Timing Analysis and Optimization Considering Process Variation

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Timing Analysis and Optimization Considering
Process Variation

• UCLA EE201C
• Spring 2006, Professor Lei He

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Outline

• Static Timing Analysis (STA)
• Motivation
• Modeling
• Algorithm
• Summary
• Process Variation
• Trends
• Modeling
• Statistic Static Timing Analysis (SSTA)
• Monte Carlo simulation
• Path-based and block-based SSTA
• Optimization
• Student presentations

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Static Timing Analysis (STA)

• Part I

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Motivation

• Given a gate-level netlist
• Does the netlist meet the timing spec?
• How fast can it be clocked?
• Where is the critical path?
• Can the design be optimized?
• Its all about timing!
• Performance
• Timing analysis
• Quite complicated in the real world
• Well touch some of the basics

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Dynamic v.s. Static Simulation

• Dynamic timing analysis
• Start time ? end time
• Inputs are fully specified (input patterns)
• SPICE!
• Pro accurate
• Con super slow for large circuits
• Static timing analysis
• Separate timing analysis and function
  verification
• Characterize circuit timing independent of inputs
  and time periods
• Worst case analysis
• Work at the extremes ? guarantee work always

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STA Assumptions

• Design is synchronous
• Signal can be arbitrary
• Signal may take any value (0/1)
• Signal can switch at any time within the cycle
• Worst case assumption
• Signal becomes stable at the latest possible time
• Signal becomes unstable at the earliest possible
  time

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STA Close Lookup

• All storage elements are latch or FF
• Cycles cut by clocked storage elements
• STA characterizes timing of combinational parts
  between storage elements

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Single Path Delay Computation

• Path delay sum of 50 propagation delay

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Device Delay Models

• Unit delay model
• Delay through ANY gate is 1 time unit
• Const gate delay model
• Different gates have different delays

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Device Delay Models

• Delay is a function of fanout/slew
• Table based
• Input pins are different

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Interconnect Delay Model

• Interconnect delay becomes a dominant portion of
  total delay
• Lumped RC model
• Distributed RC model
• As frequency increases, RLC model may be necessary

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Interconnect Delay Model

• Delay for RC trees
• Rubinstein-Penfield Theorem
• For our purpose, we assume point-to-point wiring
  delays are pre-characterized as cell delays
• Different interconnect have different delays

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Putting things together

• Interconnect delay in red
• Gate delay in blue
• Arrival time in green

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Combinational Block Modeling Delay Graph

• Labeled directed acyclic Graph (DAG)
• Vertices gates, PI, PO
• Edges Interconnect
• Labels delays

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Path-based STA

• Path enumeration
• List all paths, output path delays
• Algorithms
• It works, but whats the problem?

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Path Enumeration

• Problem is the number of paths
• Can be exponential growth
• Total paths 2n

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Block Based STA

• Arrival times (AT) at a node
• Time when signal arrives at the node
• Latest time signal can become stable
• Determined by longest path from source

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Block Based STA

• Required arrival time (RAT) at a node
• Time before which signal must arrive to avoid
  timing violation
• Latest time signal is allowed to become stable
• Determined by longest path to sink

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AT Computation Kirkpatrick 1966

• AT at node n
• Longest path delay from source to node n
• Longest path Forward-prop(source)
• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W
• Linear algorithm operates on DAG

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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AT Computation illustrated

• Forward-prop(W)
• For each vertex v in W
• For each edge ltv,wgt from v
• Final-delay(w)max(Final-delay(w),
  delay(v)delay(w)delay(ltv,wgt))
• If all incoming edges of w have been traversed
• Add w to W

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How about RAT?

• RAT required arrival time
• Backtrack from sinks

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Timing Slack

• Each node AT RAT
• Timing slack RAT-AT
• Reflects criticality of a node
• Negative timing violation, node is no critical
  path
• Positive extra timing budget
• Optimize for power/area/robustness
• Slack distribution is KEY for timing
  optimization!!
• An example assuming RAT(f)5.80

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Logical v.s. Topological

• So far we have ignored the logic details for each
  vertices
• Topological timing analysis
• We dont care the logic functionality
• Major assumption for STA
• Major flaw due to this false path
• In contrast, for logical timing analysis
• We care the logic functionality

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False Path in STA

• A major (critical) assumption hidden behind STA!
• All paths are sensitizable
• There always exists a set of inputs that will
  cause the logic propagating along any chosen path
• But this is NOT true in general

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False Path Example Carry Bypass Adder

• Our previous algorithm will identify the red path
  as the longest path!
• Is this path sensitizable?

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False Path Example Carry Bypass Adder

• To sensitize red path we need
• ai XOR bi ai1 XOR bi1
• But red path is false
• When above condition is true, MUX selects 1
  input, i.e. directly from ci-1
• Instead shorter green paths are sensitized
• Hence, red path is not the critical path of the
  circuit!

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False Path Aware STA

• False path is universal for STA
• Causes a lot of pain for STA
• Determining sensitization path is very hard
• As hard as Boolean Satisfiability (SAT) problem,
  which is proven NP-hard
• Fortunately, there exist a lot of effective
  heuristics
• Beyond the scope

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Mini Summary

• STA is a very powerful tool for chip sign-off
• Delay modeling (device interconnect)
• DAG for timing analysis
• AT/RAT/Slack computation algorithms
• False path in STA

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Midterm Presentation

• Analysis and optimization of power supply network
  using RC model or RCL model
• variation aware circuit tuning (e.g.,
  criticality, gate sizing)
• variation aware clock design (clock tree or
  network)
• chip-level variation analysis on timing and power
• parameterized model order reduction
• new development on statistical static timing
  analysis (SSTA)

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Bet on Topics

• Submit proposal by May 8th
• One page per topic
• Two topics a team
• Proposal content
• Theme of presentation
• Papers to be presented
• Likelihood for a programming project