DIGITAL CIRCUIT
DESIGN
EEE122 A
Ref. Morris MANO & Michael D. CILETTI
DIGITAL DESIGN 4th edition
Fatih University- Faculty of Engineering- Electric and Electronic Dept.
Signed Numbers
Digital systems, such as computer,
must be able to handle both positive
and negative numbers.
A signed binary number consists of
both sign and magnitude information.
• The sign indicates whether a number is
positive or negative.
• The magnitude is the value of the
number.
Signed Numbers
There are 3 forms in which signed
integer numbers can be represented
in binary:
• Sign-magnitude (least used)
• 1’s complement
• 2’s complement (most used)
The left-most bit in a signed binary number is the sign bit.
It tells whether the number is positive (sign bit = 0) or
negative (sign bit = 1).
Sign-Magnitude Form
The left-most bit is the sign bit and the remaining bits
are the magnitude bits.
• The magnitude bits are in true binary for both positive
and negative numbers.
ex: the decimal number +25 is expressed as an 8-bit
signed binary number as:
00011001
While the decimal number -25 is expressed as
10011001
“ In the sign-magnitude form, a negative number has
the same magnitude bits as the corresponding
positive number but the sign bit is a 1 rather than a 0.
“
1’s Complement Form
Positive numbers in 1’s complement
form are represented the same way
as the positive sign-magnitude.
Negative numbers are the 1’s
complements of the corresponding
positive numbers.
ex: the decimal number +25 is expressed as:
00011001
While the decimal number -25 is expressed as
11100110
2’s Complement Form
Positive numbers in 2’s complement
form are represented the same way as
the positive sign-magnitude and 1’s
complement form.
Negative numbers are the 2’s
complements of the corresponding
positive numbers.
ex: the decimal number +25 is expressed as:
00011001
While the decimal number -25 is expressed as
11100111
Decimal Value of Signed Numbers
Sign-magnitude: Both positive and negative
numbers are determined by finding the equivalent
decimal to the binary number excluding the sign bit.
The sign bit will determine the final sign.
Sign-magnitude
ex: decimal values of these numbers (expressed
in sign-magnitude)
1) 10010101
2) 01110111
magnitude
magnitude
26 25 24 23 22 21 20
0 0 1 0 1 0 1
= 16+4+1 = 21
sign
= 1 negative
Hence: 10010101 = -21
26 25 24 23 22 21 20
1 1 1 0 1 1 1
= 64+32+16+4+2+1 = 119
sign
= 0 positive
Hence: 01110111 = 119
Decimal Value of Signed Numbers
1’s complement:
• Positive – determined by summing the
weights in all bit positions where there
are 1s and ignoring those positions where
there are 0s.
• Negative – determined by assigning a
negative value to the weight of the sign
bit, summing all the weights where there
are 1’s, and adding 1 to the result.
Decimal Value of Signed Numbers
1’s complement (by example)
ex: decimal values of these numbers (expressed
in 1’s complement)
1) 00010111
2) 11101000
-27 26 25 24 23 22 21 20
0 0 0 1 0 1 1 1
= 16+4+2+1 = +23
-27 26 25 24 23 22 21 20
1 1 1 0 1 0 0 0
= (-128)+64+32+8 = -24
+1
Hence: 00010111 = +23
Hence: 11101000 = -23
Decimal Value of Signed
Numbers
2’s complement:
• Positive – determined by summing the
weights in all bit positions where there
are 1s and ignoring those positions
where there are 0s.
• Negative – the weight of the sign bit in
a negative number is given a negative
value.
Decimal Value of Signed Numbers
2’s complement (by example)
ex: decimal values of these numbers (expressed
in 2’s complement)
1) 01010110
2) 10101010
1) 01010110
2) 10101010
-27 26 25 24 23 22 21 20
0 1 0 1 0 1 1 0
-27 26 25 24 23 22 21 20
1 0 1 0 1 0 1 0
= 64+16+4+2 = +86
Hence: 01010110 = +86
= (-128)+32+8+2 = -86
Hence: 10101010 = -86
Range of Signed Integer Numbers
The magnitude range of a binary number depends on the number of bits (n).
Total combinations = 2n
• 8 bits = 256 different numbers
• 16 bits = 65,536 different numbers
• 32 bits = 4,294,967,296 different numbers
For 2’s complement signed numbers:
• Range = -(2n-1) to +(2n-1-1)
• where there is one sign bit and n-1 magnitude
ex:
Negative
Boundary
Positive
Boundary
4 bits
-(23) = -8
(23-1) = +7
8 bits
-(27) = -128
(27-1) = +127
16 bits
-(215) = -32,768
(215-1) = +32767
Arithmetic Operations with
Signed Numbers
Because the 2’s complement form for
representing signed numbers is the
most widely used in computer
systems. We’ll limit to 2’s
complement arithmetic on:
• Addition
• Subtraction
Multiplication (same as multiple addition)
Division (same as continuous subtraction)
Addition
4 cases may occur when 2 signed
numbers are added:
• Both numbers positive
• Positive number with magnitude larger
than negative number
• Negative number with magnitude larger
than positive number
• Both numbers negative
Addition
Both numbers positive:
ex:
00000111
+00000100
00001011
7
+4
11
The sum is positive and is therefore in true
(un complemented) binary.
Positive number with magnitude larger than
negative number:
ex:
00001111
15
+11111010
+ -6
Discard
carry
1 00001001
9
The final carry bit is discarded. The sum is positive
and is therefore in true (un complemented)
binary.
Addition
Both numbers negative:
ex:
11111011
+11110111
Discard
1 11110010
carry
-5
+ -9
-14
The final carry bit is discarded. The sum is negative
and therefore in 2’s complement form.
Negative number with magnitude larger than
positive number:
ex:
00010000
16
+11101000
+ -24
11111000
-8
The sum is negative and therefore in 2’s
complement form.
Addition
Overflow condition:
• When two numbers are
added and the number of
bits required to represent
the sum exceeds the
number of bits in the two
numbers, an overflow
results as indicated by an
incorrect sign bit.
• An overflow can occur only
when both numbers are +
or -.
ex: 01111101
+00111010
10110111
Magnitude
incorrect
Sign
incorrect
125
+ 58
183
Addition
Numbers are added two at a time:
• Computer add strings of numbers two
numbers at a time.
ex:
add the signed numbers: 01000100, 00011011,
00001110, and
00010010
68
01000100
+ 27
+ 00011011
Add 1st two numbers
95
01011111
1st sum
+ 14
+ 00001110
Add 3rd number
109
01101101
2nd sum
+ 18
+ 00010010
Add 4th number
127
01111111
Final sum
Subtraction
Subtraction is a special case of
addition.
• Subtraction is addition with the sign of
the subtrahend changed.
• The result of a subtraction is called the
difference.
The sign of a positive or negative
binary is changed by taking it’s 2’s
complement.
Subtraction
Since subtraction is simply an
addition with the sign of the
subtrahend changed, the process is
stated as follows:
• To subtract two signed numbers, take
the 2’s complement of the subtrahend
and add. Discard any final carry.
Subtraction
ex: Perform each of the following subtraction of the signed
numbers:
(a) 00001000 – 00000011
(c) 11100111 – 00010011
(b) 00001100 – 11110111
(d) 10001000 - 11100010
(a) 00001000
+11111101
100000101
+
(b) 00001100
+00001001 +
00010101
(c) 11100111
+11101101
111010100
-25
+ -19
-44
8
-3
5
12
9
21
(d) 10001000
-120
+00011110 +
30
10100110
-90
The Digital Codes
Binary Coded Decimal (BCD)
BCD is a numerical code that normally
used as interface between user and
computer system to simplify data
handling.
The 8421 code is a well known type of
BCD to deal with the situation.
Thus
BCD code advantages include providing
an excellent interface to :
• Keypad inputs
• Digital readouts
Binary Coded Decimal
Decimal
Digit
0
1
2
3
4
5
6
7
8
9
BCD
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
Note: 1010, 1011, 1100, 1101, 1110, and 1111 are not used in BCD
DEC-to-BCD
(a) 35
00110101
(b) 170
000101110000
BCD-to-DEC
(a)10000110
(86)
(b)1001010001110000 (9470)
BCD Addition
BCD is a decimal number coded in binary.
