# ch7 - SPECIFICATION OF SEQUENTIAL SYSTEMS 1 SYNCHRONOUS...

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1 SPECIFICATION OF SEQUENTIAL SYSTEMS SYNCHRONOUS SEQUENTIAL SYSTEMS MEALY AND MOORE MACHINES TIME BEHAVIOR STATE MINIMIZATION Introduction to Digital Systems 7 – Specification of Sequential Systems

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2 DEFINITION z ( t ) = F ( x (0 , t )) x(t) z(t) t x z time Figure 7.1: INPUT AND OUTPUT TIME FUNCTIONS. Introduction to Digital Systems 7 – Specification of Sequential Systems
3 SYNCHRONOUS AND ASYNCHRONOUS SYSTEMS time Clock CLK x(t) z(t) 0 1 2 3 4 5 6 (a) x(t) z(t) time (b) Figure 7.2: a) SYNCHRONOUS BEHAVIOR. b) ASYNCHRONOUS BEHAVIOR. CLOCK I/O SEQUENCE x ( t 1 , t 2 ) x (2 , 5) = aabc z (2 , 5) = 1021 Introduction to Digital Systems 7 – Specification of Sequential Systems

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4 Example 7.1: SERIAL DECIMAL ADDER x 1638753 y 3652425 z 5291178 LEAST-SIGNIFICANT DIGIT FIRST (at t=0) t 0 1 2 3 4 5 6 x(t) 3 5 7 8 3 6 1 y(t) 5 2 4 2 5 6 3 z(t) 8 7 1 1 9 2 5 Introduction to Digital Systems 7 – Specification of Sequential Systems
5 STATE DESCRIPTION t time Clock CLK x(t) z(t) s(t) t+1 t-1 s(t+1) H G s(t) Input x State s Output z Figure 7.3: OUTPUT AND STATE TRANSITION FUNCTIONS State-transition function s ( t + 1) = G ( s ( t ) , x ( t )) Output function z ( t ) = H ( s ( t ) , x ( t )) Introduction to Digital Systems 7 – Specification of Sequential Systems

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6 Example 7.3: STATE DESCRIPTION OF SERIAL ADDER Input: x ( t ) , y ( t ) ∈ { 0 , 1 , ..., 9 } Output: z ( t ) ∈ { 0 , 1 , ..., 9 } State: s ( t ) ∈ { 0 , 1 } (the carry) Initial state: s (0) = 0 Functions: The transition and output functions are s ( t + 1) = 1 if x ( t ) + y ( t ) + s ( t ) 10 0 otherwise z ( t ) = ( x ( t ) + y ( t ) + s ( t )) mod 10 EXAMPLE: t 0 1 2 3 4 5 6 x(t) 3 5 7 8 3 6 1 y(t) 5 2 4 2 5 6 3 s(t) 0 0 0 1 1 0 1 z(t) 8 7 1 1 9 2 5 Introduction to Digital Systems 7 – Specification of Sequential Systems
7 Example 7.4: ODD/EVEN TIME-BEHAVIOR SPECIFICATION: Input: x ( t ) ∈ { a, b } Output: z ( t ) ∈ { 0 , 1 } Function: z ( t ) = 1 if x (0 , t ) contains an even number of b 0 s 0 otherwise I/O SEQUENCE: t 0 1 2 3 4 5 6 7 x, z a, 1 b, 0 b, 1 a, 1 b, 0 a, 0 b, 1 a, 1 Introduction to Digital Systems 7 – Specification of Sequential Systems

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8 Example 7.4: STATE DESCRIPTION OF ODD/EVEN Input: x ( t ) ∈ { a, b } Output: z ( t ) ∈ { 0 , 1 } State: s ( t ) ∈ { even , odd } Initial state: s(0)= even Functions: Transition and output functions PS x ( t ) = a x ( t ) = b even even , 1 odd , 0 odd odd , 0 even , 1 NS, z ( t ) Introduction to Digital Systems 7 – Specification of Sequential Systems
9 MEALY AND MOORE MACHINES Mealy machine z ( t ) = H ( s ( t ) , x ( t )) s ( t + 1) = G ( s ( t ) , x ( t )) Moore machine z ( t ) = H ( s ( t )) s ( t + 1) = G ( s ( t ) , x ( t )) EQUIVALENT IN CAPABILITIES Introduction to Digital Systems 7 – Specification of Sequential Systems

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10 Example 7.5: MOORE SEQUENTIAL SYSTEM Input: x ( t ) ∈ { a, b, c } Output: z ( t ) ∈ { 0 , 1 } State: s ( t ) ∈ { S 0 , S 1 , S 2 , S 3 } Initial state: s (0) = S 0 Functions: Transition and output functions: PS Input a b c S 0 S 0 S 1 S 1 0 S 1 S 2 S 0 S 1 1 S 2 S 2 S 3 S 0 1 S 3 S 0 S 1 S 2 0 NS Output Introduction to Digital Systems 7 – Specification of Sequential Systems
11 REPRESENTATION OF STATE-TRANSITION AND OUTPUT FUNCTIONS STATE DIAGRAM S k S j input/output state x/z (a) S k S j a/p b/q Complete state diagram (b) S k S j a/p, b/q Simplified state diagram Figure 7.4: (a) STATE DIAGRAM REPRESENTATION. (b) SIMPLIFIED STATE DIAGRAM NOTATION.

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• Fall '07
• ERCEGOVAC
• Input/output, Transition function

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