Maximum-length sequence (m-sequence) generator

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Key focus: Model and simulate m-sequence generator using Galois linear feedback shift registers (LFSR) that implement linear recursion. Plot correlation properties.

Maximum-length sequences (also called as m-sequences or pseudo random (PN) sequences) are constructed based on Galois field theory which is an extensive topic in itself. A detailed treatment on the subject of Galois field theory can be found in references [1] and [2].

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● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Maximum length sequences are generated using linear feedback shift registers (LFSR) structures that implement linear recursion. There are two types of LFSR structures available for implementation – 1) Galois LFSR and 2) Fibonacci LFSR. The Galois LFSR structure is a high speed implementation structure, since it has less clock to clock delay path compared to its Fibonacci equivalent. What follows in this discussion is the implementation of an m-sequence generator based on Galois LFSR architecture (Figure 1).

Galois LFSR architecture
Figure 1: A basic LFSR architecture (m-sequence generator)

The basic Galois LFSR architecture for an L^{th}-order generating polynomial in GF(2) is given in Figure 1. The integer value L denotes the number of delay elements in the LFSR architecture and g_0,g_1,...,g_{L} represent the coefficients of the generator polynomial. The generator polynomial of the given LFSR is

Generator polynomial for Galois LFSR

where, g_0,g_1,...,g_{L-1} \in GF(2), i.e, they take only binary values. The first and last coefficients are usually unity: g_0=g_{L}=1.

For generating an m-sequence, the characteristic polynomial that dictates the feedback coefficients, should be a primitive polynomial. Table 1 lists some of the primitive polynomials of degree upto L=12. More comprehensive tables of primitive polynomials can be found in reference [3].

Table showing Primitive polynomials up to degree L=12
Table 1: Primitive polynomials up to degree L=12 used in m-sequence generator

For implementation in Matlab, the LFSR structure can be coded in a straightforward manner that involves at least two for loops. If we could exploit the linear recursion property of LFSR and its equivalent matrix model, the LFSR can be implemented using only one for loop as shown in the Matlab function given in the book (click here). The function implements the LFSR structure Figure 1 by using the following equivalent matrix equation:

\mathbf{x}(n+1) = \mathbf{A} \mathbf{x}(n), \quad n >=0 \quad (2)

with the initial state of the shift registers represented by the vector \mathbf{x}(0), and \mathbf{x}(n) is an L dimensional vector given by

\mathbf{A}= \begin{bmatrix} \mathbf{x}_0(n) & \mathbf{x}_1(n) & \cdots & \mathbf{x}_{L-1}(n)\end{bmatrix}^T \quad \quad(3)

The matrix \mathbf{A} is of dimension L \times L , given by

\mathbf{A}= \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & g_0\\ 1 & 0 & 0 & \cdots & 0 & g_1 \\ 0 & 1 & 0 & \cdots & 0 & g_2\\ 0 & 0 & 1 & \cdots & 0 & g_3\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & g_{L-1} \end{bmatrix} \quad\quad (4)

Correlation properties of m-sequences are demonstrated here. It uses the sequence_correlation function defined in section 12.2.1 in the book (click here).

Let’s generate an m-sequence of period N=31 using the 5^{th} order characteristic polynomial g(x)=x^5+x^2+1. The coefficients of the characteristic polynomial is G=[g_5,g_4,g_3,g_2,g_1,g_0]=[1,0,0,1,0,0] and let’s start the generator with initial seed X_0=[0,0,0,0,1]. Typically, the autocorrelation function of m-sequences are two valued. The normalized autocorrelation of an m-sequence of length N, takes two values [1,-1/N]. This is demonstrated in Figure 2. We observe that that autocorrelation of m-sequence carries some similarities with that of a random sequence. If the length of the m-sequence is increased, the out-of-peak correlation -1/N reduces further and thereby the peaks become more distinct. This property makes the m-sequences suitable for synchronization and in the detection of information in single-user Direct Sequence Spread Spectrum systems.

autocorrelation of m-sequence generated using the polynomial
Figure 2: Normalized autocorrelation of m-sequence generated using the polynomial g(x) = x^5 + x^2 + 1

To demonstrate cross-correlation properties of two different m-sequences, consider two different primitive polynomials g_1(x)=x^5+x^4+x^2+x+1 and g_2(x)=x^5+x^4+x^3+x+1 that could generate two different m-sequences of length N=31. The normalized cross-correlation of the aforementioned m-sequences, shown in Figure 3, is given by the Matlab script given in the book (click here).

Normalized cross correlation of two m-sequences generated using the polynomials
Figure 3: Normalized cross_correlation of two m-sequences generated using the polynomials g_1(x) = x^5 + x^4 + x^2 + x + 1 and g_2(x) = x^5 + x^4 + x^3 + x + 1

The cross-correlation plot contains high peaks at certain lags (as high as 35\%) and hence the m-sequences causes multiple access interference↗ (MAI), leading to severe performance degradation. Hence, the m-sequences are not suitable for orthogonalization of users in multi-user spread spectrum systems like CDMA.

Continue reading on generation of Gold codes using preferred pair m-sequences…

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References

[1] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, vol. 68, no. 5, pp. 593–619, May 1980.↗
[2] S. W. Golomb, Shift-register sequences and spread-spectrum communications, Proceedings of IEEE 3rd International Symposium on Spread Spectrum Techniques and Applications (ISSSTA’94), Oulu, Finland, 1994, pp. 14-15 vol.1.↗
[3] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread Spectrum Communications, Prentice Hall, Inc., 1995.↗

Topics in this chapter

Spread spectrum techniques
● Introduction
● Code sequences
 □ Sequence correlations
 □ Maximum-length sequences (m-sequences)
 □ Gold codes
● Direct Sequence Spread Spectrum
 □ Simulation of DSSS system
 □ Performance of Direct Sequence Spread Spectrum over AWGN channel
 □ Performance of Direct Sequence Spread spectrum in the presence of Jammer
● Frequency Hopping Spread Spectrum
 □ Simulation model
 □ Binary Frequency Shift Keying (BFSK)
 □ Allocation of frequency channels
 □ Frequency hopping generator
 □ Fast and slow frequency hopping
 □ Simulation code for BFSK-FHSS

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