1. Digital Logic and Computer Organization
Fatma A. El-Licy ISSR_CS504
Digital Logic and Computer
Organization
Lecture One

2. Fatma A. El-Licy ISSR_CS504
Outline
Introduction
Number System
Truth tables
Design process for combinational systems
Fatma A. El-Licy ISSR_CS504
May 31, 2018

3. Fatma A. El-Licy ISSR_CS504
Introduction
Digital Logic and Computer Organization is the science of utilizing logic design in organizing digital systems, including computer systems.
A Digital System is a one in which all signals are represented by discrete values. Computers, calculators and most electronic systems contains large amount of digital logic.
Digital systems usually operate with two-valued signals (0 & 1).
Fatma A. El-Licy ISSR_CS504
May 31, 2018

4. Fatma A. El-Licy ISSR_CS504
Digital
System .
n Inputs A B W X
m outputs
Fatma A. El-Licy ISSR_CS504
May 31, 2018

5. Digital Systems are Binary
Inputs and outputs of a digital system represent real quantities, binary (two-valued), or otherwise encoded multi-valued.
Multi-valued inputs (decimal, character,…,etc.) must be represented by a set of binary digits (bits), which we call coding the inputs into binary.  
Fatma A. El-Licy ISSR_CS504
May 31, 2018

6. Fatma A. El-Licy ISSR_CS504
Example
A System with three inputs A, B and C and one output Z, such that: Z =1 iff two of the inputs are 1.
Fatma A. El-Licy ISSR_CS504
May 31, 2018

7. Typical solution DesignB C Z 1
0/1
Fatma A. El-Licy ISSR_CS504
May 31, 2018

8. Typical solution DesignB C Z 1
1/0
Fatma A. El-Licy ISSR_CS504
May 31, 2018

9. Fatma A. El-Licy ISSR_CS504
Examples
A system with eight inputs representing two 4-bit binary numbers and one 5-bit output, representing the sum.
A system with one input A and a clock, and one output Z, which is 1 iff the input was one at the last three consecutive clock times.
A traffic controller, in the simplest case, there are just two streets. The light is green on each street for a fixed period of time. It then goes to yellow for another fixed period and finally it goes o red. There are no inputs to this system other than the clock. There are six output, one for each color in each direction (or three for each street).
Fatma A. El-Licy ISSR_CS504
May 31, 2018

10. Fatma A. El-Licy ISSR_CS504
Fill in the Spaces
are Combinational examples, the output depends only on the present values.
are Sequential (needs memory), the output depends on values/states at earlier time.
Fatma A. El-Licy ISSR_CS504
May 31, 2018

11. Natural Language vs Logic
Natural Language: Is not very precise!!! See example 1
We need a more precise description for logic systems, and that is what we will develop for combinational systems.
Fatma A. El-Licy ISSR_CS504
May 31, 2018

12. Fatma A. El-Licy ISSR_CS504
Number Systems
Integers are normally written using positional number system, where, each digit represents the coefficient in a power series
N = dn* Rn + dn-1 * Rn-1 +…+ d0 * R0 (1)
Where
n is number of digits,
R is the radix/base and di are the coefficients such that  di  R
Fatma A. El-Licy ISSR_CS504
May 31, 2018

13. Fatma A. El-Licy ISSR_CS504
Examples
The decimal number 2473)10
= 2* * * * 100 =
= 2473)10
The Binary Number )2
= 125+ 0  24 +1 23 +122 +12+1 =
= 47)10
Fatma A. El-Licy ISSR_CS504
May 31, 2018

14. Fatma A. El-Licy ISSR_CS504
An n-bit number can represent the positive integers from 0 to 2n –1.
A 4-bit numbers have the range
A 8-bit numbers have the range
Let us try to use the power series to convert decimal to binary
745) 10 = 111  (1010)  = ?!!!
Fatma A. El-Licy ISSR_CS504
May 31, 2018

15. Decimal to Binary Conversion
Anther method to compute the binary equivalent of a decimal number
Divide the decimal number by the radix (2 in this case)
Keep the remainder,
Repeat this process until the result of the division is zero
The sequence of digits obtained (the remainders) arranged from right to left is the number in the radix (binary) system.
Fatma A. El-Licy ISSR_CS504
May 31, 2018

16. Fatma A. El-Licy ISSR_CS504
Example 1
Convert decimal to decimal: )10
  )10
Fatma A. El-Licy ISSR_CS504
May 31, 2018

17. Fatma A. El-Licy ISSR_CS504
Example 2
Convert decimal to binary: 9)10 = ??) ) ) ) )2 0 9)10 = 1001)2
Fatma A. El-Licy ISSR_CS504
May 31, 2018

18. Fatma A. El-Licy ISSR_CS504
Example 3
Convert decimal to binary: 41)10 = ???)2 )2 )2 )2 )2 )2 )2 00
41)10 = )2
Fatma A. El-Licy ISSR_CS504
May 31, 2018

19. Fatma A. El-Licy ISSR_CS504
Example 4
Convert decimal to binary: )10 = ???)2 )2
000
Fatma A. El-Licy ISSR_CS504
May 31, 2018

20. Fatma A. El-Licy ISSR_CS504
Example 4
Convert decimal to binary: )10 = ???)2 )2 1
Fatma A. El-Licy ISSR_CS504
May 31, 2018

21. Fatma A. El-Licy ISSR_CS504
Exercise
Convert the following decimal into Binary numbers:
a. 1000
b. 512
c. 217
d. 177
e. 105
Fatma A. El-Licy ISSR_CS504
May 31, 2018

22. Conversion from Binary to Decimal
Use formula (1), with the radix equals to 2.
Example (Binary to Decimal): 1001)2= ???)10
1001)2= 1     20
= 1     1 =
= 9)10
Convert the following Binary numbers to Decimal:
a b c d
Fatma A. El-Licy ISSR_CS504
May 31, 2018

23. Conversion from Binary to Decimal
Use formula (1), with the radix equals to 2.
Example (Binary to Decimal): 1001)2= ?)10
)2= 1     20
= 1     1 =
= 9)10
Fatma A. El-Licy ISSR_CS504
May 31, 2018

24. Fatma A. El-Licy ISSR_CS504
Octal and Hexadecimal
Octal (r=8) and Hex (r=16) are a shorthand notations for binary numbers that are commonly used in computer documentations.
In Octal, binary digits are grouped in threes.
Example: a 9-bit number N
Fatma A. El-Licy ISSR_CS504
May 31, 2018

25. Octal to Binary Conversion
A 9-bit number N N = (b b b6 26)+ (b b b3 23 )+ (b2 22+ b b0 ) = 26 (b b7 2 + b6)+ 23(b b4 2 + b3)+ (b2 22+ b b0 ) = 82o2 + 8 o1 + o0 where oi is an octal digits such that 0  oi  8
Fatma A. El-Licy ISSR_CS504
May 31, 2018

26. Fatma A. El-Licy ISSR_CS504
Examples
)2 = = )8
= 1352)8
1352)8 = 1    = = 746)10
746)10 = ???)8
either, first convert decimal to binary then group up the binary to octal numbers, or
(simpler) convert to octal by repetitive division by 8.
Fatma A. El-Licy ISSR_CS504
May 31, 2018

27. Fatma A. El-Licy ISSR_CS504
Example
) ) ) )8 746)10= 1352)8 = )2
Fatma A. El-Licy ISSR_CS504
May 31, 2018

28. Fatma A. El-Licy ISSR_CS504
Hexadecimal Numbers
Hexadecimal (R=16) groups bits by 4’s.
Hex is the most common representation, since most computer word size are multiple of 4.
Each digit is in the range 0-15 (0..9, A, B, C, D, E, F)
Examples:
)2= )2 = 2EA)16
2EA)16
= 2  E  16 + A = 
= 746)10
Fatma A. El-Licy ISSR_CS504
May 31, 2018

29. Fatma A. El-Licy ISSR_CS504
Binary Addition  
Fatma A. El-Licy ISSR_CS504
May 31, 2018

30. Fatma A. El-Licy ISSR_CS504
One-bit Adder
Cin a b Cout S
a b Cin
Cout S
Fatma A. El-Licy ISSR_CS504
May 31, 2018

31. Fatma A. El-Licy ISSR_CS504
FULL ADDER
A device that does this 1-bit computation is referred to as a Full Adder.
Fatma A. El-Licy ISSR_CS504
May 31, 2018

32. Fatma A. El-Licy ISSR_CS504
FULL ADDER a4 b4 a3 b3 a2 b2 a1 b1
Full
Adder S4 c4 c3 S3 c2 S2 c1 S1
Fatma A. El-Licy ISSR_CS504
May 31, 2018

33. Full Adder a4 b4 a3 b3 a2 b2 a1 b1 S4 c4 c3 S3 c2 S2 c1 S1
If ‘1’, it Means Subtraction as we get the 2’s Complement a4 b4 a3 b3 a2 b2 a1 b1
Full
Adder S4 c4 c3 S3 c2 S2 c1 S1
Fatma A. El-Licy ISSR_CS504
May 31, 2018

34. Fatma A. El-Licy ISSR_CS504
Signed Numbers
Singed magnitude: -5 = 1101, 5= 0101
3+5 = = =
add  design an Adder
Subtract  design a subtracter
which is bigger?  design a comparator
Fatma A. El-Licy ISSR_CS504
May 31, 2018

35. Fatma A. El-Licy ISSR_CS504
Add/Subtract Numbers
Better find a simpler method to represent signed numbers, and better yet, to perform all these computations using one design!!!
Signed numbers are stored in tow’s complement
Positive numbers have the leading bit zero.
A Negative number, -a, is stored as the binary equivalent of 2n - a (n = word size)
Fatma A. El-Licy ISSR_CS504
May 31, 2018

36. Negative Numbers and two’s Complement
A Negative number, -a, is stored as the binary equivalent of 2n - a (n = word size)
Let n= 5 (word size = 5)
Let a= 6)10 = 110)2
a = - 110)2 = 11010)2’s omplement
Apply the rule: 2n - a = 25 – 110
100000
11010
Fatma A. El-Licy ISSR_CS504
May 31, 2018

37. Fatma A. El-Licy ISSR_CS504
RADIX COMPLEMENT
5: : :
-5: : : 0000
Fatma A. El-Licy ISSR_CS504
May 31, 2018

38. Fatma A. El-Licy ISSR_CS504
Carry Out Vs Overflow
The Carry out of the most significant bit is ignored in two’s complement addition
Overflow occurs when the sum is out of range, i.e., -2n-1 > sum> 2n-1 –1
Fatma A. El-Licy ISSR_CS504
May 31, 2018

39. Fatma A. El-Licy ISSR_CS504
Code Systems
BCD: Binary Coded Decimal
EBCDIC: Extended Binary Coded Decimal Interchangable Code
Unicode: Unified Code
Gray Code
Hamming Code
Fatma A. El-Licy ISSR_CS504
May 31, 2018

40. Fatma A. El-Licy ISSR_CS504
BCD
Decimal
Digit
8421
Code
5421
2421
Excess 3
2 of 5
0000
0011
11000 1
0001
0100
10100 2
0010
0101
10010 3
0110
10001 4
0111
01100 5
1000
1011
01010 6
1001
1100
01001 7
1010
1101
00110 8
1110
00101 9
1111
00011
Fatma A. El-Licy ISSR_CS504
May 31, 2018

41. Fatma A. El-Licy ISSR_CS504
Gray Code
Number
Gray Code
0000 8
1100 1
0001 9
1101 2
0011 10
1111 3
0010 11
1110 4
0110 12
1010 5
0111 13
1011 6
0101 14
1001 7
0100 15
1000
Fatma A. El-Licy ISSR_CS504
May 31, 2018

42. Fatma A. El-Licy ISSR_CS504
Hamming Code
It is a single error-correction code.
Check bits are added to the information bits, so that if, at most, one error occurs during transmission or storage, the original value can be restored.
The value of the check bit is chosen so that the total number of ones in the bits selected (row) is even.
Check bits are the bits that are power of 2, starting from 1 (1, 2, 4, 8)
The pattern of checking is as follows:
Fatma A. El-Licy ISSR_CS504
May 31, 2018

43. Fatma A. El-Licy ISSR_CS504
D D D D
C C D C D D D a1 a2 a3 a4 a5 a6 a7
Bit 1
check x
Bit 2
Bit 4
Hamming Code
Fatma A. El-Licy ISSR_CS504
May 31, 2018

44. Fatma A. El-Licy ISSR_CS504
a1= a3  a5  a7
a2= a3  a6  a7
a4 = a5  a6  a7
a1 a a3 a4 a5 a6 a7
Example:
Data Hamming Code a1 a2 a3
0100
1011
1111
C C C
Fatma A. El-Licy ISSR_CS504
May 31, 2018

45. Fatma A. El-Licy ISSR_CS504
a1= a3  a5  a7
a2= a3  a6  a7
a4 = a5  a6  a7
a1 a2 a3 a4 a5 a6 a7
Example:
Data Hamming Code a1 a2 a3
1011
1111
C C C
Fatma A. El-Licy ISSR_CS504
May 31, 2018

46. Fatma A. El-Licy ISSR_CS504
a1= a3  a5  a7
a2= a3  a6  a7
a4 = a5  a6  a7
a1 a2 a3 a4 a5 a6 a7
Example:
Data Hamming Code a1 a2 a3
1111
Fatma A. El-Licy ISSR_CS504
May 31, 2018

47. Fatma A. El-Licy ISSR_CS504
a1= a3  a5  a7
a2= a3  a6  a7
a4 = a5  a6  a7
a1 a2 a3 a4 a5 a6 a7
Example:
Data Hamming Code a1 a2 a3
C C C
Fatma A. El-Licy ISSR_CS504
May 31, 2018

48. Fatma A. El-Licy ISSR_CS504
when a word is received, the same bits are checked:
e1= a1  a3  a5  a7 = 0 iff no error
e2= a2  a3  a6  a7 = 0 iff no error
e4= a4  a5  a6  a7 = 0 iff no error
If one error occurs, the check produce the bit number, calculated as follows:
4*e4 + 2*e2 +e1
With n check bits (n  2) there can be 2n – n-1 bits of info.
Fatma A. El-Licy ISSR_CS504
May 31, 2018

49. Fatma A. El-Licy ISSR_CS504
Examples
Received:  
e1= 0; e2 = 1; e4= 0
4*0 + 2 *1 + 0= 2;
Bit 2 in error
The correct word is
The data is 1011
Received:
e1= 1; e2 = 0; e4= 1; 4*1 + 2* = 5
Bit 5 is in error
The correct word is
The data is 0001
Fatma A. El-Licy ISSR_CS504
May 31, 2018