Question

Consider a 2-bit saturating up/down counter that performs the saturating up count
when the input P is 0, and the saturating down count when P is 1. The Next State
table of the counter is as shown. The counter is built as a synchronous sequential
circuit using D flip-flops.
Input
Current
State
Next
State
𝑃
𝑄1
𝑄0
𝑄1
+
𝑄0
+
0
0
0
0
1
0
0
1
1
0
0
1
0
1
1
0
1
1
1
1
1
0
0
0
0
1
0
1
0
0
1
1
0
0
1
1
1
1
1
0
Which one of the following options corresponds to the expressions for the inputs of
the D flip-flops, 𝐷1 and 𝐷0?

• A. 𝐷1 = 𝑃 𝑄1 + 𝑃̅𝑄0 + 𝑄1𝑄0 𝐷0 = 𝑃 𝑄0 + 𝑃̅ 𝑄1 + 𝑄1 𝑄0 ̅̅̅
• B. 𝐷1 = 𝑃̅ 𝑄1 + 𝑃̅𝑄0 + 𝑄1𝑄0 𝐷0 = 𝑃̅ 𝑄0 ̅̅̅ + 𝑃 ̅𝑄1 + 𝑄1 𝑄0 ̅̅̅
• C. 𝐷1 = 𝑃̅ 𝑄1 ̅̅̅ + 𝑃 ̅𝑄0 + 𝑄1 𝑄0 𝐷0 = 𝑃̅ 𝑄0 + 𝑃̅ 𝑄1 + 𝑄1 𝑄0 ̅̅̅
• D. 𝐷1 = 𝑃 𝑄1 ̅̅̅ + 𝑃̅ 𝑄0 + 𝑄1 𝑄0 𝐷0 = 𝑃 𝑄0 ̅̅̅ + 𝑃̅ 𝑄1 + 𝑄1 𝑄0 ̅̅̅

Hint:

Since D flip-flops directly store the next state, write $D_1$ and $D_0$ as Boolean functions of $P, Q_1, Q_0$ from the next-state columns, then simplify each using a K-map.

Answer 1

The Correct Option is B

Alternative approach - Karnaugh map by cell. Step 1: Build a 3-variable K-map for \(D_1\) with variables \(P, Q_1, Q_0\). The eight cells (in order \(PQ_1Q_0\) = 000,001,010,011,100,101,110,111) hold values \(0,1,1,1,0,0,0,1\) read directly from the \(Q_1^+\) column. Step 2: Cells 010 and 011 merge into \(\bar{P}Q_1\); cells 011 and 111 merge into \(Q_1Q_0\); cell 001 merges with 011 into \(\bar{P}Q_0\). These three groups cover all four 1-cells, giving \(D_1 = \bar{P}Q_1 + \bar{P}Q_0 + Q_1Q_0\). Step 3: Build a second K-map for \(D_0\) using the \(Q_0^+\) column: values at 000,001,010,011,100,101,110,111 are \(1,0,1,1,0,0,1,0\). Step 4: Cells 000 and 010 merge into \(\bar{P}\bar{Q_0}\); cells 010 and 011 merge into \(\bar{P}Q_1\); cells 010 and 110 merge into \(Q_1\bar{Q_0}\), giving \(D_0 = \bar{P}\bar{Q_0} + \bar{P}Q_1 + Q_1\bar{Q_0}\). Step 5: Cross-check with \(P=1, Q_1Q_0=11\): \(D_1 = 0+0+1=1\), \(D_0=0+0+0=0\), giving next state 10, matching the table entry for 11 with P=1. Final answer: \(D_1 = \bar{P}Q_1+\bar{P}Q_0+Q_1Q_0\) and \(D_0=\bar{P}\bar{Q_0}+\bar{P}Q_1+Q_1\bar{Q_0}\), option B.