Consider the digital circuit shown below with two input lines A and B, two select lines S0 and S1, and an output line Y. The blocks Q and M represent active high 2:4 decoder and 4-to-1 multiplexer, respectively. Out of 16 possible input combinations, the number of combinations that produce Y=1 is ____________. (answer in integer)
Note: One input combination is an instance of [A B S1 S0].
Problem Analysis
The circuit contains two main functional blocks: an active-high 2:4 decoder (Q) and a 4-to-1 multiplexer (M). We need to determine how many of the 16 possible combinations of the input vector \([A, B, S_1, S_0]\) result in the output \(Y = 1\).
1. Functional Logic of the Blocks
A. 2:4 Decoder (Block Q)
An active-high decoder activates a specific output line based on the binary input \((A, B)\). The standard boolean expressions for the relevant outputs are:
- \(D_0 = \overline{A} \cdot \overline{B}\) (High only when \(A=0, B=0\))
- \(D_3 = A \cdot B\) (High only when \(A=1, B=1\))
B. 4-to-1 Multiplexer (Block M)
The multiplexer selects one of its four data inputs based on the select lines \(S_1\) and \(S_0\). From the diagram, the inputs are:
- Input 0 (\(I_0\)) = \(D_0\)
- Input 1 (\(I_1\)) = \(0\) (Logic Low)
- Input 2 (\(I_2\)) = \(1\) (Logic High)
- Input 3 (\(I_3\)) = \(D_3\)
The boolean expression for the output \(Y\) is:
$$ Y = \overline{S_1}\overline{S_0} \cdot D_0 + \overline{S_1}S_0 \cdot 0 + S_1\overline{S_0} \cdot 1 + S_1S_0 \cdot D_3 $$
2. Case-by-Case Evaluation
To find when \(Y = 1\), we evaluate the output for each combination of the select lines \((S_1, S_0)\):
- Case 1: \((S_1, S_0) = (0, 0)\)
\(Y = D_0\). For \(Y\) to be 1, \(D_0\) must be 1. This occurs only when \((A, B) = (0, 0)\).
Combination: \([0, 0, 0, 0]\) → 1 combination
- Case 2: \((S_1, S_0) = (0, 1)\)
\(Y = 0\). Output is always low regardless of \(A\) and \(B\).
→ 0 combinations
- Case 3: \((S_1, S_0) = (1, 0)\)
\(Y = 1\). Output is tied to logic high, meaning \(Y=1\) for any value of \(A\) and \(B\).
Combinations: \([0,0,1,0], [0,1,1,0], [1,0,1,0], [1,1,1,0]\) → 4 combinations
- Case 4: \((S_1, S_0) = (1, 1)\)
\(Y = D_3\). For \(Y\) to be 1, \(D_3\) must be 1. This occurs only when \((A, B) = (1, 1)\).
Combination: \([1, 1, 1, 1]\) → 1 combination
3. Total Count
Summing the combinations from all cases:
$$ \text{Total Combinations} = 1 (\text{Case 1}) + 0 (\text{Case 2}) + 4 (\text{Case 3}) + 1 (\text{Case 4}) = 6 $$
Summarized Solution
The multiplexer output \(Y\) is logic high in three distinct scenarios based on the select lines \(S_1\) and \(S_0\). When \((S_1, S_0) = (0, 0)\), the output depends on the decoder's \(D_0\) pin (1 combination). When \((S_1, S_0) = (1, 0)\), the output is connected to a constant logic '1', which covers 4 combinations of \(A\) and \(B\). When \((S_1, S_0) = (1, 1)\), the output depends on the decoder's \(D_3\) pin (1 combination). Together, there are 6 input combinations that produce \(Y=1\).
Important Questions from Multiplexer
The device which changes from serial data to parallel data is:
Consider a logic gate circuit. with 8 input lines (D 0, D 1..... D 7) and 3 output lines (A 0, A 1, A 2) specified by following operations
A 2= D 4+ D 5+ D 6+ D 7
A 1= D 2+ D 3+ D 6+ D 7
A 0= D 1+ D 3+ D 5+ D 0
Where + indicates logical OR operation. This circuit is
Which of the following statements are true?
A. Frequency division multiplexing technique can be handled by digital circuits.
B. Time division multiplexing technique can be handled by analog circuits
C. Wavelength division multiplexing technique is used with optical fiber for combining two signals.
D. Frequency division multiplexing technique can be applied when the bandwidth of a link is greater than the bandwidth of the signals to be transmitted.
Choose the correct answer from the options given below :
Consider the sentence below.
There is a country that borders both India and Pakistan.
Which of the following logical expressions express the above sentence correctly when the predicate Country(x) represents that x is a country and Borders(x, y) represents that the countries x and y share the border?
A digital computer has a common bus system for 8 registers 16 bits each. How many multiplexers are required to implement common bus? What size of multiplexers is required?
