1. Introduction

This paper is the second paper of the sequel devoted to the methods of space-time signal processing (STSP) against the background of combined interference in coherent-pulse programmed and surveillance radars. By a combined interference is meant an additive mixture of internal noise, masking noise jamming and clutter.

The first paper [1] gives an overview of optimal systems of joint STSP and non-optimal systems of separate space-time and time-space processing of signals and also the combined STSP systems. This overview provides an analysis of their potential (ultimate) capabilities under the hypothetical conditions of exact knowledge of statistical characteristics of input actions. As was shown in paper [1], the energy losses of nonoptimal systems of separate processing of signals as compared to the optimal joint STSP depend on the level of strange interference (the clutter for noise jamming or noisy for clutter) in the first stage of sequential processing and also on the radial speed of target movement in the viewing direction and can be as high as 30 dB.

This paper analyzes the efficiency of adaptive sequential systems with separate space-time signal processing [2]–[16] against the background of combined interference under conditions of the parametric a priori uncertainty of their statistical characteristics. They make use of separate adaptive systems of spatial or ICP (inter-channel processing) and time or IPP (inter-period processing) of signals. The adaptive tuning of systems of ICP and IPP signals, i.e., the estimation of weight vectors of spatial and inter-period processing, involves the need of classified samples of noise jamming (NJ) and clutter (CL), the absence of which can significantly reduce the efficiency of processing in general.

However, it is difficult to generate such samples under conditions of combined interference [17]–[21], and sometimes it is not possible, especially in a surveillance radar with mechanical azimuth rotation of antenna. For example, the hope of collecting a classified NJ sample in such radar at sections of the range, where CL is not present, cannot be justified in the interference environment of Fig. 1 in the north and south directions, where CL is present practically along the entire distance.

Fig. 1.
figure 1

Screen of PPI under action of clutter.

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Therefore, next, we performed the quantitative analysis of the impact of decorrelating factors, such as the lack of classification of training sample of interference and fluctuations of the estimate of spatial weight vector of separate adaptive STSP from period-to-period of sounding, on the quality of interference suppression.

2. Generalized Structure of Adaptive Sequential STSP against Backfround of Combined Interference

Since the adaptive separate STSP involves the need to obtain the classified training samples of interference in estimating their parameters, in practice, the spatial inter-channel NJ compensation usually precedes the inter-period compensation of clutter (Fig. 2) [1], [3]–[8], [22], [25]. In this case, for the programmed surveillance radar, unlike the all-round surveillance radar, the possibilities of obtaining the classified (without clutter and useful signals) training sample of continuous noise jamming of required size for estimating the ICP weight vector increase at the first stage of processing. In such radars, the classified NJ training sample is obtained, for example, at the beginning of the operating cycle during time Tts (Fig. 2(a)) after the electronic switching of beam into the specified angular direction without emission of sounding signal. Therefore, neither the reflected useful signals, nor the reflections from CL sources, in principle, cannot get into the training sample.

Fig. 2.
figure 2

Functional block diagram of adaptive sequential STSP system against background of combined interference (b) and time diagram of its operation (a).

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However, such method of obtaining the classified training sample of continuous NJ is not possible in all-round surveillance radars with mechanical azimuth rotation of antenna. Therefore, the research workers and designers of such radars are forced to look for other methods of obtaining an NJ classified training sample [17]–[21] in considering the adaptive sequential space-time signal processing against the background of combined interference.

For eliminating the influence of fluctuations of spatial weight vector \( {{\mathbf{\hat{w}}}^{(s)}} \) estimated on the basis of the classified sample from period-to-period of sounding that stipulates an undesirable inter-period clutter decorrelation, this estimate \( {{\mathbf{\hat{w}}}^{(s)}} \) is fixed (“frozen”) [22]–[25] during the NJ compensation for the time of inter-period clutter compensation. The impact of fluctuations of the estimate of spatial weight vector of adaptive sequential STSP on the quality of clutter suppression in the absence of fixation of this vector is analyzed below.

One of the possible variants of the structure for the system of adaptive sequential space-time signal processing against the background of combined interference is shown in Fig. 2(b).

The parameter estimation unit (PEU) of NJ (NJ PEU) performs processing of the classified training sample U of Kjam independent N-dimensional NJ vectors during time Tts (Fig. 2(a)) specified by the control time gate “Time gate ts” (Fig. 2(b)). Here N is the number of spatial receive channels. The result of processing is estimate \( \mathbf{\hat{\Psi }}_{\rm a} \) of N×N matrix \( {{\mathbf{\Psi }}_{\text{a}}}={{({{\mathbf{\Phi}}_{\text{a}}})}^{-1}} \) that is inverse to the N×N correlation NJ matrix \( {{\mathbf{\Phi}}_{\text{a}}} \) . The latter is used to generate the estimating spatial weight vector \( \mathbf{\hat{w}}^{(s)}={{\mathbf{\hat{\Psi }}}_{\text{a}}}\cdot {{\mathbf{x}}_{s}} \) , where \( {{\mathbf{x}}_{s}} \) is the column-vector of expected spatial signal.

This vector is fixed, i.e., stored in random-access memory (RAM) and used for NJ compensation (NJC) and inter-channel accumulation of useful signal in ICP weight adder during time Tcom (Fig. 2(a)) of clutter inter-period compensation. To this end, according to control instruction “Gate com jam” (Fig. 2(b)), the fixed estimate of spatial weight vector (SWV) \( {{\mathbf{\hat{w}}}^{(s)}} \) is fed to the multipliers of weight adder of ICP signals.

Its output signal after the delay line (DL) unit \( {{\mathbf{V}}_{\text{rest}}}=\mathbf{\hat{w}}_{{}}^{(s)*}\cdot {{\mathbf{U}}_{s\tau }} \) representing an M-dimensional vector of complex amplitudes of additive mixture of internal noise, clutter, NJ compensation residues and, possibly, the signal is processed next in the adaptive IPP unit. Here M is the number of time channels. Horizontal line of M×KK size in Fig. 2(b) designates the M-element trains of complex amplitude of reflections from CL sources in KK range elements of the analyzed azimuth direction. A sliding \( M\times (K+1) \) -dimensional window in terms of range (time) is separated inside this band that is used to generate the training sample of volume K:

$$ \begin{aligned} {\mathbf{V'}}_{\text{rest}}&=\left[ {\mathbf{V}}_1, {\mathbf{V}}_2 \right]=\left[ {\mathbf{v}}_i \right]_{i=1}^K, \\ {\mathbf{v}}_i&=\left[ v_{l }^{(i)} \right]_{l =1}^{M}= \left\{ \begin{array}{ll} \mathbf{v}_{i}^{(1)},& i\le K/2, \\ \mathbf{v}_{i-K/2}^{(2)}, & i>K/2, \end{array} \right. \end{aligned} $$
(1)

As a result of its processing, an estimate of inverse matrix \( {{\mathbf{\hat{\Psi }}}_{\text{rest}}} \) is generated in the parameter estimation unit of CL (CL PEU) with due regard for the available a priori information. This estimate is used to obtain an estimate of IPP weight vector \( {{\mathbf{\hat{w}}}_{\tau }}={{\mathbf{\hat{\Psi }}}_{\text{rest}}}\cdot {{\mathbf{x}}_{\tau }} \) , where \( {{\mathbf{x}}_{\tau }} \) is the column-vector of expected time signal. This estimate is used in the weight adder of IPP signals where at the same time (simultaneously) the clutter is compensated with specific efficiency and pulses of the useful signal train are accumulated.

3. Analysis of Influence of SWV Estimate Fluctuations of Sequential STSP and Nonclassified Nature of Training Sample on Quality of Interference Suppression

3.1. In the process of analysis, we shall determine whether it is efficient in the adaptive sequential STSP system against the background of combined interference (Fig. 2) to estimate SWV \( {{\mathbf{\hat{w}}}^{(s)}} \) in each period of sounding rather than to fix (store) it for time Tcom (Fig. 2(a)). To this end, using an example of separate STSP with spatial (N = 6) and time (M = 16) channels (and employing 16-tap DL in Fig. 2(b)), we will analyze the influence of fluctuations of SWV estimate \( \mathbf{\hat{w}}_{m}^{(s)}=\,\,{{\mathbf{\hat{\Psi }}}_{\text{a}m}}\cdot {{\mathbf{x}}_{s}} \) in each m-th (m ∈ 1, M) period of pulse repetition on the quality of CL suppression using the IPP weight vector \( {{\mathbf{\hat{w}}}_{\tau }}={{\mathbf{\hat{\Psi }}}_{\text{rest}}}\cdot {{\mathbf{x}}_{\tau }} \) .

Since the SWV estimate \( \mathbf{\hat{w}}_{m}^{(s)}=\,\,{{\mathbf{\hat{\Psi }}}_{\text{a}m}}\cdot {{\mathbf{x}}_{s}} \) in each m-th (m ∈ 1, M) period is determined by its own training sample of finite size, it is a random quantity, i.e., different in various repetition periods that can decorrelate the clutter.

At the same time, we shall analyze the impact of the non-classified feature of NJ training sample at the ICP stage on STSP efficiency.

During simulation we will assume that the NJ sources are stationary, for example, such as throwable transmitters reaching the earth surface, while the radar antenna does not scan during time Tts + Tcom (Fig. 2(a)), i.e., the beam of programmed surveillance radar is set into the specified angular position during this time.

At the stages of spatial and time processing in generating the estimates of ICP \( \mathbf{\hat{w}}_{m}^{(s)}=\,\,{{\mathbf{\hat{\Psi }}}_{\text{a}m}}\cdot {{\mathbf{x}}_{s}} \) and IPP \( {{\mathbf{\hat{w}}}_{\tau }}=\mathbf{\hat{\Psi }}_{\text{rest}}^{\text{b}}\cdot {{\mathbf{x}}_{\tau }} \) weight vectors, respectively, we will use diagonally regularized [26], [27] estimates of matrices \( {{\mathbf{\hat{\Psi }}}_{\text{a}m}} \) and \( \mathbf{\hat{\Psi }}_{\text{rest}}^{\text{b}} \) that are inverse to the maximum likelihood estimates of correlation matrices:

$$ {{\mathbf{\hat{\Psi }}}_{\text{a}m}}={{\left[ \frac{1}{{{K}_{sp}}}\cdot \left( \text{re}{{\text{g}}_{\mathbf{a}}}\cdot \mathbf{I}_{N}+{{\mathbf{U}}_{m}}\cdot \mathbf{U}_{m}^{*} \right) \right]}^{-1}},\,\,\,\,\,\,m\in 1,\,M, $$
(2)
$$ \begin{aligned} {{\mathbf{U}}_{m}}&=\left[ \mathbf{u}_{i}^{(m)}\right]_{i=1}^{{{K}_{sp}}}, \\ \mathbf{u}_{i}^{(m)}&\sim CN(0,{{\mathbf{R}}_{sp}}),\\ {{\mathbf{R}}_{sp}}&={{\mathbf{\Phi }}_{\text{a}}}+ \sigma_{\rm cl}^2 \cdot \mathbf{I}_{N}, \end{aligned} $$
(3)
$$ \begin{aligned} \mathbf{\hat{\Psi }}_{\text{rest}}^{\text{b}}&=K{{\left[ \text{re}{{\text{g}}_{{\rm rest}}}\cdot {{\mathbf{I}}_{M}}+{{\mathbf{V}}_{\text{rest}}}\cdot \mathbf{V}_{\text{rest}}^{*} \right]}^{-1}}, \\ {{\mathbf{V}}_{\text{rest}}}&=\mathbf{\hat{w}}_{{}}^{(s)*}\cdot {{\mathbf{U}}_{s\tau }}, \end{aligned} $$
(4)

where \( {{\mathbf{U}}_{m}} \) is the ICP spatial training sample of \( K_{sp} \) volume in the m-th (m ∈ 1, M –1) repetition period, rega and regrest are the regularization parameters of ICP and IPP; \( \mathbf{{\Psi }}_{\text{rest}}^{\text{b}} \) is the Hermitian band matrix with band width \( z=2zz-1 \) , \( zz\in 1,...,M \) ; K is the size of IPP signal training sample.

If the spatial sample is classified, \( K_{sp}=K_{\rm jam} \) , while the clutter dispersion in training sample \( \sigma_{\rm cl}^2 =0 \) . In the event that the clutter, completely or partially, enters into the ICP training sample, the dispersion \( \sigma_{\rm cl}^2 \ge 0 \) and the non-classified NJ sample is formed, if the sounding signal is radiated on interval Tts (Fig. 2). Here Kjam is the size of the training sample of the mixture of NJ and internal noise.

Let us analyze two cases. The first case (sl = 1) corresponds to the situation when the ICP weight vector \( \mathbf{\hat{w}}_{m}^{(s)} \) \( (m\in 1,\,\,M) \) is calculated only before the first repetition period (first pulse) \( \mathbf{\hat{w}}_{1}^{(s)} \) , and then it is fixed for the rest of periods \( \mathbf{\hat{w}}_{m}^{(s)}=\mathbf{\hat{w}}_{1}^{(s)} \) \( (m\in 1,\,\,M) \) for eliminating the influence of fluctuations of SWV estimate \( \mathbf{\hat{w}}_{m}^{(s)} \) on the NJ suppression quality.

The second case (sl = 2) corresponds to the situation when the ICP weight vector \( \mathbf{\hat{w}}_{m}^{(s)} \) \( (m\in 1,\,\,M) \) is calculated in each of M periods; therefore, the fluctuations of SWV estimate \( \mathbf{\hat{w}}_{m}^{(s)} \) occur from period-to-period of sounding. As can be seen from Figures 3–6 below, the curves of corresponding relationships for the first case (sl = 1) and the second case (sl = 2) are solid and dashed, respectively.

The signal-to-(interference+noise) ratio (SINR) μ at the output of adaptive STSP system was calculated by using the following relationships:

for the case sl = 1 \( (\mathbf{\hat{w}}_{m}^{(s)}=\mathbf{\hat{w}}_{s}^{{}},\,\,m\in 1,M) \) :

$$ \mu (K)={{h}_{\text{sign}}}\frac{{{\left| \left( \mathbf{\hat{w}}_{s}^{*}\cdot {{\mathbf{x}}_{s}} \right) \right|}^{2}}\cdot {{\left| \mathbf{\hat{w}}_{\tau }^{*}(K)\cdot {{\mathbf{x}}_{\tau }} \right|}^{2}}}{\mathbf{\hat{w}}_{s}^{*}\cdot {{{\mathbf{\hat{w}}}}_{s}}\cdot \mathbf{\hat{w}}_{\tau }^{*}(K)\cdot \left( \frac{\mathbf{\hat{w}}_{s}^{*}\cdot \mathbf{\Phi }_{\text{a}}^{{}}\cdot \mathbf{\hat{w}}_{s}^{{}}}{\mathbf{\hat{w}}_{s}^{*}\cdot {{{\mathbf{\hat{w}}}}_{s}}}\cdot {{\mathbf{I}}_{M}}+{{\mathbf{R}}_{c}} \right)\cdot {{{\mathbf{\hat{w}}}}_{\tau }}(K)} , $$
(5)

for the case sl = 2 \( (\mathbf{\hat{w}}_{m}^{(s)},\,\,m\in 1,M) \) :

$$ \mu (K)={{h}_{\text{sign}}}\frac{{{\left| \mathbf{\hat{w}}_{\tau }^{*}(K)\cdot {{\mathbf{x}}_{\tau }}\circ \boldsymbol{\hat{\omega }} \right|}^{2}}}{\mathbf{\hat{w}}_{\tau }^{*}(K)\cdot \left( {\rm diag}\left[ \mathbf{\hat{w}}_{m}^{*}\cdot \mathbf{\Phi }_{\rm{a}}^{{}}\cdot \mathbf{\hat{w}}_{m}^{{}} \right]_{m=1}^{\,M}+\mathbf{\hat{\Omega }}\circ {{\mathbf{R}}_{c}} \right)\cdot \mathbf{\hat{w}}_{\tau }(K)} , $$
(6)

where K is the size of time training sample; \( \boldsymbol{\hat{\omega }}=\left[ \mathbf{\hat{w}}_{m}^{*}\cdot {{\mathbf{x}}_{s}} \right]_{m=1}^{M} \) is the M-dimensional vector distorting the M-dimensional vector of time signal \( {{\mathbf{x}}_{\tau }} \) ; \( \mathbf{\hat{w}}_{m}^{*}=\mathbf{w}_{m}^{(s)*}= [ \hat{w}_{n}^{(s)*} ]_{n=1}^{N},\,m\in 1,M \) , \( {{\mathbf{\hat{w}}}_{\tau }}=\mathbf{\hat{\Psi }}_{\text{rest}}^{\text{b}}\cdot {{\mathbf{x}}_{\tau }} \) , \( \mathbf{\hat{\Omega }}=\,\,\left[ \mathbf{\hat{w}}_{i}^{*}\cdot {{{\mathbf{\hat{w}}}}_{j}} \right]_{i,j=1}^{M} \) is the M×M matrix of decorrelations introduced into the CL correlation matrix (CM) \( {{\mathbf{R}}_{c}} \) by the difference of estimating SWV \( \mathbf{\hat{w}}_{m}^{(s)}={{\mathbf{\hat{\Psi }}}_{\text{a}m}}\cdot {{\mathbf{x}}_{s}} \) in different periods of sounding; hsign is the relative dispersion of random amplitude multiplier; “ \( \circ \) ” is the symbol of componentwise multiplication (Schur–Hadamard product) used in paper [28] for the analysis of radar protection efficiency against the noise jamming under conditions of non-identical characteristics of spatial receive channels. This makes it possible to assume that the considered inter-period difference of random SWV estimates (sl = 2) leads to a similar effect of non-identity of characteristics of time receive channels.

3.2. Let us analyze the influence of fluctuations of SWV estimate \( \mathbf{\hat{w}}_{m}^{(s)} \) and the non-classified feature of spatial sample of noise jamming on the efficiency of only adaptive inter-channel signal processing of sequential STSP (Fig. 2(b)).

Figure 3 presents the relationships of SINR \( {{\mu }_{sp}} \) at the output of the 6-channel ICP system of sequential STSP (Fig. 2(b)) averaged over 100 realizations as a function of the size of spatial training sample Ksp under the action of NJ from n = 1, 3, and 5 sources (Fig. 3) with the total relative power h = 50 dB and autoregressive (AR) NJ of about p = 10 with power \( {\sigma }_{\text{cl}}^{2}= \) 50 dB and inter-period correlation coefficient ρτ = 0.99 at the value of spatial parameter of regularization: rega = 10. In this case, the power was determined in relation to the power of internal noise \( {\sigma }_{\text{noise}}^{2}=1 \) , while h was determined as the interference-to-noise ratio (INR).

Fig. 3.
figure 3

Relationships of SINR μsp as function of size of training spatial sample Ksp at n = 1 (a), n = 3 (b) and n = 5 (c) sources of NJ at output of spatial part of sequential STSP system.

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The analysis of Fig. 3 reveals the following features:

a) In the absence of spatial training sample of clutter (cl = 0), when SWV \( \mathbf{\hat{w}}_{m}^{(s)} \) \( (m\in 1,\,M) \) was estimated in terms of the classified NJ sample of size Ksp = Kjam in each of M periods (curves sl = 2, cl = 0), the efficiency of ICP adaptive system of STSP is close to potential obtained at the known NJ CM (curves POT, cl = 0). It means that NJ is compensated actually to the noise level, while the useful signal is effectively separated against the NJ background. This is an obvious result since the continuous correction of SWV value involves the need to process the signal against the NJ background.

The similar results for adaptive ICP of signals take place also at the fixed SWV (curves sl = 1, cl = 0), but only because the NJ jammers are stationary, while the radar did not scan the antenna directional pattern (DP). The influence of “aging” of weight coefficients during their fixation on the efficiency of signal processing under conditions of movement of NJ jammers with respect to the programmed surveillance radar or rotation of antenna DP of all-round surveillance radar was analyzed in papers [22], [24].

b) If the clutter even partially gets into the NJ training sample, for example, 30 dB of \( {\sigma }_{\text{cl}}^{2}= \) 50 dB (curve cl = 30 dB), this causes the formation of NJ non-classified sample, and the quality of estimating the ICP weight vector \( {{\mathbf{\hat{w}}}^{(s)}} \) degrades, because the clutter uncorrelated between the channels behaves as an additional hampering white noise. As a result, the efficiency of adaptive ICP decreases.

For both cases (sl = 1, sl = 2, cl = 30 dB) the losses in SINR can reach 6.5 dB at n = 1 and 18.2 dB at n = 5. The rise of losses in SINR at the number of jammers n = 5 as compared to the case of n = 1 is stipulated by a more complex mechanism of forming an estimate of spatial weight vector for creating valleys in the adaptive DP simultaneously in five directions.

3.3. Let us analyze the influence of fluctuations of SWV estimate \( \mathbf{\hat{w}}_{m}^{(s)} \) and the non-classified feature of spatial sample of noise jamming on the efficiency of adaptive separate STSP as a whole.

Under conditions of Fig. 3, Figure 4 shows the relationships of SINR μ in accordance with expressions (5) and (6) at the STSP resultant output after IPP (Fig. 2(b)) as a function of the size of training spatial sample Ksp of system of ICP signals.

Fig. 4.
figure 4

Relationships of SINR μ at output of sequential STSP system as function of size of training spatial sample.

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In this case, temporally, for eliminating the influence of estimate quality of the M-dimensional IPP weight vector \( {{\mathbf{\hat{w}}}_{\tau }}=\mathbf{\hat{\Psi }}_{\text{rest}}^{\text{b}}\cdot {{\mathbf{x}}_{\tau }} \) on the efficiency of separate STSP in relationships (5) and (6), we will use the M-dimensional vector \( {{\mathbf{w}}_{\tau }}={{\mathbf{\Psi }}_{\text{rest}}}\cdot {{\mathbf{x}}_{\tau }} \) (M = 16) generated from the exact M×M CM \( {{\mathbf{\Psi }}_{\text{rest}}} \) of the mixture of clutter and NJ residues after ICP (at the input of IPP system):

$$ {{\mathbf{\Psi }}_{\text{rest}}}=\mathbf{R}_{\text{rest}}^{-1}={\rm diag}\left[ \mathbf{w}_{m}^{*}\cdot \mathbf{\Phi }_{\rm{a}}^{{}}\cdot \mathbf{w}_{m}^{{}} \right]_{m=1}^{\,M}+\mathbf{\Omega }\circ {{\mathbf{R}}_{c}} . $$
(7)

The analysis of Fig. 4 reveals the following features:

a) In the case of the classified training spatial NJ sample, fluctuations of SWV estimate from period-to-period of sounding degrade the IPP efficiency and the sequential STSP as a whole (curves sl = 2, cl = 0). SINR losses at one NJ source (n = 1) (Fig. 4(a)) amount to about 20 dB. As the number of NJ sources increases, these losses decrease and at n = 5 they amount to 3 dB.

Such degradation of STSP efficiency occurs at the expense of degradation of IPP signal efficiency due to inter-period clutter decorrelation caused by fluctuations of the estimate of ICP weight vector from period-to-period of sounding that was obtained from its own NJ training sample of finite size in each period. The degree of inter-period clutter decorrelation varies at different number of NJ jammers.

This fact compels the designers to reject the option of SWV estimation in each sounding period and fix it for the time of inter-period clutter compensation. At the same time, such fixation decreases the NJ compensation efficiency owing to the SWV “aging” during time Tcom (Fig. 2(a)) due to the rotation of radar antenna or/and displacement of NJ sources. The admissible interval of SWV fixation was estimated in paper [22]. This interval determines the ICP energy losses of signals that do not exceed the specified level.

b) The non-classified feature of training spatial sample (curves sl = 2, cl = 30 dB) increases the specified losses; moreover, this increase gets bigger with the increasing number of NJ sources. Hence, at n = 5 the losses increase to 23 dB. In the large, the behavior of blue dashed lines (Fig. 4) corresponding to SWV fluctuations and non-classified sample at the number of NJ sources equal to: n = 1, n = 3 and n = 5 is about the same.

3.4. In addition, we shall take into account the influence of the quality of estimate of the M-dimensional IPP weight vector \( {{\mathbf{\hat{w}}}_{\tau }}=\mathbf{\hat{\Psi }}_{\text{rest}}^{\text{b}}\cdot {{\mathbf{x}}_{\tau }} \) on the efficiency of separate STSP and also the value of regularization parameter rega of ICP of signals and the number of stages (band width) zz of signal IPP, on the efficiency of completely adaptive sequential STSP (adaptive ICP + adaptive IPP) in accordance with relationships (5) at the fixed SWV (sl = 1) and relationships (6) in case of estimating SWV in each period (sl = 2).

Hence, Fig. 5 displays the relationships of SINR μ at the output of adaptive separate STSP (Fig. 2(b)) as a function of the size of time training sample K of signal IPP system under exposure to NJ from one (Fig. 5(a–c)), three (Fig. 5(d–f)) and five (Fig. 5(g–i)) sources with INR h = 50 dB and AA (antenna array) clutter of about p = 10 with power \( {\sigma }_{\text{cl}}^{2}= \) 50 dB and coefficient of inter-period correlation ρτ = 0.99 at different values of regularization parameter during the spatial signal processing: rega = 10 (Fig. 5(a, d, g)), rega = 100 (Fig. 5(b, e, h)), and rega = 1000 (Fig. 5(c, f, i)). The size of training spatial sample is equal to Ksp = 32.

Fig. 5.
figure 5

Relationships of SINR μ at output of adaptive sequential STSP system as function of size of training time sample K of IPP system under action of NJ from one (a–c), three (d–f) and five (g–i) sources at different values of rega and zz = 4.

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The analysis of Fig. 5 revealed the following features.

a) With the classified training samples of interference, fixed SWV (sl = 1) and stationary NJ jammers and nonrotating radar antenna, the efficiency of adaptive STSP system (red solid curves) is close to the potential (red dash-dotted straight lines), and NJ is compensated actually to the noise level. The high-speed adaptation is provided at rega = 10 and rega = 100. This speed decreases at rega = 1000 that is in good agreement with the results obtained in paper [27] for adaptive spatial signal processing against the background of noise jamming.

b) With such samples, SWV fluctuations from period-to-period of sounding (sl = 2) degrade the IPP efficiency and the sequential adaptive STSP as a whole (red dashed curves). What is more, with an increase of the number of NJ sources and the increased value of spatial regularizer from rega = 10 to rega = 1000 at n = 1 and n = 3, these losses decrease. However, at n = 5 the losses at rega = 1000 are larger than at rega = 10 and rega = 100.

c) The non-classified feature of the training spatial sample, as was noted above (Fig. 4), increases the losses of adaptation efficiency in both cases sl = 1 and sl = 2. However, under the considered conditions, the value of losses in comparison with Fig. 4 is larger, because they are also stipulated by the fact that in completely adaptive sequential STSP, the non-classified feature of spatial NJ sample at the stage of signal ICP gives rise to the non-classified feature of time clutter sample at the IPP stage owing to the ill-quality compensation (large residues of compensation) of NJ at ICP. This, in turn, degrades the quality of estimation of weight vector of signal IPP \( {{\mathbf{\hat{w}}}_{\tau }} \) .

