ETE-2343 (Spring 2024) (DELD)

Q1. a) Short Notes & Parity Check

Reflective Code (Gray Code):

• Gray Code changes only one bit at a time when counting.

• Example: 000 → 001 → 011 → 010 → 110 → 111 → 101 → 100

Even Parity Check:

• Total number of 1’s must be even.

  • (a) 10101010 → 4 ones (Even) ✅

  • (b) 1110110 → 5 ones (Error) ❌

  • (c) 10111001 → 5 ones (Error) ❌

Odd Parity Check:

• Total number of 1’s must be odd.

  • (d) 10101111 → 6 ones (Error) ❌

  • (e) 10011010 → 4 ones (Error) ❌

  • (f) 11101010 → 5 ones (Correct) ✅

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Q1. b) Advantages & Disadvantages of Digital Systems

Advantages:

• More accurate, less noise.

• Easy to store and process data.

• Can be encrypted for security.

Disadvantages:

• Needs more power than analog.

• Complex design.

• Can lose data if error occurs.

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OR

Q1. a) Universal Gate (NAND) & XOR/XNOR Gates

• NAND is Universal: It can make NOT, AND, OR gates.

  • NOT: NAND with same input (A NAND A = NOT A).

  • AND: NAND + NOT.

  • OR: NAND with inverted inputs.

XOR Gate (⊕):

• Outputs 1 if inputs are different.

• Function: $A\oplus B=A^{\mathrm{\prime }}B+A{B}^{\mathrm{\prime }}$

XNOR Gate (⊙):

• Outputs 1 if inputs are same.

• Function: $A\odot B=AB+A^{\mathrm{\prime }}{B}^{\mathrm{\prime }}$

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Q1. b) Gate Implementation & Complement

Function: $F1=x+y^{\mathrm{\prime }}z$

• Gate Implementation: OR gate with $x$ and $y^{\mathrm{\prime }}z$ (AND of NOT y and z).

• Truth Table:

x	y	z	y'	y'z	F1 = x + y'z
0	0	0	1	0	0
0	0	1	1	1	1
... (Fill for all 8 combinations)

Complement of $F(x,y,z)=x{y}^{\mathrm{\prime }}{z}^{\mathrm{\prime }}+x^{\mathrm{\prime }}yz$:

• Use De Morgan’s Law:
  $G=(x{y}^{\mathrm{\prime }}{z}^{\mathrm{\prime }}+x^{\mathrm{\prime }}yz{)}^{\mathrm{\prime }}=(x{y}^{\mathrm{\prime }}{z}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}\cdot (x^{\mathrm{\prime }}yz{)}^{\mathrm{\prime }}=(x^{\mathrm{\prime }}+y+z)(x+y^{\mathrm{\prime }}+z^{\mathrm{\prime }})$

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Q2. a) Sum of Minterms for $F=A+B^{\mathrm{\prime }}C$

• First, make truth table for $A,B,C$.

• Minterms where $F=1$:

  • $A=1$ (any $B,C$) → Minterms: 4,5,6,7

  • $B^{\mathrm{\prime }}C=1$ → Minterms: 1,3

• Final: $F=\sum (1,3,4,5,6,7)$

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Q2. b) Product of Maxterms for $F=xy+x^{\mathrm{\prime }}z$

• Find where $F=0$:

  • $x=0,y=0$ → $F=0$

  • $x=0,z=0$ → $F=0$

  • $x=1,y=0,z=0$ → $F=0$

• Maxterms: $(x+y+z)(x+y+z^{\mathrm{\prime }})(x^{\mathrm{\prime }}+y+z)(x^{\mathrm{\prime }}+y^{\mathrm{\prime }}+z)$

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Q3. a) Simplify $F(x,y,z)=\sum (0,2,4,5,6)$

• K-Map Simplification:

  • Group: $0,2,4,6$ → $z^{\mathrm{\prime }}$

  • Group: $4,5$ → $x{y}^{\mathrm{\prime }}$

• Simplified: $F=z^{\mathrm{\prime }}+x{y}^{\mathrm{\prime }}$

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Q3. b) Simplify $F(w,x,y,z)=\sum (0,1,2,4,5,6,8,9,12,13,14)$

• K-Map Simplification:

  • Group: $0,1,4,5,8,9,12,13$ → $y^{\mathrm{\prime }}$

  • Group: $0,2,4,6,8,12$ → $x^{\mathrm{\prime }}{z}^{\mathrm{\prime }}$

  • Group: $0,1,2$ → $w^{\mathrm{\prime }}{x}^{\mathrm{\prime }}$

• Simplified: $F=y^{\mathrm{\prime }}+x^{\mathrm{\prime }}{z}^{\mathrm{\prime }}+w^{\mathrm{\prime }}{x}^{\mathrm{\prime }}$