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ch2 - SPECIFICATION OF COMBINATIONAL SYSTEMS HIGH-LEVEL AND...

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1 SPECIFICATION OF COMBINATIONAL SYSTEMS HIGH-LEVEL AND BINARY-LEVEL SPECIFICATIONS REPRESENTATION OF DATA ELEMENTS BY BINARY VARIABLES; STANDARD CODES FOR POSITIVE INTEGERS AND CHARACTERS REPRESENTATION BY SWITCHING FUNCTIONS AND SWITCHING EXPRESSIONS not , and , or , nor , xor , and xnor SWITCHING FUNCTIONS TRANSFORMATION OF SWITCHING EXPRESSIONS USING SWITCHING ALGEBRA USE OF VARIOUS SPECIFICATION METHODS USE OF THE μ vhdl DESCRIPTION LANGUAGE Introduction to Digital Systems 2 – Specification of Combinational Systems
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2 COMBINATIONAL SYSTEM z ( t ) = F ( x ( t )) or z = F ( x ) Time t x z Figure 2.1: Combinational system. Introduction to Digital Systems 2 – Specification of Combinational Systems
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3 BINARY LEVEL z b = F b ( x b ) High-level specification F Coding C Decoding D Binary specification F b x b z b n m z x Figure 2.2: High-level and binary-level specification. Introduction to Digital Systems 2 – Specification of Combinational Systems
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4 Example 2.1: Input: x ∈ { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } Output: z ∈ { 0 , 1 , 2 } Function: F is described by the following table x 0 1 2 3 4 5 6 7 8 9 z = F ( x ) 0 1 2 0 1 2 0 1 2 0 or by the arithmetic expression z = x mod 3, x 0 1 2 3 4 5 6 7 8 9 x b = C ( x ) 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 z b 00 01 10 z = D ( z b ) 0 1 2 Input: x b = ( x 3 , x 2 , x 1 , x 0 ) , x i ∈ { 0 , 1 } Output: z b = ( z 1 , z 0 ) , z i ∈ { 0 , 1 } Function: F b is described by the following table x b 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 z b = F b ( x b ) 00 01 10 00 01 10 00 01 10 00 Introduction to Digital Systems 2 – Specification of Combinational Systems
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5 HIGH-LEVEL SPECIFICATION SET OF VALUES FOR THE INPUT, input set; SET OF VALUES FOR THE OUTPUT, output set; and SPECIFICATION OF THE input-output function. Introduction to Digital Systems 2 – Specification of Combinational Systems
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6 INPUT AND OUTPUT SETS { UP, DOWN, LEFT, RIGHT, FIRE } { x | (5 x 10 4 ) and ( x mod 3 = 0) } Examples of vectors Vector type Example Digit x = ( x n - 1 , x n - 2 , . . . , x 0 ) x = (7 , 0 , 6 , 3) x i ∈ { 0 , 1 , 2 , . . . , 9 } Character c = ( c n - 1 , c n - 2 , . . . , c 0 ) c = ( B, O, O, K ) c i ∈ { , A, B, . . . , Z } Set s = ( s n - 1 , s n - 2 , . . . , s 0 ) s = (red , blue , blue) s i ∈ { red , blue , white } Bit y = ( y n - 1 , y n - 2 , . . . , y 0 ) y = (1 , 1 , 0 , 1 , 0 , 0) y i ∈ { 0 , 1 } y = 110100 Introduction to Digital Systems 2 – Specification of Combinational Systems
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7 INPUT-OUTPUT FUNCTION 1. TABLE x z A 65 B 66 C 67 D 68 E 69 2. ARITHMETIC EXPRESSION z = 3 x + 2 y - 2 3. CONDITIONAL EXPRESSION z = a + b if c > d a - b if c = d 0 if c < d Introduction to Digital Systems 2 – Specification of Combinational Systems
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8 INPUT-OUTPUT FUNCTION (cont.) 4. LOGICAL EXPRESSION z = ( SWITCH 1 = CLOSED ) and ( SWITCH 2 = OPEN ) or ( SWITCH 3 = CLOSED ) 5. COMPOSITION OF SIMPLER FUNCTIONS max ( v, w, x, y ) = GREATER ( v, GREATER ( w, GREATER ( x, y ))) in which GREATER ( a, b ) = a if a > b b otherwise Introduction to Digital Systems 2 – Specification of Combinational Systems
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9 Example 2.2 Inputs: x = ( x 3 , x 2 , x 1 , x 0 ) , x i ∈ { A , B , . . . , Z , a , b , . . . , z } y ∈ { A , B , . . . , Z , a , b , . . . , z } k ∈ { 0 , 1 , 2 , 3 } Outputs: z = ( z 3 , z 2 , z 1 , z 0 ) , z i ∈ { A , B , . . . , Z , a , b , . . . , z } Function: z j = x j if j 6 = k y if j = k Input: x = (c,a,s,e) , y = r , k = 1 Output: z = (c,a,r,e) Introduction to Digital Systems
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