Diss / 10

of multiplicity of interperiod subtractions: ISF-1, ISF-2, ISF-3, and ISF-4—a single, twofold, threefold, and fourfold interperiod subtraction filters.

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

where

νs is the sample size of the target return signal equal to Ns − 1 (Ns is the number of target return pulses in train) in the case of recursive filter and the interperiod subtraction multiplicity ν in the case of nonrecursive filter

νin is the sample size of interference determined by analogous way as for νs ρs is the coefficient of correlation of the target return signal

ρpi 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 ftg 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 fpiiT (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.

FIGURE 3.4  Figure of merit as a function of fpiT (i = 1) for a set of digital filters: 1—the digital nonrecursive filter with twofold interperiod subtractions; 2—the digital recursive filter of the second order; 3—the digital nonrecursive filter with the threefold interperiod subtractions; 4—the digital composite filter: the digital nonrecursive filter with the single interperiod subtraction plus the digital recursive filter of the second order.

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, νin ≥ 20, and νin ≥ 30, if we use the digital recursive filters of the third order. In doing so, the sample size of the target return signal νs does not affect the efficacy of the digital recursive filters of the second and third orders already if νs ≥ 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 ν = 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:

Zout[n] = a0Z in[n] + D1 (a1Z in[n] − b1Zout[n])+ + Di (aiZ in[n] − biZout[n])+ + DN (aN Z in[n] − bN Zout[n])

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

. . . . . . . .

FIGURE 3.5  Digital recursive filter based on the iteration network.

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 bj = 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 x based on the condition x/σn ≤ 1, where σ2n 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 σ2qn = x2 /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 σan2 = 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 fD = ±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 Ns, where Ns 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 ± T. In this case, if Ns 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 choos-

–

ing the values of T and T. The minimal repetition period Tmin 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 Tmax is defined from conditions that are not associated with the

Then, knowing the value of Ns, from (3.25) we can define T.

In general, the wobble procedure is defined by the criterion of figure of merit η 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 ± T from pulse to pulse in the train

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

•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 η 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 nonstationarity 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, = 2π fDpi 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, ϕˆ 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

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 ρpi of passive interference interperiod correlation along with the estimation ϕˆ 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 ρpi of passive interference interperiod correlation by unit leads to negligible losses.

3.2  DGD FOR COHERENT IMPULSE SIGNALS WITH KNOWN PARAMETERS

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:

where S(ti, 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 (ti ) is the noise. The signal parameters are the delay td 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

where

φdd 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 VA 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:

a.The train of nonfluctuated pulses

b.The train of independently fluctuating pulses obeying to the Rayleigh pdf with zero mean and the variance σS2, that is,