Sequential Machine Design and Implementation

Somayyeh KoohiDepartment of Computer Engineering

Sharif University of TechnologyAdapted with modifications from lecture notes prepared by author

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Topics

n Retimingn Time borrowingn Sequential machine design

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Clock period

n For each phase, phase period must be longer than sum of:vCombinational delayvLatch propagation delay

n Phase period depends on longest path

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Unbalanced delays

§ Logic with unbalanced delays leads to inefficient use of logic:§ Lower clock frequency

long clock periodshort clock period

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Retiming

§ Retiming moves memory elements through combinational logic:§ Changes circuit’s timing § Keep the functionality

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Retiming properties

n Retiming changes encoding of values in registersv But proper values can be reconstructed with combinational

logicn Retiming may increase number of registers requiredn Retiming must preserve number of latches around a

cyclevMay not be possible with reconvergent fanout

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Topics

n Retimingn Time borrowingn Sequential machine design

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Advanced performance analysis

n Latch-based systems always have some idle logicvWaiting for the next latch to become active

n Can increase performance by blurring phase boundaries (time borrowing)vResults in cycle time closer to average of phasesvHigher frequency

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Example with unbalanced phases

§ One phase is much longer than the other:

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Spreading out a phase

n Compute only part of long paths in one phase (Ф1):

n Use other phase (Ф2) for end of long logic block and all of short logic block:

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Example of time borrowing

Without time borrowing With time borrowing40

40

Φ1

Φ2

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Problems

n Feasible for latches, not FFsv Since FFs are edge triggered

n Hard to debugv Can’t stop the system

n Hard to initialize system state v Since some of that state is stored on wires in the

combinational logicn More sensitive to process variationsv Process variation changes combinational delaysv So, rarely used

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Topics

n Retimingn Time borrowingn Sequential machine design

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Sequential machine design

n Two ways to specify sequential machine:vStructure: interconnection of logic gates and

memory elementsvFunction: Boolean description of next-state and

output functionsn Best way depends on type of machine being

described

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Counter

n Easy to specify as one-bit countern Harder to specify n-bit counter behaviorvCan specify n-bit counter as structure made of 1-

bit counters

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One-bit counter

Truth table:count Cin next Cout

0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1

next = count ⊕ CinCout = count & Cin

⇒

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One-bit counter (implementation and operation)

n All operations are performed as sφ2

n XOR computes next value of this bit of countern NAND/inverter compute carry-out

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One-bit counter stick diagramCout

VDD

VSSl1(latch) n(NAND) i(INV) x(XOR) l2(latch)

Cinφ1 φ1’ φ2 φ2’

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n-bit counter structure

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State transition graphs/tables

n Basic functional description of FSMn Symbolic truth table for next-state, output

functions:vNo structure of logicvNo encoding of states

n State transition graph and table are functionally equivalentvTransition graph ó Transition table

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01 string recognizer

Behavior of machine which recognizes “01” in continuous stream of bits:

time 0 1 2 3 4 5input 0 0 1 1 0 1state bit1 bit2 bit2 bit1 bit1 bit2next bit2 bit2 bit1 bit1 bit2 bit1output 0 0 1 0 0 1

n Waits for 0 to appear in state bit1n Goes into separate state bit2 when 0 appearsn If 1 appears immediately after 0, can’t have a 01 on next cycle

v So, can go back to wait for 0 in state bit1

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State transition table and graph§ Symbolic state transition table:input present next output0 bit1 bit2 01 bit1 bit1 00 bit2 bit2 01 bit2 bit1 1

§ State transition graph (equivalent to state transition table):

v Mealy FSM

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State assignment

n State assignment: must find binary encoding for symbolic states

n Choice of state assignment directly affects both the next-state and output logic:v Areav Delayv Power

Ø Efficient encoding reduces number of bit transition between adjacent states

n May also encode some machine inputs/outputs

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01 recognizer encoding

Choose bit1= 0, bit2 = 1:input present next output 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1

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Logic implementation

§ After encoding, truth table can be implemented in gates: