# Spread spectrum and Pseudonoise Sequences

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2 2 Frequency-hopping spread spectrum A signal is broadcast over a pseudo-random sequence of frequencies, a sequence obtained by a pseudo-random generator. The pseudo-random sequence is referred to as the spreading code or a pseudo-noise sequence. The energy of the signal is equally divided among different frequencies. An FHSS system consists of a modulator that uses a standard modulation technique to produce a signal centered around some base frequency. This signal is next modulated again to produce a signal centered at a different frequency, which is determined by the pseudo-noise sequence. This second modulation is done using a chip signal c(t) which characterizes the pseudo-noise sequence. The spacing between the carrier frequencies used in the chip sequence is usually close to the bandwidth of the input signal. The interval of time that an FHSS system spends at a hop frequency is referred to as the chip duration. As we will see shortly, this is a parameter that is independent of the data rate of the original signal and can be varied to obtain different FHSS schemes. FHSS using BFSK. Consider using BFSK as the first data modulation scheme. The output of the first modulation step is the signal s d (t) = A cos (2π(f (b i + 1) f)t), for it < t < (i + 1)T, where f 0 is the base frequency, f is the frequency separator in the BFSK scheme, b i is the ith bit, and T is the duration of a single bit. Now, suppose the next frequency as determined by the pseudo-noise sequence is f i. We obtain a product signal of p(t) = A cos (2π(f (b i + 1) f)t) cos(2πf i t) = 0.5A (cos(2π(f (b i + 1) f + f i )t) + cos(2π(f (b i + 1) f f i )t)). We can eliminate the second half of the above sum and obtain signal with frequency centered around f 0 + f i. When the signal is received at the receiver, the original bit value b i can be recovered by following the reverse process, as long as the sender and the receiver are synchronized with respect to the pseudo-noise sequence. FHSS using MFSK. A natural generalization of the above is to use multi-level signaling technique such as MFSK. If the MFSK modulated signal is given by s d (t) = A cos(2πf i t), where f i = f c + (2i 1 M)f d is the frequency used for the ith signal element, f c is the center frequency, and M is the number of signal elements. This MFSK signal is then translated to a new frequency every T c seconds by modulating it with the FHSS carrier. For a data rate R, the duration of a bit T = 1/R, so the duration of a signal element is T s = log 2 MT. If T c T s, the system is referred to as a slow FHSS; otherwise, it is a fast FHSS. In FHSS, the bandwidth of a hop is W d = MF d, where M is the number of different signal elements. The total transmission bandwidth is W s = NW d if N is the number of different hop-channels. FHSS is used in one of the RF interfaces for IEEE , in combination with two- or four-level Gaussian FSK. FHSS-CDMA. In an FHSS/CDMA system, each user has its own hopping sequence. In such a system, interference occurs only when two users land on the same frequency at the same time. If 2

4 Multiple users can now simultaneously access the channel by simply encoding the data using their own spreading code and having the interference canceled by the orthogonality of the codes. There are several important factors to discuss in connection with the workings of CDMA. First, it is clear that one does not need strict orthogonality; approximate orthogonality will suffice as long as it clearly separates out the different user data. In fact, guaranteeing strict orthogonality is also not practically feasible. Second, synchronization of the users to the chipping process is important for the orthogonality condition to be useful; one can settle for limited sychronization if the codes are approximately orthogonal, even with small shifts, but this may limit the number of users. Finally, some kind of power control is required to ensure that the strength of a signal is not overwhelmed by the strengths of other signals. A problem arising due to lack of power control is the near/far problem : the signal of a user that is far from a base station is dominated by that from a nearby user, both transmitting at the same signal strengths, since the attenuation of the first signal is much more than that of the second. As a result, when the combined signal is decoded using the code of the farther user (and both the codes are not perfectly orthogonal to each other), the decoded value is not correct. 5 Pseudo-noise Sequences Linear feedback shift register (LFSR). LFSRs are one of the simplest ways to generate pseudorandom sequences. In an LFSR, any bit is determined by a linear combination of the previous n bits, for a suitable choice of n. In particular, we may have an LFSR in which B n = A 0 B 0 A 1 B 1 A 2 B 2... A n 1 B n 1. Since any bit is a function of the previous n bits, we can show that every LFSR has a period of 2 n 1. Consider any window of n bits. As we slide 2 n 1 times forward, we cover 2 n 1 n-bit strings. If any two of the strings repeat, then we already have the period at most 2 n 1. Furthermore, since the all 0s cannot be generated (unless the initial value is all 0s in which all values remain all 0s), within 2 n steps at least one window is identical to another n-bit window encountered earlier in the process; this the period of the sequence is 2 n 1. M-Sequences. M-sequences, or maximal length sequences, are pseudonoise sequences generated by LFSR that have maximum period. That is, any sequence that is generated by an n-bit LFSR and has period 2 n 1 is an m-sequence. An m sequence has several desirable properties. It has an almost uniform distribution of 0s and 1s. Specifically, in any window of 2 n 1 bits, it has 2 n 1 ones and 2 n 1 1 zeros. Moreover, in any window of 2 n 1 bits, it has one run of ones of length n, one run of zeros of length n 1, and in general, 2 n i 2 runs of ones and 2 n i 2 runs of zeros of length i, 1 i n 2. For spread spectrum applications, one important measure of the randomness of a pseudonoise sequence is its autocorrelation function, which gives the correlation of a period of the sequence with its cyclic shifts. The smaller the correlation, the easier is it for the sender-receiver pair to synchronize; furthermore, interference effects are also somewhat alleviated. For a sequence B 1,..., B N of ±1, the periodic autocorrelation function R is given by R(τ) = 1 N N B k B k τ. k=1 4

