5

Discrete spread spectrum signals

5.1 Spread spectrum modulation

Let us revisit the general model (2.37) of a bandpass signal:

sðtÞ ¼ Re½S_ðtÞ expðj2 f0tÞ&; S_ðtÞ ¼ SðtÞ exp½ j ðtÞ&

It is quite comprehensible that the spreading of a signal spectrum is accomplished by appropriate controlling of a signal complex envelope S_(t), i.e. modulation of its instant amplitude S(t) and instant initial phase (t). As was observed in Chapter 1, ‘pure’ amplitude modulation cannot be an efficient tool for spectrum spreading, since it may remarkably widen the bandwidth only at the cost of concentrating the signal energy within short time intervals. As a matter of fact, this implies operating with short plain signals. On the contrary, angle (phase or frequency) modulation is capable of unlimited (at least in theory) widening of the spectrum with no effect on the distribution of the signal energy in time, i.e. signal duration, due to which its role in spread spectrum technology is fundamental. Amplitude modulation is just an auxiliary instrument, which sometimes appears to be useful in combination with angle modulation.

Depending on the character of the modulation involved all spread spectrum signals may be classified into continuous and discrete ones. For the first, the modulation law, i.e. complex envelope S_(t), is a continuous function of time, while the modulated parameters (amplitude, frequency, initial phase) of the second are piecewise constant and change by hops only at discrete time moments. The example of a continuous spread spectrum signal will be discussed briefly in Section 6.2; however, the main attention will be focused on discrete signals, owing to their predominant role in the majority of modern and forward-looking commercial systems.

Spread Spectrum and CDMA: Principles and Applications Valery P. Ipatov

2005 John Wiley & Sons, Ltd

5.2 General model and categorization of discrete signals

Discrete signals considered in this book may be covered by the following description, generalizing the one already presented in Section 2.7.3: a discrete signal is a sequence of elementary pulses of a fixed form, recurring at some fixed time interval. The elementary pulse is called a chip. The complex envelope S_0(t) defines its shape and internal angle modulation, if any. The time interval D between consecutive chips typically, but not compulsorily, equals or exceeds chip duration Dc. Modulation of the whole signal consists in manipulating the amplitudes, phases and, possibly, frequencies of individual chips. Accordingly, formal representation of the complex envelope of a discrete signal is given by the equation:

where, in addition to the designations explained, ai and Fi are, respectively, complex amplitude and frequency (in terms of shift against the fixed central frequency) of the ith chip. It is obvious that the sequence fjaij, i ¼ . . . , 1, 0, 1, . . .g determines real amplitudes of chips, i.e. their amplitude modulation. Similarly, sequences f i ¼ arg ai, i ¼ . . . , 1, 0, 1, . . .g and fFi, i ¼ . . . , 1, 0, 1, . . .g define the laws of modulation of chip phases and frequencies. Figure 5.1 may be helpful in understanding some of the definitions above.

Suppose that in the model (5.1) real amplitudes jaij may take on non-zero values only for 0 i N 1 and jaij ¼ 0 for i < 0 and i N. Putting it differently, the signal is a burst of a finite number N of manipulated chips. Such a signal we will call pulse or aperiodic. The duration of an aperiodic signal is T ¼ (N 1)D þ Dc. Another important case is a signal for which the modulation law repeats itself with a period of N chips: ai ¼ aiþN , Fi ¼ FiþN , i ¼ . . . , 1, 0, 1, . . .. For natural reasons, this sort of discrete signal is called periodic. Its real-time period is T ¼ ND and any periodic signal is just a repetition with period ND of an aperiodic one, the latter being a one-period segment of the periodic signal. In both cases we will call parameter N the length of a code sequence (see Section 2.7.3).

Within the described general model we distinguish between several categories of discrete signals in accordance with a specific chip modulation mode.

ith chip

|ai |

t

∆∆c

Figure 5.1 Example of a discrete signal

for both aperiodic and periodic signals, where E ¼ kak2E0 is complete energy for the first and per-period energy for the second. Here E0 stands for chip energy and kak is a geometric length (Euclidean norm) of the code vector a ¼ (a0, a1, . . . , aN 1); in other

words, kak2¼ PN 1 jaij2 is the energy of the N-element sequence fa0, a1, . . . , aN 1g.

i¼0

Substituting (5.2) into (5.3) gives:

where the last equality follows from vanishing the integral, whenever i is beyond the set f0, 1, . . . , N 1g.

Introducing the ACF of a single chip:

is the ACF of the code sequence fa0, a1, . . . , aN 1g characterizing its resemblance to its replica shifted by m positions.

Equation (5.5) has quite an eloquent implication. Comparing it with the model (5.2) allows the observation that the ACF of an APSK signal is the APSK signal itself! The chip of the latter is the ACF c( ) of the original chip, while the code sequence is the ACF (5.6) of the code sequence fa0, a1, . . . , aN 1g of the original signal. Therefore, given the chip, the ACF of the APSK signal is entirely determined by the ACF (m) of the code sequence (or code ACF), and designing APSK signals with good autocorrelation properties means searching for sequences with good ACF. Note that, like any ACF, (m) at m ¼ 0 equals 1 and is even: (m) ¼ ( m).