In multiuser CDMA systems families of discrete signals are necessary with special cross-correlation properties (see Section 4.5 and Chapter 7). Repeating accurately the derivation above but this time for two different (kth and lth) APSK signals having identical chips and lengths results in the following equation for their cross-correlation function (CCF):

1

X

the two signals, which shows the resemblance of the first to an m-shifted replica of the second. Certainly, CCF (5.7) is again the APSK signal whose code sequence is just the CCF of the two original codes (code CCF). Designing families with the necessary crosscorrelation properties means searching for the families of sequences having appropriate CCF. Equations (5.7) and (5.8) are most general, since the ACF of the kth signal iskk( ) and the same is true for the code sequences.

In what follows we will widely use the results obtained, sometimes omitting front factors in (5.6) and (5.8), i.e. operating with non-normalized correlation functions of code sequences:

5.4 Calculating correlation functions of code sequences

Consider the code sequence fa0, a1, . . . , aN 1g. If it is used to generate a pulse signal, in the general model (5.2) ai ¼ 0 for negative i and i N, so that according to (5.6) aperiodic or pulse ACF is calculated as:

>

>

:

i¼0

The second row here is somewhat redundant since any ACF features evenness, and in particular, a( m) ¼ a(m). As is seen, ignoring the normalizing factor, aperiodic ACF is an inner product of the vector a ¼ (a0, a1, . . . , aN 1) and its m-position non-cyclically shifted version. The latter is a shifted to the right (m 0) or to the left (m < 0) and only

the overlapping components of a and its shifted replica enter the sum in (5.10), all the rest are regarded as being forced to zero. For example, to calculate a(1) we first write, one above the other, the initial sequence and its copy conjugated and shifted to the right by one position, then compute all of their component-wise products and sum all the products:

Let us assume now that the signal is periodic, i.e. aiþN ¼ ai, i ¼ . . . , 1, 0, 1, . . .. Then (5.6) defines the periodic ACF p(m), the sum in which always contains N summands, since a 1 ¼ aN 1, a 2 ¼ aN 2, etc:

In this case the inner product is computed for the original code sequence and its cyclically shifted copy, where under m 0 m left ‘empty’ positions are filled with symbols pushed out rightward. For instance, a scheme of calculating p(1) looks like this:

Since p(m) is calculated under the assumption of periodicity of a code sequence, it is periodic itself with period N, i.e. p(m) ¼ p(m þ N), m ¼ . . . , 1, 0, 1, . . ., which stems directly from (5.11) and, in its turn, reformulates the evenness property as:

This equation shows that p(m) is entirely characterized by its values at only shifts

m¼ 1; 2; . . . ; N2 ;

where b c symbolizes rounding towards zero. Another important property of the periodic ACF follows from (5.11) after splitting its sum into two:

a(m N), which is again verified immediately from (5.10) by calculating a(m N) according to its second row. This produces the equality associating the periodic and aperiodic ACF:

where use is made of the fact that all chips with non-zero amplitudes now have equal energies E0, i.e. kak2¼ N and E ¼ NE0.

It is typical of FSK modulation to use a uniform frequency alphabet so that Fi 2 f0, F, 2F, . . .g, where the frequency step F is no smaller than the chip bandwidth. Thus, the spectra of two chips having frequencies Fi and Fk do not overlap and the chips are orthogonal independently of their time mismatch (see Section 4.4), whenever Fi 6¼Fk. Allowing for this fact, we arrive at:

where c( ) is, as before, chip ACF and:

Unlike APSK signals, ACF (5.18) in the general case cannot be further simplified to a form similar to (5.5). It is common practise to analyse the behaviour of the ACF of FSK signals primarily at the delays, which are multiples of the chip duration: ¼ mD, where m is integer. Assuming that the integer number of periods l of each frequency fits in the chip duration (FD ¼ l) and taking into account that c(0) ¼ 1, c( ) ¼ 0, j j D,

When the signal is finite and m 0, all summands possessing indexes beyond the range m, m þ 1, . . . , N 1 disappear, and the aperiodic ACF of the FSK signal(mD) ¼ a(m) assumes the form:

where complex conjugation in the second equation (expressing evenness) is needless, since (mD) is always real-valued.

In the case of signal periodicity no zero products aiai m enter the sum of (5.19) and the periodic ACF of the signal in question (mD) ¼ p(m) looks as follows:

The sums in (5.20) and (5.21) just accumulate the number of coincident frequencies in the FSK signal and its replica time-shifted by m chip positions. Therefore, to compute ACF of an FSK signal at the point mD, it is quite enough to count the number of pairs fFi, Fi mg with equal Fi and Fi m, where i runs over the ranges fm, m þ 1, . . . , N 1g (aperiodic ACF, m 0) or f0, 1, . . . , N 1g (periodic ACF). Clearly, equation (5.13), linking together periodic and aperiodic ACF, remains valid for FSK signals.

One widely used way to represent an FSK signal is as an M N array where the horizontal and vertical directions are assigned to time and frequency, respectively, and M is the size of the frequency alphabet (i.e. the number of frequencies used in modulation). In the ith vertical column of this array only a single entry is labelled (e.g. by a point or shading), which corresponds to a frequency of an ith chip. Then to calculate aperiodic ACF at any specific m we just sum the number of labelled pairs along all rows having distance m and normalize the result if necessary. If periodic ACF is of interest, the sums above obtained for m and N m should be summed together.

Example 5.5.1. Figure 5.3 shows the modulation law of an FSK signal of length N ¼ 8 with

7

P

N ¼ 8, M ¼ 5. Its non-normalized aperiodic ACF Ra(m) ¼ (Fi Fi m ) has values 8, 1, 1, 0,

i¼m

Figure 5.3 FSK signal with N ¼ 8, M ¼ 5

1, 0, 0, 0 corresponding to m ¼ 0, 1, . . . , 7, since there is one labelled pair along one line at distance 1, one such pair at distance 2, etc. Directly from (5.13), the values of non-normalized periodic ACF may be found as 8, 1, 1, 0, 2, 0, 1, 1.

Generalization of results (5.19) and (5.20) onto CCF can be done without trouble merely by adjusting the designations: