# ch5 - DESIGN OF GATE NETWORKS 1 DESIGN OF TWO-LEVEL...

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1 DESIGN OF GATE NETWORKS DESIGN OF TWO-LEVEL NETWORKS: and - or and or - and NETWORKS MINIMAL TWO-LEVEL NETWORKS KARNAUGH MAPS MINIMIZATION PROCEDURE AND TOOLS LIMITATIONS OF TWO-LEVEL NETWORKS DESIGN OF TWO-LEVEL nand - nand and nor - nor NETWORKS PROGRAMMABLE LOGIC: pla s and pal s. Introduction to Digital Systems 5 – Design of Two-Level Gate Networks

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2 DESIGN OF TWO-LEVEL NETWORKS IMPLEMENTATION: Level 1 (optional) not GATES Level 2 and GATES Level 3 or GATES LITERALS (uncomplemented and complemented variables) not GATES (IF NEEDED) PRODUCTS: and gates SUM: or gate MULTIOUTPUT NETWORKS: ONE or GATE USED FOR EACH OUTPUT PRODUCT OF SUMS NETWORKS - SIMILAR Introduction to Digital Systems 5 – Design of Two-Level Gate Networks
3 MODULO-64 INCREMENTER Input: 0 x 63 Output: 0 z 63 Function: z = ( x + 1) mod 64 x 0 10101 z 0 10110 x 0 01111 z 0 10000 RADIX-2 REPRESENTATION z i = 1 if ( x i = 1 and there exists j < i such that x j = 0) or ( x i = 0 and x j = 1 for all j < i ) 0 otherwise z 5 = x 5 ( x 0 4 x 0 3 x 0 2 x 0 1 x 0 0 ) x 0 5 x 4 x 3 x 2 x 1 x 0 = x 5 x 0 4 x 5 x 0 3 x 5 x 0 2 x 5 x 0 1 x 5 x 0 0 x 0 5 x 4 x 3 x 2 x 1 x 0 z 4 = x 4 x 0 3 x 4 x 0 2 x 4 x 0 1 x 4 x 0 0 x 0 4 x 3 x 2 x 1 x 0 z 3 = x 3 x 0 2 x 3 x 0 1 x 3 x 0 0 x 0 3 x 2 x 1 x 0 z 2 = x 2 x 0 1 x 2 x 0 0 x 0 2 x 1 x 0 z 1 = x 1 x 0 0 x 0 1 x 0 z 0 = x 0 0 Introduction to Digital Systems 5 – Design of Two-Level Gate Networks

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4 z 5 x 5 x' 4 x' 3 x' 2 x' 1 x' 0 x 4 x' 5 x 3 x 2 x 1 x 0 z 4 x 4 x' 3 x' 2 x' 1 x' 0 x' 4 x 3 x 2 x 1 x 0 z 3 x 3 x' 2 x' 1 x' 0 x' 3 x 2 x 1 x 0 z 2 x 2 x' 1 x' 0 x' 2 x 1 x 0 z 1 x 1 x' 0 x 0 x' 1 z 0 x 0 x 4 x 3 x 2 x 1 x 0 x 5 x' 4 x' 3 x' 2 x' 1 x' 0 x' 5 Figure 5.1: not-and-or MODULO-64 INCREMENTER NETWORK. Introduction to Digital Systems 5 – Design of Two-Level Gate Networks
5 UNCOMPLEMENTED AND COMPLEMENTED INPUTS AVAILABLE TWO TYPES OF TWO-LEVEL NETWORKS: and - or NETWORK SUM OF PRODUCTS ( nand - nand NETWORK) or - and NETWORK PRODUCT OF SUMS ( nor - nor NETWORK) z (a) x 0 x 2 x 1 x’ 1 x’ 2 x 1 x’ 0 x’ 2 x 1 x 1 x’ 0 x 2 x’ 1 x 0 z (b) Figure 5.2: and - or and or - and NETWORKS. E ( x 2 , x 1 , x 0 ) = x 0 2 x 0 1 x 0 x 2 x 1 x 1 x 0 0 E ( x 2 , x 1 , x 0 ) = ( x 0 2 x 1 )( x 1 x 0 0 )( x 2 x 0 1 x 0 ) Introduction to Digital Systems 5 – Design of Two-Level Gate Networks

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6 MINIMAL TWO-LEVEL NETWORKS 1. INPUTS: UNCOMPLEMENTED AND COMPLEMENTED 2. FANIN UNLIMITED 3. SINGLE-OUTPUT NETWORKS 4. MINIMAL NETWORK: MINIMUM NUMBER OF GATES WITH MINIMUM NUMBER OF INPUTS (minimal expression: min. number of terms with min. number of literals) Introduction to Digital Systems 5 – Design of Two-Level Gate Networks
7 NETWORKS WITH DIFFERENT COST z Network A x’ 0 x 1 x 2 x 2 x 0 z x 0 x’ 2 x 0 x 1 x 2 x 1 x 1 Network B Figure 5.3: NETWORKS WITH DIFFERENT COST TO IMPLEMENT f ( x 2 , x 1 , x 0 ) = one-set (3,6,7). Introduction to Digital Systems 5 – Design of Two-Level Gate Networks

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8 MINIMAL EXPRESSIONS EQUIVALENT BUT DIFFERENT COST E 1 ( x 2 , x 1 , x 0 ) = x 0 2 x 1 x 0 0 x 0 1 x 0 x 2 x 0 E 2 ( x 2 , x 1 , x 0 ) = x 2 x 1 x 0 x 0 2 x 1 x 0 0 x 0 2 x 0 1 x 0 x 2 x 0 1 x 0 BOTH MINIMAL SP AND PS MUST BE OBTAINED AND COMPARED BASIS: ab ab 0 = a (for sum of products) ( a b )( a b 0 ) = a (for product of sums) Introduction to Digital Systems
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