Decoding SFN

This one is interesting piece of puzzle.
Normally the puzzle is MIB contains 8 bit info about SFN, then how does UE decode the 10 bit SFN out of 8 bit information.

Let's have a look at the MIB contain's first.

MIB:

• 3 bits for system bandwidth
• 3 bits for PHICH information,
  • 1 bit to indicate normal or extended PHICH
  • 2 bit to indicate the PHICH Ng value
• 8 bits for system frame number
• 10 bits are reserved for future use

If we look carefully then apart from 8bit SFN other information are fixed and will not change over the time.
Only SFN can change here over time.
Which means in every 40ms of MIB transmission the content of the MIB will differ and that is mainly due to 8bit SFN number.

For an example...
SFN changes every 10 ms as frame is of 10ms.

0000 0000 00    ---->> 1st frame at 0th to1ms (Original MIB)
0000 0000 01    ---->> 2nd frame at  10th ms (First copy of MIB)
0000 0000 10    ---->> 3rd frame at 20th ms (Second copy of MIB)
0000 0000 11    ---->> 4th frame at 30th ms (Third copy of MIB)
0000 0001 00    ---->> 5th frame at 40th ms (Original MIB)
0000 0001 01    ---->> 6th frame at 50th ms (First copy of MIB)
0000 0001 10    ---->> 7th frame at 60th ms (Second copy of MIB)
0000 0001 11    ---->> 8th frame at 70th ms (Third copy of MIB)

Now in MIB we receive the 8bit info except the last to bit.
MIB re-transmission has been designed carefully at every 10ms.
Which means after every 40 ms the content of MIB changes.

If UE syncs at second frame on the above example , UE will keep an check on the last bit received from MIB SFN number. Once it changes which is in 5th frame on the above example UE understand that the last two bit is 00 now. 
The decoded frame shall be then 0000 0001 00 

The longest time shall be taken by UE is 40 ms to sync. This happens when UE sync at the very first transmission of the MIB or the original MIB. UE shall wait for the next changed MIB content to realise the last two bit which will be happening in another 40 ms.

The shortest time shall be taken by UE is 10 ms to sync. This happens when UE sync at the third copy of the MIB. UE shall wait for the next changed MIB content to realise the last two bit which will be changing in another 10 ms.