Frequency correction of timestamp events

[pyuxiang]

Issue by franciumxzf
Monday Jan 30, 2023 at 09:29 GMT
Originally opened as franciumxzf/fpfind#6

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1. frequency correction done on rawevents instead of epochs
2. pfind will find the frequency differency: if it is large, will perform the frequency correction on rawevents before chopper
3. store the frequency difference in the units of 1e-12
4. how to apply the frequency correction in integer multiplication

[pyuxiang]

Comment by pyuxiang
Wednesday Feb 01, 2023 at 14:28 GMT
Originally commented as franciumxzf/fpfind#6 (comment)

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The equation to compute for each timestamp $t_i$ is,

$$ (t_i - T_{\text{start}}) ( 1 + \Delta{}f) + T_{\text{start}} = t_i + (t_i - T_{\text{start}}) \Delta{}f $$

Some observations first:

1. The pfind algorithm is currently hardcoded to terminate when the frequency offset correction is < 1e-10 (or 2^-34)
2. Frequency differences larger than 250ppm cannot be used as an input to the PLL in the clock chip (or 2^-12)
   • The range of accepted frequency offsets thus encompasses only 22 bits (excl. 1 sign bit)
3. The timestamp is 54-bits wide in units of 4ps, and thus cycles every 20 hours

To avoid expensive multiplication with a floating point, we express $\Delta{}f$ as a signed integer.

Assume we use units of 1e-12 (or 2^-40) for the $\Delta{}f$, and consider two different offsets, (1) 2^-40 (or 1-bit) and (2) 2^-12 (or 28 bits). We enumerate the two cases,

1. Only the 54 - 40 = 14 MSb timestamp will have a direct impact on the timing correction
2. The required resolution is 54 - 12 = 42 MSb timestamp.

The minimum required timing correction is thus at most on the order of 28 + 42 = 70 bits (excl. sign bit).
A naive implementation suggests we simply use a 128-bit integer to hold the multiplication result of $(t_i - T_{\text{start}}) \Delta{}f$, before the 40-bit right-shift.

[pyuxiang]

Comment by pyuxiang
Wednesday Feb 01, 2023 at 14:37 GMT
Originally commented as franciumxzf/fpfind#6 (comment)

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In considering a possible optimization, maybe easier if we think of it in base-10?
Suppose the timestamp is 987656789 and the offset is 989 in units of 1e-6.

A full multiplication yields 976792.564321, with the unnecessary fractional part.
A representative multiplication is 9876567 * 989 * 1e-4 = 976792.4763, since the remainder (89 * 989 / 1e6) < 10^6 / 1e6 = 1 is entirely fractional.

In other words, the larger the multiplied value, the smaller the irrelevant part. In the worst case scenario with a 2^-12 (or 28-bit) offset in units of 2^-40, only 40-28-1=11 bits of the timestamp can be cut out, i.e. the maximum size of the resulting multiplication that must be performed is actually 28 + (54 - 11) = 71 bits (again excluding a sign bit), which has the following algorithm:

$$ \left( \left[ (t_i - T_{\text{start}}) \gg 11 \right] \times \Delta{}f \right) \gg (40-11) $$

This size can be reformulated as (40-12) + (54 - (40-(40-12)-1)) = (40-12) + (54 - 12 + 1) = 55 + 40 - 2*12 = 31 + 40 = 71 bits.

[pyuxiang]

Comment by pyuxiang
Wednesday Feb 01, 2023 at 14:46 GMT
Originally commented as franciumxzf/fpfind#6 (comment)

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To fit everything into a 64-bit signed multiplication result is then clear: if the maximum frequency difference is 2^-12, then the finest frequency offset resolution we can accept is 32 bits.

Alternatively, we can have a maximum $\Delta{}f$ of 2^-13 = 122 ppm, and still support a frequency resolution of 34 bits = 0.6e-10. In this scheme, we finally have

$$ \left( \left[ (t_i - T_{\text{start}}) \gg 12 \right] \times \Delta{}f \right) \gg 22 $$

where $\Delta{}f$ is in units of 2^-34. Whew!

[pyuxiang]

Comment by pyuxiang
Wednesday Feb 01, 2023 at 14:50 GMT
Originally commented as franciumxzf/fpfind#6 (comment)

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One small note, due to the overflow of the timestamp after every 20 hours, the timestamp correction will become erroneous within 20 hours. We might need to consider a fix to this as well.

[pyuxiang]

Comment by pyuxiang
Friday Feb 03, 2023 at 00:45 GMT
Originally commented as franciumxzf/fpfind#6 (comment)

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Closed with commit franciumxzf/fpfind@82b84bd.