As can be seen from the comparison of Fig. 4 and Fig. 5, due to the last factor, the energy losses for sl = 2 at n = 5 increased from 23 dB (Fig. 4) to 30 dB (Fig. 5, blue dashed curves). At the fixed SWV (sl = 1) these losses amounted to 24 dB (blue solid curves).

In these conditions, the value of spatial regularizer rega hardly affects the STSP efficiency, unlike the estimation of weight vector based on the classified sample.

Figure 6 makes it possible to analyze the impact of the number of IPP stages (multiplicity factor of compensation) zz on the efficiency of adaptive sequential STSP.

Fig. 6.
figure 6

Relationships of SINR μ as function of size of training time sample of IPP system for different multiplicity of compensation at output of adaptive sequential STSP system.

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For conditions in Fig. 5, Figure 6 displays the relationships of SINR μ at the output of adaptive separate STSP (Fig. 2(b)) as a function of the size of time training sample K of IPP system under exposure to NJ from one (Fig. 6(a, b)), three (Fig. 6(c, d)) and five (Fig. 6(e, f)) sources for sl = 1 (Fig. 6(a, c, e)) and sl = 2 (Fig. 6(b, d, f)).

The analysis of Fig. 6 makes it possible to draw the following conclusions.

a) With the classified training samples of interference, fixed SWV (sl = 1), stationary NJ jammers and nonrotating radar antenna, the efficiency of adaptive STSP system (red, blue and black solid curves) is close to the potential (red dash-dotted straight lines). In this case, the best speed of response is achieved at the number of stages zz = 4 as compared with zz = 8 and zz = 16 at the number of pulses in the train M = 16 that agrees with the results obtained in paper [27] for adaptive time processing of signals against the background of clutter.

b) With such samples, SWV fluctuations from period-to-period of sounding (sl = 2) degrade the IPP efficiency and the sequential adaptive STSP in general. What is more, the minimum losses are ensured at zz = 16.

c) The lack of classification of training samples of interference increases the losses of adaptation efficiency in both cases sl = 1 and sl = 2. Moreover, the number of stages of signal IPP zz actually does not affect this efficiency.

4. Conclusions

1. The quantitative analysis of the influence of decorrelating factors, in particular, fluctuations of the estimate of spatial weight vector (SWV) of signal inter-channel processing (ICP) system against the background of noise jamming on the efficiency of adaptive sequential systems with separate time-space signal processing (STSP) against the background of combined interference (additive mixture of noise jamming and clutter) was (has been) performed on condition that the spatial processing with compensation of active noise jamming precedes the inter-period processing (IPP) with compensation of clutter.

It has been shown that fluctuations of SWV random estimate from period-to-period of sounding stipulated by the finite size of classified training sample of noise jamming in each period in the absence of SWV fixation leads to the degradation of compensation quality of clutter at the stage of signal IPP due to the emerging inter-period decorrelation of clutter and, correspondingly, to the energy losses in adaptive sequential STSP system.

Under exposure to one NJ source with relative power of 50 dB and the same power of clutter, these losses for the analyzed adaptive sequential STSP system containing 6 spatial (N = 6) and 16 temporal (M = 16) channels amounted to 20 dB.

This fact compels the designers to reject the option of the spatial weight vector (SWV) estimation in each sounding period and fix it for the time of inter-period clutter compensation. At the same time, such fixation decreases the compensation efficiency of noise jamming owing to the SWV “aging” that gives rise to the contradiction, the resolution of which is considered in the fourth paper of this sequel.

2. This paper analyzed the influence of such a decorrelating factor as the non-classified feature of spatial training sample of noise jamming on the efficiency of adaptive system of sequential STSP against the background of combined interference riving rise to the lack of classification of the training time sample of clutter at the IPP stage owing to the expense of large compensation residues of noise jamming after ICP.

It was shown that with such non-classified samples of interference in the analyzed adaptive system of sequential STSP, the energy losses at the fixed SWV amounted to 24 dB in the case when part of the clutter (30 of 50 dB) was in the training spatial sample of ICP system. In case of additional fluctuations of SWV estimate, these losses amounted to 30 dB. The fourth paper of this sequel is devoted to the elimination of these losses.