In arithmetic operations, it behave as
normal decimal numbers with some
correction rules. The steps below show how
to add two BCD numbers:
• Add the two BCD numbers, using the rules for
basic binary addition.
• If a 4-bit sum is equal to or less than 9, it is a
valid BCD number.
• If a 4-bit sum > 9, or if a carry out of the 4-bit
group is generated it is an invalid result. Add 6
(0110) to the 4-bit sum in order to correct
result and return the code to 8421. If a carry
results when 6 is added, simply add the carry to
the next 4-bit group.
BCD Addition
Example:carry out the following BCD
addition
(a) 0101 5
+0010 +2
0111 7
(c) 1000 8
+1001 +9
Binary sum 10001 17
Add 6
+ 0110
BCD sum 10111
(b)00100011 23
+ 00010101 +15
00111000 38
1
(d) 000110000100
+010101110110
011100001010
01100110
011101101000
184
+576
760
The Gray Code
The Gray code is unweighted and is
not an arithmetic code.
• There are no specific weights assigned
to the bit positions.
Important: the Gray code exhibits
only a single bit change from one
code word to the next in sequence.
• This property is important in many
applications, such as shaft position
encoders.
The Gray Code
Decimal Binary Gray Code
0
1
2
3
4
5
6
7
0000
0001
0010
0011
0100
0101
0110
0111
0000
0001
0011
0010
0110
0111
0101
0100
Decimal
Binary
Gray Code
8
1000
1100
9
1001
1101
10
1010
1111
11
1011
1110
12
1100
1010
13
1101
1011
14
1110
1001
15
1111
1000
The Gray Code
Binary-to-Gray code conversion
• The MSB in the Gray code is the same as
corresponding MSB in the binary number.
• Going from left to right, add each adjacent
pair of binary code bits to get the next Gray
code bit. Discard carries.
ex: convert 101102 to Gray code
1 + 0 + 1 + 1 + 0
binary
1
Gray
1
1
0
1
The Gray Code
Gray-to-Binary Conversion
• The MSB in the binary code is the same as
the corresponding bit in the Gray code.
• Add each binary code bit generated to the
Gray code bit in the next adjacent
position. Discard carries.
ex: convert the Gray code word 11011 to
binary
1
+
1
1
+
0
0
+
0
1
+
1
1
Gray
0
Binary
ASCII
ASCII has 128 characters and
symbols represented by a 7-bit
binary code.
• It can be considered an 8-bit code
with the MSB always 0. (00h-7Fh)
00h-1Fh (the first 32) – control
characters
20h-7Fh – graphics symbols (can be
printed or displayed)
ASCII
CODE
Table
Binary Storage and Registers
A binary cell: is a device that have the ability to
store two stable binary information (0 and 1), the
input to the cell receives an excitation signal and
set its output to either 0 or 1.
Register: is a group of n digital storage elements
capable of storing n bits of binary information
Register Transfer:
For large systems, digital circuit are designed using
modular approach, so the system is partitioned into
subsystems, that performs some functional task.
Registers are assumed to be the basic element of
the digital system, the information flow and the
processing performed are through register
Binary
Storage
and
Registers
Binary
Storage
and
Registers
Binary
Information
processing
through
registers
Binary Logic
Binary Logic deals with variables that take two discrete values and
with operations that assume logical meaning.
It consist of binary variables (x,y,z,...A,B,C,..) and a set of binary
operations. Each variable can have only one of two possible values
(True/False) or (Yes/No) or (1/0)
There are three basic operations: AND, OR, and NOT.
AND: z=x AND y or x.y or xy; Z is true only if both x and y are true
OR:
z=x OR y, x+y; Z is true if at least one of the variables is true
NOT: z=NOT x, x ,x’; its a complement operation
Logic Operation is not
mathematic operation
Truth tables are used to
express all possible
combination of the input
variables and their
corresponding output logic
Logic Gates
Logic gates are electronic circuits that
operate on one or more input signals to
produce an output signal
Logic Gates
Inpu/Output signal for gates: