Introduction to Digital Electronics

1. Introduction to Digital Electronics

2. Learning Objectives • To introduce analog and digital system • Combinational circuit • Identify the basic gates and describe the behavior of each • Combine basic gates into circuits • Adders • Multiplexer and de multiplexer • Encoder and decoder • Sequential circuit • Latch and flip flop • Types of flip flop

3. Analog and Digital To be transmitted, data must be transformed to electromagnetic signals. • Data can be analog or digital. • The term analog data refers to information that is • continuous; digital data refers to information that has discrete states.

4. Analog and Digital Data and Signal • Analog and digital data • Data can be analog or digital. • Analog data are continuous and take continuous values. • Digital data have discrete states and take discrete values. • Signals can be analog or digital. • Analog signals can have an infinite number of values in a range. • Digital signals can have only a limited number of values.

5. Comparison of Analog and Digital Signals

6. Digital Electronics • Digital Electronics represents information (0, 1) with only two discrete values. • Ideally “no voltage” (e.g., 0v) represents a 0 and “full source voltage” (e.g., 5v) represents a 1 • Realistically “low voltage” (e.g., <1v) represents a 0 and “high voltage” (e.g., >4v) represents a 1 • We achieve these discrete values by using switches. • We use transistor switches, which operates at high speed, electronically, a small in size.

7. Electronic Aspects of Digital Design • How we represent digital information in electronic devices? • By discrete voltages.

8. What is the Basic Digital Elementin Electronics? • a Switch

9. Using Switch to Represent Digital Information

10. Digital Abstraction • It is difficult to make ideal switches means a switch is completely ON or completely OFF. • So, we impose some rules that allow analog behavior to be ignored in most cases, so circuits can be modeled as if they really did process 0s and 1s, known as digital abstraction. • Digital abstraction allows us to associate a noise margin with each logic values (0 and 1).

11. Logic levels • Undefined regionis inherent • digital, not analog • Switching threshold varies with voltage, temp • need “noise margin” • Logic voltage levels decreasing with new processors. • 5 , 3.3 , 2.5 , 1.8 V

12. Analog versus Digital • Analog systems process time-varying signals that can take on any value across a continuous range of voltages (in electrical/electronics systems). • Digital systems process time-varying signals that can take on only one of two discrete values of voltages (in electrical/electronics systems). • Discrete values are called 1 and 0 (ON and OFF, HIGH and LOW, TRUE and FALSE, etc.)

13. Representing Information Electronically • “Analog electronics” deals with non-discrete values • “Digital electronics” deals with discrete values

14. Benefits of Digital over Analog • Reproducibility • Not effected by noise means quality • Ease of design • Data protection • Programmable • Speed • Economy

15. Digital Revolution • Digital systems started back in 1940s. • Digital systems cover all areas of life: • still pictures • digital video • digital audio • telephone • traffic lights • Animation

16. Basic terminology • Gate • A device that performs a basic operation on electrical signals • Circuits • Gates combined to perform more complicated tasks • How do we describe the behavior of gates and circuits? • Boolean expressions • Uses Boolean algebra, a mathematical notation for expressing two-valued logic • Logic diagrams • A graphical representation of a circuit; each gate has its own symbol • Truth tables • A table showing all possible input value and the associated output • values

17. circuit Input(s) Output(s) memory Circuits • Circuits can be Combinational or Sequential • Combinational logic circuits produce a specified output (almost) at the instant when input values are applied. • The addition of a memory device to a combinational circuit allows the output to be fed back into the input: Sequential circuit Combinational circuit Sequential circuit

18. Digital Devices • Combinational circuit • Gates • Multiplexer • Demultiplexer • Adders • Encoder • Decoder • Sequential circuit • Flip-Flops • Registers • Counters

19. Combinational Circuits

20. Overview • Gates • Iterative combinational circuits • Binary adders • Half and full adders • Ripple carry • Binary subtraction • Binary adder-subtractors • Signed binary numbers • Signed binary addition and subtraction • Overflow

21. Combinational Circuits Combinational logic circuits produce a specified output (almost) at the instant when input values are applied.

22. Gates • The most basic digital devices are called gates. • Gates got their name from their function of allowing or blocking (gating) the flow of digital information. • A gate has one or more inputs and produces an output depending on the input(s). • A gate is called a combinational circuit. • Three most important gates are: AND, OR, NOT

23. Digital Logic • Binary system -- 0 & 1, LOW & HIGH, negated and asserted. • Basic building blocks -- AND, OR, NOT

24. NOT Gate • A NOT gate accepts one input signal (0 or 1) and returns the o • pposite signal as output

25. AND Gate • An AND gate accepts two input signals • If both are 1, the output is 1; otherwise the output is 0

26. OR Gate • An OR gate accepts two input signals • If both are 0, the output is 0; otherwise, the output is 1

27. XOR Gate • An XOR gate accepts two input signals • If both are the same, the output is 0; otherwise, the output is 1

28. XOR Gate • Note the difference between the XOR gate and the OR • gate; they differ only in one input situation • When both input signals are 1, the OR gate produces a • 1 and the XOR produces a 0 • XOR is called the exclusive OR

29. NAND Gate • The NAND gate accepts two input signals • If both are 1, the output is 0; otherwise, the output is 1

30. NOR Gate • The NOR gate accepts two input signals • If both are 0, the output is 1; otherwise, the output is0

31. De Morgan again • A NAND gate: • Y = A.B = A + B • is the same as an OR gate with two NOT gates • Similarly a NOR gate is the same as an AND gate with two inverters • Y = A + B = A.B

32. Dual gates

33. Truth Tables and Boolean Notation • NAND Gate Representation • It is possible to implement any boolean expression using only NAND gates NOT X X AND A.B A A.B B OR A A+B B

34. Truth Tables and Boolean Notation • NAND Gate representation • Implement the following circuit using only NAND gates x2 x4 x3

35. x2 x4 x3 Solution • Dual the gates, remember two nots together can be removed. A A.B A A.B A+B B B AND feeding OR

36. Exercise • Implement NOT, AND and OR using NOR gates • Example AND gate dual circuit:

37. X X Solution • Similar pattern to using NAND gates (not surprising) • NOT • AND • OR X X A.B A A A.B B A.B B A A+B A.B A A+B B B

38. Logic Gates NAND and NOR are known as universal gates because they are inexpensive to manufacture and any Boolean function can be constructed using only NAND or only NOR gates.

39. Truth Tables and Boolean Notation • NOR Gate representation • It is also possible to implement any boolean expression using only NOR gates • Implement the following circuit using only NOR gates X4 X3 X 2

40. Solution • Two NOR gates in sequence acting as NOT’s can be eliminated: X4 X3 X 2

41. Logic Gates Gates can have multiple inputs and more than one output. A second output can be provided for the complement of the operation.

42. Computers are implementations of Boolean logic. Boolean functions are completely described by truth tables. Logic gates are small circuits that implement Boolean operators. The basic gates are AND, OR, and NOT. The XOR gate is very useful in parity checkers and adders. The “universal gates” are NOR, and NAND. Conclusion

43. Implementation • F=X.Y.+X’.Y’.Z

44. Iterative Combinational Circuits • Arithmetic functions • Operate on binary vectors • Use the same subfunction in each bit position • Can design functional block for subfunction and repeat to obtain functional block for overall function • Cell - subfunction block • Iterative array - a array of interconnected cells • An iterative array can be in a single dimension (1D) or multiple dimensions

45. Block Diagram of a 1D Iterative Array • Example: n = 32 • Number of inputs = ? • Truth table rows = ? • Equations with up to ? input variables • Equations with huge number of terms • Design impractical! • Iterative array takes advantage of the regularity to make design feasible

46. Functional Blocks: Addition • Binary addition used frequently • Addition Development: • Half-Adder (HA), a 2-input bit-wise addition functional block, • Full-Adder (FA), a 3-input bit-wise addition functional block, • Ripple Carry Adder, an iterative array to perform binary addition, and • Carry-Look-Ahead Adder (CLA), a hierarchical structure to improve performance. *(Details not required)

47. X 0 0 1 1 + Y + 0 + 1 + 0 + 1 C S 0 0 0 1 0 1 1 0 X Y C S 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 Functional Block: Half-Adder • A 2-input, 1-bit width binary adder that performs the following computations: • A half adder adds two bits to produce a two-bit sum • The sum is expressed as a sum bit , S and a carry bit, C • The half adder can be specified as a truth table for S and C 

48. S C Y Y 1 0 1 0 1 1 1 X X 2 3 2 3 = × + × = Å S X Y X Y X Y = + × + S ( X Y ) ( X Y ) = × C X Y = C ( ) × ( X Y ) Logic Simplification: Half-Adder • The K-Map for S, C is: • This is a pretty trivial map!By inspection: • and • These equations lead to several implementations. 

49. = + × ( d ) S ( X Y ) C = × + × ( a ) S X Y X Y = + = × C ( X Y ) C X Y = Å ( e ) S X Y = + × + ( b ) S ( X Y ) ( X Y ) = × C X Y = × C X Y = ( c ) S ( ) + × C X Y = × C X Y C Five Implementations: Half-Adder • We can derive following sets of equations for a half-adder: • (a), (b), and (e) are SOP, POS, and XOR implementations for S. • In (c), the C function is used as a term in the AND-NOR implementation of S, and in (d), the function is used in a POS term for S.

50. X S Y C = Å S X Y = × C X Y C X S = + × S ( X Y ) C Y = C ( ) × ( X Y ) Implementations: Half-Adder • The most common half adder implementation is: (e) • A NAND only implementation is: