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I want to tap into a set of serial data that is transferred in binary pulses of 100 nanoseconds of width. Can Arduino catch this signal, or is the signal too fast? What can be recommended?

The pulses swing from +3volts to -3volts, with each half having a width of 50 nanoseconds.

Ron
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This one may be up to the task...
http://arduino.cc/en/Main/arduinoBoardDue
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http://jeelabs.org/2010/01/06/pin-io-performance/

I found this 2 weeks ago, i hope it helps.
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That 100ns means you have 10 million pulses per second and 20 million changes per second.  The Duo is running at 84MHz.  You can't do it with 4 clock cycles.
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You assume the Due would be polling.  @Ronduino has not provided a definition for "serial data" so it is too early to say if your assumption holds water.
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If I can guarantee to capture one measurement of each pulse, then I would be satisfied. Messages are 34 bit and that's what I want to capture and then process. First step is to be able to capture the pulse, as they are very fast. I'm wanting to work toward creating a type of network analyzer to search for certain messages and then activate an LED as an indicated that a given message has come across the line. That's my overall aim.
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The pulses swing from +3volts to -3volts, with each half having a width of 50 nanoseconds.

Manchester encoding?
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Yes, Manchester Encoding.
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If you're feeling brave, the Due processor has a "Manchester Decoder".  I have no idea what it is or how to use but it sounds promising.  A snippet from the datasheet to tease you...

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36.7.3.4 Manchester Decoder
When the MAN field in US_MR register is set to 1, the Manchester decoder is enabled. The decoder performs both preamble and start frame delimiter detection. One input line is dedicated to Manchester encoded input data.
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