3.1.1 PRINCIPLES OF CONSTRUCTION AND EFFICIENCY INDICES

When signal processing is carried out under conditions of correlated passive interferences, an initial optimal signal processing of pulse train from N coherent pulses is reduced to v-fold (v ≤ N) interperiod subtraction of complex amplitude envelopes of the pulse train with subsequent accumulation of uncompensated remainders. A procedure of the interperiod subtraction (equalization) of fixed-position correlated interferences (interferences generated by fixed-position correlated sources) allows us to indicate targets moving relative to radar system. As a rule, this procedure is called the moving-target indication.

About 25–30 years ago, the moving-target indication was carried out by analog wireless devices such as delay circuit, analog filters, and so on. Currently, digital moving-target indicators are widely used for the equalization of passive interferences. The digital moving-target indicators consist of memory devices and digital filters [5–8].

The designing of digital moving-target indicators and the evaluation of their efficiency must take into consideration the following peculiarities:

• Using the digit capacity N_b ≥ 8 of the target return signal amplitude samples in the digital moving-target indicator, quantization errors can be considered as the white Gaussian noise that is added to the receiver noise; on account of this, a synthesis of the digital moving-target indicator is carried out using an analog prototype, that is, by the digitization of well-known analog algorithms.

• Quantization of the target return signal amplitude samples leads to additional losses under the equalization of passive interferences, compared to that observed with the use of analog moving-target indicators.

The digital moving-target indicators can be presented using both two-channel and single-channel forms. The block diagram of the two-channel digital moving-target indicator is shown in Figure. 3.1. After processing by DGD, the in-phase and quadrature components of the DGD output signal come in at the inputs of two similar digital filters realizing the interperiod subtraction operation. Under single subtraction, the output signals of quadrature channels of the filter can be presented in the following form:

In a similar manner, we can obtain the formulas for the filter output signals in the case of any ν-fold (ν ≤ N) interperiod subtraction when ν = 2, 3,… There is a need to take into consideration that the operations are realized on the signals obtained within the limits of the same sampling interval under sampling in time and the index i denotes the current value of the sweep. Thus, the signals ZIout[i] and ZQout[i] are either accumulated at first and united after accumulation or are united at first and subjected to noncoherent accumulation before coming in at the input of decision circuit. Possible ways of packing of the in-phase and quadrature signal components are shown in Figure 3.1, the outputs 1, 2, and 3. The single-channel digital moving-target indicator can be obtained from a two-channel digital moving-target indicator version if only one of the channels of the two-channel digital moving-target indicator shown in Figure 3.1 is used. In this case, the block diagram of the digital moving-target indicator is significantly simplified, but additional losses in the SNR do occur.

In practice, the following indices are widely used for the evaluation of the efficacy of digital moving-target indicators:

• The coefficient of interference cancellation G_can that is defined as a ratio of the passive interference power at the equalizer input to the passive interference power at the equalizer output

  Gcan=PpiinPpiout=σpiin2σpiout2atσpiin2≫σn2,⁢(3.2)

  where

  • Ppiin is the power of the passive intereference at the equalizer input

  • σpiin2 is the variance of the passive interference at the equalizer input

  • Ppiout is the power of the passive interference at the equalizer output

  • σpiout2 is the variance of the passive interference at the equalizer output

  • σn2 is the receiver noise

• Figure of merit for the case of linear interperiod cancellation of the passive interferences can be defined in the following manner:

  η=Psout/PpioutPsin/Ppiin=PsinPpiout×PsoutPsin=Gcan×Gs,⁢(3.3)

  where G_s is the coefficient characterizing a signal passing through the passive interferences equalizer. In general, when a nonlinear signal processing is used, the figure of merit η indicates to what extent the interference power at the equalizer input can be increased, without losses in detection performance.

• The coefficient of improvement is a characteristic of the response of the digital moving-target indicator on passive interferences with respect to the averaged response on target return signals:

  Gim=Psout/Ppiout[Psin/Ppiin]Vtg,⁢(3.4)

  where [Psin/Ppiin]Vtg¯ is the SNR at the equalizer input while deciding on the average values with respect to all target velocities.

The aforementioned Q-factors can be determined both for passive interferences equalizer systems using the interperiod subtraction and for equalizer systems with subsequent accumulation of residues after the passive interference cancellation.

3.1.2 DIGITAL REJECTOR FILTERS

The main element of the digital moving-target indicator is the digital rejector filter that can guarantee a cancellation of correlated passive interference. In the simplest case, the digital rejector filter is constructed in the form of the filter with the v-fold (v < N) interperiod subtraction that corresponds to the structure of the nonrecursive filter. The recursive filters are also widely used for the cancellation of passive interferences. Consider briefly the problems associated with the analysis and synthesis of rejector filters.

The signal processing algorithm of the digital nonrecursive filter takes the following form:

where

• h_i is the weight coefficient of the filter

• ZIQnrin[n−i] are in-phase and quadrature components of signal at the filter input (the DGD output)

At ν = 2, (3.5) defines a signal processing algorithm for the nonrecursive filter with the double interperiod subtraction. To define the coefficients of this filter we can set up the following equations:

Consequently, the filter coefficients are equal to h₀ = h₂ = 1, h₁ = −2, h_i = 0 at i > 2. The flowchart of the digital filter with the interperiod subtraction at v = 2, in the case of the single quadrature channel, is shown in Figure 3.2. Similarly, we can derive the filter coefficients at v > 2. So, at v = 3 we obtain h₀ = 1, h₁ = −3, h₂ = 3, h₃ = −1, h_i = 0 at i > 3; and the signal processing algorithm takes the following form:

The digital nonrecursive filters are very simple under realization, but they have very low-angle fronts of the amplitude-frequency performance, which severely affects the effectiveness of passive interference cancellation.

In addition to the nonrecursive filters, the recursive filter can be used as the rejector filter with the following signal processing algorithm:

where a_i and b_j are the coefficients of the digital recursive filter.

Methods of recursive filter synthesis are considered and discussed in numerous literatures [9-21]. Without going into detail, we note that a synthesis procedure based on a squared amplitude– frequency filter response in accordance with a technique described in Ref. [22] can be considered the most effective for the digital moving-target indicators constructed based on the digital rejector filters. As a first step, and on the basis of the given parameters of environment conditions and initial premises, an order is chosen and the amplitude-frequency characteristic of analog filter prototype is defined. After that we define an approximation of squared amplitude-frequency characteristic of digital filter to the squared amplitude-frequency characteristic of analog filter-prototype, taking into consideration the restrictions arising from structural complexity. The elliptic filter, also known as Zolotaryov-Cauer filter [23], is best suited for these requirements.

Under realization of the digital rejector filter in accordance with (3.10), it is convenient that a_i and b_j are simple binary numbers. In this case, there is no need to use ROM to store these coefficients, and the multiplication operations are changed by simple shift and summation operations. For example, in the case of the recursive elliptic filter of the second order, synthesized using the squared amplitude-frequency characteristic, we obtain the following values of coefficients [22]: a₀ = a₂ = 1; a₁ = −1.9682; b₁ = −0.68; and b₂ = −0.4928. To simplify the realization of the digital rejector filter, the coefficients a₁, b₁, and b₂ can be rounded in the following order: a₁ = −1.875 = −2¹ + 2⁻³, b₁ = −0.75 = 2⁰ + 2⁻³, b₂ = −0.5 = −2⁻¹. Investigations showed that rounding off of the coefficients a₁, b₁, and b₂ can give a very small effect on the cancellation of passive interferences; however, this effect is negligible.

The digital rejector filter ensures the best cancellation of passive interferences due to an improvement in the amplitude-frequency characteristic shape, compared with the nonrecursive rejector filter of the same order. However, the degree of correlation of the passive interference remainders at the digital recursive filter output is higher compared to the correlation degree of the passive interference remainders at the digital nonrecursive filter. Moreover, the presence of positive feedback leads to an increase in the duration of transient process and corresponding losses in efficacy if the number of target return pulses in the train is less 20, that is, N < 20.

Now, consider some results of the comparison between the efficiency levels of the digital filter and the analog filter. At first, transform (3.10) to nonrecursive form, that is, present the signal processing algorithm of the digital recursive filter in the following form:

where

• N is the number of pulses in train (sequence)

• h_irf are the coefficients of digital recursive filter impulse response defined by [23,24]

For example, in the case of the digital recursive second-order filter (ν = k = 2) we obtain h_{0rf = 1, h1rf = a1 + h2rf = a2 + b1h1rf + b2, at i > 2 we have hirf} = b₁h_i−1rf + b₂h_i−2rf. As follows from (3.12), the impulse response of recursive filter tends to approach infinity and the number of used coefficients is defined by the sample length of the input signal.

Efficiency of interperiod subtraction at different multiplicities of ν can be compared using the coefficient of passive interference cancellation [25]:

where

• ρ_pi[·] is the normalized coefficient of interperiod correlation of the passive interferences T is the radar bang period

At determination of G_can for nonrecursive filters using (3.13) we think that ν is the multiplicity of the interperiod subtraction and for the recursive filters ν = N_pi − 1, where N_pi is the sample size of passive interferences.

To make calculations using (3.13) there is a need to define a model of interferences. The energy spectrum of passive interferences, that is, the signals reflected from, for example, extended territorial objects, clouds, or other types of reflectors, can be approximated by the Gaussian distribution law given by

where

• S_pi is the energy spectrum of passive interferences

• Δf_pi is the bandwidth of passive interference spectrum at the level 0.5

The normalized coefficient of correlation corresponding to this spectrum is given as

Figure 3.3 represents the coefficient of passive interference cancellation G_can as a function of Δ f_piiT(i = 1) for various values of multiplicity of interperiod subtractions. As we can see from Figure 3.3, the use of the twofold interperiod subtraction brings a win in cancellation for about 15.dB compared to a single interperiod subtraction. In the case of the threefold interperiod subtraction, a win in cancellation in comparison with the twofold interperiod subtraction is less than 15.dB. In the case of the fourfold interperiod subtraction, a win in cancellation in comparison with the threefold interperiod subtraction is still less. The coefficient of cancellation of passive interferences for recursive filters depends on the sample size N_pi of the passive interference. Thus, it is of no use to compare the recursive and nonrecursive filters by the coefficient of passive interference cancellation G_can. Comparative efficacy of the interperiod subtraction and recursive filters can be evaluated by the figure of merit η given by (3.3), which corresponds to a win in SNR in the case of digital linear filters.

Assume that a passive interference source is stationary and fixed, that is, the Doppler shift is zero, ϕ_D = 0, and the target velocity is optimal, that is, ϕ_D = π in such a scenario, the figure of merit η can be determined in the following form [24]:

where

• v_s is the sample size of the target return signal equal to N_s − 1 (N_s is the number of target return pulses in train) in the case of recursive filter and the interperiod subtraction multiplicity v in the case of nonrecursive filter

• v_in is the sample size of interference determined by analogous way as for v_s

• ρ_s is the coefficient of correlation of the target return signal

• ρ_pi is the coefficient module of passive interference interperiod correlation

There is a need to define a model of the target return signal for calculations. As a rule, we can think that the target return signal amplitude envelope is subjected to Rayleigh distribution law with the coefficient of correlation defined by

where Δf_tg is the spectrum bandwidth of the fluctuated target return signal.

The figure of merit η for a set of the digital nonrecursive filters with the twofold (the curve 1) and threefold (the curve 3) interperiod subtractions, the digital recursive filter of the second order (the curve 2), and the digital composite filter consisting of the nonrecursive filter with the single interperiod subtraction and the recursive filter of the second order connected in cascade (the curve 4) is shown in Figure 3.4 as a function of Δf_piiT (i = 1). As follows from Figure 3.4, the use of digital recursive filters provides a win in 10 dB in the figure of merit in comparison with the use of filters with the interperiod subtraction. The digital recursive filter order and the multiplicity of the interperiod subtraction of the digital nonrecursive filters are the same under comparison.

In addition, we can see that the time of transient process in the digital recursive filters is much more in comparison with the time of transient process of the digital nonrecursive filters. Because of this, the interference sample size under the use of the digital recursive filters of the second order must be greater than 20, that is, v_in ≥ 20, and v_in ≥ 30, if we use the digital recursive filters of the third order. In doing so, the sample size of the target return signal v_s does not affect the efficacy of the digital recursive filters of the second and third orders already if v_s ≥ 10.

To reduce computer cost under realization of the signal processing algorithm (3.10) using the digital recursive filter, we can implement various computational processes in parallel. As an example, consider a hardware implementation by iteration network [14,26,27]. For this purpose, we can rewrite (3.10) in the following form at v = k = N

where Dⁱ is the operator delaying the input data on i cycles Equation 3.18 can be presented in the following detailed form:

After elementary transformations we obtain

As follows from (3.19), the signal processing algorithm based on the recursive digital filter can be realized using the iteration network (see Figure 3.5) consisting of homogenous elementary blocks with three inputs and a single output and realizing the signal processing algorithm that can be presented in the following form:

Operating speed of the iteration network presented in Figure 3.5 is defined by the operating speed of a single filter cell carrying out two multiplications on constant factors and two summations. Because of this, further increase in the operating speed is made possible using specific structural designs that help with making calculations based on (3.20). When in the considered iteration network all b_j = 0 we can obtain a realization of the digital nonrecursive filter. Under the realization of rejector digital filters some specific losses in efficacy occur. The main sources of these losses are quantization of the input signals, rounding-off of the filter weight coefficients, and results of calculations.

As we noted in Section 2.1, under choosing the quantization step Δx based on the condition Δx/σ_n ≤ 1, where σn2 is the variance of receiving path noise, a mutual correlation of quantization noise and quantized process noise is absent and the variance of the quantization noise is given by σqn2=Δx2/12. Digital filter apparatus errors caused by round-off of the calculation results do not depend on the quantization noise and have zero mean and the variance determined by σqn2=mδ2/12, where m is the number of multiplications on fractional weight coefficients in the course of single realization of signal processing algorithm by the digital filter and δ is the value of the arithmetic unit low order. Taking into consideration the error sources identified, the total error variance at the digital filter output can be presented in the following form:

In this case, the resulting value of the figure of merit, taking into consideration the quantization and round-off errors, will be determined as

3.1.3 DIGITAL MOVING-TARGET INDICATOR IN RADAR SYSTEM WITH VARIABLE PULSE REPETITION FREQUENCY

Under indication of moving targets by a complex radar system (CRS) with a constant repetition period of radar searching signal, there take place the so-called “blind” velocities at Doppler frequencies f_D = ±k/T, k = 0, 1, 2,… because the phase of the target return signal from moving target varies by 2kπ times within the limits of the period T. To avoid this phenomenon the wobble (modulation) procedure for the repetition period of radar searching signals is used, which leads to the spreading of velocity response of moving-target indication and, finally, to a decrease in the number and depth of dips of resulting velocity response.

Implementation of analog moving-target indicators with the wobble procedure within the limits of the interval T is very difficult, since in this case there is a need to employ various individual delay circuits for each value of repetition period T and complex switching system for these circuits. Under realization of the digital moving-target indicator with the wobble procedure within the limits of the interval T, it is enough to realize only synchronization between a sample of delayed data from memory device and radar searching signal instants. In doing so, the size of memory device does not change and does not depend on the number of discrete values and the wobble procedure function within the limits of the repetition period T. The best speed performance of the digital moving-target indicators with the wobble procedure within the limits of the repetition period T can be obtained if each pulse from the target return pulse train is measured N_s, where N_s is the number of target return pulses in the train adjusted with a new individual repetition period. The individual repetition period T must vary with respect to the average value T¯ on the multiple value of the fixed time interval ±ΔT. In this case, if N_s an odd sequence of the repetition periods T within the limits of the train takes the following form:

The problem of design and construction of the digital moving-target indicators with the wobble procedure within the limits of the repetition period T is to make a correct selection of a value T¯ and define the wobble function for the pulse train in sequence given by (3.23).

In the case of coherent pulse radar, we should take into consideration the following conditions choosing the values of T¯ and ΔT. The minimal repetition period T_min must satisfy the condition of the unique radar range determination given before. Because of this, the following condition must be satisfied:

The maximal repetition period T_max is defined from conditions that are not associated with the digital moving-target indicator operation. Under the given and known values T_min and T_max and symmetric disposition of the wobble interval with respect to the value T¯, we obtain

Then, knowing the value of N_s, from (3.25) we can define ΔT.

In general, the wobble procedure is defined by the criterion of figure of merit r) maximization, taking into consideration the minimization of amplitude-frequency characteristic ripple in the digital filter bandwidth. As a rule, this problem is solved by simulation methods. The following wobble procedures are used:

• Linear—a consecutive increasing or decreasing T on ±ΔT from pulse to pulse in the train

• Cross-sectional analysis, for example, following the procedure

  {T2i=T¯+iΔT,i=0,2,…,0.5(Ns−1)T2i+1=T¯−(i+1)ΔT,i=1,3,…,0.5(Ns−2)⁢(3.26)

• Random, for example, by realization of the “bowl” model, restored from the total value of a prior given set T.

Calculations and simulations demonstrate that the wobble procedures for the repetition period T lead to a decrease in the depth of amplitude-frequency response dips of the nonrecursive and recursive digital filters. However, at the same time, a stop band of the nonrecursive and recursive digital filters is narrowed simultaneously with the frequency band extension and distortion of the interference frequency spectrum. By this reason, the effectiveness in cancellation of passive interferences is considerably reduced. Absolute losses in the figure of merit r) for the recursive digital filters of the second order vary from 0.3 to 4.3 dB and from 4 to 19 dB for the recursive digital filters of the third order, in comparison with the digital moving-target indicators without wobble procedures for the repetition period T(at optimal velocity) [28-34].

3.1.4 ADAPTATION IN DIGITAL MOVING-TARGET INDICATORS

In practice, spectral-correlation characteristics of passive interferences are unknown a priori and, moreover, are heterogeneous in space and not stationary in time. Naturally, the efficacy of passive interference cancellation becomes essentially inadequate. To ensure high effectiveness of the digital moving-target indication radar subsystems under conditions of a priori uncertainty and non-stationarity of passive interference parameters, the adaptive digital moving-target indication radar subsystems are employed [4]. In a general sense, the problem of adaptive moving-target indication is solved based on an implementation of the correlation feedback principle [35]. Correlation automatic equalizer of passive interferences represents a closed tracker adapting to noise and interference environment without taking into consideration the Doppler frequency. Parallel with high effectiveness, the tracker has a set of the following imperfections:

• Poor cancellation of area-extensive interference leading edge that is a consequence of high time constant value (for about 10 resolution elements) of adaptive feedback

• Decrease in passive interference cancellation efficiency over the powerful target return signal

• Very difficult realization, especially using digital signal processing

The problem of moving-target indicator adaptation can be solved using the so-called empirical Bayes approach using of which we at first, define the maximal likelihood estimation (MLE) of passive interference parameters and after that we use these parameters to determine the impulse response coefficients of rejector digital filter. In this case, we obtain an open-loop adaptation system, a transient process that is carried out within the limits of transient process of the digital filter.

The first simplest adaptation level in the open-loop adaptation system is to cancel the average Doppler frequency of passive interference caused by relocating to an interference source relative to radar system. In this case, an estimator of the average Doppler frequency of passive interference fDpi¯ or the equivalent Doppler shift in phase within the limit of the period T, that is, Δφ¯^Dpi=2πfDpi¯T, must be included in the adaptive rejector digital filter. Determination of the average Doppler frequency of passive interference fDpi¯ or the equivalent Doppler shift in phase within the limits of the period T, that is, Δφ¯^Dpi, must be carried out in real time. To estimate the average Doppler phase shift of passive interference within the limits of the searching period T implementing the MLE, the sample consisting of k readings of the in-phase and quadrature component pairs of passive interference is used, which are related to the adjacent elements of sampling in radar range from two adjacent searching periods. The algorithm to estimate the average Doppler phase shift of passive interference takes the following form [35]:

where ZI1in,ZQ1in and ZI2in,ZQ2in are the in-phase and quadrature components of the input signal within the limits of two adjacent searching periods. In (3.27) we assume that a correlation of interference in the adjacent elements of sampling in radar range is absent and the interference is a stationary process in k adjacent elements of sampling. To obtain an acceptable accuracy of computer calculation the value k must be of the order 5-10. The obtained estimations Δϕ¯^Dpi are used to make corrections in the delayed in-phase and quadrature components corresponding to the rotation of complex amplitude envelope sum vector given by

on the angle Δφ¯^Dpi. Determination of the corrected in-phase and quadrature components of the vector given by (3.28) is carried out in accordance with the following formulas:

Flowchart of digital signal processing algorithm used by the digital moving-target indicator under adaptation to the displacement of passive interference source is shown in Figure 3.6. The block diagram consists of memory devices to store the four (k − 1)-fold samples of input signals for each element of radar range along with normal elements; digital signal processing circuits to determine Δφ¯^Dpi,cosΔφ¯^Dpi,sinΔφ¯^Dpi,Z′Iin[n−1],Z′Qin[n−1]; and the single-order digital filter with the interperiod subtraction. This block diagram can be used for digital filters of other types. For this purpose, there is no need to include new elements in the block diagram, but this certainly adds to the computational costs. More complete adaptation to the correlation features of passive interferences involves an estimation of the coefficient module ρ_pi of passive interference interperiod correlation along with the estimation Δφ¯^Dpi. However, as calculations showed when the passive interference spectrum has a form of a single hump and the digital filter with interperiod subtraction possesses a multiplicity, the approximation of the coefficient module ρ_pi of passive interference interperiod correlation by unit leads to negligible losses.

3.2.1 INITIAL CONDITIONS

In digital signal processing systems, a process of accumulation and signal detection is realized, as a rule, at video frequency, after the union of in-phase and quadrature channels. Henceforth, the signal detection problems will be solved taking into consideration the following initial conditions:

1. A single input signal can be presented in the following form:

   Xi=X(ti)=S(ti,α)+N(ti),⁢(3.30)

   where S(t_i, α) is the target return signal (information signal, i.e., the signal containing an information about parameters of the target), α is the function of time and signals parameters, and N(ti) is the noise. The signal parameters are the delay t_d and direction of arrival θ The total target return signal is a sequence of periodically repeated pulses (the pulse train). At uniform rotation of radar antenna in the searching plane, the pulse train is modulated by directional diagram envelope of the radar antenna. The number of pulses in train is given as

   Np=φddfVA,⁢(3.31)

   where

   • ϕ_dd is the width of radar antenna directional diagram in the searching plane at the given power P level

   • f is the pulse repetition frequency of searching signals

   • V_A is the scanning speed of antenna beam

   Under discrete scanning in radar systems with phase array the pulse train envelope of the target return signals has a square shape and the number of pulses in train is defined based on the predetermined probability of detection at the searching target zone edge with minimal effective scattering surface. As for the statistical characteristics of the target return signal (information signal), as usual, we consider two cases:

   1. The train of nonfluctuated pulses

   2. The train of independently fluctuating pulses obeying to the Rayleigh pdf with zero mean and the variance σs2, that is,

      p(Si)=SiσS2exp{−Si22σs2}.⁢(3.32)

2. Under synthesis of signal detection algorithms and evaluation of target return signal parameters we use, as a rule, a noise model in the form of Gaussian random process with zero mean and the variance σn2. When the correlated time interferences (namely, passive interferences) are absent, noise samples modeled by the Gaussian process do not have an interperiod correlation. When the passive interferences or their remainders are present after cancellation by the digital moving-target indicators, a sequence of passive interference samples are approximated by Markov chain. To describe statistically the Markov chain we, in addition to the variance, should know the coefficient module ρ_pi of passive interference interperiod correlation. The interperiod correlation coefficient of the uncorrelated noise with the variance σn2 and correlated passive interference with the variance σpi2 can be presented in the following form:

   ρij=σpi2ρpiij+σn2δijσΣ2,⁢(3.33)

   where

   σΣ2=σpi2+σn2;δij=1 at i=jandδij=0 at i≠j.⁢(3.34)

   As an example of additional non-Gaussian noise, we can consider a random pulse interference generated by other sources of radiation. This interference is characterized by the off-duty ratio Qinpulse and amplitude Zinpulse that are random variables. Analysis of the stimulus of random pulse interference under signal processing is carried out, as a rule, by simulation methods.

3. Samples of the input signal X_i in the case of absence of target reflecting surface fluctuations are subjected to the general Rayleigh pdf (the hypothesis ℋ₁—a “yes” signal)

   pSNℋ1(Xi)=XiσΣ2exp{−Xi2+Si22σΣ2}I0(XiSiσΣ2),Xi>0,⁢(3.35)

   where I₀(x) is the zero-order Bessel function of the first kind. In the case of presence of target reflecting surface fluctuations, we can write

   p′SNℋ1(Xi)=XiσΣ2+σSi2exp{−Xi22(σΣ2+σSi2)},⁢(3.36)

   where σSi2 is the variance of target return signal amplitude Introduce the following notations: x_i = X_i/σ_∑ is the relative envelope amplitude; q_i = S_i/σ_E is the SNR by voltage; ki2=σSi2/σΣ2 is the ratio of the target return signal amplitude variance to the interference amplitude variance. Using these notations, we can rewrite (3.35) and (3.36) in the following form:

   pSNℋ1(xi)=xiexp{−xi2+qi22}I0(xi,qi),⁢(3.37)
   p′SNℋ1(xi)=xi1+ki2exp{−xi22(1+ki2)}.⁢(3.38)

   If the target return signal is absent in the input signal, the pdf is the same for the considered cases, that is,

   pNℋ0(xi)=xiexp{−0.5xi2}.⁢(3.39)

4. The joint pdf of pulse train from uncorrelated normalized samples in the case of absence of target reflecting surface fluctuations (uniform radar antenna scanning) takes the following form:

   pSNℋ1(x1,x2,…,xN)=pSNℋ1{x}N=∏i=1N{xiexp{−xi2+qi22}I0(xi,qi)},⁢(3.40)

   where

   • q_i = q₀g_i, g_i are the weight coefficients depending on the radar antenna directional diagram shape

   • q₀ is the SNR at the maximum power of radar antenna directional diagram

   Similarly, for the pulse train from N samples in the case of Rayleigh fluctuations of the target return signal (the target reflecting surface fluctuations), we can write

   pSNℋ1{x}{xi1+ki2exp{−xi22(1+ki2)}}=∏i=1N{xi1+ki2exp{−xi22(1+ki2)}}.⁢(3.41)

   In the case of discrete radar antenna scanning, (3.40) and (3.41) have the same form under the conditions

   x1=x2=…=xN,ai=a0,k1=k0,gi=1.⁢(3.42)

5. Digital signal detection algorithm is considered for two versions:

   1. The target return signal is quantized by amplitude in such manner that a low-order bit value does not exceed the root-mean-square value σ_n of receiver noise. In this case, a stimulus tracking of quantization by amplitude is added up to an addition of independent quantization noise to the input noise and a synthesis of digital signal processing algorithms is reduced to digital realization of optimal analog signal processing algorithms [36]

   2. The target return signal is quantized on two levels—binary quantization. In this case, there is a need to carry out a direct synthesis of signal processing algorithms and decision-making networks to process digital binary quantized signals. The required probability of signal detection P_D and the probability of false alarm P_F take the following form [37–40]:

      PD{di}=∏i=1NPSNidibSNi(1−di);⁢(3.43)
      PF{di}=∏i=1NPNidibNi(1−di);⁢(3.44)

   where

   • PSNidi is the probability of detection

   • PNidi is the probability of false alarm of the i-th target return signal from the pulse train that can be presented in the following form:

     PSNi=∫c0∞PSNℋ1(xi)dxi,bSNi=1−PSNi;⁢(3.45)
     PNi=∫c0∞PNℋ1(xi)dxi,bNi=1−PNi;⁢(3.46)

   and

   di={1,ifxi≥c00,ifxi<c0,⁢(3.47)

   where c₀ is the normalized threshold for binary signal quantization by amplitude.

3.2.2 DGD FOR TARGET RETURN PULSE TRAIN

DGD is discussed in detail for a variety of applications in Refs. [39,40,42-69]. In this section we present a brief analysis of the digital signal processing algorithms based on the generalized approach to signal processing in noise. Our main goal in this section is to compare the work content of DGD realization with other digital signal processing algorithms. First, consider the case when the parameters of target return signals—the pulse train from N nonfluctuating pulses with the additive receiver noise with known statistical characteristics—are known.

Based on the theoretical analysis carried out in Refs. [41,44,45], we can write the likelihood ratio for the generalized signal processing algorithm in the following form:

where x˜i is the reference noise with a priori information “a no signal” and the same statistical characteristics as the noise at the receiver input of radar system. Consequently, the generalized signal detection algorithm takes the following form:

Taking logarithm and making some mathematical transformations with respect to (3.49), we can write

Further mathematical transformations of (3.50) are associated with an approximation of the function ln I₀(x). In the case of weak signals, that is, q_i ≪ 1, we use the following approximation

Consequently, taking into consideration that q_i = q₀g_i and g_i are the weight coefficients depending on the radar antenna directional diagram shape, the detection algorithm of weak target return pulse train will have the following form:

Now, consider the case of powerful signals, that is, q_i ≫ 1. In this case, we use the following approximation:

Consequently, taking into consideration that q_i = q₀g_i and g_i are the weight coefficients depending on the radar antenna directional diagram shape, the detection algorithm of powerful target return pulse train will have the following form:

Thus, in the case of the target return pulse train with completely known parameters and modulated by the radar antenna directional diagram, the generalized signal detection algorithm comes to weight summation of normalized samples of the target return pulse train and implementation of energy detector at the output of quadrature or linear detector within the limits of target return pulse train bandwidth and comparison of accumulated statistic with the threshold K′g.

In practice, when the real radar system is employed, the target return pulse train contains the unknown parameters, such as SNR, that is, q₀, at the maximum point of radar antenna directional diagram, the delay t_d of the target return pulse train relative to searching/scanning signal, and angular defection of the target return pulse train center θ in the scanning plane with respect to the fixed direction θ₀. Because of this, to process information completely inside the radar coverage a realization of generalized signal detection algorithm should be organized within the limits of each interval of time sampling or radar range discretization. Accumulation of processed signals must be carried out within the limits of “tracking/moving window.” The length of “tracking/moving window” must be equal to the number of target return pulses in train. In this case, at the low SNR (for rth sampling or discretization interval) the generalized detection algorithm of nonfluctuating target return pulse train (there are no fluctuations of the target reflecting surface) can be presented in the following form:

The generalized detection algorithm of fluctuating target return pulse train (there are fluctuations of the target reflecting surface) can be presented in the following form (at q₀ ≪ 1):

In the case of powerful signal, the generalized signal detection algorithm within the limits of the “tracking/moving window” is obtained by analogous way. Thus, the DGD of target return pulse train with unknown parameters represents a composition of the “tracking/moving window” summator and energy detector with the threshold network and signal generator indicating a detection of the target return pulse train.

3.2.3 DGD FOR BINARY QUANTIZED TARGET RETURN PULSE TRAIN

Now, consider the case when the input sequences are quantized on two levels by amplitude. The generalized detection algorithm of binary quantized target return pulse train is obtained from a synthesis of the likelihood ratio comparing it with the threshold. In doing so, we use the probability of signal detection P_D and the probability of false alarm P_F given by (3.43) and (3.44). The obtained generalized signal detection algorithm has the following form in its final version:

where the weight coefficients and the threshold are given by

and

is the probability to get the unit on the jth position of the target return pulse train;

is the probability to get the unit in noise region (a “no” signal);

where c₀ is the normalized threshold for binary signal quantization by amplitude.

Thus, the generalized signal detection algorithm for the binary quantized target return pulse signals comes to summation of the weight coefficients χ_j corresponding to positions of the target return pulse train where dμ−j(r)=1. The generalized signal detection algorithm of nonmodulated target return pulse train (in the case of scanning with the fixed antenna) takes the following form:

In other words, the generalized signal detection algorithm is reduced to accumulation of units within the limits of the target return pulse train length (within the limits of length of the “moving/tracking window” with the predetermined sampling increment) and comparison of the obtained end sum with the threshold.

3.2.4 DGD BASED ON METHODS OF SEQUENTIAL ANALYSIS

The use of methods of the sequential analysis takes a very important place in signal detection theory. Detectors constructed based on methods of the sequential analysis allow us to determine the logarithm of likelihood ratio by the following recurrence formula [70-75]

where

• ℒ_gμ−1 is the accumulated likelihood ratio over μ − 1 steps

• Δℒ_gμ is the likelihood ratio increment at the μ-th step of sequential analysis

The accumulated step-to-step statistic ln{ℒ_gμ} is compared with the upper ln A and lower ln B thresholds

where

• P_D is the predetermined probability of detection

• P_F is the predetermined probability of false alarm, correspondingly

If following a comparison we have

the decision about signal detection is accepted and we stop analysis. If following a comparison we obtain

Then the decision about a “no” signal is accepted and we also stop analysis. If the following condition is satisfied

we continue an analysis, that is, we observe a new sample and determine the likelihood increment. The process of accumulation of ln{ℒ_gμ} and making a decision is presented in Figure 3.7.

Logarithm of likelihood increment is determined by the following formula:

In doing so, in the case of nonfluctuating target return pulse train model (there are no fluctuations of the target reflecting surface), we use the following logarithm of likelihood increment:

In the case of rapidly fluctuating target return pulse train model (there are fluctuations of the target reflecting surface), we use the following logarithm of the likelihood increment:

where the parameter k is given by (3.42).

Thus, to realize a procedure of sequential signal detection there is a need to first define the expected SNR by voltage, for example, in the case of nonfluctuating target return pulse train model (there are no fluctuations of the target reflecting surface), and by power in the case of rapidly fluctuating target return pulse train model (there are fluctuations of the target reflecting surface). The main characteristic of sequential analysis procedure is the average number of steps n¯ to make a final decision a “yes” or a “no” signal in the input process. Consider how the average number of steps depends on a “yes” signal and threshold under decision making.

In the case of a “no” signal in the input process, the procedure of sequential analysis is finished by the likelihood ratio logarithm ln{ℒ_gμ} overrunning the lower threshold ln B. Value of the lower threshold ln B does not depend on the predetermined probability of false alarm P_F since the probability of false alarm P_F varies within the limits of 10⁻³ ÷ 10⁻¹¹ and is only defined by an acceptable value of the probability of miss, that is, P_M = 1 − P_D, where P_D is the probability of signal detection. By this reason, the average cycle of sequential analysis procedure, in the case of a “no” signal in the input process, depends only on the predetermined probability of signal detection P_D and increases with increasing in P_D.

In the case of a “yes” signal at the DGD input, the procedure of sequential analysis is finished, as a rule, by overrunning the upper threshold ln A. A value of the upper threshold ln A is defined by the predetermined probability false alarm P_F. Consequently, the average cycle of analysis in the case of a “yes” signal at the DGD input depends on the predetermined probability of false alarm P_F and SNR.

An advantage of the DGD based on the sequential analysis in comparison with the well-known Neyman–Pearson detector consists of the average number of target return signal samples n¯, which is necessary to make a decision based on the predetermined probability of false alarm P_F and probability of signal detection P_D is less under sequential analysis procedure than the fixed number of samples N required for the Neyman–Pearson detector. Consequently, it is possible to reduce the computer cost and power energy of a CRS under the detection of target. This possibility can be realized by a radar system with programmable radar antenna scanning when the radar antenna beam can be delayed in scanning direction until the end decision making. However, this advantage has been proved rigorously only in the case of a single channel, that is, under signal detection within the limits of a single radar range resolution interval. In the time of radar signal processing, there is a need to make a decision taking into consideration all elements of radar range resolution simultaneously (multichannel case). In this case, the effectiveness of sequential analysis is defined by the average number of searching radar signals required to make the end decision in all elements of radar range resolution with the predetermined probability of false alarm P_F and probability of signal detection P_D. This number is determined by the following formula:

where

• m is the number of analyzed resolution elements in radar range

• n¯k is the average cycle of sequential analysis at the kth radar range resolution element

Optimality of sequential procedure in the multichannel radar systems is not proved so far. There are no analytical and theoretical methods and procedures to determine the efficacy of detectors constructed based on the sequential analysis and employed by multichannel radar systems. This is also particularly true for the DGD. For this reason, an analysis of the multichannel radar systems implementing the sequential analysis procedures is carried out using simulation techniques in each particular case. In what follows, we present some results of this analysis.

In the time of sequential analysis in the multichannel radar system there are two possible types of procedures:

• Decision making is an independent procedure: In this case, a test for the given channel is finished after reaching one of the thresholds—the upper threshold or the lower threshold.

• Simultaneous decision making: This decision making is made if all values of partial likelihood ratios overrun one of the thresholds—the upper threshold or the lower threshold. In this case, multiple crossings of the thresholds are possible.

The average number of tests both for the first case and for the second cases is defined by (3.72). However, the procedure of the second kind is more cost-effective in terms of the required average number of tests since this procedure allows us to level out the lower threshold and, consequently, decrease an uncertainty zone between the upper and lower thresholds. This statement is illustrated in Figure 3.8, where the lower threshold values are demonstrated to make a decision at the predetermined probability of false alarm P_F = 10⁻³ and the probability of signal detection P_D = 0.7 for various values of SNR under various ratios of the rapidly fluctuating target return pulse train model (there are fluctuations of the target reflecting surface) to the noise by power as a function of the number m (the number of analyzed resolution elements in radar range). As we can see from Figure 3.8, the value of the lower threshold increases with a corresponding increase in the SNR and m. In this case, at fixed values of the lower threshold, the probability of missing P_M of the target return signal decreases inversely to the number m of analyzed resolution elements in radar range; however, in the case of sequential analysis procedure of the first kind, the probability of missing P_M of the target return signal is independent of the number m of analyzed resolution elements in radar range.

In the case of a “no” signal at the DGD input, the average number of tests at sequential analysis procedure in a single receiver antenna radar system depends only on the predetermined probability of signal detection P_D. In the multichannel radar system, there is a dependence of the average number of tests at sequential analysis procedure both on the predetermined probability of signal detection P_D and on the number m of analyzed resolution elements in the radar range. Dependences between the average number of tests at sequential analysis procedure and the predetermined probability of signal detection P_D, the probability of false alarm P_F, and (SNR)² = 1.5 obtained by simulation are presented in Figure 3.9. The average number of tests for a Neyman-Pearson detector are denoted by horizontal lines for each pair of the predetermined probability of signal detection P_D and the probability of false alarm P_F. The points of intersections of corresponding curves allow us to define the number of analyzed resolution elements in radar range at which the average number of tests for a Neyman-Pearson detector and the average number of tests for the DGD based on sequential analysis procedure is the same. Evidently, the DGD based on sequential analysis procedure is more effective in comparison with the Neyman-Pearson detector if the following condition in the case of a “no” signal

is satisfied. As follows from Figure 3.9, in the case of a “no” signal at the DGD input the efficacy of the DGD based on sequential analysis procedure is higher, and the predetermined probability of false alarm P_F is lower. For a fixed value of the probability of false alarm P_F the efficacy of the DGD based on sequential analysis procedure for m receiver antennas increases concomitant to decrease in the probability of signal detection P_D.

In the multichannel radar systems, the average number of tests at sequential analysis procedure in the direction of target presence becomes commensurable with the average number of tests at sequential analysis procedure in the direction of target absence even at small numbers of resolution elements. Consequently, the average number of tests at sequential analysis procedure before a decision making is mainly defined by the average number of tests at sequential analysis procedure in the direction of target absence. In the multichannel radar systems, the average number of tests at sequential analysis procedure can be prolonged to such limits that are not acceptable by energy or other tactical considerations. The only way to avoid this prolonged delay of antenna beam in scanning direction is to introduce a truncation procedure of sequential analysis at the n_tr-th step and to make a decision about exceeding some fixed threshold C. In this case, the probability of error at the truncation procedure can be presented in the following form:

where

• Per(n¯<ntr) is the probability of wrong decision at sequential analysis procedure before truncation

• P_er(C) is the probability of wrong decision at sequential analysis procedure under comparison of decision statistic with the threshold C

Obviously, we can choose such values of n_tr and C that additional errors caused by truncation will be satisfied to the following conditions:

Algorithm to choose n_tr and C satisfying to the conditions (3.75) and (3.76) comes to the following procedure: At n = n_tr1 we can choose C such that the first condition given by (3.75) is satisfied. If, at the same time, the second condition given by (3.76) is also satisfied, we stop the procedure of choice. If the (3.76) is not satisfied, we select n_tr2 = n_tr1 + 1 and test conditions (3.75) and (3.76) again. As was proved by the theory of sequential analysis [25,71], there are the required n_tr and C. Introduction of truncation does not effect essentially the average number of tests at sequential analysis procedure in the direction of target absence and slightly reduces the average number of tests at sequential analysis procedure in the direction of target presence.

At binary quantization of the target return pulse train samples, the generalized detection algorithm based on the sequential procedure becomes essentially simple, but the efficacy of detection procedure is decreased. In this case, additional losses in the average number of tests at the sequential analysis procedure can be 15% ÷ 30%. Losses caused by quantization can be reduced to 2%/5% even at 7-8 quantization levels. This fact gives us a reason to design and construct the DGDs based on the sequential analysis procedure with small digit capacity of the target return pulse train samples.

Comparative analysis of effectiveness of the DGD based on the sequential analysis procedure and a detector with the fixed sample volume, for example, the Neyman-Pearson detector, gives rise to a consideration of two-stage signal detection procedure. The two-stage signal detection procedure consists of the following. At first, the target return pulse train sample with the fixed volume n₁ is subjected to the generalized signal processing algorithm with the predetermined probability of signal detection P_D1 and the probability of false alarm P_P1. If the decision a “no” signal is made, the procedure is stopped. Otherwise, an additional procedure is carried out with respect to “suspect data,” and we use for this purpose n₂ samples of the target return pulse train and obtain the probability of signal detection P_D2 and the probability of false alarm P_F2. In doing so, using results of the two stages, we obtain

To solve the problem of optimization for the two-stage procedure we can choose such values of n₁ and n₂ that the average number of tests at sequential analysis procedure would be minimal. In doing so,

where at P_F1 ≪ 1 the probability of duplicate scanning in the a “no” signal direction is defined in the following form:

where

• P_F1 is the probability of false alarm at the first stage

• n¯ℋ1 is the average number of tests at sequential analysis procedure in the direction of target presence

• n¯ℋ0 is the average number of tests at sequential analysis procedure in the direction of target absence

• n¯2ℋ1 is the average number of tests at sequential analysis procedure in the direction of target presence at the second stage

• n¯2ℋ0 is the average number of tests at sequential analysis procedure in the direction of target absence at the second stage

• m₁ is the number of analyzed resolution elements in radar range at the first stage.

Two-stage DGD based on the sequential analysis procedure allows us to obtain an advantage from 25% to 40% in the average number of tests at sequential analysis procedure in comparison with the DGD or Neyman-Pearson detector with the fixed number of tests subjected to the predetermined probability of signal detection P_D and the probability of false alarm P_F. Under target detection at high values of the probability of false alarm P_F, for example, P_F = 10³, and at large number of resolution elements, the two-stage DGD can be more effective in comparison with the DGD based on the sequential analysis procedure.

3.2.5 SOFTWARE DGD FOR BINARY QUANTIZED TARGET RETURN PULSE TRAIN

In practice, while designing the detectors for binary quantized target return pulse train, many heuristic methods are used that make a decision about signal detection based on the definition of a unit density within the limits of each sampling interval of the target return pulse train at the envelope detector output. The best-known digital signal processing algorithms from this class of detectors are software algorithms fixing an initial instant of target return pulse train appearance by the presence of l units on m adjacent positions, where l ≤ m, m ≤ 5…10. This is the so-called criterion “ l from m” or “ l/m.” Criterion of the initial instant fixation of target return pulse train appearance is, at the same time, a criterion of target return pulse train detection. To eliminate ambiguity, a criterion of target return pulse train end is defined under an angular coordinate reading. As a rule, the target return pulse train end is fixed by presence of series from k skipping (zeros) one by one (k = 2 … 3). To determine positions between the origin and the end of target return pulse train the binary scalers are employed.

Each of the considered operations can be realized by the digital finite-state automaton. Composition of these digital finite-state automata gives us an example of the software detector block diagram presented in Figure 3.10, where A1 is the digital finite-state machine realizing the detection criterion “ l/m”; A2 is the digital finite-state automaton realizing the criterion of target return pulse train end definition based on series from k zeros; and A3 is the digital finite-state automaton (counter) to determine the number of positions from the instant of target return pulse train detection to the instant of erasing the stored information. Using the well-known methods of abstract digital automata composition [76,77], we are able to obtain the transition matrix and graph of associated automaton realizing a program “l/m - k” for each data pattern l, m, and k. This graph is a starting point to design tools to realize the required digital signal detection algorithm, and the transition matrix allows us to determine rigorously the quality characteristics of software detectors.

Digital storage devices of binary quantized target return pulse train are considered as other groups of detection algorithms for software detectors. The origin of target return pulse train is fixed by the digital storage device by the first unit obtained after counter storage erasing. The target return pulse train end is fixed when a series of k zeros appears, that is, the same principle as for the software detectors. The signal detection is fixed at the instant of counter erasing if the number of storage units is equal or greater than the threshold N_thr. Storage devices of other types are designed in such a way that the counter stores the number of positions (cycles) within the limits of the target return pulse train bandwidth, not the number of units between series from k and more skipping one by one. In doing so, skipping of units at intrinsic positions of target return pulse train is restored. These storage devices are very convenient to realize a digital signal processing algorithm to define the angle coordinate reading of target return pulse train.

Analysis of quality characteristics of the digital software detectors and storage devices can be carried out both by analytic-theoretical method and by simulation. As an example, Figure 3.11 represents the detection performance of software detectors “3/3 - 2,” “ 3/4 - 2,” and “4/5 - 2” and digital storage device of units at the predetermined probability of false alarm P_F = 10⁻⁴ for the target return pulse train consisting of 15 pulses modulated by the radar antenna directional diagram envelope of the following form:

Thresholds of binary quantization are determined by procedures discussed in Ref. [77,78]. The storage device detection threshold (the counter threshold) N_thr is determined in the following form:

• In the case of absence of target reflecting surface fluctuations under detection of the target return pulse train, the storage devices are more effective in comparison with the software detectors and vice versa; in the case of presence of target reflecting surface fluctuations under detection of the target return pulse train, the software detectors with the detection criterion “l/m” or “l < m” are more effective in comparison with the storage devices.

• Threshold losses are low when the target return pulse train is detected by “tracking/moving window” using both the software detectors and the storage devices. Digital realization of these detectors is sufficiently simple, which makes it advantageous to use these detectors when requirements for simplicity of realization prevail over requirements in minimization of energy losses.

• It is appropriate to employ the software detectors on the first stage under the two-stage detection procedure.

3.3.1 PROBLEM STATEMENTS OF DIGITAL DETECTOR SYNTHESIS

The synthesis problems of optimal detectors are solved with an assumption that there is a priori all information about the noise and interference and their statistical parameters and features under a predetermined energy and statistical characteristics of the target return signal (the information signal). The optimal signal processing and detection algorithms obtained under these conditions possess the best performance for established initial conditions. Changes in noise environment or deviation from characteristics considered under synthesis of detectors leads, as a rule, to a drastic decrease in the efficacy of signal processing and detection algorithms and even to losses in capacity for work.

Detection of target return signals by CRSs in practice is characterized by ambiguity, to a greater or lesser extent, in energy and statistical parameters of signals and noise. In addition, these characteristics are varied in time or are functions of time. In real practice, noise jumping can reach several tens of dB. For example, jumping on 2 ÷ 3 dB only under the fixed detection threshold reduces to changes in the probability of false alarm P_F on four orders by magnitude [78]. Thus, a synthesis of robust signal processing and detection algorithms possessing a sufficiently stable quality of service (QoS) under changes in environment conditions is a special interest. As a rule, there is a need to ensure the stability of important QoS only or the probability of false alarm P_F only. In this case, we must solve the problem of constant false alarm rate or, in other words, the CFAR problem.

Depending on a priori information about the target return signals and noise, we distinguish the parametric and nonparametric uncertainty. For the parametric uncertainty, the pdf pXNℋ1{xi} under the hypothesis ℋ₁ (a “yes” signal) and the pdf pNℋ0{xi} under the hypothesis ℋ₀ (a “no” signal) are considered as known pdf, and only some parameters of these pdf are considered as unknown. Changes in environment and external operation conditions are changes in the noise statistical characteristics such as the mean, variance, covariance function, and so on. The robust signal detection algorithms ensuring CFAR are considered, in this case, as adaptive algorithms allowing us to obtain estimations of unknown statistical characteristics of the noise and to employ these estimations to normalize the input signal or to control the threshold. In the second case, the target return signal and noise pdf shape is, as a rule, unknown both at the hypothesis ℋ₁ (a “yes” signal) and at the hypothesis ℋ₀ (a “no” signal). In this case, a synthesis of robust signal detection algorithms is carried out based on methods of inspection of nonparametric statistical hypotheses. The obtained nonparametric signal detection algorithms bring about the probability of false alarm P_F that is independent (or invariant) of the noise envelope pdf pNℋ0{xi}. However, a statistical independence of the target return signal sample values is the indispensable condition of invariance of the nonparametric signal detection algorithms. If there is a correlation between sample values of the target return signal, the mixed signal detection algorithms using parametric and nonparametric statistic are implemented.