Focus of this article is to discuss the details of Gold code generator using preferred pair m-sequences, implemented using linear feedback shift registers (LFSR). Finally we plot and investigate correlation properties of the generated Gold codes.
Introduction
In a multi-user environment (like spread spectrum, CDMA ) large number of codes with good correlation properties, is a necessity. Gold codes are suited for this application, since a large number of codes with controlled correlation can be generated by a simple time shift of two preferred m-sequences.
Gold sequences belong to the category of product codes where two m-sequences of same length are XOR’ed to produce a Gold sequence. The two m-sequence must maintain the same phase relationship till all the additions are performed. A slight change of phase even in one of the m-sequences, produces a different Gold sequence altogether. Gold codes are non-maximal and therefore they have poor auto-correlation property when compared to that of the underlying m-sequences.
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A typical implementation of Gold code generator is shown in Figure 1. Here, the two linear feedback shift registers (LFSR), each of length 
, are configured to generate two different m-sequences. As we cannot start the LFSRs with all zero values, there need to be some values in the LFSRs. These initial values are controlled by ‘seed 1‘ and ‘seed 2‘. The phase of the m-sequence loaded into an LFSR can be controlled by shifting the initial seed (initial value of the shift registers in the LFSR) there by providing an option to generate a new Gold sequence.
Note that there exists numerous choice of Gold sequences based on the m-sequence configuration of the LFSRs and what is initially loaded into the two LFSRs. Not all the Gold sequences possess good correlation properties. In order to have a Gold code sequence with three valued peak cross-correlation magnitudes, that is both bounded and uniform, the m-sequences generated by the two LFSRs need to be of preferred type.
Algebraic details on generating two preferred m-sequence for Gold code generation is detailed in Gold’s work [1]. It warrants the need for searching the preferred pairs, instead, the configurations given in Table 1 can be used for generating Gold codes that provide optimal correlation properties (three valued correlation as mentioned in Reference [2]). Following points must be noted when choosing the length of LFSR (
) when generating the Gold code.
● Three valued cross-correlation values occur when is odd or when 
.
● The peak cross-correlation magnitude achieves the minimum possible value only when is odd (![n=[5,7,9,11,...]](https://s0.wp.com/latex.php?latex=n%3D%5B5%2C7%2C9%2C11%2C...%5D&bg=ffffff&fg=000&s=0&c=20201002)
). Therefore, this is the best choice here.
● When , the m-sequences listed in table are still of preferred type (three valued cross-correlation), but the lower bound is weaker when compared to the lower bound when is odd.
Hardware implementation
A sample hardware implementation for generating a length 127 Gold code – using the preferred pairs ([7,3,2,1],[7,3]) is shown in Figure 2. Here each register in the LFSR is a D-flip flop,all connected in cascaded fashion and operating at a given synchronous clock. Changing the initial seed values into shift registers produces a different set of Gold codes. Matlab code for implementing a Gold code generator is available in this book.
Correlation properties
Figure 3 on this post shows the periodic cross-correlation of two m-sequences that are not a preferred pair. Let’s take a preferred pair from the Table 1 for N = 31 having the feedback connections – [2,3,4,5] and [2,5]. The correlation of two sequences can be computed using the function given in section 12.2.1 of this book.
![Normalized cross-correlation of preferred pair m-sequences using the feedback connections [2,3,4,5] and [2,5]](https://i0.wp.com/www.gaussianwaves.com/gaussianwaves/wp-content/uploads/2019/05/Normalized-cross-correlation-of-preferred-pair-m-sequences.png?resize=543%2C208&ssl=1)
![Autocorrelation of Gold code sequence generated using the preferred pair feedback connections [2,3,4,5] and [2,5]](https://i0.wp.com/www.gaussianwaves.com/gaussianwaves/wp-content/uploads/2019/05/Autocorrelation-of-Gold-code-sequence.png?resize=543%2C197&ssl=1)
The normalized cross-correlation of the chosen preferred pair and the auto-correlation of the corresponding Gold code is shown in Figure 3 and Figure 4 respectively. The auto-correlation and cross-correlation plots reveal that the Gold code sequence does not possess the excellent auto-correlation property as that of individual m-sequences, but it possess good cross-correlation properties in comparison with the individual m-sequences.
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References:
[1] Gold, R. (1967), Optimal binary sequences for spread spectrum multiplexing, IEEE Transactions on Information Theory, 13 (4), pp. 619-621.↗
[2] Sarwate D.V, Pursley, M.B., “Crosscorrelation Properties of Pseudorandom and Related Sequences”, Proceedings of the IEEE, Volume 68, Issue 5, pp 593 – 619, May 1980.↗
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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 |