), and it holds its current state if the T input is low (0).

Let's represent the 3 bits as Q2Q1Q0Q_2Q_1Q_0, where Q2Q_2 is the most significant bit (MSB) and Q0Q_0 is the least significant bit (LSB). The sequence we want to achieve is:

0:0002:0104:1006:110
\begin{align*}
0 & : 000 \\
2 & : 010 \\
4 & : 100 \\
6 & : 110 \\
\end{align*}


We will now create a state table that includes the current state, the next state, and the required T inputs to achieve the transition from the current state to the next state.

State Table:

Q2Q1Q0T2T1T0000010010100100010110000
\begin{array}{ccc|ccc}
Q_2 & Q_1 & Q_0 & T_2 & T_1 & T_0 \\
\hline
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 \\
\end{array}


From the state table, we can derive the logic for the T inputs:

T2=Q1Q0T1=Q0T0=0
\begin{align*}
T_2 & = Q_1 \cdot Q_0 \\
T_1 & = Q_0 \\
T_0 & = 0 \\
\end{align*}


Now, we can use this logic to design the counter using T flip-flops.

Explanation
The state table is used to determine the T inputs for each flip-flop based on the current state and the desired next state. The logic equations for the T inputs are derived from the state table.
Step-by-step Instruction
Create a state table that includes the current state, the next state, and the T inputs required for the transition
Derive the logic equations for the T inputs based on the state table
Design the counter using T flip-flops and the derived logic for the T inputs
Time Complexity
Not applicable as this is a hardware design problem and time complexity refers to algorithmic time efficiency.
Space Complexity
Not applicable as this is a hardware design problem and space complexity refers to algorithmic space efficiency.

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