Use of pseudo-random binary sequence (Gold, Kasami, etc.) in frequency hopped code-division multiple access (FH/CDMA)

Question

I have a clear idea of how a PN-sequence is used in a Direct Sequence Code-Division Multiple Access(DS-CDMA) system with BPSK. In such a system, a different pn-sequence is assigned to each user and each transmitted bit is XORed with the binary pn-sequence.

Now my question is:

For each symbol time, how are this binary pn-sequences used in a FH/CDMA system to chose which hop frequency to use if there are more than two possible hop frequencies?

If I have 2 possible hop frequencies to use, the answer is trivial, because each bit of the pseudo-random sequence would tell which of the two hop frequencies must be used. But when having more than two frequencies... How is this selection done?

Answer 1

There are many ways to do this. A simple method is to take the current value of the register (not just its output) as the carrier frequency selector. For instance, a (6,1) LFSR with 63 states can select between 63 different channels.

Other methods are based on cubic and quadratic congruence codes. I don't know much about them myself, but look up the papers by Titlebaum et al.

Answer 2

As the answer by @MBaz points out, one can use the contents of a $n$-bit maximal-length binary LFSR to select from $2^n-1$ different frequencies to hop to, thus creating a frequency-hopping sequence of period $2^n-1$. The difficulty is in creating more than one frequency-hopping sequence for use in a multiple-access scheme: the FH/CDMA scheme that the OP refers to. A lot of useful information about this problem and its various solutions can be found in Chapter 8 "Reed-Solomon Codes and the Design of Sequences for Spread-Spectrum Multiple-Access Communications" of the book Reed-Solomon Codes and Their Applications, S. B. Wicker and V. K Bhargava, Eds., IEEE Press, 1994. The works of Titlebaum et al. mentioned by MBaz are related to the Reed-Solomon constructions described in the book chapter.

The chapter mentioned above also points out that the Gold sequences and the Kasami sequences can be viewed as mappings of codewords of low-rate Reed-Solomon codes to binary sequences (essentially, throw away all the bits except one from each codeword symbol of the Reed-Solomon codeword, and what is left is a Gold sequence or a Kasami sequence). As a concrete example, consider that the output of a $n$-bit maximal-length LFSR is a maximal-length sequence (a special case of a Gold sequence) while the entire contents of the LFSR are the frequency-hopping sequence suggested by MBaz.