1. Overview of Digital Electronics
Embedded Systems
Overview of Digital Electronics

2. Binary Numbers and Others
System Radix Allowable Digits
Binary ,1
Octal ,1,2,3,4,5,6,7
Decimal ,1,2,3,4,5,6,7,8,9
Hexadecimal ,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F
(4021.2)5 = 4 x x x x x 5-1
4 x x (1/5)
= (511.4)10
(B65F)16 = x x x x 160
11 x x x
= (46687)10

3. Number Base Conversion
Conversion from Decimal fraction (0.6875)10 to Binary:
Integer Fraction Coefficient
x 2= a_1=1
x 2= a_2=0
x 2= a_3=1
x 2= a_4=1
(0.6875)10=(0.1011)2
The conversion from decimal fraction to any base-r system is similar to the example. Multiplication is by r instead of 2, and the coefficients found from the integers may range in value from 0 to r-1 instead of 0 and 1.
Example: (0.513)10=(?)8

4. Binary Numbers
Operations work similarly in all bases

5. Octal and Hexadecimal Numbers
The conversion from binary to octal and hexadecimal plays an important part in digital systems (computers). Since 23 = 8 and 24 = 16

6. Octal and Hexadecimal Numbers
Conversion from Octal to binary:
( ) 8 = ( ) 2
 Conversion from Hexadecimal to binary:
(306.D) 16 = ( ) 2 
Conversion from Hexadecimal to Decimal:
(37B) x x x 160
= 3 x x x 1 = = (891) 10

7. Complements Simplifies Two types exist the subtraction operation
Logical operations
Two types exist
The radix complement (r’s complement)
10’s complement, 2’s complement
The diminished radix complement ((r-1)’s complement)
9’s complement, 1’s complement

8. Diminished Radix (r-1) complement
Given a number N in base r having n digits:
(r-1)’s complement of N is (rn-1)-N
When r=10, (r-1)’s complement is called 9’s complement.
10n-1 is a number represented by n 9’s.
9’s complement of is (n=6)
=453299
9’s complement of is (n=6)
=987601

9. Diminished Radix (r-1) complement
Given a number N in base r having n digits:
(r-1)’s complement of N is (rn-1)-N
When r=10, (r-1)’s complement is called 9’s complement.
10n-1 is a number represented by n 9’s.
9’s complement of is (n=6)
=453299
9’s complement of is (n=6)
=987601

10. 1’s complement For binary numbers, r=2 and r-1=1.
1’s complement of N is (2n-1)-N
If n=4, 2n= So, 2n-1=1111.
To determine the 1’s complement of a number, subtract each digit from 1.
Or, bit flip!!! Replace 0’s with 1’s, 1’s with 0’s!!!
Example:
If N= , 1’s comp.=
If N= , 1’s comp.=

11. Radix (r’s) complement
Given a number N in base r having n digits:
r’s complement of N is rn-N
When r=10, r’s complement is called 10’s complement.
10’s complement of is (n=6)
=453300
10’s complement of is (n=6)
=987602

12. Radix (r’s) complement
The 10’s complement of decimal
2389 = (104 –1) – = 7611.
The 2’s complement of binary
= (26 –1) – =
012389= (106 –1) – =
246700= (106 –1) – =

13. Signed Binary Numbers 00001001 = 9
Example: Represent +9 and -9 in eight bit system
= 9
Signed magnitude  =-9
Signed-1’s complement 
Signed-2’s complement 

18. Two digital circuit types
Combinational digital circuits:
Consist of logic gates
Their current outputs are determined from the present combination of inputs.
Their operations can be specified logically by sets of Boolean functions.
Sequential digital circuits:
Employ storage elements in addition to logic gates.
Their outputs are a function of the inputs and the state of the storage elements.
Their outputs depend on current inputs and past input.
They have feedback connections.
Digital System Design
May 2, 2019

19. Combinational circuits
2n possible combinations of input values
Specific functions
Adders, subtractors, comparators, decoders, encoders, and multiplexers
MSI (Medium Scale Integration) circuits or standard cells
Digital System Design
May 2, 2019

20. Example Combinational Circuit (1/2)
Circuit controls the level of fluid in a tank
inputs are:
HI - 1 if fluid level is too high, 0 otherwise
LO - 1 if fluid level is too low, 0 otherwise
outputs are:
Pump - 1 to pump fluid into tank, 0 for pump off
Drain - 1 to open tank drain, 0 for drain closed
input to output relationship is described by a truth table
Digital System Design
May 2, 2019

21. Example Combinational Circuit (2/2)HI 1 LO
Pump x
Drain
Tank level is OK
Low level, pump more in
High level, drain some out
inputs cannot occur
Truth Table
Representation HI
Pump
Schematic
Representation
Drain LO
Digital System Design
May 2, 2019

22. Analysis of A Combinational Circuit
make sure that it is combinational not sequential
No feedback path
derive its Boolean functions (truth table)
design verification
a verbal explanation of its function
Ex. What is the output function of this circuit?
Digital System Design
May 2, 2019

23. Example Analysis Analysis steps Label all gate outputs with symbols
Find Boolean functions for all gates
Express functions in terms of input variables + simplify
Substitution:
F = (T2T3)’ = ((xT1)’(yT1)’)’ = (xT1)+(yT1) = x(xy)’+y(xy)’=
=(x(x’+y’)) + (y(x’+y’)) = xx’+xy’+yx’+yy’ = xy’+yx’ = x  y
T1=(xy)’
T2=(x T1)’
F=(T2T3)’
T3=(yT1)’
Digital System Design
May 2, 2019

24. Example (1/3) What are the output functions F1 and F2?
Digital System Design
May 2, 2019

25. Example (2/3)
Start with expressions that depend only on input variables:
T2 = ABC
T1 = A+B+C
F2 = AB + AC + BC
Express other outputs that depend on already defined internal signals
T3 = F2’T1
F1 = T3 + T2
Digital System Design
May 2, 2019

26. Example (3/3) Simplify: F1 = T3+T2 = F2’T1+ABC
= (AB+AC+BC)'(A+B+C)+ABC
= (A'+B')(A'+C')(B'+C')(A+B+C)+ABC
= (A'+B'C')(AB'+AC'+BC'+B'C)+ABC
= A'BC'+A'B'C+AB'C'+ABC
A full-adder
F1: the sum
F2: the carry
Digital System Design
May 2, 2019

27. Truth Table
Digital System Design
May 2, 2019

28. Binary Adders Addition is important function in computer system
What does an adder have to do?
Add binary digits
Generate carry if necessary
Consider carry from previous digit
Binary adders operate bit-wise
A 16-bit adder uses 16 one-bit adders
Binary adders come in two flavors
Half adder : adds two bits and generate result and carry
Full adder : also considers carry input
Two half adders make one full adder
Digital System Design
May 2, 2019

29. Binary Half Adder Specification: Outputs:
Design a circuit that adds two bits and generates the sum and a carry
Outputs:
Two inputs: x, y
Two output: S (sum), C (carry)
0+0=0 ; 0+1=1 ; 1+0=1 ; 1+1=10
The S output represents the least significant bit of the sum.
The C output represents the most significant bit of the sum or (a carry).
Digital System Design
May 2, 2019

30. Implementation of Half Adder
the flexibility for implementation
S=x  y
S = (x+y)(x'+y')
S' = xy+x'y'
S = (C+x'y')'
C = xy = (x'+y')'
S = x'y+xy'
C = xy
Half
Adder X Y S C
Digital System Design
May 2, 2019

31. Full-Adder Specification: Inputs: Truth table:
A combinational circuit that forms the arithmetic sum of three bits and generates a sum and a carry
Inputs:
Three inputs: x,y,z
Two outputs: S, C
Truth table:
Full
Adder X Y S Z C
Digital System Design
May 2, 2019

32. Implementation of Full Adder
S=x’y’z+ x’yz’ + xyz’ + xyz
C= xy + xz + yz
Digital System Design
May 2, 2019

33. Alternative Implementation of Full Adder
S = z (x  y)= z’(xy’+x’y) + z(xy’+x’y)’
= z’(xy’+x’y) + z(xy+x’y’)
=xy’z’+x’yz’+ xyz +x’y’z
C = x y + (x  y) z
=z(xy’ + x’y) + xy= xy’z+ x’yz+ xy
= xy + xz + yz
Digital System Design
May 2, 2019

34. Binary Adder
A binary adder is a digital circuit that produces the arithmetic sum of two binary numbers.
A binary adder can be implemented using multiple full adders (FA).
Digital System Design
May 2, 2019

35. Example: Add 2 binary numbers
Subscript i: 3 2 1
Input carry
Augend
Addend Ci Ai Bi
Sum
Carry Si
Ci+1
Digital System Design
May 2, 2019

36. Example:4-bit binary adder
4-bit Ripple Carry Adder
Classical example of standard components
Would require truth table with 29 entries! C A B S
Digital System Design
May 2, 2019

37. Carry Propagation
In any combinational circuit, the signal must propagate through the gates before the correct output is available in the output terminals.
Total propagation time = the propagation delay of a typical gateX the number of gate levels
The longest propagation delay time in a binary adder is the time it takes the carry to propagate through the full adders. This is because each bit of the sum output depends on the value of the input carry. This makes the binary adder very slow.
Digital System Design
May 2, 2019