bit and f1 þ Fi for a data bit equal to one. Figure 7.12b shows this for the bit stream 00101101. The principal difference between fast and slow FH is now seen: the latter does not spread the spectrum of an individual data symbol, widening just the total bandwidth occupied by the system. It looks like a system that merely switches from one operational frequency to another from time to time but within a fixed group of data symbols no switching happens. At the receiving end down-conversion to an intermediate frequency fi is accomplished with the aid of a reference signal repeating the signature frequency pattern (with an appropriate delay) on the carrier f0 fi (Figure 7.12c). This returns the waveform to the bandwidth inherent to a plain (non-frequency-hopping) FSK data modulation (Figure 7.12d), so that an ordinary FSK demodulator may restore the transmitted data (Figure 7.12c).

The techniques illustrated in the examples above for binary data transmission are easily extended to a general FSK data modulation (see Problems 7.5–7.7).

FH spreading has some features that make it especially attractive for military applications, in particular in various antagonistic scenarios of games against jamming systems [3,6]. At the same time, its commercial use had not until recently been significant, at least as regards fast FH. However, the advent of Bluetooth technology [55] indicates that this kind of spread spectrum may also possess good commercial prospects.

7.2 Designing signature ensembles for synchronous DS CDMA

7.2.1 Problem formulation

Consider a K-user DS CDMA system where all user datastreams and all signatures are strictly synchronized, i.e. have zero mutual time shifts, at the receiver input. As was pointed out in Section 4.4, a classical example of such a system is the downlink of CDMA mobile radio, where the base station controls entirely the timing of signals addressed to all users within the cell. Certainly, the group signal arrives at the mobile receiver preserving the initial synchronism between the signals sent to different individual users. In our current analysis we will operate with an idealized channel model, in which multipath delay spread max is smaller than the chip period D of users’ signatures, or an efficient equalizing is used, eliminating all the multipath components whose delays exceed D. This allows us to ignore any potential violations of perfect synchronism of components in a received signal.

In accordance with the concept of DS spreading, the complex envelope of a received group signal S_(t; b1, b2, . . . , bK ) is the sum in k of signature complex envelopes manipulated by users’ datastreams, each of them being defined by (7.6). Generally, each user’s signals may have its individual amplitude; however, we will restrict ourselves to the simplest case of equal intensities. Since the assumption of perfect synchronism permits us to set all delays k and initial phases k in (7.7) equal to zero, we then arrive at the expression of the received complex envelope (subscript r discarded as needless now):

k¼1

Figure 7.13 Data symbol and chip alignment in synchronous CDMA

Let us focus on a single data symbol interval of duration T. Again, due to the complete synchronism the current data symbols of all users start and end strictly simultaneously. With bk being the kth user’s current data symbol, (7.10) during a single symbol interval may be written as follows:

k¼1

where b ¼ (b1, b2, . . . , bK ) is the K-dimensional vector of current data symbols of all users. Remember now that every signature in the DS CDMA system is an APSK signal

described by the model (5.2):

i¼0

where fak, 0, ak, 1, . . . ak, N 1g is a code sequence, manipulating chips of the kth signature, and N is a spreading factor, i.e. the number of chips per one data symbol. Figure 7.13 emphasizes the strong mutual alignment between signature chips and boundaries of transmitted data symbols characteristic of synchronous CDMA.

Based on (7.11) and (7.12), several approaches to designing signature ensembles for synchronous DS CDMA networks may be formulated. Among the main factors influencing the procedure and results of the signature set optimization is the relation between the number of users K and spreading factor N, as well as the receiver algorithm (multiuser or conventional).

7.2.2 Optimizing signature sets in minimum distance

Suppose that the receiver of any complexity is admissible and, therefore, we are allowed to use the optimal (multiuser) algorithm of estimating the data vector b based on the search for the value of b minimizing the distance between observation y(t) and the candidate group signal s(t; b) (see Section 4.1). In terms of the complex envelope this means

a K-user data vector b ¼ (b1, b2, . . . , bK ). Returning to the material of Section 2.3, let us recollect that asymptotically (with SNR sufficiently high), minimizing the error probability is equivalent to maximizing the minimum distance in the constellation of M transmitted signals. In the studied case the alternative signals to be distinguished are copies of (7.11) corresponding to different data vectors b. Hence, we may formulate the problem of optimizing the signature set as maximization of the minimum squared distance:

Let us study in more detail binary data transmission when the data symbols are bits transmitted directly by BPSK so that bk, b0k ¼ 1, k ¼ 1, 2, . . . , K. This narrowing of the scope makes the analysis a bit easier, subsequent spreading to a general PSK being straightforward. Then using (7.11), (2.41) and (2.42) in (7.14) results in:

is the correlation coefficient between the complex envelopes of the kth and lth signatures. Using properties of the correlation coefficients seen from its definition,kk ¼ 1, kl ¼ lk (7.15) takes the form explicitly showing that distance is always real:

Take two data vectors (bit patterns) b, b0 differing in only one, for instance the first, component. Then "k ¼ 0, k ¼ 2, 3, . . . , K, "1 ¼ 2 and from (7.16) d2(b, b0) ¼ 4Eb. Since dmin2 is never greater than the squared distance for any specific pair of b, b0:

This upper bound tells us that a signature ensemble, for which dmin2 ¼ 4Eb, should be treated as optimal according to the criterion of maximum minimum distance (7.13). One of the sufficient conditions of achieving the bound (7.17) is the weak orthogonality of complex envelopes of signatures:

The reason why complex envelopes satisfying (7.18) are called weakly orthogonal becomes clear after comparison of (7.18) and (2.46) ( kl ¼ kl). The latter is much more demanding and forces the signals sk(t), sl(t) with the complex envelopes S_k(t), S_l(t) to preserve orthogonality under any mutual phase shifts. At the same time, two signals modulated by S_(t) and S_(t) exp [j( /2)] ¼ jS_(t), i.e. just quadrature (phase shifted by /2) replicas of the same signal, are orthogonal, but lose orthogonality if their mutual phase shift differs from /2. Hence, S_(t) and jS_(t) are only weakly orthogonal. Of course, any orthogonal (in terms of (2.46)) signatures are weakly orthogonal, but not vice versa.

For the signatures meeting (7.18), equation (7.16) becomes d2(b, b0) ¼ Eb PK¼ "2. At

k 1 k

least one of the summands in this sum is non-zero, so that dmin2 4Eb, whereby along with (7.17) dmin2 ¼ 4Eb. The implication of this is that the ensemble of K weakly

orthogonal signatures is optimal in minimum distance, and hence (asymptotically) in probability of confusion between different users’ bit patterns.

A lot of techniques exist for generating orthogonal (meeting (2.46)) spread spectrum signals for various lengths (spreading factors) N. One example is Walsh functions or, more generally, Hadamard matrices, discussed in Section 2.7.3 and providing binary orthogonal codes. Another possible option is cyclically shifted replicas of any sequence with perfect periodic ACF, e.g. ternary, polyphase etc. (see Section 6.11). Any ensemble of K0 orthogonal signatures is trivially transformed into the set of weakly orthogonal signatures containing 2K0 signals by adding quadrature replicas of any signal—a fact repeatedly referred to before (see Sections 2.5 and 4.1).

Under any specific choice of orthogonal signatures the signal space dimension strictly limits their number (and hence, number of users K) (see Section 2.5). According to (7.12), given the chip, N-dimensional vector ak ¼ (ak, 0, ak, 1, . . . , ak, N 1) of the kth code sequence exhaustively determines the kth signature, and the orthogonality of the kth and lth signatures is equivalent to the orthogonality of vectors ak, al. Indeed, repeating the derivation of (2.52) for complex envelopes (alternatively using (5.7)) allows the inner

confirming that the orthogonality of ak, al is necessary and sufficient for the orthogonality of S_k(t), S_l(t). Dimension N of the space of code sequence vectors ak is evidently a maximal number K0 of orthogonal signatures S_k(t). Let us stress again that when quadrature splitting of every signature is allowed, the maximal number of users accommodated within the signature ensemble is defined as K ¼ 2K 0 ¼ 2N. If, however, for some reason the accurate phase shift /2 between the quadrature copies of the same

Since the bit patterns b, b0 are different, at least one of "k, k ¼ 1, 2, . . . , N þ 1 is nonzero, i.e. equals 2. If "Nþ1 ¼ 0 then such an "k is present among "1, "2, . . . , "N and

Comparing this with (7.17) shows the possibility of adding one extra signature to the N orthogonal ones without sacrificing the minimum distance, whenever N 4. Generalization of this idea underlies the following procedure of building an optimal oversaturated signature ensemble [56,57]. Let vectors a00, a01, . . . , a0N 1 be an orthonormal basis of N-dimensional space where N ¼ 4l, l is natural. Let us use them as codes of N primary signatures. We arrange oversaturating supplementary signatures as an l-layer procedure. Supplementary signature code sequences of the sth layer are:

In other words, in the first layer of supplementary signatures we perform splitting of

the basic set a00, a01, . . . , a0N 1 into 4l 1 groups each containing four primary signatures. The linear combination (7.20) (where N ¼ 4) of these four basic signatures is added to their group, producing the total of N þ N/4 signatures. At the second layer we split all supplementary signatures of the first layer the same way into groups of four and introduce again in each group linear combinations (7.20), and so forth. The tree in Figure 7.14 illustrates the whole procedure. Then there are 4l 1 supplementary signatures at the first layer, 4l 2 at the second and generally 4l s at the sth layer totalling:

4l 1 þ 4l 2 þ þ 4 þ 1 ¼ 4l 1 ¼ N 1 3 3

supplementary signatures, or—together with the primary ones identified with a layer zero—an overall number of signatures:

Since norms of all vectors (7.23) remain equal to one, supplementary signatures preserve the same energy per bit Eb as the primary ones.

Figure 7.14 Constructing an oversaturated signature set

Let S_sk(t) and bsk be the complex envelope of the kth signature on the sth layer and the user’s bit transmitted by this signature, respectively. Then composing a group signal similarly to (7.21) leads to:

The double sum in k, m here contains N summands independently of s. It can be rearranged into a single sum after changing the summation index as:

Primary signatures are just rows of this matrix, i.e. in the normalized form:

a00; a01; a02; a03; a04; a05; a06; a07; a08; a09; a010; a011; a012; a013; a014; a015 T¼

It may be quite a challenge for the reader to check the minimum distance property of this oversaturated ensemble.

Let us remind ourselves that the criterion of minimum distance is adequate (at least asymptotically) whenever multiuser reception is affordable. Up to this point we have not worried about the multiuser receiver complexity. For the case of non-oversaturated systems (K N) this is not a critical matter, since—the orthogonal ensemble being optimal—multiuser reception degenerates in this case to a single-user one (see Section 4.1). On the other hand, when an oversaturated system is analysed, the opportunities for simplifying the multiuser algorithm at the cost of proper design of signatures are very important. One way to realize this approach again exploits the idea of splitting the overall N-dimensional signal space into orthogonal subspaces of smaller dimension n. However, in contrast with what was discussed above, every subspace is further