The paper focuses on the problems of the use of pseudonoise phase modulation broadband signals in the radar survey. The average correlation characteristics of these signals in the Doppler frequency range and the structure of the processing system, including a multi-channel frequency matched filter, are determined. Methods of protecting the signal detection systems from the effects of narrow-band impulse and passive interference are proposed

Рассмотрены вопросы применения широкополосных сигналов с псевдошумовой модуляцией фазы в РЛС обзора. Определены среднестатистические корреляционные характеристики данных сигналов в диапазоне доплеровских частот и структура системы обработки, включающая многоканальный по частоте согласованный фильтр. Предложены способы защиты систем обнаружения данных сигналов от воздействия узкополосных импульсных и пассивных помех

At the present time, a tendency can be seen for expanding the bandwidth of probing signals in radars of various purpose [

A signal with phase PNM can be represented by the following general expression:

Here, rk - evenly distributed random numbers;

B - signal base.

The autocorrelation function (ACF) of the signal with PNM is expressed by the following formulas:

Since signals with PNM are random by their nature, it is possible to determine their statistical characteristics. In particular, the ACF will have, on the average, zero side lobes:

The ACF dispersion can be calculated by the formula

Proceeding from this, root-mean-square deviation (RMSD) of the ACF is equal to with the 1st side lobe (SL) relative to the main lobe being approximately equal to and the last one to - 1/B. In Fig. 1, for B = 10,000 and a signal with the duration of 100 µs, statistical characteristics of the ACF are given in the form of dependencies on time t (µs) of the following quantities:

where z – matched filter (MF) output;

Sz – fluctuation scattering of individual SL.

The lower SLL boundary Mz − Sz is not shown, as it is of less interest. It should be pointed out that the value of SL average level (SLL) remains virtually the same in different implementations, being in this case equal to approximately –45 dB.

For broadband signals (BBS) with PNM with large base, fairly low SLL is achievable. Among the negative properties of these signals we could mention the fundamental impossibility to decrease SLL through application of Hamming, Dolph - Chebyshev window functions, and the like. Weighing does not affect the SLL in practical terms, as only the far lobes are decreased. Fig. 2 shows an example of weighing as per Dolph - Chebyshev.

Cross-correlation function (CCF) of signals Si, Sj and its dispersion are determined similarly to (2)–(4):

An example of CCF calculation is given in Fig. 3.

To complete the analysis of signal characteristics, let us find the spectrum of such signal:

The obtained result expectedly corresponds to uniform distribution of signal unit energy across the entire frequency band.

A matter of substantial interest is the analysis of Doppler frequency effect on the shape and power of the discussed signals at the MF output. A partial result is shown in Fig. 4 for a signal with PNM, duration Ti =100 µs, at Doppler frequency fd = 2 kHz, with losses L amounting to about 0.6 dB. However, with Dopper frequency increased to fd = 5 kHz, the losses will be unacceptably high: L = 3.9 dB. At the same time, the SLL remains virtually unchanged (the average value of Mz and SLL upper boundary Mz + Sz in a set of 104 statistically independent signals increased by less than 1 dB), which is determined by the switchboard form of the uncertainty function (UF) of a given signal.

The section of the UF of a signal with PNM at τ = 0 is the same as that of a rectangular pulse of the corresponding duration Ti :

It should be pointed out that, in practical terms, SLL does not depend on Doppler frequency, i. e., outside of the main lobe the UF forms a plane with level:

The dependence of detection losses on Doppler frequency for Ti = 100 µs is given in Fig. 5. Its shape is determined by the UF (9), in particular, at it makes ρ(0, f) = 0, = with losses L → ∞ .

Fig. 5. Dependence of energy losses of signal with PNM on Doppler frequency at B = 10,000

To end up with sufficiently low losses in the detection of signals with PNM in a broad band of Doppler frequencies, a multichannel matched filter (MMF) will apparently have to be applied. The amplitude-frequency response (AFR) of the MMF for signals with PNM is defined by the expression

Here,

fdm - Doppler frequency modulo maximum value;

Nk - total number of channels for positive and negative frequencies.

For Nk = 16 an acceptable result is obtained (Fig. 6), which also entails, of course, substantial increase in equipment quantity or power of the computing device on which digital processing of signals is performed.

Fig. 6. Dependence of energy losses of signal with PNM on Doppler frequency in MMF for Nk = 16

When a train of N signals with PNM is detected (each signal having a different modulation), with the repetition period Tp , the following AFR is obtained:

Here, - centre frequencies of the Doppler filters under their doubled quantity.

An estimated AFR for an 8-pulse train and Nk = 16, with the number of Doppler channels in coherent integrator (CI) of the train Nkp = 16, is shown in Fig. 7. The AFR peaks follow with frequency which corresponds to the tuning increments of MMF channels. An AFR fragment is shown in Fig. 8, where AFR peaks and lows can be seen, alternating with frequency (train pulses repetition period Tp = 1 ms) and associated with tuning frequencies of the CI channels. The AFR dips made upto ~1 dB.

Fig. 7. AFR of processing in MMF with CI at Nk = 16, N = 8, Nkp = 16

Fig. 8. AFR of processing in MMF with CI (fragment) at Nk = 16, N = 8, Nkp = 16

The BBS detection systems, in particular those intended for detection of signals with PNM, can be exposed to various sources of narrow-band impulse interference (NII) with noise filling, which for different reasons may have remained unsuppressed at the stage of spatial processing of signals. The NII is specified by the following parameters:

Pui – power relative to noise;

τui – duration and frequency band ∆fui.

