2. Presumptions

The behavior of a hopper is assumed to be random: the channels of the hop range ( MFH ) are used in random sequence, the current channel being selected independently of all previously and subsequently selected channels. The probability of selection should be the same for all hop channels (1/ MFH ), although there is no certainty of any channel appearing at all within a given period of time. The burst time (dwell time of the hopper on an emitted frequency) is Th . In the case of variable hop times, Th is the average value. For the sake of simplicity, regular, contiguous channel spacings are assumed for the frequency range. The frequency range occupied by the hopper is also taken as being free from any other signals.

It is further assumed that the receiver (monitoring receiver or DF receiver) systematically scans the defined frequency range with the same frequency spacing as used by the hopper. The center frequencies of the scanned channels ( MSc ) may come in any sequence, eg linear with time (linear staircase) or as a pseudo-random sequence. It must however be certain with all hop sequences that each of the selected MSc channels is used exactly once in a scan. In this case the random sequence is not a stochastic process like the hop sequence but a systematic search, even though with complex timing. A full number of complete scans is assumed to be performed each time. The duration of a scan corresponds to TSc. Irrespective of the type of scan, the probability of a receiver operating at a certain channel during a randomly chosen time is 1/ MSc for each of the MSc channels.

Each frequency position of the receiver is assigned a dwell time T_d (FIG 4) consisting of the integration time Ti and a remaining period T_Syn . T_Syn contains the settling time of the receiver synthesizer and the time required for signal processing (eg with direction finders the time required to calculate a bearing):

T_Syn = T_d – Ti (1)

The integration time Ti depends on the settling time of the filters used. Signals with a duration of Th ² Ti are considered not detectable, ie a signal must have a length of at least T_h > T_i to be detected. The received bursts should have sufficient power to be detected when at least one integration time Ti falls within the burst interval Th and the hop frequency and the receive frequency are identical. This is a very simple way of approaching the intercept procedure.
In this consideration the relationship between the probability of a false alarm, intercept threshold, S/N ratio, averaging and probability of intercept is not taken into account. Therefore, the fact will also be ignored that, to obtain constant S/N ratio at the input of the detection device, field strength should be increased proportionally to EDDD (1/Ti ) at short integration times T_i .

The frequency ranges of hopper and receiver should at least partially overlap by a number of channels Mg in the common frequency band (FIG 5). With digital signal processing, receivers using parallel, identical filter/detector channels can be implemented (multi-channel receiver), in the case of many channels by means of fast Fourier transform for instance. Whenever the receiver locks to a specific frequency, K filters/detectors are simultaneously active.

FIG 5 Frequency bands of hopper and receiver: a) general overlap; b) search range fully inside hop range; c) hop range fully inside search range; d) search range = hop range