Data Representation and Digital Logic

The assignment content is about Data Representation and Digital Logic, specifically converting floating-point numbers to decimal and exploring digital logic gates. It discusses IEEE 754 single precision format, signed magnitude, one's complement, and two's complement representations of binary numbers. The content also covers a logic diagram for a 24-hour clock timing system and proves a Boolean expression using algebraic identities.

Assignment 1: Data Representation and Digital Logic
ITC544
NAme:
ID:

1.
a. To convert floating point number represented by IEEE – 754 single precision format
to the decimal:
0 01111110 10100000000000000000000
Sign bit Exponent Mantissa
Positive number
Exponent --- 0*27 + 1*26 + 1*25 + 1*24 + 1*23 + 1*22 + 1*21 + 0*20
64 + 32 + 16 + 8 + 4 + 2
= 126
 126 – 127
-1
Mantissa --- 1.101
As the exponent is -1, therefore, number becomes 0.1101
0.1101 --- 0*20 + 1*2-1 + 1*2-2 + 0*2-3 + 1*2-4
0.5 + 0.25 + 0.0625
= 0.8125
Decimal number is 0.8125
b.
Binary number Decimal
Low High Low High
i. Signed
magnitude
01111 11111 -15 15
ii. One’s
Complement
10000 01111 -16 15
iii. Two’s
Complement
10001 01111 -17 15
2.
a. For the logic diagram, we have assumed that X denotes the input for 24-hour clock
timing and Y denotes the switch.
Y gives 1 as an input only when the time in 24-hour format is 9.00 to 12.00 or 13.00
to 16.00. For all other timings, 0 will be input.
Hence, the Boolean expression is:
X.Y = Z
Page 1 of 3

Logic Gate diagram:
Minimization:
No minimization is required. But if we assume that Y is always 1, i.e. switch is
always on, then expression
Z = X.1
Z = X
b. Prove that:
X’Y + XYZ’ + Y’ + XZ (Y+Y’) = 1
Considering L.H.S
 X’Y + XYZ’ + Y’ + XZ (Y+Y’)
 X’Y + Y’ + XYZ’ + XZ.1 (1 and 2)
 X’ + Y’ + XYZ’ + XZ (3 and 5)
 X’ + XZ + Y’ + XYZ’ (1)
 X’ + Z + Y’ + XZ’ (3)
 X’ + XZ’ + Z + Y’ (1)
 X’ + Z’ + Z + Y’ (3)
 X’ + 1 + Y’ (2)
 1
Hence Proved
Boolean Algebra Identities
1. Associative law A + B + C = A + C + B
2. Complementary law A + A’ = 1
3. Common Identity law A + A’B = A + B
5. Identity law A.1 = A
7. Null law 1 + A = 1
Page 2 of 3