[4G&5G Topic-34]: Physical Layer-On the principle of m-sequence and its application in NR PSS

If the object of the arrangement is a binary bit 0 or 1, it is called a binary sequence. The binary sequence is a sequence composed of two points 0 and 1 on the coordinate axis, and different points can overlap.

If the object of the arrangement is an integer, it is called a sequence of integers . An integer sequence is a sequence of points on the horizontal or vertical axis, and the points can overlap.

If the object of the arrangement is a complex number, it is called a complex number sequence. The complex number sequence is a sequence of points in the plane coordinates, and the points can overlap.

It may be pre- first determine and may be repeated to achieve the sequence is referred to determine the sequence ;

A sequence that can neither be predetermined nor reproduced is called a random sequence ;

A sequence that cannot be determined in advance but can be repeatedly generated is called a pseudo- random sequence .

1.2 Overview of M sequence

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Model of n-stage cyclic sequence generator

m sequence is a widely used pseudo-random sequence , which has a wide range of applications in the communication field, such as spread spectrum communication, satellite communication code division multiple access, encryption, scrambling, synchronization, bit error rate measurement in digital data, etc. field.

Among all pseudo-random sequences, m-sequence is the most important and basic kind of pseudo-random sequence. It is easy to produce, has strong regularity, has good autocorrelation and good cross-correlation characteristics.

In the 4G LTE system, physical layer signals such as PSS, SSS, cellRS, DMRS, SRS, PRACH, PUCCH, etc., basically involve ZC (Zadoff-Chu) sequence signals.

In the 5G NR system, in addition to using the M sequence to generate PSS and SSS signals that resist large frequency offset scenarios, other signals also involve the Zadoff-Chu sequence.

1.3 Definition of m-sequence

The m sequence is the abbreviation of the longest linear shift register sequence , which is a pseudo-random sequence , pseudo-noise (PN) code or pseudo-random code.

As the name suggests, m sequence is: an n-stage shift register or a delay element, by a linear feedback, the maximum generated $2^{n}-1$ bits code binary sequence , the binary code sequence is a m-sequence.

In a binary shift register, if the number of stages of the n-bit shift register, the n-stage shift register has a total of 2 n states, except for all 0 states, there is still a $2^{n}-1$ middle state, so the maximum length code sequence it can generate is $2^{n}-1$ Bit.

In other words, the longest period generated by an n-stage linear feedback shift register is equal to $2^{n}-1$ .

1.4 Features of m-sequence

(1) Balance

In a cycle of the m sequence, the number of 0 and 1 is basically the same. The number of ones is one more than the number of zeros. This property can be seen from the m sequence 1000010010110011111000110111010: there are 16 1s and 15 0s in total.

(2) Run length distribution

m sequence of successive same values as those elements collectively referred to as a "run."

The number of elements in the run is called the run length. In the n-level m sequence, there are 2n-1 runs in total.

Among them, the run length of 1 occupies 1/2 of the total number of runs, the length 2 runs 1/4 of the total number of runs, and the length k occupies 2k of the total number of runs.

And in the run length of k, the number of runs of consecutive 0s and consecutive 1s each occupies half. For example, in the sequence 1000010010110011111000110111010, the total number of runs is 25-1=16, and the run lengths of this sequence are distributed as follows:

The number of runs of length 1 is 8, of which 4 are 1 run and 4 are 0 run;

The number of runs of length 2 is 4, 2 runs of 11, and 2 runs of 00;

The number of runs of length 3 is 2, 1 is 111 runs, and 1 is 000 runs;

The number of consecutive 0 runs of length 4 is 1;

The number of runs of length 5 is 1.

(3) Shift and add characteristics

An m sequence m1 is added modulo 2 to another sequence m2 generated by any delayed shift, and the result is still a certain delayed shift sequence m3 of m1, that is, the exclusive OR of m1 and m2 is m3.

(4) Correlation characteristics
We can verify the autocorrelation characteristics of m-sequences based on the shift-and-add characteristics. Because the result of shifting and adding is still m sequence, so the number of 0 is 1 less than the number of 1,

1.5 Comparison of M sequence and ZC sequence

(1) Code content: ZC sequence is a complex number sequence, and M sequence is a binary sequence.

(2) The randomness of the code: ZC sequence is a known sequence, the output value of the sequence is determined by an accurate mathematical function, and the M sequence is a pseudo-random sequence.

1.6 Why does 5G NR change PSS to m-sequence?

The ZC sequence is a sub-carrier phase sequence, which is a phase modulation at any angle.

5G application scenarios include the high frequency band 5G-60GHz, and the use of phase modulation at any angle results in a ZC sequence, which has a greater frequency deviation in the high frequency band, for example, 5ppm is up to 300kz at 60GHz.

The correlation is affected, and the correlation peak-to-peak value decreases and the false detection increases, so the m-sequence is used instead.

