6

Spread spectrum signals for time measurement, synchronization and time-resolution

6.1 Demands on ACF: revisited

Let us return to the problems of estimating time delay and time-resolution studied in Sections 2.12 and 2.15, and recollect the demands imposed on a signal if high measuring accuracy and resolution capability are required. The principal condition to be met in both these problems is a response of the matched filter to a signal that is highly concentrated in time, or, equivalently, a ‘sharp’ signal ACF corresponding in the frequency domain to a wide spectrum. The attractiveness of spread spectrum as opposed to just shortening the signal is that with high processing gain WT 1 it is possible to put into a signal the energy dictated by the necessary SNR controlling only the duration of T rather than peak power, which typically has a strict upper limit. Then involvement of an appropriate angle modulation allows widening the signal bandwidth to the extent which provides time compression of the signal by the matched filter, so that the duration of the filter response (correlation spread c 1/W) appears to be many (about WT) times smaller than the duration T of the signal itself.

Let us specify what sort of ACF we may treat as sharp or ‘good’ concerning the reception problems in question. Actually, the ACF (see definitions (2.66) and (2.67)) of any physically realizable signal cannot equal strict zero at all beyond the range [ c, c], if the correlation spread c is smaller than the signal duration T. Thus, along

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

2005 John Wiley & Sons, Ltd

Figure 6.1 Central peak and sidelobes of ACF

rd (τ)

t

Figure 6.2 ACF sidelobes and abnormal errors

with the so-called mainlobe or central peak within the interval [ c, c], the ACF has also sidelobes outside this range (see Figure 6.1). The effect of sidelobes is predominantly obstructive in both delay measuring and time-resolution. Indeed, to measure in an optimal (ML) manner the signal delay one should fix the time position of the maximum of the matched filter output envelope rd (t) (Section 2.12), and the ACF envelope is exactly the matched filter-detector response to a noiseless signal. In a real situation of noisy observation there is always some risk that somewhere beyond the ‘body’ of the ACF central peak a false maximum appears higher than a true (i.e. located within the body) one, as the dashed line in Figure 6.2 shows. In this case, an abnormal estimation error occurs, which is a deviation " of the estimate ^ from a genuine value exceeding c. It is obvious that confusion of the mainlobe with a false peak emerging in the vicinity of a high sidelobe is more probable than with the false peak located at the ‘empty place’. Actually, the closer the levels of mainlobe and sidelobe, the ‘easier’ it is for the Gaussian noise to raise the second over the first.

To illustrate a harmful effect of sidelobes on time-resolution consider the superposition of two replicas of a bandpass signal, which are time-shifted and scaled as shown in Figure 6.3a. After processing by a matched filter the mainlobe of the weaker replica proves to be fully hidden under the sidelobe of the stronger replica (Figure 6.3b). In these circumstances the observer cannot confidently extract necessary information from both signal copies, or even tell how many copies are received. A situation of this sort is a typical case of non-resolved signals, despite the mainlobe of the ACF being much shorter than the signal duration.

We may now summarize in the most general terms the requirements placed on spread spectrum signals by delay estimation and time-resolution problems: the ACF of the signal should have a sharp enough central peak and as low as possible level of sidelobes. In the remainder of this chapter we study ways and instruments of approaching this fundamental objective.

t

Figure 6.3 Non-resolution due to ACF sidelobes

6.2 Signals with continuous frequency modulation

Historically, among the first discovered signals possessing the matched filter timecompression feature was the linear frequency modulated (LFM) pulse. As the name tells us, the carrier frequency of this signal changes linearly throughout its duration. Consider a bandpass pulse whose instantaneous frequency f(t) grows with time according to the equation:

f ðtÞ ¼ f0 þ WTd t ; jtj T2

where Wd is the frequency deviation, i.e. the entire range of frequency variation, and f0 is, as usual, the central frequency. A complete phase F(t) of a signal is the integral of a momentary frequency, therefore for an LFM pulse the phase obeys a square law:

FðtÞ ¼ 2 Zt f ðtÞdt ¼ 2 f0t þ WTd t2; jtj T2

0

Assuming a rectangular real envelope, the complex envelope of the LFM signal takes the form:

>

> T

>

: 0; jtj > 2

.

~ ρ (τ) s(f)

Figure 6.4 Approximation of the spectrum and ACF of an LFM pulse

Substituting this equation into the definition of ACF (2.66), the latter can be calculated formally without any special difficulties. Less formal and more physically transparent logic, however, allows faster achievement of the result. It is well known [1] that when the modulation index ¼ Wd T is sufficiently large ( 1), the spectrum of a fre- quency-modulated signal contains components of all momentary frequencies, the shape of the spectrum approaching the signal real envelope. Thus, in our case, the spectrum spans the range [f0 Wd /2, f0 þ Wd /2] and has a form close to rectangular (Figure 6.4a). Now, ACF (2.66) may be found from the inverse Fourier transform, as was done in Section 2.12.2. Since the energy spectrum is again rectangular, its inverse Fourier transform is the function of view sinx x, so that the normalized ACF of the LFM signal:

which is shown in Figure 6.4b. To compare this result with the exact one, the reader may turn to Problem 6.41.

As is seen, the complete (i.e. measured between zeros closest to the origin) duration of the ACF mainlobe equals 2 c ¼ 2/Wd . As a matter of convention, the duration of the mainlobe on some non-zero level may be set equal to c ¼ 1/Wd , so that a matched filter time-compresses the LFM signal T/ c Wd T WT times.

A substantial deficiency of the LFM signal is a high level of ACF sidelobes. The one nearest to the origin has intensity 2/3 ( 13:5 dB) versus the mainlobe independently of the processing gain WT, i.e. its level cannot be reduced by increasing deviation Wd . To lower the sidelobes, smoothing of the signal envelope is an effective method (Problem 6.41) as well as weighting by a special window or mismatched processing in the receiver. In all of these methods, gain in the sidelobes is obtained in exchange for widening of the mainlobe or/and loss in output SNR.

Example 6.2.1. Take a rectangular LFM pulse with deviation Wd ¼ 20/T . Using the program of Problem 2.55 produces waveforms of the signal itself and the matched filter response given in Figure 6.5. Compare the time-compression ratio and level of the first sidelobe to what is expected theoretically.