The use of pseudonoise phase modulation broadband signals in radar surveys

At the present time, a tendency can be seen for expanding the bandwidth of probing signals in radars of various purpose [1][2][3][4]. The advan­tage of using signals with pseudonoise phase modulation (PNM) in surveillance radars in­stead of conventional linear frequency-modu­lated (LFM) or nonlinear frequency-modulated (NFM) signals consists, primarily, in much higher protection of surveillance radars from the effects of imitating interference. This paper considers the basic properties of signals with PNM of the phase in the absence and in the presence of Doppler frequency shift, as well as the methods of processing these signals under the influence of narrow-band and passive in­terference. The issues of technical implemen­tation of the schemes for generating and pro­cessing these signals, while undoubtedly being of practical interest, go beyond the scope of this paper.

Determination and basic properties of signals with phase PNM

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

Here, r_k - 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;

S_z – fluctuation scattering of individual SL.

The lower SLL boundary M_z − S_z 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 S_i, S_j 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.

Processing of signals with phase PNM in the Doppler 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 T_i =100 µs, at Doppler frequency f_d = 2 kHz, with losses L amounting to about 0.6 dB. However, with Dopper frequency increased to f_d = 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 M_z and SLL upper boundary M_z + S_z 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 T_i :

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 T_i = 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 → ∞ .

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,

f_dm - Doppler frequency modulo maximum value;

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

For N_k = 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.

When a train of N signals with PNM is detected (each signal having a different modulation), with the repetition period T_p , 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 N_k = 16, with the number of Doppler channels in coherent integrator (CI) of the train N_kp = 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 T_p = 1 ms) and associated with tuning frequencies of the CI channels. The AFR dips made up
to ~1 dB.

Methods for processing signals with phase PNM under narrow-band and passive interference 

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:

P_ui – power relative to noise;

τ_ui – duration and frequency band ∆f_ui.

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,..., X_n. samplings. For confident detection of signals with PNM under NII conditions, we use the method of generalised median filtering (GMF) proposed in [5]. The stages of signal processing at the MF input will be described below.

1. Processing in GMF1:

Threshold C_m 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 X_im. As a result, spectral samplings Yi. are obtained.
3. Processing in GMF2:

Processing result:

4. IDFT of samplings Y_im. As a result, temporal sequence X_imm is determined.

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

The values obtained were 0.1; 0.2; 0.4.

NII power P_ui = 0.. .40 dB. An example of calculation of threshold signals P_c 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 N_i – 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 V_c , resulting in Doppler frequency shift f_c . For simplification, we consider a 2-pulse train first. A scatter signal can be written as follows:

Here, ψ₁ - initial phase;

τ_d – sampling interval;

σ_i - 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 PI
signal 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, ρ_s (0, f_c ) - UF of the signal with PNM in the point with coordinates (0, f_c).

With f_c = 1 kHz, value ρ_s (0, f_c) = 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 

P_c - PI power;

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

It is obvious that with PI parameters P_c, p_{k, m}, f_c 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 [6].

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

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

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 [5] as follows:

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:

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.

Conclusions

1. The ACF of signals with PNM has the mean SLL somewhat lower than Application of the window functions does not result in any significant decrease of the SLL. Mean level of the CCF of signals with PNM having the same base is equal to SLL of the ACF. Due to it, for efficient application of BBS with PNM, implementation of larger signal bases is required.
2. The SLL of signals at the MF output vir­tually does not depend on Doppler frequency shift. However, to achieve fairly low losses in detection of signals with PNM in a wide range of Doppler frequencies, owing to the switchboard form of their uncertainty function, it will be necessary to apply a multichannel matched filter with the num­ber of channels at least 8-16.
3. In radar survey it is possible to apply the procedures of probing with a coherent pulse train with different PNM laws in different train periods. Signal processing in this case is performed in MMF retunable in compliance with modulation law changes and DFT with the doubled num­ber of channels. In so doing, effective suppres­sion of imitating interference to the CCF level of approximately -40 dB is achieved.
4. Application in radar survey of BBS with PNM of the phase changing from pulse to pulse renders unlikely and inefficient the imitating de­ception of the given radar.
5. A system for detection of BBS with PNM which employs pre-processing of signals under temporal and frequency GMF ensures high tol­erance of detection characteristics to the NII pa­rameters. Threshold signal increase amounted to no more than 7...4.2 dB. At the same time, in a system without pre-processing of signals the threshold signal would increase to a far greater degree: 9.8...35.7 dB.
6. When probing angular directions with a coherent train of BB S with PNM varying in differ­ent train periods, application of a retunable MMF and adaptive rejection filter against PI enables to suppress PI as efficiently as under the conventional method using a train of LFM pulses with the same modulation law, or even more efficiently in case of a point PI source.