LTE uses ZC because of its good auto-correlation, cross-correlation, and 4G is mainly in the 2GHz frequency band, so the time-frequency offset is relatively small.

5N adopts m-sequence, and the bottom layer adopts PSK modulation. The phase of each subcarrier is two deterministic values, not arbitrary values. Therefore, the problem of difficult phase difference detection caused by high frequency is overcome by binary PSK modulation.

Chapter 2 NR PSS m-sequence generation process

2.1 Composition of 5G NR physical cell

The 5G physical cell ID is the same as LTE, consisting of $^{N_{ID}^{2}}$ and $^{N_{ID}^{1}}$ ; among them $^{N_{ID}^{2}}$ , it is carried in the PSS and carried $^{N_{ID}^{1}}$ in the SSS.

$^{N_{ID}^{2}}$ There are three types of 4G physical cell IDs carried in PSS, and there $^{N_{ID}^{1}}$ are 168 types carried in SSS signals. The total number of physical cell ID numbers is 504=3 * 168;

$^{N_{ID}^{2}}$ There are three types of 5G physical cell IDs carried in the PSS, and there $^{N_{ID}^{1}}$ are 336 types carried in the SSS signal. The total number of physical cell ID numbers is 1008=3 * 336; it is exactly twice the number of LTE physical cells.

2.2 The physical cell ID in PSS is mapped to m sequence parameters

The m sequence is a pseudo-random number. Given the cyclic shift position and the number of sequences n, the content of the m sequence is actually known. Although the numbers of 0 and 1 are arranged randomly, this This sorting relationship is actually deterministic and repeatable.

In 5G NR, it is specified that the physical cell number $^{N_{ID}^{2}}$ {0,1,2} in the PSS is mapped to the cyclic shift position parameters of the cell m sequence as {0,43,86} respectively ;

2.3 Generation of m sequence

When the cyclic shift position parameters are {0, 43, 86} respectively, through the m-sequence hardware circuit, 3 different pseudo-random m-sequences (in binary format) with a total of 127 bits are generated .

During network deployment, when the physical cell ID is determined, its $^{N_{ID}^{2}}$ value is also determined, which is one of {0,1,2} .

In a physical cell, the m-sequence of the device can only have one of three m-sequences.

2.4 M-sequence to sub-carrier mapping

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The PSS signal occupies 127 subcarriers, the m sequence has 127 bits, and each bit is mapped to a subcarrier.

There are 8 and 9 sub-carriers SC on both sides as guard bands, occupying a total of 144 sub-carriers=144=127+8+9.

2.5 m sequence of sub-carrier modulation

Each sub-carrier adopts binary phase modulation BPSK.

(1) BPSK time domain waveform

(2) BSPK constellation diagram

2.6 The essence of NR PSS m sequence

(1) NR PSS m sequence itself is a pseudo-random binary sequence

(2) The number of bits in the NR PSS m sequence is 127 bits

(3) After the NR PSS m sequence is mapped to sub-sub-carriers, it becomes a sub-carrier sequence composed of 127 consecutive sub-carriers.

(4) The 127 sub-carriers in the frequency domain undergo OFDM transformation together with other sub-carriers, and are modulated into OFDM time-domain signals.

Chapter 3 UE's Detection Process of PSS m Sequence

This is different from the LTE PSS detection process. The specific steps are as follows:

(1) The UE will try to receive the PSS signal near the center frequency of the NR frequency it supports

(2) Use fast Fourier transform to obtain the frequency domain signal of each subcarrier

(3) The binary bit corresponding to each sub-carrier is demodulated by BPSK.

(4) After demodulation, the binary bit sequence in the m sequence is obtained.

(5) The three m-sequences in NR PSS are known, and $^{N_{ID}^{2}}$ the corresponding relationship between m-sequences and m-sequences is also consistent.

(6) The key now: How to determine which m-sequence is the binary sequence demodulated by Fourier transform and BPSK?

The specific method is to use the known m-sequence and the demodulated m-sequence to perform correlation operations (sequence inner product operations, that is, bitwise multiplication and summation):

If the result of the operation is 0, it means that the received binary value sequence is not the m sequence selected for comparison.

If the result of the operation is not 0, it means that the received binary value sequence is not the m sequence selected for comparison, but may be other interfering data.

If the result of the operation is not 0 and the result of the operation is the predetermined maximum value, it means that the received binary sequence is a known binary sequence.

(7) Map the known binary sequence to $^{N_{ID}^{2}}$ one of {0,1,2} .

So far, the mobile terminal obtains the physical cell ID number of the cell by receiving and demodulating the PSS signal $^{N_{ID}^{2}}$ .

The complete physical cell ID number requires further demodulation of the SSS signal.

reference:

http://ziyubiti.github.io/2018/02/07/5gnrpss/