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CS 150 - Fall 2000 - Combinational Implementation - 1 Combinational Logic Implementation zTwo-level logic yImplementations of two-level logic yNAND/NOR zMulti-level logic yFactored forms yAnd-or-invert gates zTime behavior yGate delays yHazards zRegular logic yMultiplexers yDecoders yPAL/PLAs yROMs

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CS 150 - Fall 2000 - Combinational Implementation - 2 Implementations of Two-level Logic zSum-of-products yAND gates to form product terms (minterms) yOR gate to form sum zProduct-of-sums yOR gates to form sum terms (maxterms) yAND gates to form product

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CS 150 - Fall 2000 - Combinational Implementation - 3 Two-level Logic using NAND Gates zReplace minterm AND gates with NAND gates zPlace compensating inversion at inputs of OR gate

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CS 150 - Fall 2000 - Combinational Implementation - 4 Two-level Logic using NAND Gates (contd) zOR gate with inverted inputs is a NAND gate yde Morgan's:A' + B' = (A B)' zTwo-level NAND-NAND network yInverted inputs are not counted yIn a typical circuit, inversion is done once and signal distributed

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CS 150 - Fall 2000 - Combinational Implementation - 5 Two-level Logic using NOR Gates zReplace maxterm OR gates with NOR gates zPlace compensating inversion at inputs of AND gate

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CS 150 - Fall 2000 - Combinational Implementation - 6 Two-level Logic using NOR Gates (contd) zAND gate with inverted inputs is a NOR gate yde Morgan's:A' B' = (A + B)' zTwo-level NOR-NOR network yInverted inputs are not counted yIn a typical circuit, inversion is done once and signal distributed

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CS 150 - Fall 2000 - Combinational Implementation - 7 OR NAND OR AND NOR AND Two-level Logic using NAND and NOR Gates zNAND-NAND and NOR-NOR networks yde Morgan's law:(A + B)'= A' B' (A B)' = A' + B' ywritten differently: A + B= (A' B') (A B) = (A' + B')' zIn other words –– yOR is the same as NAND with complemented inputs yAND is the same as NOR with complemented inputs yNAND is the same as OR with complemented inputs yNOR is the same as AND with complemented inputs

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CS 150 - Fall 2000 - Combinational Implementation - 8 A B C D Z A B C D Z NAND Conversion Between Forms zConvert from networks of ANDs and ORs to networks of NANDs and NORs yIntroduce appropriate inversions ("bubbles") zEach introduced "bubble" must be matched by a corresponding "bubble" yConservation of inversions yDo not alter logic function zExample: AND/OR to NAND/NAND

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CS 150 - Fall 2000 - Combinational Implementation - 9 Z = [ (A B)' (C D)' ]' = [ (A' + B') (C' + D') ]' = [ (A' + B')' + (C' + D')' ] = (A B) + (C D) Conversion Between Forms (contd) zExample: verify equivalence of two forms A B C D Z A B C D Z NAND

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CS 150 - Fall 2000 - Combinational Implementation - 10 Step 2 conserve "bubbles" Step 1 conserve "bubbles" NOR \A \B \C \D Z NOR A B C D Z Conversion Between Forms (contd) zExample: map AND/OR network to NOR/NOR network A B C D Z

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CS 150 - Fall 2000 - Combinational Implementation - 11 Z = { [ (A' + B')' + (C' + D')' ]' }' = { (A' + B') (C' + D') }' = (A' + B')' + (C' + D')' = (A B) + (C D) Conversion Between Forms (contd) zExample: verify equivalence of two forms A B C D Z NOR \A \B \C \D Z

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CS 150 - Fall 2000 - Combinational Implementation - 12 ABCDEFGABCDEFG X Multi-level Logic zx = A D F + A E F + B D F + B E F + C D F + C E F + G yReduced sum-of-products form – already simplified y6 x 3-input AND gates + 1 x 7-input OR gate (may not exist!) y25 wires (19 literals plus 6 internal wires) zx = (A + B + C) (D + E) F + G yFactored form – not written as two-level S-o-P y1 x 3-input OR gate, 2 x 2-input OR gates, 1 x 3-input AND gate y10 wires (7 literals plus 3 internal wires)

