Why is Logical root sequence number in Root Zadoff-Chu needed?

Started by guyholtzman 5 years ago7 replieslatest reply 5 years ago1423 views


I have been investigating RACH preamble planning and the properties of Root Zadoff-Chu sequences as used in 3GPP #LTE .

From Wikipedia, Zadoff-Chu sequences have the following properties:

  • The auto correlation of a Zadoff–Chu sequence with a cyclically shifted version of itself is zero, i.e., it is non-zero only at one instant which corresponds to the cyclic shift.
  • The cross-correlation between two prime length Zadoff–Chu sequences, i.e. different values of u = u1, u = u2 is constant 1/√NZC, provided that u1- u2 is relatively prime to NZC


  • NZC is chosen to be the prime number 839 in preamble formats 0-3
  • The physical root sequence number u is between 1 to 838

Since any u1- u2 from the allowed u set are always relatively prime to NZC= 839, I can understand that the cross correlation between any u1, u2 , always equals to 1/√NZC given u1 is not equal to u2 and they are both chosen from 1 - 838.

in 3GPP, 36.211 Table 5.7.2-4 the standard further defined a logical root sequence number. in particular, it also partitioned different groups ( rows), which every group has a different amount of root sequence numbers. 

The intent behind this mapping (logical to physical) is not understood to me. Working with only the physical root sequence numbers seems to be sufficient for RACH planning given the properties above.

My questions are:

  • Why is this table needed? 
  • Why is the logical to pyhsical mapping done this way?
  • Why is it partitioned to different rows? 

Thanks in advance,


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Reply by djmaguireJanuary 31, 2019

This is not my area of expertise.  Some questions catch my interest, though, and I use them to learn something new to which I wouldn't otherwise be exposed.

After a little poking around, it appears to me that the logical root sequence number is a concept that:

  • ...was arrived at in committee.
  • ...is a compromise proposed by LG that balanced proposals of TI and Panasonic.
  • ...follows an equation whose parameters dictate the irregularity of output.
  • ...involved beer.  Kidding.

I was going to link documents, but you are probably better off drawing your own conclusions.  I recommend that you look at these two batches of 3GPP Technical Working Group contribution documents:



Just search the title names for relevant starting points and then use the references in those documents to direct you to other less clearly named sources.  

You may want to go earlier or later for a more complete story.

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Reply by guyholtzmanJanuary 31, 2019

Thanks djmaguire!,

I have followed the trail of documents until I figured it out. there are  two properties: 

  • CM - Cubic metric, which is similar to PAR (Peak to average). it is correlated wit the efficiency of the PA of the UE
  • S_max - which is associated with the maximum radius of a cell supporting high speed

these 2 parameters are different from one u to another. the committee chose to device the logical entries into  subgroups where each subgroup is limited to an increasing maximum, (depending on the group). here is one of graphs from the drafts trying to mitigate between the properties:rach_94721.png 

for more information:

R1 - 074692



R1 - 080761



there are many others if you ever get bored :)

Regards, Guy

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Reply by kazJanuary 31, 2019
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Reply by guyholtzmanJanuary 31, 2019

Thanks, kaz

I have read chapter 6. while this paper does seem to present a very good overview on RACH and Zadoff–Chu sequence, it does not refer to the logical sequence numbers or to any of my questions on them. 

Regards, Guy

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Reply by guyholtzmanJanuary 31, 2019

Post deleted by author

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Reply by kazJanuary 31, 2019

Well this one explains it better(scroll down to middle of page), sorry I am not myself focused on details of preamble detection but I do implement Prach extraction on FPGA:


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Reply by guyholtzmanJanuary 31, 2019


Yes, I understand that 0 is translated to 129, 1 is translated to 710 and so on. I also understand that the logical entries are the ones that are broadcasted using SIB2. however, this does not answer the question  why did the 3GPP standard find it necessary to create this translation table (mapping) and why did it partition it in the way it is partitioned. for example, the first row translates logical entries 0-23, and the second row translates entries 24 - 29. why does the first row translate 24 entries and the second row only 5? the SIB2 could have easily send the physical root sequence number instead of the logical ones. but for some reason there is a translation table between. these questions are not answered by the link.