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# DIGITAL LOGIC DESIGN E E 316

UT

GPA 3.57

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This 43 page Class Notes was uploaded by Tate Monahan PhD on Monday September 7, 2015. The Class Notes belongs to E E 316 at University of Texas at Austin taught by Nur Touba in Fall. Since its upload, it has received 67 views. For similar materials see /class/181891/e-e-316-university-of-texas-at-austin in Electrical at University of Texas at Austin.

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Date Created: 09/07/15

EE 316 Digital Logic Design Lecture 2 Nur A Touba University of Texas at Austin BOOLEAN ALGEBRA E Switching Devices 0 TwoStates 0 Special Case of Boolean Algebra Used Twovalued switching algebra E George Boole o Developed Boolean Algebra in 1847 E Claude Shannon 0 Applied to switching circuits in 1939 BOOLEAN VARIABLE El Boolean Variable X X 1 o Takes two values 0 or 1 X 0 switch open X 1 switch closed El BasIc Operations 0 Inversion 0 AND 00R BOOLEAN EXPRESSIONS E Simplest 0 Single Constant 0 1 0 Single Variable X Y E More Complicated Expressions 0 AB C o ACD BE E No Parentheses 0 Order is Complementation AND OR EVALUATING EXPRESSION E Substitute value of 0 or 1 for each variable 0 ABD C E LITERAL E Literal 0 Each appearance of variable or its complement o Corresponds to gate input when using logic gates TRUTH TABLE E Truth Table Table of Combinations o Specifies value of Boolean expression for every possible combination of values of variables 0 Expression with n variables 2quot rows in truth table oo x xoo x xw gt W DOA OOOO EXAMPLE AB C AC B C ACB C ooo oo oo ooo BASIC THEOREMS E Operations with 0 and 1 o X 0 X X1 X o X 1 1 X0 0 E Idempotent Laws 0 X X X XX X E lnvolution Law 0 X X E Laws of Complementarity o X X XX LAWS E Commutative Laws 0 XY YX XY YX E Associative Laws 0 XYZ XYZ XYZ o XYZXYZXYZ E Distributive Law 0 XYZXYXZ o XYZXYXZ PROOF OF ASSOCIATIVE LAW l XYZ XYZ YZ XYZ l XY 000 010 01 100 101 I gtuu0gtOZ O gtmm00gtltm gtltlt CO gt 0 Ocugt HY my SIMPLIFICATION THEOREMS E Useful Theorems for Simplifying Boolean Expressions 0 XY XY X XYXY X XXYX XXYX oXY YXY XY YXY SIMPLIFICATION EXAMPLES E Simplify z ABC A E Simplify z A B C D EF A B C D EF E Simplify z AB CB D C E AB C MULTIPYING OUT AND FACTORING E SumofProducts o All Products are products of single variables MULTIPYING OUT AND FACTORING E Product of Sums o All Sums are sums of single variables DeMORGAN s LAWS E DeMorgan s Laws 0 X Y X Y Complement of sum is product of complements 0 XY X Y Complement of product is sum of complements DUAL E Dual of Boolean Expression o Replace AND with OR OR with AND 0 with 1 1 with 0 0 Variables and Complements left unchanged EE 316 Digital Logic Design Lecture 1 Nur A Touba University of Texas at Austin DIGITAL SYSTEMS E Digital Signals Two Values eg 0 and 5 Volts EAnalog Signals Range of Values eg 10 to 10 Volts E Digital Systems 0 Discrete Quantities o Represent Exact Output Analog multiplier has error range DESIGN OF DIGITAL SYSTEMS E System Design 0 Breaking overall system to subsystems E Logic Design 0 lnterconnecting basic logic building blocks to perform function E Circuit Design 0 lnterconnecting transistors resistors diodes to form gate flipflop etc SWITCHING CIRCUIT ElCombinational Depends only on present input EISequential Depends only present and past inputs has memory Inputs X1 gt X2 gt Xm 39gt Switching Circuit 21 22 Outputs gtZn COMBINATIONAL CIRCUIT DESIGN E Basic Building Blocks Logic Gates E Determine how to interconnect to convert input signals to appropriate output signals POSITIONAL NOTATION E Base 10 0 9537810 9x102 5x101 3x10o 7x10391 8x102 EBase 2 01011112 1x23 0x22 1x211x20 1x23911x2392 802105025117510 EBase R o a3R3 a2R2 a1R1 aoR a1R391 a2R392 POWER SERIES EXPANSION E14138 POWER SERIES TO CONVERT BASE E14710 convert to Base 3 c 14710 1x1012 11x1011 21x101o POWER SERIES TO CONVERT BASE E14710 convert to Base 2 c 14710 1x10102 100x10101 111x10100 288 u 2 S 83 u 02me 23 52 u SEE emvuu eium oavuo eSuo azum esult o ZltI mmltmm0 mwltm CONVERTING DECIMAL TO BASE R E N a3R3 a2R2 a1R1 aoRo E Division Method 0 NIR a3xR3quot1 a2xR2391 a1xR1391 Q1 remainder a0 0 Q1IR a3xR3quot2 a2xR2392 Q2 remainder a1 o 0le a3xR3393 Q3 remainder a2 0 Q3IR 0 remainder a3 EXAMPLE E Convert 47 to Binary DECIMAL FRACTION TO BASE R E F a1R391 a2R392 a3R393 E Multiply by R 0 FR a1R3911 a2R3921 a3R3931 a1 F1 0 F1R a2R3922 a3R3932 a2 F2 0 F2R a3R3933 a3 F3 EXAMPLE E Convert 037510 to Binary EXAMPLE E Convert 0710 to Binary 0 May not terminate but results in repeating fraction CONVERTING BETWEEN OTHER BASES E Converting between bases other than decimal 0 Can be done with same procedures Requires arithmetic operations in bases other than 10 o Easier to convert to decimal first and then to new base BINARY AND HEX CONVERSION E Each Hex Digit Corresponds to 4 Binary Digits E Example 0 Convert 1101101110112 to Hex BINARY ARITHMETIC EArithmetic in Digital Systems 0 Usually carried out in Binary Logic circuits simpler E000 E011 E101 E1 1 0 and carry1 to next column EXAMPLE EAdd1310 and 1110 in Binary E0 E0 E1 E1 A O A 0 II II II II 0 A A O BINARY SUBTRACTION and borrow from next column EXAMPLES MULTIPLICATION EXAMPLE DIVISION EXAMPLE REPRESENTING NEGATIVE NUMBERS E Sign and Magnitude 0 First bit is sign 0011 3 1011 3 0 Two representations of 0 0000 0 1000 1

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