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CS 150 - Fall 2000 - Combinational Implementation - 13 Level 1Level 2Level 3Level 4 original AND-OR network A C D B B \C F introduction and conservation of bubbles A C D B B \C F redrawn in terms of conventional NAND gates A C D \B B \C F Conversion of Multi-level Logic to NAND Gates zF = A (B + C D) + B C'

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CS 150 - Fall 2000 - Combinational Implementation - 14 Level 1Level 2Level 3Level 4 A C D B B \C F original AND-OR network introduction and conservation of bubbles A C D B B \C F redrawn in terms of conventional NOR gates \A \C \D B \B C F Conversion of Multi-level Logic to NORs zF = A (B + C D) + B C'

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CS 150 - Fall 2000 - Combinational Implementation - 15 A X B C D F (a) original circuit A X B C D F (b) add double bubbles at inputs \D A \X B C F (c) distribute bubbles some mismatches \D A X B C F \X (d) insert inverters to fix mismatches Conversion Between Forms zExample

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CS 150 - Fall 2000 - Combinational Implementation - 16 & & + 2x2 AOI gate symbol & & + 3x2 AOI gate symbol NAND Invert possible implementation A B C D Z AND ORInvert logical concept A B C D Z AND-OR-Invert Gates zAOI function: three stages of logicAND, OR, Invert yMultiple gates "packaged" as a single circuit block

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CS 150 - Fall 2000 - Combinational Implementation - 17 01100110 A B & & + A' B' A B F Conversion to AOI Forms zGeneral procedure to place in AOI form yCompute complement of the function in sum-of-products form yBy grouping the 0s in the Karnaugh map zExample: XOR implementation––A xor B = A' B + A B' yAOI form:F = (A' B' + A B)'

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CS 150 - Fall 2000 - Combinational Implementation - 18 each implemented in a single 2x2 AOI gate Examples of using AOI gates zExample: yF = B C' + A C' + A B yF' = A' B' + A' C + B' C yImplemented by 2-input 3-stack AOI gate yF = (A + B) (A + C') (B + C') yF' = (B' + C) (A' + C) (A' + B') yImplemented by 2-input 3-stack OAI gate zExample: 4-bit equality function yZ = (A0B0+A0'B0')(A1B1+A1'B1')(A2B2+A2'B2')(A3B3+A3'B3') 0100010001 11101110 C B A

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CS 150 - Fall 2000 - Combinational Implementation - 19 high if A0 B0 low if A0 = B0 if all inputs are low then Ai = Bi, i=0,...,3 output Z is high conservation of bubbles Examples of Using AOI Gates (contd) zExample: AOI implementation of 4-bit equality function

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CS 150 - Fall 2000 - Combinational Implementation - 20 Summary for Multi-level Logic zAdvantages yCircuits may be smaller yGates have smaller fan-in yCircuits may be faster zDisadvantages yMore difficult to design yTools for optimization are not as good as for two-level yAnalysis is more complex

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CS 150 - Fall 2000 - Combinational Implementation - 21 Time Behavior of Combinational Networks zWaveforms yVisualization of values carried on signal wires over time yUseful in explaining sequences of events (changes in value) zSimulation tools are used to create these waveforms yInput to the simulator includes gates and their connections yInput stimulus, that is, input signal waveforms zSome terms yGate delaytime for change at input to cause change at output xMin delay–typical/nominal delay–max delay xCareful designers design for the worst case yRise timetime for output to transition from low to high voltage yFall timetime for output to transition from high to low voltage yPulse widthtime an output stays high or low between changes

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CS 150 - Fall 2000 - Combinational Implementation - 22 F is not always 0 pulse 3 gate-delays wide D remains high for three gate delays after A changes from low to high F ABCD Momentary Changes in Outputs zCan be usefulpulse shaping circuits zCan be a problemincorrect circuit operation (glitches/hazards) zExample: pulse shaping circuit yA' A = 0 ydelays matter in function

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CS 150 - Fall 2000 - Combinational Implementation - 23 initially undefined close switch open switch + resistor A B C D Oscillatory Behavior zAnother pulse shaping circuit

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CS 150 - Fall 2000 - Combinational Implementation - 24 Hazards/Glitches zHazards/glitches: unwanted switching at the outputs yOccur when different paths through circuit have different propagation delays xAs in pulse shaping circuits we just analyzed yDangerous if logic causes an action while output is unstable xMay need to guarantee absence of glitches zUsual solutions y1) Wait until signals are stable (by using a clock): preferable (easiest to design when there is a clock – synchronous design) y2) Design hazard-free circuits: sometimes necessary (clock not used – asynchronous design)

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CS 150 - Fall 2000 - Combinational Implementation - 25 1 00 11 00 11 000 11 Types of Hazards zStatic 1-hazard yInput change causes output to go from 1 to 0 to 1 zStatic 0-hazard yINput change causes output to go from 0 to 1 to 0 zDynamic hazards yInput change causes a double change from 0 to 1 to 0 to 1 OR from 1 to 0 to 1 to 0

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CS 150 - Fall 2000 - Combinational Implementation - 26 F A B S S' F hazard static-0 hazardstatic-1 hazard A S B S' Static Hazards zDue to a literal and its complement momentarily taking on the same value yThru different paths with different delays and reconverging zMay cause an output that should have stayed at the same value to momentarily take on the wrong value zExample: multiplexer

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CS 150 - Fall 2000 - Combinational Implementation - 27 B2 A C B1 F hazard dynamic hazards B3 A C B F 1 2 3 Dynamic Hazards zDue to the same versions of a literal taking on opposite values yThru different paths with different delays and reconverging zMay cause an output that was to change value to change 3 times instead of once zExample:

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CS 150 - Fall 2000 - Combinational Implementation - 28 multiplexerdemultiplexer4x4 switch control Making Connections zDirect point-to-point connections between gates yWires we've seen so far zRoute one of many inputs to a single output --- multiplexer zRoute a single input to one of many outputs --- demultiplexer

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CS 150 - Fall 2000 - Combinational Implementation - 29 Mux and Demux zSwitch implementation of multiplexers and demultiplexers yCan be composed to make arbitrary size switching networks yUsed to implement multiple-source/multiple-destination interconnections A B Y Z A B Y Z

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CS 150 - Fall 2000 - Combinational Implementation - 30 multiple input sources multiple output destinations MUX A B Sum Sa Ss Sb B0 MUX DEMUX Mux and Demux (cont'd) zUses of multiplexers/demultiplexers in multi-point connections B1A0A1 S0S1

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CS 150 - Fall 2000 - Combinational Implementation - 31 two alternative forms for a 2:1 Mux truth table functional form logical form AZ0I01I1AZ0I01I1 I1I0AZ00000010010101101000101111011111I1I0AZ00000010010101101000101111011111 Z = A' I 0 + A I 1 Multiplexers/Selectors zMultiplexers/Selectors: general concept y2 n data inputs, n control inputs (called "selects"), 1 output yUsed to connect 2 n points to a single point yControl signal pattern forms binary index of input connected to output

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CS 150 - Fall 2000 - Combinational Implementation - 32 2 -1 I0 I1 I2 I3 I4 I5 I6 I7 A B C 8:1 mux Z I0 I1 I2 I3 A B 4:1 mux Z I0 I1 A 2:1 mux Z k=0 n Multiplexers/Selectors (cont'd) z2:1 mux:Z = A' I0 + A I1 z4:1 mux:Z = A' B' I0 + A' B I1 + A B' I2 + A B I3 z8:1 mux:Z = A'B'C'I0 + A'B'CI1 + A'BC'I2 + A'BCI3 + AB'C'I4 + AB'CI5 + ABC'I6 + ABCI7 zIn general, Z = (m k I k ) yin minterm shorthand form for a 2 n :1 Mux

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CS 150 - Fall 2000 - Combinational Implementation - 33 Gate Level Implementation of Muxes z2:1 mux z4:1 mux

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CS 150 - Fall 2000 - Combinational Implementation - 34 control signals B and C simultaneously choose one of I0, I1, I2, I3 and one of I4, I5, I6, I7 control signal A chooses which of the upper or lower mux's output to gate to Z alternative implementation C Z A B 4:1 mux 2:1 mux I4 I5 I2 I3 I0 I1 I6 I7 8:1 mux Cascading Multiplexers zLarge multiplexers implemented by cascading smaller ones Z I0 I1 I2 I3 A I4 I5 I6 I7 B C 4:1 mux 2:1 mux 8:1 mux

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CS 150 - Fall 2000 - Combinational Implementation - 35 C AB 0123456701234567 1010001110100011 S2 8:1 MUX S1S0 F Multiplexers as General-purpose Logic z2 n :1 multiplexer implements any function of n variables yWith the variables used as control inputs and yData inputs tied to 0 or 1 yIn essence, a lookup table zExample: yF(A,B,C) = m0 + m2 + m6 + m7 = A'B'C' + A'BC' + ABC' + ABC = A'B'(C') + A'B(C') + AB'(0) + AB(1)

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CS 150 - Fall 2000 - Combinational Implementation - 36 ABCF00010010010101101000101011011111ABCF00010010010101101000101011011111 C' C' 0 1 AB S1S0 F 01230123 4:1 MUX C' C' 0 1 F C AB 0123456701234567 1010001110100011 S2 8:1 MUX S1S0 Multiplexers as General-purpose Logic (contd) z2 n-1 :1 mux can implement any function of n variables yWith n-1 variables used as control inputs and yData inputs tied to the last variable or its complement zExample: yF(A,B,C) = m0 + m2 + m6 + m7 = A'B'C' + A'BC' + ABC' + ABC = A'B'(C') + A'B(C') + AB'(0) + AB(1)

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CS 150 - Fall 2000 - Combinational Implementation - 37 n-1 mux control variables single mux data variable four possible configurations of truth table rows can be expressed as a function of I n choose A,B,C as control variables multiplexer implementation I 0 I 1...I n-1 I n F....00011....10101 0I n I n '1 Multiplexers as General-purpose Logic (contd) zGeneralization zExample: F(A,B,C,D) implemented by an 8:1 MUX C AB 0123456701234567 1D01DDDD1D01DDDD S2 8:1 MUX S1S0 10101010 10 D A1 11011101 01100110 B C

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CS 150 - Fall 2000 - Combinational Implementation - 38 1:2 Decoder: O0 = G S O1 = G S 2:4 Decoder: O0 = G S1 S0 O1 = G S1 S0 O2 = G S1 S0 O3 = G S1 S0 3:8 Decoder: O0 = G S2 S1 S0 O1 = G S2 S1 S0 O2 = G S2 S1 S0 O3 = G S2 S1 S0 O4 = G S2 S1 S0 O5 = G S2 S1 S0 O6 = G S2 S1 S0 O7 = G S2 S1 S0 Demultiplexers/Decoders zDecoders/demultiplexers: general concept ySingle data input, n control inputs, 2 n outputs yControl inputs (called selects (S)) represent binary index of output to which the input is connected yData input usually called enable (G)

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CS 150 - Fall 2000 - Combinational Implementation - 39 active-high enable active-low enable active-high enable active-low enable O0 G S O1 O0 \G S O1 Gate Level iImplementation of Demultiplexers z1:2 Decoders z2:4 Decoders

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CS 150 - Fall 2000 - Combinational Implementation - 40 demultiplexer generates appropriate minterm based on control signals (it "decodes" control signals) Demultiplexers as General-purpose Logic zn:2 n decoder implements any function of n variables yWith the variables used as control inputs yEnable inputs tied to 1 and yAppropriate minterms summed to form the function A'B'C' A'B'C A'BC' A'BC AB'C' AB'C ABC' ABC C AB 0123456701234567 S2 3:8 DEC S1S0 1

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CS 150 - Fall 2000 - Combinational Implementation - 41 F1 F2 F3 Demultiplexers as General-purpose Logic (contd) zF1 = A' B C' D + A' B' C D + A B C D zF2 = A B C' D + A B C zF3 = (A' + B' + C' + D') AB 0A'B'C'D' 1A'B'C'D 2A'B'CD' 3A'B'CD 4A'BC'D' 5A'BC'D 6A'BCD' 7A'BCD 8AB'C'D' 9AB'C'D 10AB'CD' 11AB'CD 12ABC'D' 13ABC'D 14ABCD' 15ABCD 4:16 DEC Enable CD

