Unit-1 Digital Design and Binary Numbers:

UNIT-1
Mohammad Asif Iqbal
Assistant Professor,
Deptt of ECE,
JETGI, Barabanki

Digital Signals
• Digital Signals have two basic states:
1 (logic “high”, or H, or “on”)
0 (logic “low”, or L, or “off”)
• Digital values are in a binary format.
Binary means 2 states.
• A good example of binary is a light (only
on or off)

Binary
Base 2 = Base 10
000 = 0
001 = 1
010 = 2
011 = 3
100 = 4
101 = 5
110 = 6
111 = 7
In Binary, there are only 0’s and 1’s. These numbers are
called “Base-2” ( Example: 0102)
We count in “Base-10”
(0 to 9)

Binary as a Voltage
• Voltages are used to represent logic values:
• A voltage present (called Vcc or Vdd) = 1
• Zero Volts or ground (called gnd or Vss) = 0
A simple switch can provide a logic high or a logic low.

A Simple Switch
• Here is a simple switch used to provide a logic value:
Vcc
Gnd, or 0
Vcc
Vcc, or 1
There are other ways to connect a switch.

• Converting to decimal from binary:
– Evaluate the power series
• Example
1 0 1 1 1 12
5 4 3 02 1
0*241*25 + 1*23 ++ 1*22 +
1*21 + 1*20 = 4710
Number systems

• Convert to decimal from binary:
– 1011011
a. 27
b. 91
c. 109
d. -109
e. 551
Number systems

Memorize the first ten powers of two
Review of Number systems

Copyright © 2008 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Review of Number systems

• Converting to binary from decimal:
– Divide the decimal number by 2 repeatedly.
– The remainder gives the digits of the binary number
7462
2
2
2
2
2
2
2
2
373 R 0
186 R 1
93 R 0
46 R 1
23 R 0
11 R 1
5 R 1
2
2 R 1
1 R 0
Number systems
10111010102

• Convert to binary from decimal:
–65
a.110101
b.101110
c.100001
d.100000
e.1000001
Number systems

• Shorthand for binary
• Binary digits are grouped into 4
– Start at the least significant
– If number of digits is not a multiple of 4, add zeros
• Each group is interpreted in decimal
• Digits above 9 are represented by the first six letter of the alphabet:
– 10: A; 11: B; 12: C; 13: D; 14: E; 15: F
• Example:
10111010102 = 0010 1110 10102
= 2EA16
Hexadecimals – Base 16

• Convert to hexadecimal from binary:
–1111111
a.771
b.177
c.F7
d.7F
e.127
Number systems

• Converting to decimal from hex:
• Example
2 E A 16
02 1
14*161
2*162 + 10*160+
= 74610
Hexadecimals – Base 16

• Convert to decimal from hexadecimal:
–65
a.65
b.101
c.86
d.100001
e.41
Number systems

• Same steps as for conversion as binary and hexadecimal and any other
base
• Converting to octal from decimal:
– Divide the decimal number by 8 repeatedly.
– The remainder gives the digits of the binary number
• Example: Convert 15310 to base 8.
Octal – Base 8

• Convert to octal from decimal:
15
a.71
b.177
c.F7
d.17
e.27
Number systems

• Converting to decimal from hex:
• Example
2 0 7 8
02 1
0*81
2*82 + 7*160+
= 13510
Octal – Base 16

Binary Addition
• Add one digit at a time
• Obtain a sum and a carry
• Similar to decimal addition – but pay attention to the
base

Binary Addition
• Add the following binary number
• 10011+11111
a. 110010
b. 001100
c. 101110
d. 021120
e. 010011

Binary Addition

Signed Numbers
• Signed numbers are mostly stored in two’s complements form
• Leading bit is 0 for positive numbers and 1 for negative
• For n bits, the range of numbers that can be stored is:
• -2n-1: 2n-1-1
• To derive the binary negative (two’s complement) of a number:
• Determine the magnitude (how many bits)
• Find the binary equivalent of the magnitude
• Complement each bit
• Add 1

Signed Numbers
• Example:
• Derive the 6-bit binary negative (two’s complement) of 17
• Determine the magnitude (how many bits)
• 6bits
• Find the binary equivalent of the magnitude
• 010001
• Complement each bit
• 101110
• Add 1
• 101111

Signed Numbers
• Derive the 5-bit binary negative (two’s
complement) of 17
a. 0101111
b. 101111
c. 10000
d. 01111
e. 01110

Overflow
• This occurs when the sum is out of range
• Example: for 4-bit numbers, the range is [- 8:7]
• Find the sum of +4 and +5
• Find the sum of -4 and -5
• Addition of two numbers of the opposite sign never produces
overflow
• Adding two same-signed numbers and obtaining a result of the
opposite sign indicates overflow

Overflow
• For each of the following problems, enter A if the result is an
overflow and B if it’s not. Assume the number of bits is 6
1. 15 + 17
2. -15 + 17
3. -15 -17
4. 2 - 3

Binary Subtraction
• Take the two complement of the second operand
• Then add
• For signed numbers:
• Ignore the carry-out of the higher order
• If two numbers of the same sign are added, and a result of the opposite sign
is obtained, there’s an overflow
• Ex: 7 – 5; -7 – 5
• For Unsigned number
• A carry-out of zero in the higher-order bit indicates overflow
• Ex: 5 - 7

Binary Subtraction
• What is the 5-bit binary representation of 8 -
15
a. 10111
b. 11000
c. 01001
d. 11001
e. overflow

Fractions
• Converting fractions to decimal from binary:
• Example
. 1 0 1 2
0*2-21*2-1 + 1*2-3+
= .62510
...3
1
2
1
1
1





  rarara

Fractions
• Convert .01112 to decimal
a. .875
b. .375
c. .4375
d. .0700
e. 4.375

• Converting to binary from decimal:
– Multiply the decimal number by 2 repeatedly.
– Use the integer part as the next digit each time, and then discard the
integer
– When the fraction part is zero, we have an exact conversion
– Add trailing zeroes to obtain the desired size
– For some fractions, we never get an exact conversion because the
fraction parts repeats, example: .3
.1.625*2 = 1.25
.25*2 = 0.50 .10
.101.5*2 = 1.00
Fractions

Examples
• Convert the following to base 2 : .7510
a. .111000
b. .000011
c. .110000
d. .111111
e. .101000

• Covert the integer and the fraction separately
• Example:
– 5.75 = 101.11
Mixed Numbers

Examples
• Convert the following to base 10 : 11.011002
a. 3.7500
b. 3.0300
c. 3.1875
d. 3.0300
e. 3.3750

• Computer storage
– The standard notation (IEEE Standard 754) for 32 bit numbers is:
• A sign bit: 1 for negative and 0 for positive
• An 8-bit exponent
– Stored as the binary version of 127+exponent
– Can store -126:127 as 1:254
• 23 bits for the significant digits
• The first significant digit is always a binary 1 so this is not stored
• Example: -27.875
• 27.875 = 11011.111 = 1.1011111*24
One sign bit – 1 if –ve, 0 otherwise
8 exponent bits 32 bits for significant digits
1 1000011 10111110000000000000000
Mixed Numbers

Computer Storage
• How would the number 2.1 be stored in
IEEE Standard 754 for 32 bit numbers
a. 1 10000001 01100110011001100110000
b. 0 10000000 00001100110011001100110
c. 0 00000001 10000110011001100110011
d. 1 10000000 10000000000000000000000
e. Can’t be stored

Logic Gates
• Basic Digital logic is based on 3 primary functions
(the basic gates):
• AND
• OR
• NOT

The AND function
• The AND function:
• If all the inputs are high is the output is high
• If any input is low, the output is low
• “If this input AND this input are high, the
output is high”

AND Logic Symbol
Inputs
Output
If both inputs are 1, the output is 1
If any input is 0, the output is 0

AND Logic Symbol
Inputs
Output
Determine the output
0
0
0

AND Logic Symbol
Inputs Output
0
1
0

AND Logic Symbol
Inputs
Output
1
1
1

AND Truth Table
• To help understand the function of a digital device, a
Truth Table is used:
Input Output
0 0 0
0 1 0
1 0 0
1 1 1
AND Function
Every possible input
combination

AND Gates
• It is possible to have AND gates with more than
2 inputs. The same logic rules apply – “if any
input…”

The OR function
• The OR function:
• if any input is high, the output is high
• if all inputs are low, the output is low
• “If this input OR this input is high, the
output is high”

OR Logic Symbol
Inputs
Output
If any input is 1, the output is 1
If all inputs are 0, the output is 0

OR Logic Symbol
Inputs
Output
0
0
0

OR Logic Symbol
Inputs
Output
0
1
1

OR Logic Symbol
Inputs
Output
1
1
1

OR Truth Table
• Truth Table
Input Output
0 0 0
0 1 1
1 0 1
1 1 1
OR Function

The NOT function
• The NOT function:
• If any input is high, the output is low
• If any input is low, the output is high
• “The output is the opposite state of the input”
• The NOT function is often called INVERTER

NOT Logic Symbol
Input
Output
If the input is 1, the output is 0
If the input is 0, the output is 1

NOT Logic Symbol
Input
Output
0 1

NOT Logic Symbol
Input
Output
1 0

OR (written as +)1
a + b (read a OR b) is 1 if and only if a = 1 or b = 1 or both
AND (written as  or simply two variables catenated)
a  b = ab (read a AND b) is 1 if and only if a = 1 and b = 1.
NOT (written)
a (read NOT a) is 1 if and only if a = 0
Summary

Unit-1 Digital Design and Binary Numbers:

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