Notes on Synchronizers and Probability of Failure

Introduction to Synchronizers

• Synchronizers manage asynchronous inputs.
• Common in user interfaces (e.g., smartphones, computers).
• Goal: Minimize probability of failure, but cannot eliminate it completely.

Structure of a Synchronizer

• Constructed using two back-to-back flip-flops:
  • First Flip-Flop: Sampling flip-flop that samples asynchronous input (D).
  • Second Flip-Flop: Allows for signal regeneration by providing feedback.

Timing Considerations

• Important timing parameters:
  • Tc: Cycle time
  • T_setup: Setup time
• Regeneration time:
  • Defined as: T_waiting = Tc - T_setup
• This waiting time is crucial for reducing the probability of failure.

Calculating Probability of Failure

• Probability of Failure: Can be calculated based on the waiting time.
• Probability of Failure per Second:
  • If a button is pressed n times per second, overall probability = Probability of failure × n.
• Example Calculation:
  • If the probability of hitting a target is 0.1 and you hit it 5 times per second, the probability of failure per second = 0.1 × 5 = 0.5.

Mean Time Between Failures

• Calculated as:
  • Mean Time Between Failures = 1 / Probability of Failure per Second
• Example with Coin Toss:
  • Probability of Heads = 0.5, if tossed once per second,
  • Probability of failure per second = 0.5, thus Mean Time = 2 seconds.

Example Synchronizer Parameters

• Example values:
  • Cycle time = 2 ns
  • Frequency = 500 MHz
• Calculation results in:
  • Probability of failure ≈ 5.6 x 10^-6
  • Probability of failure per second and mean time between failures calculated accordingly.
  • Example: Mean time between failures = 5 hours.

Application Context

• Application of synchronizer affects acceptable failure rates:
  • Computer with a 5-hour failure rate may be unacceptable.
  • A child's toy may be more forgiving with similar failure rates.

Improving Reliability

• To decrease failure rate:
  • Add additional flip-flops to the synchronizer.
  • Each added flip-flop doubles the regeneration time, exponentially increasing reliability.
• Output is sent to the system only after confirming high stability of the output value (either high or low).

Summary: Understanding the role and design of synchronizers, including the calculation of probability of failure and mean time between failures, is key for ensuring system reliability in various applications.