10/14/2009. Reading: Hambley Chapters

Transcription

1 EE40 Lec 14 Digital Signal and Boolean Algebra Prof. Nathan Cheung 10/14/2009 Reading: Hambley Chapters Slide 1

2 Analog Signals Analog: signal amplitude is continuous with time. Amplitude Modulated Signal Signal in microvolts Time in microseconds Slide 2

3 Digitalized signal Digital: signal amplitude is represented by a restricted set of discrete numbers. Slide 3

4 Why Digital? (For example, why CDROM audio vs. vinyl recordings?) Digital i signals can be transmitted, received, amplified, and re-transmitted with far less degradation. Digital information is easily and inexpensively stored (in RAM, ROM, etc.), with arbitrary accuracy. Complex logical functions are easily expressed as binary functions (e.g. in control applications). Digital signals are easy to manipulate (as we shall see). Slide 4

5 Digital Signal Representations Binary numbers can be used to represent any yquantity. We generally have to agree on some sort of code, and the dynamic range of the signal in order to know the form and the number of binary digits ( bits ) required. Example : To encode the signal to an accuracy of 1 part in 64 (1.5% precision), i 6 binary digits it ( bits ) are needed Slide 5

6 Digital Signals For a digital signal, the voltage must be within one of two ranges in order to be defined: V DD 1 undefined region Positive Logic: 0 low voltage logic state 0 high voltage logic state 1 V OH V IH V IL V OL 0 Volts increasing voltage Slide 6

7 Number Base Number Base B B symbols per digit: Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base 2 (Binary): 0, 1 Number base representation: d 31 d d 1 d 0 is a 32 digit number value = d 31 B 31 + d 30 B d 1 B 1 + d 0 B 0 Example : Binary (B=2): 0,1 (In binary digits it called bits ) = = = 26 Here 5 digit binary number turns into a 2 digit decimal number Slide 7

8 Decimal-Binary Conversion Decimal Integer to Binary Repeated Division By 2 Example: Decimal integer 343 Slide 8

9 Decimal-Binary Conversion Decimal Fraction to Binary Fraction Repeated Multiplication By 2 Example: Decimal Fraction Other examples: = Slide 9

10 Binary to Decimal conversion =1x2 5 +1x2 4 +0x2 3 +0x2 2 +0x2 1 +1x2 0 +0x x x2-3 = = Slide 10

11 Adding Binary Numbers Addition Rules Example Slide 11

12 Two complement of Binary Numbers Slide 12

13 Subtracting Binary Numbers Find signed two s complement of substrahend Subtraction = add the signed binary number to the signed two s complement of substrahend Note: If two numbers to be added have the same sign bit but the result have the opposite sign bit, overflow or underflow has occurred Slide 13

14 Hexadecimal Numbers: Base 16 Hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Normal digits + 6 more from the alphabet Conversion: Binary Hex 1 hex digit represents 16 decimal values 4binary digits i represent 16d decimal values 1 hex digit replaces 4 binary digits Example: = =F Slide 14

15 Digital Representations of Logical Functions Digital signals offer an easy way to perform logical functions, using Boolean algebra. Variables have two possible values: true or false usually represented by 1 and 0, respectively. All modern control systems use this approach. Example: Hot tub controller with the following algorithm Turn on the heater if the temperature is less than desired (T < Tset) and the motor is on and the key switch hto activate t the hot ttub bis closed. Slide 15

16 Combinatorial Logic gates Combine several logic variable inputs to produce a logic variable output Memoryless: output at a given instant depends the input values of that instant. Slide 16

17 Logic Functions, Symbols, & Notation TRUTH NAME SYMBOL NOTATION TABLE NOT A F F = A A F A OR F F = A+B B A B F AND A B F F = AB A B F Slide 17

18 3-Input Gates Slide 18

19 Boolean algebra The operators of fboolean algebra may be represented in various ways. Often they are simply pywritten as AND, OR and NOT. In describing circuits, NAND (NOT AND), NOR (NOT OR) and XOR (exclusive OR) may also be used. An excellent web site to visit Mathematicians often use + for OR and for AND (since in some ways those operations are analogous to addition and multiplication in other algebraic structures) and represent NOT by a line drawn above the expression being negated. Note: In Hambley text, there is no dot for AND opeartion Slide 19

20 A A = A A A A = 0 A 1 = A A0 A 0 = 0 A B = B A Boolean Algebra Relations A (B C) = (A B) C A+A = A A+A A = 1 A+1 = 1 A+0 = A A+B = B+A A (B+C) = A B +A C A B = A + B A B = A + B Slide 20 A+(B+C) = (A+B)+C De Morgan s laws

21 Boolean Expression Example F = ABC + ABC + (C + D)(D + E) AC (B + B) = AC C(D + E) + D(D + = CD + CE + DE E) F = C(A + D + E) + DE Slide 21

22 Logic Functions, Symbols, & Notation 2 NOR A B F F F = A+B B A B NAND A B F F F = AB B A B XOR (exclusive OR) A B F F = A + B A B F Slide 22

23 NAND Gate Implementation De Morgan s law tells us that is the same as By definition, is the same as Slide 23

24 Graphic Representation AA = 0 A A A + A = 1 Venn Diagram Full square = complete set =1 Yellow part = NOT(A) =A White circle = A Slide 24

25 Graphic Representation of XOR A AB B A + B A B= AB + AB = (A+ B)(A+ B) = AB+ A+ B Exclusive OR=yellow and blue part intersection part = exactly when only one of the input is true Slide 25

26 Circuit Realization of XOR Gate A B= AB + AB = (A+ B)(A+ B) = AB+ A+ B A A AB B A B B AB Slide 26

27 Logical Sufficiency of NAND Gates If the inputs to a NAND gate are tied together, an inverter results From De Morgan s laws, the OR operation can be realized by inverting the input variables and combining the results in a NAND gate. Since the basic logic functions (AND, OR, and NOT) can be realized by using only NAND gates, NAND gates are sufficient to realize any combinational logic function. Slide 27

28 Logical Sufficiency of NOR Gates Show how to realize the AND, OR, and NOT functions using only NOR gates Since the basic logic functions (AND, OR, and NOT) can be realized by using only NOR gates, NOR gates are sufficient to realize any combinational logic function. Slide 28