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CS 150 - Fall 2000 - Combinational Implementation - 42 0A'B'C'D'E' 1 2 3 4 5 6 7 S2 3:8 DEC S1S0 AB 01230123 S1 2:4 DEC S0 F 0 1 2A'BC'DE' 3 4 5 6 7 S2 3:8 DEC S1S0 E CD 0AB'C'D'E' 1 2 3 4 5 6 7AB'CDE Cascading Decoders z5:32 decoder y1x2:4 decoder y4x3:8 decoders 3:8 DEC 0 1 2 3 4 5 6 7ABCDE E CD S2S1S0S2 3:8 DEC S1S0

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CS 150 - Fall 2000 - Combinational Implementation - 43 inputs AND array outputs OR array product terms Programmable Logic Arrays zPre-fabricated building block of many AND/OR gates yActually NOR or NAND yPersonalized" by making or breaking connections among gates yProgrammable array block diagram for sum of products form

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CS 150 - Fall 2000 - Combinational Implementation - 44 example: F0 = A + B' C' F1 = A C' + A B F2 = B' C' + A B F3 = B' C + A personality matrix 1 = uncomplemented in term 0 = complemented in term – = does not participate 1 = term connected to output 0 = no connection to output input side: output side: productinputsoutputs termABCF0F1F2F3 AB11–0110 B'C–010001 AC'1–00100 B'C'–001010 A1––1001 reuse of terms Enabling Concept zShared product terms among outputs

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CS 150 - Fall 2000 - Combinational Implementation - 45 Before Programming zAll possible connections available before "programming" yIn reality, all AND and OR gates are NANDs

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CS 150 - Fall 2000 - Combinational Implementation - 46 After Programming zUnwanted connections are "blown" yFuse (normally connected, break unwanted ones) yaAnti-fuse (normally disconnected, make wanted connections)

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CS 150 - Fall 2000 - Combinational Implementation - 47 notation for implementing F0 = A B + A' B' F1 = C D' + C' D AB+A'B' CD'+C'D AB A'B' CD' C'D ABCD Alternate Representation for High Fan-in Structures zShort-hand notation--don't have to draw all the wires ySignifies a connection is present and perpendicular signal is an input to gate

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CS 150 - Fall 2000 - Combinational Implementation - 48 ABCF1F2F3F4F5F6 000001100 001010111 010010111 011010100 100010111 101010100 110010100 111110011 full decoder as for memory address bits stored in memory Programmable Logic Array Example zMultiple functions of A, B, C yF1 = A B C yF2 = A + B + C yF3 = A' B' C' yF4 = A' + B' + C' yF5 = A xor B xor C yF6 = A xnor B xnor C

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CS 150 - Fall 2000 - Combinational Implementation - 49 a given column of the OR array has access to only a subset of the possible product terms PALs and PLAs zProgrammable logic array (PLA) yWhat we've seen so far yUnconstrained fully-general AND and OR arrays zProgrammable array logic (PAL) yConstrained topology of the OR array yInnovation by Monolithic Memories yFaster and smaller OR plane

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CS 150 - Fall 2000 - Combinational Implementation - 50 0 1 X 0 0 1 X 0 0 0 X X 0 0 X X D A B C minimized functions: W = A + B D + B C X = B C' Y = B + C Z = A'B'C'D + B C D + A D' + B' C D' ABCDWXYZ00000000000100010010001100110010010001100101111001101010011110111000100110011000101–––––11––––––ABCDWXYZ00000000000100010010001100110010010001100101111001101010011110111000100110011000101–––––11–––––– 0 0 X 1 0 1 X 1 0 1 X X 0 1 X X D A B C K-map for WK-map for X 0 1 X 0 0 1 X 0 1 1 X X 1 1 X X D A B C K-map for Y PALs and PLAs: Design Example zBCD to Gray code converter K-map for Z 0 0 X 1 1 0 X 0 0 1 X X 1 0 X X D A B C

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CS 150 - Fall 2000 - Combinational Implementation - 51 not a particularly good candidate for PAL/PLA implementation since no terms are shared among outputs however, much more compact and regular implementation when compared with discrete AND and OR gates minimized functions: W = A + B D + B C X = B C' Y = B + C Z = A'B'C'D + B C D + A D' + B' C D' PALs and PLAs: Design Example (contd) zCode converter: programmed PLA

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CS 150 - Fall 2000 - Combinational Implementation - 52 4 product terms per each OR gate AB CD PALs and PLAs: Design Example (contd) zCode converter: programmed PAL

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CS 150 - Fall 2000 - Combinational Implementation - 53 PALs and PLAs: Design Example (contd) zCode converter: NAND gate implementation yLoss or regularity, harder to understand yHarder to make changes

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CS 150 - Fall 2000 - Combinational Implementation - 54 PALs and PLAs: Another Design Example zMagnitude comparator 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 D A B C 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 D A B C 0 0 1 0 0 0 1 1 0 1 1 1 0 0 D A B C 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 0 D A B C K-map for EQ K-map for NE K-map for GT K-map for LT

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CS 150 - Fall 2000 - Combinational Implementation - 55 decoder 0n-1 Address 2 -1 n 0 1111 word[i] = 0011 word[j] = 1010 bit lines (normally pulled to 1 through resistor – selectively connected to 0 by word line controlled switches) j i internal organization word lines (only one is active – decoder is just right for this) Read-only Memories zTwo dimensional array of 1s and 0s yEntry (row) is called a "word" yWidth of row = word-size yIndex is called an "address" yAddress is input ySelected word is output

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CS 150 - Fall 2000 - Combinational Implementation - 56 F0 = A' B' C + A B' C' + A B' C F1 = A' B' C + A' B C' + A B C F2 = A' B' C' + A' B' C + A B' C' F3 = A' B C + A B' C' + A B C' truth table ABCF0F1F2F3 0000010 0011110 0100100 0110001 1001011 1011000 1100001 1110100 block diagram ROM 8 words x 4 bits/word addressoutputs ABCF0F1F2F3 ROMs and Combinational Logic zCombinational logic implementation (two-level canonical form) using a ROM

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CS 150 - Fall 2000 - Combinational Implementation - 57 ROM Structure zSimilar to a PLA structure but with a fully decoded AND array yCompletely flexible OR array (unlike PAL) n address lines inputs decoder 2 n word lines outputs memory array (2 n words by m bits) m data lines

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CS 150 - Fall 2000 - Combinational Implementation - 58 ROM vs. PLA zROM approach advantageous when yDesign time is short (no need to minimize output functions) yMost input combinations are needed (e.g., code converters) yLittle sharing of product terms among output functions zROM problems ySize doubles for each additional input yCan't exploit don't cares zPLA approach advantageous when yDesign tools are available for multi-output minimization yThere are relatively few unique minterm combinations yMany minterms are shared among the output functions zPAL problems yConstrained fan-ins on OR plane

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CS 150 - Fall 2000 - Combinational Implementation - 59 Regular Logic Structures for Two-level Logic zROM – full AND plane, general OR plane yCheap (high-volume component) yCan implement any function of n inputs yMedium speed zPAL – programmable AND plane, fixed OR plane yIntermediate cost yCan implement functions limited by number of terms yHigh speed (only one programmable plane that is much smaller than ROM's decoder) zPLA – programmable AND and OR planes yMost expensive (most complex in design, need more sophisticated tools) yCan implement any function up to a product term limit ySlow (two programmable planes)

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CS 150 - Fall 2000 - Combinational Implementation - 60 Regular Logic Structures for Multi-level Logic zDifficult to devise a regular structure for arbitrary connections between a large set of different types of gates yEfficiency/speed concerns for such a structure yXilinx field programmable gate arrays (FPGAs) are just such programmable multi-level structures xProgrammable multiplexers for wiring xLookup tables for logic functions (programming fills in the table) xMulti-purpose cells (utilization is the big issue) zUse multiple levels of PALs/PLAs/ROMs yOutput intermediate result yMake it an input to be used in further logic

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CS 150 - Fall 2000 - Combinational Implementation - 61 Combinational Logic Implementation Summary zMulti-level Logic yConversion to NAND-NAND and NOR-NOR networks yTransition from simple gates to more complex gate building blocks yReduced gate count, fan-ins, potentially faster yMore levels, harder to design zTime Response in Combinational Networks yGate delays and timing waveforms yHazards/glitches (what they are and why they happen) zRegular Logic yMultiplexers/decoders yROMs yPLAs/PALs yAdvantages/disadvantages of each

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