CS223-Homework 1 Solved

Q1. Given the Boolean function  𝑭(𝑨, 𝑩, 𝑪, 𝑫) = ∑(𝟎, 𝟐, 𝟑, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎,𝟏𝟐, 𝟏𝟑)

(a)     Find a minimal Boolean equation for the function by using Karnaugh map method only (don’t use Boolean algebra).

(b)    Draw a circuit diagram implementing the function 𝐅 using logic gates (the complements of input variables are available).

(c)     Implement the function 𝐅 by using only a 8:1 multiplexer and a 2:1 multiplexer (the complements of input variables are NOT available, do not use any other logic gate. Logic levels 1 and 0 are available)

 

Q2. Simplify the following equations using the theorems of Boolean algebra.

          (a)     𝒀                                      
           (b)     𝒀 = 𝑨𝑩𝑪                                𝑨𝑩𝑪𝑫 

 

Q3. Write the following Boolean equations in product-of-sums (POS) canonical form.

(a)     𝒀(𝑨, 𝑩, 𝑪, 𝑫) = 𝑨 𝑩 + 𝑩𝑪 + 𝑩 𝑪 𝑫 

(b)    𝒀(𝑨, 𝑩, 𝑪) = 𝑨 𝑩 𝑪 + 𝑨 𝑩 𝑪 + 𝑨 𝑩 𝑪 + 𝑨 𝑩 𝑪 

 

Q4. Simplify the following equation using the Karnaugh map method.

 

𝒀 = 𝑩 𝑪 + 𝑨 𝑪 + 𝑨 𝑩 + 𝑨 𝑩 𝑪 𝑫  

 

Q5. You are given the following circuit and the Boolean function 𝑭 defined by the truth table.

 

A 
B 
C 
F 
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
0
 
 

Find a minimal Boolean equation for the output of the circuit  𝒀(𝑨, 𝑩,𝑪, 𝑫).

                 

Q6. A sequential circuit with two state variables S1 and S0 has two inputs X and Y. The boolean equations for next state logic and output logic functions are given as follows;

 𝑺𝟎′ = 𝑿 𝒀 + 𝑿𝑺𝟎 

𝑺𝟏′ = 𝑿𝑺𝟏 + 𝑿𝑺𝟎 

𝒛 = 𝑺𝟎 ⊕ 𝑺𝟏 

(a) Write the state transition table (b) Draw state transition diagram

 

Q7. A sequential circuit design with input A and output Y is shown in the following diagram.

(a)     Write the state transition and output tables with binary encodings.

(b)    Write the Boolean equations for the next-state and output logic.

(c)     Sketch the state transition diagram.  

 

Q8. Design a sequential circuit with input B and three outputs xyz which is used as a counter that counts through the 3-bit prime numbers.

On reset circuit outputs are xyz=010.  

If B=0, output will not change, else if B=1 output will change as follows  011, 101, 111, 010, 011, 101, 111, 010, ....

Show the state diagram and state table of FSM, use binary encoding for the states.  

Write next state equations and output equations, implement the next state logic, sketch the circuit schematic.

 

Q9. Design a Finite State Machine (FSM) which has a one-bit input 𝑼 and three-bit output 𝑪. If 𝑼 = 𝟏, the circuit will count up two-by-two and if  𝑼 = 𝟎, the circuit will count down one-by-one. The circuit will count up as follows: 0, 2, 4, 6, 0, 2, 4, 6, ... or 1, 3, 5, 7, 1, 3, 5, 7, ... and it will count down as follows: 0, 7, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, ... On reset, its output will be 𝟎𝟎𝟎.

(a)     Sketch the state transition diagram.

(b)    Write the state transition and output tables with binary encodings.

(c)     Write the Boolean equations for the next-state and output logic.

(d)    Sketch the circuit schematic for the FSM.

 

Q10. Design a Finite State Machine (FSM) with one input and one output. The circuit should read one-bit data at every rising edge of the clock. According to the data input, the circuit will produce the following output. On reset, its output will be 𝟏𝟎𝟎.

If the input bit sequence is 𝟏𝟏 then the output will be 𝟎𝟎𝟎. If the input bit sequence is 𝟎𝟎 then the output will be 𝟎𝟎𝟏.

If the input bit sequence is 𝟏𝟏𝟏 then the output will be 𝟎𝟏𝟎.

If the input bit sequence is 𝟎𝟎𝟎 then the output will be 𝟎𝟏𝟏.

Otherwise, the output will be 𝟏𝟎𝟎.

(a)     Sketch the state transition diagram.

(b)    Write the state transition and output tables with binary encodings.

(c)     Write the Boolean equations for the next-state and output logic.

(d)    Sketch the circuit schematic for the FSM.