NII temporal position and position on the frequency axis are considered random and unknown terms.

We consider signal with PNM, in its mixture with additive noise and NII, represented in the form of complex envelope X1, X2,..., Xn. samplings. For confident detection of signals with PNM under NII conditions, we use the method of generalised median filtering (GMF) proposed in [

1. Processing in GMF1:

Threshold Cm was determined in the course of algorithm simulation and corresponded to the probability of false triggering of median limiter (13) under uniform noise conditions The processing result:

2. DFT of samplings Xim. As a result, spectral samplings Yi. are obtained.3. Processing in GMF2:

Processing result:

4. IDFT of samplings Yim. As a result, temporal sequence Ximm is determined.

The calculations were performed for signals with PNM over band W = 100 MHz, pulse duration Ti = 100 µs, Nyquist sampling. The coefficients of NII affection were calculated bythe formulas:

The values obtained were 0.1; 0.2; 0.4.

NII power Pui = 0.. .40 dB. An example of calculation of threshold signals Pc dependencies on the NII power is shown in Fig. 9.

For surveillance radars, an obviously important characteristic is the degree of passive interference (PI) suppression. Let the probing signal be represented by a coherent train of N pulses with PNM laws that are arbitrary and change from period to period of the train:

where Ni – number of samplings on the base of signal with PNM.

Let us consider a point scatterer of the PI, which is moving at certain speed Vc , resulting in Doppler frequency shift fc . For simplification, we consider a 2-pulse train first. A scatter signal can be written as follows:

Here, ψ1 - initial phase;

τd – sampling interval;

σi - instantaneous radar cross-section (RCS) in the i-th pulse.

Correlations of the RCS values in the first and second periods are defined as follows:

where ρ - PI inter-period correlation coefficient modulus.

It should be noted that on a pulse duration of, suppose, 100 µs, the correlation coefficient of PI signal, with spectrum width of ~50 Hz, is 0.9999, which allows not to take into account PIsignal amplitude fluctuations over the pulse time.

At the output of the MF, which is retuned in accordance with the PNM law in a given train period, after transition process completion, we have:

Expression (18) can be written, within the accuracy of the initial phase, as:

Here, ρs (0, fc ) - UF of the signal with PNM in the point with coordinates (0, fc).

With fc = 1 kHz, value ρs (0, fc) = 0,9837. The influence of this multiplier comes down to decrease by approximately 0.14 dB of the amplitude of a signal reflected from point PI at the MF output. It follows from formula (19) that the signals at the MF output have a correlation matrix with virtually the same parameters as during radiation, for instance, of a train of LFM signals with the similar modulation law. Hence, the efficiency of alternate-period compensation of passive points will be approximately the same.

In a similar way, having considered PI in an N-pulse train, the expression for signal received in the k-th period will be obtained:

where

Pc - PI power;

Pk,m - PI correlation coefficient modulus in the k-th and m-th train periods.

It is obvious that with PI parameters Pc, pk, m, fc unknown, in this case it is possible to use adaptive methods of PI rejection, developed for narrow-band signals, e. g., such as obtained in [

For multichannel processing in MMF with channels tuned to different Doppler fm, frequency we obtain, instead of (20), the expression

Depending on detuning ∆f = fc - fm the amplitude of PI signal will be dropping (see Table).

Dependence of PI signal amplitude on frequency detuning

∆f, kHz | ∆U, dB |
---|---|

5 | - 3,92 |

10 | - да |

15 | - 13,46 |

20 | - да |

In case of detunings divisible by , PI falls within UF zeros by the frequency axis. A more accurate calculation of the PI power at the outputs of MMF channels can be done using the formula for transmission coefficient for the m-th MMF channel by power:

Therewith, a correlation coefficient modulus can be calculated, for example, for the PI fractional rational spectrum [

An example of calculation according to the given formulas of the dependencies of transmission coefficient by power on the MMF channel frequency is given in Fig. 10, a. However, the obtained result corresponds to the point PI only. If the PI is extended and homogeneous, then instead of formula (22) we have:

Fig. 10. Transmission coefficients by power for a point (а) and extended (b) PI in MMF channels at B = 10,000, Nk = 16, f0 = 1000 Hz, σf = 50 Hz, Pc = 40 dB

In this case the PI power in all MMF channels is virtually equal (Fig. 10, b). Formula (24) is derived as follows. For a single PI source shifted by range by ∆ samplings, signal at the MF output:

The aggregate transmission coefficient under the impact of interference from all sources shifted by range is

Summing up (27) and (22), expression (24) is obtained.

Thus, at the outputs of MMF channels tuned to Doppler frequencies, with the MMF being retuned in accordance with the PNM law in each train period, PI signals were obtained in N periods of the train having the same inter-period covariance coefficients as during radiation of narrow-band pulse train with the same LFM law. For range point PI, the interference power in the MMF channels tuned to a Doppler frequency different from the PI noticeably decreases due to the switchboard form of the UF of signals with PNM. If the PI is distributed by range, its power is virtually the same in all MMF channels. Connecting a PI signal rejection filter with fixed coefficients or with coefficients adaptive to the PI correlation characteristics, to the outputs of MMF channels it can be possible to obtain the same degree of PI suppression, or even higher (for point PI sources), as in the conventional method, which implies radiation of a coherent pulse train with the same LFM law.

The authors declare that there are no conflicts of interest present.