5 It can be shown that the periodic autocorrelation of an m-sequence is { 1 τ = 0, N, 2N,... R(τ) = otherwise 1 N In spread spectrum applications, when the codes for different users are generated from different sources, then a similar measure of interest is the cross-correlation of the two codes. The crosscorrelation function of two sequences A 1,..., A n and B 1,..., B N of ±1s is defined as R A,B (τ) = 1 N N A k B k τ. It is intuitive that we would like the cross-correlation function of two user spread spectrum codes to be low since that would allow a receiver to discriminate among spread spectrum signals generated by different m-sequences. Gold sequences. For DSSS systems, we need to generate seperate codes for different users. A Gold sequence set provides such a collection of codes. It is obtained by repeatedly taking bit-wise XORs of two uncorrelated sequences. The starting sequence is an m-sequence a. The second sequence a is obtained by sampling in a deterministic manner every qth bit, for some appropriate q from multiple copies of a until the length of a is the same as a. The ith subsequent sequence is obtained by a bit-wise XOR of a with a i-bit shifted copy of a. In order for the Gold sequences to satisfy desirable properties such as each code being an m-sequence a and q have to satisfy certain conditions, which are described in the text. Orthogonal codes. Orthogonal codes are a set of sequences whose all pairwise cross-correlations are zero. Walsh codes are the most common orthogonal codes and are used in CDMA applications. The set of Walsh codes of length n = 2 k, k 0, consists of the n rows of an n n matrix, which is recursively defined as follows. ( ) Wn W W 1 = (0) W 2n = n, where A is the matrix A with every bit replaced by its logical NOT. We now prove that codes in a 2 n 2 n Walsh matrix are orthogonal to each other for any integer n 1. The proof is by induction on n. The claim is trivially true for the base case n = 1. For the induction step, let us assume that the claim is true for a 2 n 1 2 n 1 Walsh matrix. Consider any two distinct row vectors w i and w j of a 2 n 2 n Walsh matrix, where we replace 0s by -1s. We consider three cases. The first case is when i, j 2 n 1. In this case, w i w j is the sum of two quantities, each of which is the dot-product of the ith and jth row vectors of the 2 n 1 2 n 1 Walsh matrix. By the induction hypothesis, both these quantities are zero, thus establishing the induction step for this case. The second case is when i, j > 2 n 1. In this case, w i w j is the sum of two quantities, the first of which is the dot-product of row i 2 n 1 and row j 2 n 1 of the 2 n 1 2 n 1 Walsh matrix, and the second is the corresponding dot-product of the complement matrix. By the induction hypothesis, both these quantities are zero, thus establishing the induction step for this case. The final case is when i 2 n 1 and j > 2 n 1. Again, the dot-product is the sum of two quantities, the first being the dot-product of row i and row j 2 n 1 of the 2 n 1 2 n 1 Walsh matrix, while k=1 W n W n 5

6 the other being the dot-product of row i of the 2 n 1 2 n 1 Walsh matrix and row j 2 n 1 of the complement of the 2 n 1 2 n 1 Walsh matrix. Since these quantities complement each other, we obtain a dot-product of 0, thus completing the proof. 6

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