Spread spectrum system, originally developed for military applications, is extremely resistant to unauthorized detection, jamming, interference and noise. It converts a narrowband signal to wideband signal by the means of spreading. Standards like WiFi and bluetooth use spread spectrum technology such as direct sequence spread spectrum and frequency hopping spread spectrum respectively. Spread spectrum techniques rely on the use of one or other form of coding sequences like m-sequences, Gold codes, Kasami codes, etc., The study of properties of these codes is important in the design and implementation of a given spread spectrum technique. In this chapter, generation of spreading sequences and their main properties are covered, followed by simulation models for direct sequence spread spectrum and frequency hopping spread spectrum systems.
Spread spectrum
A spread-spectrum signal is a signal that occupies a bandwidth that is much larger than necessary to the extent that the occupied bandwidth is independent of the bandwidth of the input-data. With this technique, a narrowband signal such as a sequence of zeros and ones is spread across a given frequency spectrum, resulting in a broader or wideband signal. Spread spectrum was originally intended for military application and it offers two main benefits. First, a wideband signal is less susceptible to intentional blocking (jamming) and unintentional blocking (interference or noise) than a narrowband signal. Secondly, a wideband signal can be perceived as part of noise and remains undetected. The two most popular spread spectrum techniques widely used in commercial applications are direct sequence spread spectrum (DSSS) and frequency hopping spread spectrum (FHSS).
Bluetooth, cordless phones and other fixed broadband wireless access techniques use FHSS; WiFi uses DSSS. Given that both the techniques occupy the same frequency band, co-existence of bluetooth and Wifi devices is an interesting issue. Both FSSS and DSSS devices perceive each others as noise, i.e, the WiFi and Bluetooth devices see each other as mutual interferers. All the spread spectrum techniques make use of some form of spreading or code sequences.
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Code sequences for spread spectrum
Be it DSSS or FHSS, the key element in any spread spectrum technique is the use of code sequences. In DSSS, a narrowband signal representing the input data is XOR’ed with a code sequence of much higher bit rate, rendering a wideband signal. In FHSS, the transmitter/receiver agrees to and hops among a given set of frequency channels according to a seemingly random code sequence. The most popular code sequences used in spread spectrum applications are
● Maximum-length sequences (m-sequences)
● Gold codes
● Walsh-Hadamard sequences
● Kasami sequences
● Orthogonal codes
● Variable length orthogonal codes
These codes are mainly used in the spread spectrum for protection against intentional blocking (jamming) as well as unintentional blocking (noise or interference), to provide a provision for privacy that enables protection of signals against eavesdropping and to enable multiple users share the same resource. The design and selection of a particular code sequence for a given application largely depends on its properties. As an example, we will consider generation of m-sequences and Gold codes, along with the demonstration of their code properties.
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Sequence correlations
Given the choice of numerous spreading codes like m-sequences, Gold codes, Walsh codes etc., the problem of selecting a spreading sequence for a given application reduces to the selection of such codes based on good discrete-time periodic cross-correlation and auto-correlation properties.
The cross-correlation of two discrete sequences x and y, normalized with respect to the sequence length N, is given by
![\begin{aligned} R_{xy}(k) &= \frac{1}{N} \text{[Number of agreements - Number of disagreements when comparing one full period}\\ & \quad\quad\quad \text{of sequence x with k position cyclic shift of the sequence y]} \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D+R_%7Bxy%7D%28k%29+%26%3D+%5Cfrac%7B1%7D%7BN%7D+%5Ctext%7B%5BNumber+of+agreements+-+Number+of+disagreements+when+comparing+one+full+period%7D%5C%5C+%26+%5Cquad%5Cquad%5Cquad+%5Ctext%7Bof+sequence+x+with+k+position+cyclic+shift+of+the+sequence+y%5D%7D+%5Cend%7Baligned%7D+&bg=ffffff&fg=000&s=1&c=20201002)
Similarly, the auto-correlation of a discrete sequence refers to the degree of agreement between the given sequence and its phase shifted replica. To avoid problems due to false synchronization, it is important to select a spreading code with excellent auto-correlation properly. The periodic autocorrelation functions of m-sequences approach the ideal case of noise when the sequence length N is made very large. Hence, m-sequences are commonly employed in practice when good autocorrelation functions are required. For applications like code division multiple access (CDMA), the codes with good cross-correlation properties are desired.
Maximum-length sequences (m-sequences)
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].
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. Kindly check out this article where I detailed the generation of m-sequences using Galois LFSR.
Figure 1, depicts the auto-correlation of an m-sequence of period N=31 using the 5th order characteristic polynomial g(x)=x5+x2+1. The normalized auto-correlation of an m-sequence of length N, takes two values [1,-1/N]. We observe that that auto-correlation 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.
Figure 2, depicts the cross-correlation of two different m-sequences, consider two different primitive polynomials g1(x)=x5+x4+x2+x+1 and g2(x)=x5+x4+x3+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.
Gold codes
In applications like code division multiple access (CDMA) technique and satellite navigation, a large number of codes with good cross-correlation property is a necessity. Code sequences that have bounded small cross-correlations, are useful when multiple devices are broadcasting in the same frequency range. Gold codes, named after Robert Gold, 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 pair of m-sequences. Kindly check out this article where I detailed the generation of Gold codes using preferred pair m-sequences.
Gold sequences belong to the category of product codes where two preferred pair of m-sequences of same length are XOR’ed (modulo-2 addition) to produce a Gold sequence. The two m-sequences 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.
The auto-correlation of a Gold code sequence, plotted in Figure 3 and the normalized cross-correlation of preferred pair m-sequences (used for Gold code generation) is plotted in Figure 4.
![Auto-correlation 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=513%2C186&ssl=1)
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] 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.↗
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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 |