Figure 3.5 Clearing a signal off a barrage jammer

do to optimize its performance in more complicated situations than those considered. The idea was just to demonstrate the principal advantages of spread spectrum in countering a jammer. An interested reader may consult numerous specific works on the issue and confirm that whatever sophisticated systems and strategies are investigated, the general tendency is always the same: spread spectrum raises the jamming immunity potential.

3.2 Low probability of detection

It had been already pointed out that the potential of spread spectrum was first recognized by designers of military and intelligence systems and, as shown in Section 3.1, one reason for this is the high anti-jamming resistance of spread spectrum signals. The other reason we will discuss in this section.

In the confrontation of electronic systems effective jamming may be organized only after detecting the presence of an adversary system on the air and estimation of its parameters, such as carrier frequency and bandwidth. This entails a very popular scenario of the confrontation of two systems, when the first (call it intended) tries to operate as covertly as possible and escape unintended detection of its signal, while the

N0 /2

f W

(b)

f

wi

Figure 3.7 Spectra at the radiometer input and the bandpass filter bandwidth

Let us find the mean and variance at the envelope detector output. First of all, an output voltage ud of the square-law detector equals the input instant power. Therefore, expectation ud of ud in the absence of a signal is simply the average power of the filtered noise ud0 ¼ 2n, while when the signal is present it goes up to ud1 ¼ 2n þ 2s . It is exactly the increment of ud caused by the signal:

which allows the interceptor to hope to detect the presence of a signal. On the other hand, this useful (from the interceptor’s viewpoint) effect is masked by the random fluctuations of ud , measured by its variance varfud g. The latter can be found if it is remembered that the instant power of a bandpass process is its instant envelope Y squared and halved, and thus ud ¼ Y2/2, meaning that varfud g ¼ varfY2g/4. Variance of any random variable may be calculated as the mean square minus the mean squared [1,13,14]:

2

varfY2g ¼ Y4 Y2 ð3:5Þ

Now make use of the fact that the envelope Y of a Gaussian bandpass random process with variance 2 has Rayleigh PDF [1]:

and its even-order moments are found via elementary integration [13]:

Substituting this into (3.5) gives varfY2g ¼ 8 4 4 4 ¼ 4 4. When the signal is not present at the interceptor receiver input we should use in this expression 2 ¼ 2n. Strictly speaking, with the signal advent the filtered observation may differ from a Gaussian process, making applicability of the result just obtained doubtful. This detail, however, has no importance to the case in question, since the intended system does its best to hide its signal under the thermal noise and we have every reason to assume that a signal has a negligible effect on the variance of instant power and thus on the variance of the detector response. Therefore, independently of signal presence, variance of ud can be assumed the same:

varfud g ¼ varfY2g=4 ¼ 4n ¼ ðN0WiÞ2 ð3:6Þ

In order to estimate the mean ud and register its growth due to a signal presence, the integrator in the scheme of Figure 3.6 runs time averaging of the detector response over the observation period T. To make a constant component at the detector output distinctive enough against the random fluctuations the latter should be smoothed as a result of integration. This is possible only if fluctuations of the detector response around ud are sufficiently fast and change their polarity many times during the period T to compensate each other and produce an averaging effect. In other words, the number of statistically independent samples ns of the detector response within T should be large enough. Extension in time (correlation spread c) of the autocorrelation function of a random process is a trustworthy first approximation of a minimal time interval between samples, starting with which samples may be treated as independent. Since the filtered observation has the bandpass Wi its correlation spread is estimated as c 1/Wi, which gives the number of independent samples ns WiT.

Although practically integration may be implemented as continuous, its result is rather close to that of just summation of ns independent samples [6,9], which is even a more practicable technique, especially in digital circuitry. To perform an accurate

^

analysis, the PDF of the integrator output value E should be found for both hypotheses (signal absence and presence) and then integrated over the decision regions to obtain two probabilities, of false alarm and detection. These PDFs are subject to the chi-square law, which is a bit bulky and not quite transparent enough for a physical treatment. However, we may again exploit the fact that a signal is weak and its reliable

statistical independence of the integrated samples. Similarly, when AWGN plus signal is

^ ¼

observed, E nsud1 but the variance remains unchanged because of signal weakness. Thus PDF at the integrator output corresponding to the hypotheses H0 (signal absence) and H1 (signal presence) are: